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Khan_Academy_AP_Microeconomics | Minimum_efficient_scale_and_market_concentration_AP_Microeconomics_Khan_Academy.txt | - [Narrator] In this video, we're going to think about the concept of minimum efficient scale and then how that impacts market concentration. And we're going to make sure we understand what both of these ideas are. So first of all, minimum efficient scale, you can view it as the smallest scale at which we stop getting economies of scale, or another way of thinking about it is the minimum scale at which we are no longer, our long-run average total cost curve is declining. So in this example, let's see, when we talk about our taco trucks, when we were at about 80 units, we're not at minimum efficient scale yet because our long-run average total cost curve is still declining as we add more and more and more units. We're getting economies of scale all the way until, at least in this example, it looks like our long-run average total cost curve stops declining around 200 units. So in this example, that would be our minimum efficient scale, which is sometimes abbreviated as MES. And one way to think about it is this is the minimum scale at which an operation needs to run at in order to be very competitive, in order to be truly efficient out there in the market. Because you can imagine, if some operators are able to achieve minimum efficient scale of, let's say, getting to the 200 tacos a day while others are not, let's say they're only able to stay at 100 tacos per day, and if the market were to get very competitive and the price of a taco were to go down to, say, 55 cents per taco, the people who are at minimum efficient scale could still operate and still make money while the people who are not at minimum efficient scale, well, they're not going to be able to participate in the market. And most well-functioning markets, it gets quite competitive. And so economists like to think about what the minimum efficient scale is and compare that to the entire market size. Now, what do we mean by market size? Well, it depends on how you are defining the market. In our example of our taco trucks, it might be the market for tacos, market for tacos in our city. And of course, you can define different markets. You could define it as the market for food trucks in our city. You could define it as the market for tacos in our state or our country. But let's say if we were to say the market for tacos in our city, and let's say that that market is 10,000, 10,000 tacos per day. Well, in this reality, our minimum efficient scale is 200 tacos per day, and it's a very small fraction of the total market. And so that means that you could have many different competitors, each at that minimum efficient scale, so they're able to produce tacos at that 50 cents per taco. And because of that, you are likely to have many competitors. So when this, when our minimum efficient scale is a small fraction of the total market, that is going to lead to fragmentation. So here, we're going to have a fragmented, fragmented market. So if this circle were to represent the 10,000 tacos that are sold per day, if you're able to have a lot of competitors, each operating at minimum efficient scale, in fact, there's no real advantage to operating above minimum efficient scale because then you start getting diseconomies to scale, well, then you are going to have maybe 50 competitors who are splitting this market. So I won't take the trouble of making this into 50 different chunks. So one competitor has that part of the market. Another competitor has that part of the market. Another competitor has that part of the market. And you could imagine we're going to have this market fragmented into maybe 50 different players, and so that's why it's called a very fragmented market. But let's say that the minimum efficient scale was pretty close to the market size. So let's say instead of a market for 10,000 tacos per day, let's say that the market in our city is for 400, 400 tacos per day. Well, then the market is going to be smaller, like this. And so then it makes sense for, if someone's able to get to the minimum efficient scale, they can take up a lot of the market. In fact, they could take up half of the market. So this type of market might only be able to really support two players in this market. So this is considered to be a far more concentrated, concentrated market. And you could go to a reality where your minimum efficient scale is at the market size or is even larger than the market size. So let's say that the market size is not 400 tacos per day, but let's say we had a market, let's say we had a market of 195 tacos per day, per day. Well, in that world, whoever can get to 195 tacos is going to produce most efficiently. They're not even getting to the minimum efficient scale. But the more, the closer that you can get to that number, you're going to have the lowest average total, long-run average total cost of production, and so it's going to be very hard for anyone else to compete with you, especially if you're taking up most of the market. No one else is going to be able to get to scale, so they're going to be operating out here on the long-run average total cost curve. And so in that world, you might only have one player. And when you have one player, where the market dynamics make it so that it is actually efficient for only one player, this is sometimes referred to as a natural monopoly. There's other dynamics that could lead to a natural monopoly. But one way to think about it is, is if you keep getting economies of scale, up to the market size, well, then whoever can get to that market size first is going to be the most efficient producer. |
Khan_Academy_AP_Microeconomics | Law_of_supply_Supply_demand_and_market_equilibrium_Microeconomics_Khan_Academy.txt | We've talked a lot about demand. So now let's talk about supply, and we'll use grapes as this example. We'll pretend to be grape farmers of some sort. So I will start by introducing you-- and maybe I'll do it in purple in honor of the grapes-- to the law of supply, which like the law of demand, makes a lot of intuitive sense. If we hold all else equal-- in the next few videos, we'll talk about what happens when we change some of those things that we're going to hold equal right now-- but if you hold all else equal and the only thing that you're doing is you're changing price, then the law of supply says that if the price goes up-- I'll just say p for price-- if the price goes up, then the supply-- now, let me be careful-- the quantity supplied goes up. And then you can imagine, if the price goes down, the quantity supplied goes down. And you might already notice that I was careful to say quantity supplied. And it's just like we saw with demand. When we talk about demand going up or down, we're talking about the entire price-quantity relationship shifting. When we talk about a particular quantity demanded, we say quantity demanded. We don't just say demand. This is the exact same thing for supply. When we're talking about a particular quantity, we'll be careful to say quantity. If we talk about supply increasing, we're talking about the entire relationship shifting either up or down. So let's just make sure that this makes intuitive sense for us. And I think it probably does. Let's think about ourselves as grape farmers. And I'll make a little supply schedule right over here. So Grape Supply Schedule, which is really just a table showing the relationship between, all else equal, the price and the quantity supplied. So let's label some scenarios over here, just like we did with the demand schedule. Scenarios. And then let's put our Price over here. This will be in price per pound, the per pound price of grapes. And then this is the quantity produced over the time period. And whenever we do any of these supply or demand schedules, we're talking over a particular time period. It could be per day, it could be per month, it could be per year. But that's the only way to make some sense of, OK, what is the quantity per day going to be produced if that's the price? So if we didn't say per day, we don't know what we're really talking about. Quantity Supplied. And so let's just say Scenario A, if the price per pound of grapes is $0.50-- if it's $0.50 per pound-- actually, let me just do round numbers, but you get the idea. If the price per pound is $1, let's just say for us, we consider that to be a relatively low price. And so we'll only kind of do the easiest land, our most fertile land, where it's easy to produce grapes. And maybe the fertile-- and cheap land. So no one else wants to use that land for other things. It's only good for growing grapes. And so we will provide-- so this is price per pound. And in that situation, we can produce 1,000 pounds in this year. And I've never been a grape farmer, so I actually don't know if that's a reasonable amount or not, but I'll just go with it, 1,000 pounds. Now, let's take Scenario B. Let's say the price goes up to $2. Well now, not only would we produce what we were producing before, but we might now want to maybe buy some more land, land that might have had other uses, land that's maybe not as productive for grapes. But we would, because now we can get more for grapes. And so maybe now we are willing to produce 2,000 pounds. And we can keep going. The same dynamics keep happening. So let's say the price-- if the price were $3 per pound, now we do want to produce more. Maybe we're even willing to work a little harder or plant things closer to each other, or maybe I'll get even more land involved than I would have otherwise used for other crops. And so then I'm going to produce 2,500 pounds. And I'll do one more scenario. Let's say Scenario D, the price goes to $4 a pound. Same dynamic, I will stop planting other crops, use them now for grapes, because grape prices are so high. And so I will produce 2,750 pounds. And so we can draw a supply curve just like we have drawn demand curves. And it's the same exact convention, which I'm not a fan of, putting price on the vertical axis. Because as you see, we tend to talk about price as the independent variable. We don't always talk about it that way. And in most of math and science, you put the independent variable on the horizontal axis. But the convention in economics is to put it on the vertical axis. So price on the vertical axis. So then this is really Price per pound. And then in the horizontal axis, Quantity Produced, or-- let me just write it. Quantity Produced, I'll say in the next year. We're assuming all of this is for the next year, so next year. And it's in thousands of pounds, so I'll put it in thousands of pounds. And so let's see, we go all the way from 1,000 to close to 3,000. So let's say this is 1,000, that's 1 for 1,000, that's 2,000, and that is 3,000. And then the price goes all the way up to 4. So it's 1, 2, 3, and then 4. So we can just plot these points. These are specific points on the supply curve. So at $1, we would supply 1,000 pounds, at $1, 1,000 pounds. That's Scenario A. At $2, we would supply 2,000 pounds, $2, we'd supply 2,000 pounds. That's scenario B. At $3, we'd supply 2,500 pounds, $3-- oh, sorry. Now, when we look up-- See, now notice, I get my axes confused. This is Price. This isn't, when we talk about it this way, that we're viewing the thing that's changing. Although, you don't always have to do it that way. So at one $1, 1,000 pounds. $1, 1,000 pounds. $2, 2,000 pounds. $2, 2,000 pounds. $3-- this isn't $3, this is $3. $3, 2,500 pounds. So right about there. That's about 2,500. But I want to do it in that blue color, so we don't get confused. So $3, 2,500 pounds. That's about right. So this is Scenario C. And then Scenario D, at $4-- actually, let me be a little bit clearer with that, because we're getting close. So this is 2,500 pounds, gets us right over here. This is Scenario C. And then Scenario D at $4, 2,750. So 2,750 is like right over there. So that is $4. That is Scenario D. And if we connect them, they should all be on our supply curve. So they will all be-- it will look something like that. And there's some minimum price we would need to supply some grapes at all. We wouldn't give them away for free. So maybe that's something-- that minimum price is over here, that just even gets started producing grapes. So this right over here is what our supply curve would look like. Now remember, the only thing we're varying here is the price. So if the price were to change, all else equal, we would move along this curve here. Now, in the next few videos, I'll talk about all those other things we've been holding equal and what they would do at any given price point to this curve or, in general, what they would do to the curve. |
Khan_Academy_AP_Microeconomics | Long_run_supply_curve_in_constant_cost_perfectly_competitive_markets_Microeconomics_Khan_Academy.txt | - [Man] Alright, now let's dig a little bit more into analyzing perfectly competitive markets, and in particular we're gonna focus on the long run, and remember the long run is the time span where firms can enter and exit the market. Or, another way to think about it is, in the long run, fixed costs actually become variable, you can shutter factories, or you can build factories, in the long run. Now, in previous videos, we talked about that in the long run, in a perfectly competitive market, the firms that operate in that perfective competitive market are going to be operating at zero economic profit, and you can see that example right over here. As we've talked about it in many, many videos, in a perfectly competitive market, the firms are price takers, that price is set by that equilibrium point between the supply and demand curves, and the firms just take that. And so, their marginal revenue curve, it would just be a horizontal line that you see right over there, and zero economic profit happens when you produce a quantity where your average total cost is the same as your marginal revenue. For each unit, the amount that you get, which is that marginal unit, that's also how much it costs you to produce it, now remember, when we're talking about economic profit, that includes your opportunity costs, so, that doesn't mean that these firms are operating at zero accounting profit, they could still be making money, but, if you were to factor in their opportunity costs, that's when you get things to zero. Now, what I want to think about, what happens in the short and long run, if something say happens to market demand. Let's say that this is the market for apples, the fruit apples. So, this is the market for apples we're talking about, this is the market as a whole, this is a firm that produces apples, it could be a farm of some kind. And let's say that a new study comes out that apples actually are super good for your health, and they can be used as a performance enhancer for sports, and all sorts of positive results. Well, what is likely to happen in the short run, and this is a little bit of a review in terms of our supply and demand curves, and also what would happen to Firm A's economic profits. Pause this video and think about that. Well, in the short run, your demand curve would shift to the right, and so, because at a given price, people are willing to demand more apples because it has all these new and exciting benefits. And so, the demand curve might shift someplace like that, so, that is D Prime, and if the demand curve shifts like that, now we have a new equilibrium price in the market, our new equilibrium price in the market is right over here, we also have a new equilibrium quantity, so, our quantity is shifted from there, to now there, so, a new equilibrium quantity, and we have a new equilibrium price, let's just call this P Prime. And, at that new equilibrium price, well now, we have a higher marginal revenue curve for Firm A, and now, Firm A, it'd be rational for them to produce, remember, it's rational for them to produce up until the marginal revenue is equal to marginal cost, because each of those incremental units up to that point, they're going to be making money on those incremental units, and so, now it's rational for them to produce at this quantity, let me call that, Q Prime, and at this level, now all of a sudden Firm A is making economic profit, because at this quantity, that's revenue per unit, this is average total cost per unit, so they're making this height per unit, and then you multiply it, times your total number of units, which is the base of this rectangle, and so, this area would represent this positive economic profit. Now, you might be saying, wait, hold on a second, I thought you said in the long run, firms don't make economic profit in a perfectly competitive market. And that is true, at least based on the models that we are constructing, because what happens when you have this positive economic profit? Well, other firms will enter this market, remember, we're talking about a perfectly competitive market, there are no barriers to enter in, everyone is fairly non-differentiated, and they have similar cost structures. And so, what you could imagine is, in the long run, folks will enter the market, and then, the supply curve will also shift to the right, and assuming that that doesn't change the cost structure for the individual firms, and actually, let me show someone entering into this market, so now, Firm B is entering this market, and when Firm B enters the market, it has the same cost structure as Firm A, and it didn't shift either of their first, either of their cost structures, so, this is known as a constant cost, perfectly competitive market, where the entering or the exiting of firms does not affect the cost structures of the firms that are entering, or that are in the market. And so, this situation, these graphs look the same, but now we have more firms entering the market, the supply curve will shift to the right, and it's going to keep shifting to the right until these firms that all have identical cost structures, are no longer making economic profit again. It wouldn't shift further to the right because no one's going to enter if they're making economic loss, it'll keep shifting until no one is making that economic profit. And so, you see what happened in this constant cost, perfectly competitive market, that now, we are back to this equilibrium point where we are at a higher quantity because people like all the benefits of these apples, but we're at the price we were before, which is the same marginal revenue curve that we were before for the various players in this market. And so, when you see something like this in a constant cost, perfectly competitive market, you can actually create a long run supply curve, you could view this S and S Prime as a short run supply curve, the long run supply curve, on the other hand, for a perfectly competitive market, in which the cost structure of the participating firms do not depend on the number of firms that are in or out of the market, then you're at long run supply curve in what we could call a constant cost, perfectly competitive market, is going to be a horizontal line like this. So, I will leave you there. In other videos, we'll think about, well, what happens if the cost structure changes for these firms depending on how many firms are in or out of the market, and that will be based on, how do the costs of the inputs into their production change as you have people entering or exiting that market. |
Khan_Academy_AP_Microeconomics | Visualizing_marginal_utility_MU_and_total_utility_TU_functions.txt | - [Sal] What we're going to do is think about the graphs of marginal utility and total utility curves. And so right over here, I have a table showing me the marginal utility I get from getting tennis balls. And so it says look if I have no tennis balls and I'm not able to play tennis, so I'm pretty happy when I get that first ball. It gives me a marginal utility of 100. Now you might say 100 what Sal? And I would say to you well that's the thing about marginal utility. We're using fairly abstract units here. We're not speaking in terms of dollar or opportunity cost. We're just have this abstract unit, we could call it utility units if we want, but what really matters is the relation between these values. So for example, that second ball, it's nice, now I can lose that first ball or I don't have to chase that first ball around as frequently, but it's not as, I don't get as much incremental utility as I got from that first ball. That first ball allowed me to play tennis. Now the second ball, I was already playing tennis, now my tennis is just going to be a little bit more pleasant. So the marginal utility one way to think about it, this is 80 and this is 100, the marginal utility of that second ball is 80% of what that first ball is. Now I promised that I would graph these, so let's get a graph out here So there we go. And let's start plotting these and see what it looks like. Well that first ball gave me a utility of 100. The second ball gives me a utility of 80. Third ball, you see the trend. This is a downward sloping, if this was a line, these are discrete values here, but if I cared about 1/2 of a tennis ball or 3/4 of a tennis ball, then I could connect the dots here as well. But you can see that it is downward sloping line if you were to connect the dots. So three balls, utility of 60. If I have four balls, it's a utility of 40. Five balls, utility of 20. And once again, when I get that fifth ball, yeah it's nice, but now my pockets are getting pretty full and I was all, it's hard to play tennis with those full pockets. By the time I get that sixth ball, I get no marginal utility from that. I don't have a place to store it, it's I don't really, I'm not really into it. And then that seventh ball, I actually view it as a negative. It's one extra ball to worry about. No place to store it, it's taking up space in my life. And so if you were to connect the dots, we're talking about a discrete case, but oftentimes you could think about something that's a little bit more continuous where the quantity is more granular if you said pounds of chocolate or something like that, then you could imagine in general for a marginal utility curve, to be able to connect these dots. And you see in this case, it is downward sloping. So this is the marginal utility, the marginal utility curve. Notice that it is slopes down, slopes down. And this you're generally going to see this for any marginal utility curve because the incremental benefit of that next amount, that next unit, is seldom as good as the benefit of getting it before. You get tired of the thing, you start running out of space, you've already consumed more than you need of it. And this is consistent with what we know from the law of demand. And the law of demand, every incremental amount of quantity, people are like well I already have some, and this might not be, the law of demand is not talking about just an individual, it's talking about a market, but the market as a whole is made up of individuals, and if each individual is saying, hey you know that first unit really matters to me, but the next unit, it's nice, but not as good, so I'd pay less for it. And then the unit after that, it's nice, so I might pay less for it. And so if you aggregate all those individuals, that's where that law of demand comes from. These are consistent ideas, and that's why the demand curve, you are also, it is sloping down. Price is usually on this axis instead of utility, and you could imagine price as being a proxy for utility. And quantity of course is on this axis right over here. So this is quantity of balls. So that's our marginal utility curve. What about total utility? So let me have a table here that shows total utility. And total utility from marginal utility is pretty straightforward. All you do is say okay, well that first ball, when I have one ball, my total utility is the same as my marginal utility. And so you're going to have that same starting place at when your consumption is just beginning. But then your total utility from two balls, well I had 100 utility units from the first ball, and then I get 80 more from that second ball, so it's gonna be 180. So for two balls, my total utility is 180. All I'm doing is I just added that to that. Now for three balls I add that to that to that. I take 180 and I add the 60 extra utility units I get for that third ball, and now I'm at 240. So that third ball gives me, gets me to 240 right over there. Now the fourth ball, once again, I'm just gonna take 240 and add the incremental utility of that fourth ball, the marginal utility of that fourth ball. That gets me to 280. So that gets me to 280, which is right over, let's see, 20, 30, 20, 40, 60, 80. So this is right over there. Now that fifth ball, I'm just gonna take the 280 and then the marginal utility of the fifth ball, add 20, gets me to 300. Fifth ball gets me to 300 which would be right around there. Now the sixth ball gave me no incremental, no marginal utility, so my total utility, when I have six balls, stays the same. I'm indifferent as to whether I have five or six balls. So the sixth ball, it is now flat right over here. And now the seventh ball, I'm tired of these tennis balls. I'm being overwhelmed by them. I'm finding it stressful. And so it actually has a negative marginal utility, and so my total utility, if someone forced me to have seven balls, my total utility would now go down by 20. And so my total utility now would be 280, right over here. And you could see the marginal utilities here, if you just say look this is plus 80, this is plus 60, this is plus 40, this is plus 20, this is plus zero, and then this is minus 20. And so you see the numbers right over, right over there. Now this tennis ball example, this would be a discrete case. You wouldn't have 1/2 a tennis ball or pi tennis balls or something like that, but if we wanna speak in general terms, you could think about connecting the dots, if you had a more continuous example. And your total utility curve might look something like this. Now what's interesting is right when you're beginning consumption you're starting at the same place. Well that makes sense. Your first unit, you get marginal utility, that's gonna be your total utility. And this is upward sloping as long as you're getting some positive marginal utility from each increment. So as long as my marginal utilities were positive, well this graph is going to be increasing. But notice, because the marginal utilities are getting are decreasing right over here, the rate of increase for total utility is decreasing, the slope here is decreasing. You can view the marginal utility as the slope of the total utility curve. And then notice the total utility curve has a maximum value, it's starting to hit a maximum value right over there, when the marginal utility curve is hitting zero. Because beyond that point, where at least in this example we had negative marginal utility. And so when you add that seventh unit, well that's gonna make your total utility curve go down, and so you're gonna have a negative slope in this particular example. |
Khan_Academy_AP_Microeconomics | Changing_equilibria_from_trade_AP_Microeconomics_Khan_Academy.txt | - [Instructor] In this video, we're going to think about how trade can alter the equilibrium price and quantity in a given market. So what we see here, as we like to do, are very simplified examples of markets in various economies. So first, we have country A, and let's say it's the market for widgets. And we're going to assume that country A is not trading with anyone else. So it is an autarky, a very fancy word, which just means that this country is operating independently. It's operating in isolation. And so you can see the demand curve in this market in orange, and we can see the supply curve. And you can see, when this country is operating in isolation, this market for widgets has an equilibrium price. It looks like it's a little bit under $4. I'll just assume that the price is in dollars per widget. And the equilibrium quantity looks like it's about a little under four units per whatever time period we're looking at. Fair enough. Now let's look at country B, and let's assume that they are also operating independently. It's an autarky in this market. And so here, we can see a different demand curve than what we saw in country A and a different supply curve. And notice they have a different equilibrium price and quantity. So here, the equilibrium price seems to be a little bit over $1, and the equilibrium quantity seems to actually be not that different than what we saw in the first country. Although, in many situations, it could be very different. Now let's imagine what would happen if they opened up their economies to each other. Well, then you would essentially horizontally add these two demand curves, and you would horizontally add these two supply curves to come up with a new supply curve and a new demand curve. This is a little bit of a review of what we've seen in other videos. But notice, at a price of five, in total, no one's demanding anything, no quantity of these widgets. And then at a price of zero, country A, the market there is demanding 15, and country B is demanding five. So in aggregate, they're demanding 20. And similarly, you can see that with supply, that at a price of five, country A will supply five. And at a price of five, country B will supply 15. And so together, they will supply 20 units per time period. And so we can view this right over here as our supply and demand curves for the combined markets because now they're trading. We are not in an autarky anymore. And notice what has happened. Our equilibrium price is now someplace in between these two equilibrium prices. So it looks like it's a little bit under $3. And our equilibrium quantity, our equilibrium quantity is a little bit under 10 units. And so you might notice some interesting things that are happening. What would happen from a, someone in country A's point of view, the people who are the buyers, the people who are demanding this widget? Well, now, instead of having to pay almost four, they're paying someplace in between two and three. This is the new equilibrium price if they were to open up their economy. And so what you have is, is that this, this amount would be produced by, in theory, by the domestic manufacturers, so the first, let's call that 2 1/2 units. And then the remainder, where are they going to get those units from? Well, those are going to be imports. And let's look at the equilibrium price on country B, and country B was really the lower cost producer. And so country B, now we have a, let's put it a little bit over 2.5, so let's put it right over there. And so now you have a situation where the suppliers in country B are going to be producing a lot, a lot more than they were before. And only this amount is coming from their domestic demand. And then all of this amount right over here, that is being exported. And if we're assuming that the world economy is only made up of country A and country B or that they're only trading with each other, these exports become country A's imports right over there. And so proponents of free trade will say, hey, look, the overall consumer surplus is larger than the combined consumer surpluses that we had before. Before, you had this consumer surplus in country A. And country B, it was all of this. But still, this entire area is larger than these two combined. And you could do it mathematically if you like, calculate the area of these triangles. And if you look at the producer surplus, you'll see a similar story. The total producer surplus of the combined economies now, this is going to be larger than this producer surplus plus this producer surplus. Now, some, there will not always be winners in this. The winners here are the demanders in country A. Because instead of this little, small consumer surplus that they had before, now they have this much larger consumer surplus right over there. And then the other winners are the suppliers in country B. 'Cause instead of this producer surplus that they had before, they now have this producer surplus. But the losers in this situation are the suppliers in country A who now have a much lower producer surplus. So their triangle has shrunk to that right over there. And then the other losers in this situation are the consumers in country B who now have to pay a higher equilibrium price. And so their consumer surplus is only this small triangle, when before it was this whole thing. So anyway, the big takeaway here is, is that if you go from autarky to opening up economies, it can affect what the equilibrium prices and quantities are going to be. And oftentimes or usually, it's going to increase your total consumer and producer surplus. Although, there will be some winners, and there will be some losers. |
Khan_Academy_AP_Microeconomics | Monopolist_optimizing_price_Total_revenue_Microeconomics_Khan_Academy.txt | What I want to start thinking about in this video is, given that we do have a monopoly on something, and in this example, in this video, we're going to have a monopoly on oranges. Given that we have a monopoly on oranges and a demand curve for oranges in the market, how do we maximize our profit? And to answer that question, we're going to think about our total revenue for different quantities. And from that we'll get the marginal revenue for different quantities. And then we can compare that to our marginal cost curve. And that should give us a pretty good sense of what quantity we should produce to optimize things. So let's just figure out total revenue first. So obviously, if we produce nothing, if we produces 0 quantity, we'll have nothing to sell. Total revenue is price times quantity. Your price is 6 but your quantity is 0. So your total revenue is going to be 0 if you produce nothing. If you produce 1 unit-- and this over here is actually 1,000 pounds per day. And we'll call a unit 1,000 pounds per day. If you produce 1 unit, then your total revenue is 1 unit times $5 per pound. So it'll be $5 times, actually 1,000, so it'll be $5,000. And you can also view it as the area right over here. You have the height is price and the width is quantity. But we can plot that, 5 times 1. If you produce 1 unit, you're going to get $5,000. So this right over here is in thousands of dollars and this right over here is in thousands of pounds. Just to make sure that we're consistent with this right over here. Let's keep going. So that was this point, or when we produce 1,000 pounds, we get $5,000. If we produce 2,000 pounds, now we're talking about our price is going to be $4. Or if we could say our price is $4 we can sell 2,000 pounds, given this demand curve. And our total revenue is going to be the area of this rectangle right over here. Height is price, width is quantity. 4 times 2 is 8. So if I produce 2,000 pounds then I will get a total revenue of $8,000. So this is 7 and 1/2, 8 is going to put us something right about there. And then we can keep going. If I produce, or if the price is $3 per pound, I can sell 3,000 pounds. My total revenue is this rectangle right over here, $3 times 3 is $9,000. So if I produce 3,000 pounds, I can get a total revenue of $9,000. So right about there. And let's keep going. If I produce, or if the price, is $2 per pound, I can sell 4,000 pounds. My total revenue is $2 times 4, which is $8,000. So if I produce 4,000 pounds I can get a total revenue of $8,000. It should be even with that one right over there, just like that. And then if the price is $1 per pound I can sell 5,000 pounds. My total revenue is going to be $1 times 5, or $5,000. So it's going to be even with this here. So if I produce 5,000 units I can get $5,000 of revenue. And if the price is 0, the market will demand 6,000 pounds per day if it's free. But I'm not going to generate any revenue because I'm going to be giving it away for free. So I will not be generating any revenue in this situation. So our total revenue curve, it looks like-- and if you've taken algebra you would recognize this as a downward facing parabola-- our total revenue looks like this. It's easier for me to draw a curve with a dotted line. Our total revenue looks something like that. And you can even solve it algebraically to show that it is this downward facing parabola. The formula right over here of the demand curve, its y-intercept is 6. So if I wanted to write price as a function of quantity we have price is equal to 6 minus quantity. Or if you wanted to write in the traditional slope intercept form, or mx plus b form-- and if that doesn't make any sense you might want to review some of our algebra playlist-- you could write it as p is equal to negative q plus 6. Obviously these are the same exact thing. You have a y-intercept of six and you have a negative 1 slope. If you increase quantity by 1, you decrease price by 1. Or another way to think about it, if you decrease price by 1 you increase quantity by 1. So that's why you have a negative 1 slope. So this is price is a function of quantity. What is total revenue? Well, total revenue is equal to price times quantity. But we can write price as a function of quantity. We did it just now. This is what it is. So we can rewrite it, or we could even write it like this, we can rewrite the price part as-- so this is going to be equal to negative q plus 6 times quantity. And this is equal to total revenue. And then if you multiply this out, you get total revenue is equal to q times q is negative q squared plus 6 plus 6q. So you might recognize this. This is clearly a quadratic. Since you have a negative out front before the second degree term right over here, before the q squared, it is a downward opening parabola. So it makes complete sense. Now, I'm going to leave you there in this video. Because I'm trying to make an effort not to make my videos too long. But in the next video what we're going to think about is, what is the marginal revenue we get for each of these quantities? And just as a review, marginal revenue is equal to change in total revenue divided by change in quantity. Or another way to think about it, the marginal revenue at any one of these quantities is the slope of the line tangent to that point. And you really have to do a little bit of calculus in order to actually calculate slopes of tangent lines. But we'll approximate it with a little bit of algebra. But what we essentially want to do is figure out the slope. So if we wanted to figure out the marginal revenue when we're selling 1,000 pounds-- so exactly how much more total revenue do we get if we just barely increase, if we just started selling another millionth of a pound of oranges-- what's going to happen? And so what we do is we're essentially trying to figure out the slope of the tangent line at any point. And you can see that. Because the change in total revenue is this and change in quantity is that there. So we're trying to find the instantaneous slope at that point, or you could think of it as the slope of the tangent line. And we'll continue doing that in the next video. |
Khan_Academy_AP_Microeconomics | Law_of_demand_Supply_demand_and_market_equilibrium_Microeconomics_Khan_Academy.txt | In this video, we're going to talk about the law of demand, which is one of the core ideas of microeconomics. And lucky for us, it's a fairly intuitive idea. It just tells us that if we raise the price of a product, that will lower the quantity demanded for the product. Quantity demanded will go down. And you could imagine the other side of that. If we lower the price of a product, that will raise the quantity demanded of that product. And the law of demand says this just kind of generally. What we'll see in a few videos from now is that there are some exceptions to this. But to make this little concrete, let's think about the demand for a certain product. And one thing I want to clear here, and I'm going to go through great pains to not mess this up, is that when we talk about the word demand in a formal economic sense, we're not talking about a quantity. We're actually going to talk, all else equal, ceteris paribus, the relationship between price and quantity demanded. If we talk about an actual quantity, we should say the quantity demanded. So demand versus quantity demanded. These are two different things. And if it's a little confusing to you right now, hopefully by the end of this video, the difference between demand and quantity demanded will become a little bit clearer. And definitely over the next few videos, because in this video, we're going to focus on how the quantity demanded changes relative to the price. In future videos, we'll talk about how the entire relationship, how demand changes based on different factors. But to make things concrete, let's say I'm about to release my science fiction book, Space Whatever. I don't know, the book that I want to release. So I'm going to release some ebook. And we've done some market study, or we just know how the demand is related to price or the price is related to demand. And we're going to show that in a demand schedule, which is really just a table that just shows how the price-- and, actually I just made my first mistake. I just said how price relates to demand. I should say how price relates to quantity demanded and how quantity demanded relates to price. So demand schedule, it shows a relationship between price and quantity demanded, all else equal. So we're going to have multiple scenarios here. So this column, let me do my scenarios. In this column, let me put my price. In this column, I put my quantity demanded. So scenario, let's call this scenario A. I could price my book at $2. And I'll get a ton of people downloading it at that price. So I will get 60,000 people download my book at that price, my ebook. Scenario B, I could raise the price by $2. So it's now $4. And that kills off a lot of the demand. Now the quantity demanded goes down to 40,000 people downloading it. Then I can go to scenario C, if I raise it by another $2. So now I'm at $6. Now that lowers the quantity demanded to 30,000. I'll do a couple more of these. Scenario D, I raise another $2. So I get to $8 now. Now the quantity demanded goes down to 25,000. And I'll do one more of these. Let me see, what color have I not used yet. I haven't used yellow yet. Scenario E, if I raise it to $10, now the quantity demanded, let's just say, is 23,000. So this relationship shows the law of demand right over here. And this table that shows how the quantity demanded relates to price and vice versa, this is what we call a demand schedule. Now we can also, based on this demand schedule, draw a demand curve. And really, we're just going to plot these points and draw the curve the connects them. Because these aren't the only scenarios. Anything in between is possible. We could charge $2.01 for the book. We could charge for $4.50 for the book. And so that's what the demand curve captures a little bit better, because it's a continuous curve, not just five points. So let's do that. Let's graph it. And this is one of those conventions of economics that I am not a fan of. Because people often talk about changing the price, and how the quantity demanded changes from that. And in traditional-- in most of math and science, the thing that you're changing, you normally put on the horizontal axis. So if I was in charge of the convention of economics, I would plot price on the horizontal axis right over here. But the way it's done typically is that price is done on the vertical axis. And so you're used to seeing it in kind of a traditional class environment. I'll do the same. So we'll put price in the vertical axis, and we'll put quantity demanded in the horizontal axis. And our quantity demanded goes all the up to 60,000. So let's see, that's 10, 20, 30, 40, 50, 60. So that's 10-- this is in thousands-- 20, 30, 40-- sorry, not 45-- 40, 50, and 60. And this is in thousands. And then the price goes up to $10, from $2 to $10. So let's say this is 2, 4, 6, 8, and 10. So let's plot the scenarios. So scenario A, price is $2, 60,000 units are demanded. That is scenario A right over there. Scenario B, when the price is $4, 40,000 units are demanded. And that's right over there. That's scenario B. Scenario C, $6, 30,000 units. Right over there, scenario C. Scenario D, $8, 25,000 units. $8, 25 is right about there. That looks like 25,000, right in between. That's close enough. So that right over there is scenario D. And then finally scenario E, $10, 23,000 units. So it might be something like that. That is scenario E. And so we could actually have prices anywhere in between that. And maybe we could even go further. So this right over here. So if I were to draw the demand curve, it could look something like this. The demand curve would look something-- I'm trying to do my best to draw it as a straight continuous line-- could look something like that. And it could keep going on and on. And so these are two ways to show demand. So just going back to what I said earlier, the quantity demanded is, all else equal for a given price, how many units people are willing to download or buy of my ebook. When we talk about the demand itself, we're talking about this entire relationship. So this demand itself is this entire demand schedule. Or another way to think of it is this entire demand curve. If demand were to change, we would actually have a different curve. This curve would shift, or the entries in this table would shift. If the quantity of demand changes-- so we move along this curve when you hold everything else equal and you only change price. So hopefully that makes it clear. When everything else is equal, and you're only changing price, you're not changing demand, you're changing the quantity demanded. The demand, because everything else is equal, is this relationship. In the next few videos, we'll think about what does happen when you do change some of those other factors. |
Khan_Academy_AP_Microeconomics | Economic_profit_for_firms_in_perfectly_competitive_markets_Microeconomics_Khan_Academy.txt | - [Instructor] In this video, we're going to dig a little bit deeper into the notion of perfectly competitive markets, or we're gonna think about under what scenarios a firm would make an economic profit or an economic loss in them. Now as a reminder, these perfectly competitive markets are something of a theoretical ideal. There's few markets in the real world that are truly perfectly competitive. Some might get close, but most markets are someplace in a spectrum between perfectly competitive and at the other extreme, say something like a monopoly. But here we're talking about perfect competition, and in perfect competition, the firm's products aren't differentiated. There's no barriers to entry or exit. And so in that situation, the market supply and demand curves are gonna define the price in the market, which are also gonna define the marginal revenue for these firms. They're all going to be price takers. They're gonna be passive in terms of price. Whatever the market price is, that's the price that they are going to sell their products for. And their decision is really what quantity to produce and sell and whether to enter or exit the market. So let's look at that a little bit. So these are just your classic and supply demand curves, supply and demand curves, you might see for a market. The first few units in the market, there's a huge marginal benefit. So people are willing to pay a lot, but then each incremental unit, the marginal benefits a little bit lower and lower and lower and lower, and that's why we have that downward-sloping demand curve. And then on the supply curve, the first unit in the market might be fairly inexpensive to produce, but then the marginal cost gets higher and higher and higher. And where they meet, where the supply and demand meet, that tells us the equilibrium price and equilibrium quantity in the market. And we can show that with that line, and let's just say that equilibrium price is $10. And as I just mentioned, that's going to have to be the price that all of the firms, and these might not be all of the firms in the market, but all of the firms in the market, if we're talking about a perfectly competitive market, would just have to take that price. So given that, what quantity would firms A, B, and C produce, and which of these firms would be profitable or not? I encourage you to pause the video and think about those two questions. If you could just answer, which of these firms would be profitable or not, and we're talking about economic profit in this context. All right, well let's look at Firm A first. Well Firm A, for any of them, it is not rational to produce a quantity where the marginal cost is higher than the marginal revenue that the firm's getting. And remember, this line right over here, this line right here, which is the price line, that's also, that is price, which is equal to marginal revenue. And so, for that extra unit, if you can't sell it for more than you're producing, then you wouldn't produce that extra unit. So it's rational for them to produce more and more and more, the marginal cost goes higher and higher, until right at the point that marginal cost is equal to marginal revenue, which is equal to price, the market price, which they're just going to take. So it's rational for this firm to produce this quantity right over here. So I'll just go quantity, I'll say quantity for that firm. Now is this firm going to be profitable or not? Well at this quantity, what's its average total cost? Well its average total cost is right over there, and so, for every unit, it's going to make this difference between the price or the marginal revenue it's getting and its average total cost. And so one way to thing about the profit of this firm is, and we're talking about economic profit, it's going to be the area of this rectangle right over here. So let's say if the average total cost at that quantity is, let's say that this is $8, then this height of the rectangle is 10 minus eight. The height right over here, let me do this in a different color, this height right over here is $2. And then the width is going to be the quantity of that firm. And so let's say the quantity of that firm, let's say it's 10,000 units a year, 10,000, 10,000 units per year. And so the area right over here would be $2 times 10,000. It would be $20,000. $20,000 per time unit if we're talking all of this is say per year. Now let's go to Firm B. Using that same analysis, is Firm B making an economic profit, or is it not making an economic profit? Well, Firm B is once again going to be a price taker, and so the price right over here, the equilibrium price in the market, is going to be equal to the price that that firm has to take, which is going to be its marginal revenue curve. And that's why it's a flat marginal revenue curve because no matter what quantity they produce, they're gonna get that same price. And it wouldn't be rational for them to produce a quantity where marginal cost is higher than marginal revenue. And so they would produce right over there. Now what is their economic profit at this quantity? So this is quantity of the second firm, Firm B, I'll write it like that, maybe that is Firm A. And maybe this is also, it looks about the same. I'll make 'em a little bit different. Let's say that's 9,500 units per time period. Well here the average total cost at that quantity is equal to the marginal cost. So, which is equal to the marginal revenue. So, at that quantity, whatever that $10 they're getting per unit, they're also spending on average $10 per unit. Another way to think about it, the area of that rectangle is going to be zero because it has no height. So this situation right over here, the firm has zero, zero economic, I'll write $0 of economic profit. And then last but not least, let's think about Firm C. Pause this video and think about what its economic profit would be. Well, like we've seen, it would be rational for it to produce the quantity where marginal cost is equal to marginal revenue, which is equal to the market price. So it would produce this quantity right over here. And let's say that that quantity is 9,000 units. And what's its average total cost then? So at 9,000 units, its average total cost, let's say that that is $12 right over there. So what's its economic profit? So for every unit it's selling, it's getting $10, and it's costing $12 on average to produce it. So it's taking an economic loss of $2 per unit. So $2 per unit, so this height right over here is $2, times the units, times 9,000, you're going to have two times 9,000, you're going to have an $18,000 not economic profit, but economic, economic loss. Now one thing to think about is, why would any firm be in this situation? Well it's important to think about things in the short run versus the long run. In the short run, we've talked about this analysis right over here where a firm can decide what quantity it would produce that is rational. Its fixed costs are fixed in the short run. We've studied that in multiple videos. But in the long run, its fixed costs aren't fixed, and so the firm could decide to enter or exit the market. And so for Firm C, while they've already put in those fixed costs, it is actually rational for them to do it because they're actually able to make the marginal revenue they get up to that quantity. It's at least they're able to more than cover their marginal costs, and then they're able to eat up or I guess you could say take care of some of their fixed costs. But they're still not able to run an economic profit. So in the long run, it wouldn't be rational for this firm to stay in the market. They would likely exit the market. |
Khan_Academy_AP_Microeconomics | Marginal_revenue_and_marginal_cost_Microeconomics_Khan_Academy.txt | Let's continue with our orange juice producing example In this situation I want to think about what a rational quantity of orange juice might be what would be a rational quantity of orange juice to produce given a market price So let's say that the market price right now is 50 cents a gallon and I'm going to assume that there are many producers here so we're going to have to be price takers and obviously we want to charge as much as we can per gallon but if we charge even a penny over 50 cents a gallon then people are going to buy all of their orange juice from other people so this is the price that we can charge 50 cents per gallon So, if we think about it in terms of marginal revenue per incremental gallon well that first incremental gallon we're going to get 50 cents the next incremental gallon we're going to get 50 cents for that one and the next one we're going to get 50 cents as well. for the first thousand gallons we're going to get 50 cents for each of those gallons for the first 10 thousand gallons we'll get 50 cents per gallon So, our marginal revenue curve will look something like this Our marginal revenue is a flat curve right at 50 cents a gallon so that is our marginal revenue at 50 cents at a market price of 50 cents per gallon now in this situation what's a reasonable quantity that we would want to produce? Now there's two dynamics here we want to produce as much as possible so that we can spread our fixed cost over those gallons that's one way of thinking about it or, another way of thinking about it is we have a certain amount of fixed cost we are spending $1000 no matter what so why don't we try to get as much revenue as possible to try to make up for those fixed costs or if we think about it in terms of average fixed cost the more quantity that we produce the component of the cost for that from the fixed cost goes down and down and down so we want to have as much as possible to spread our fixed costs now the one thing that we do need to think about is especially once we kind of get beyond the little dip in the marginal cost curve and as we produce more and more units the marginal cost is going up higher and higher and higher we don't want to produce so much that the cost of producing that incremental unit the marginal cost of that incremental unit is more than the marginal cost of that actual or the marginal cost of that incremental unit is not higher than the marginal revenue that we're getting on that incremental unit so, until marginal revenue is equal to marginal cost or another way to think about it you don't want marginal cost and this is after we go to this little dip here we're trying to do as much as possible marginal cost is going higher and higher and higher we don't want to produce this much right over here because here the cost for that extra gallon is higher than what we're going to get for that extra gallon looks like that cost for that extra gallon might be 53 cents while we're only gonna get 50 cents for that extra gallon so every extra gallon we produce over here we're going to be losing money so you don't want marginal cost to be greater than marginal revenue so when you look at the curves like this and make sense to just say when does marginal revenue equal marginal cost? and that's this point right over here and that is the rational amount to produce so that is 9000 units so we're going to be at this line over here we're gonna produce 9000 gallons of juice our revenue that we're going to get is going to be the rectangle of the area that is high as the price we're getting per unit times the number of units so this is gonna be the total revenue we get if we were to shade this in I'm not gonna shade this in because it's going to make my whole diagram messy and what's our total cost? well, we have our average total cost right here this is our average total cost at 48 cents that's the little green triangle here so it's 48 cents per unit times the total number of units our cost, the area in this rectangle so if I were to shade this in this little slightly smaller rectangle and so our profits are the difference between the two our total revenue is the area under the rectangle that has this marginal revenue line as its upper bound and our cost is the rectangle that has our average total cost this line right over here as its upper bound so our profits in this circumstance are going to be the area right over here the height is the difference between our marginal cost which is the same as our marginal revenue and our total cost so the heigh is going to be this two cents right over here we're taking the difference of 50 and 48 so it's gonna be 2 cents and then, the quantity produced is going to be 9000 units so 9000 we're making 2 cents per unit remember, our average cost our average total cost is 48 cents per unit we're selling that 50 cents per unit so we're making 2 cents per unit that's not 20 we're making 2 cents per unit 2 cents times 9000 units gives us that's 18000 cents, or 180 dollars of profit now what I want you to think about and we'll answer this in the next video is does it make sense to sell units at all and if so, how many units should we sell if, and here is the question if the market price is lower than your average total cost so does it make sense and how many units does it make sense to produce let's say if the market price were 45 cents per unit does it make sense for us to produce |
Khan_Academy_AP_Microeconomics | Longrun_economic_profit_for_perfectly_competitive_firms_Microeconomics_Khan_Academy.txt | - [Instructor] Let's dig a little bit deeper into what happens in perfectly competitive markets in the long run. So what we have on the left-hand side, and we've seen this multiple times already, is our supply and our demand curves for our perfectly competitive market, and you can see the equilibrium price right over here, marked with this a dotted line, and as we've talked about in multiple videos, the firms in that perfectly competitive market, the perfectly competitive firms, they just have to price takers, so the market price is going to be their marginal revenue curve. It's gonna be this horizontal curve. And it would be rational for them to produce the quantity. So they're not going to set the price, but they can choose what quantity to produce, but it would be rational for them to keep producing while the marginal revenue is higher than the marginal cost up to including when the marginal revenue is equal to the marginal cost. So for this firm at this current state of affairs, it would be rational for them to produce this quantity right over there. And as we've talked about in other videos, at that quantity, they're going to make an economic profit. And the way that we can see that is at this quantity, this is the average total cost, that is your marginal revenue, and so you are going to get this much per unit and then you multiply, so the height is how much you get per unit and then you multiply that times the number of units, so the area of this rectangle is that positive economic profit that this firm will have. Now, that's in the short run, but now let's think about what will likely happen in the long run. If folks see other folks making a positive economic profit, remember, economic profit doesn't just account for regular cost, it also includes opportunity cost, so a lot of you will say hey, I would wanna put my resources into this market so that I can make that positive economic profit as well. But what's gonna happen as you have entrance into this market? Well, that's gonna shift the supply curve to the right. At any given price, you're gonna have more supplies, one way to think about it. So that's a supply curve. Let's just call that one. Now, you're gonna have more entrance. More entrance. And what's going to happen? Well, you might get to something like you might get to a situation like this. Let me see if I can draw it well. You might get to a situation like this where you have more entrance and you got the supply curve two. Now, what's going to be the quantity that firm A produces in that one? Remember, firm A is one of many firms. Well, in this situation, we have a new equilibrium price. So if this was P sub one, now we have this new equilibrium price, P sub two, which is going to define a new marginal revenue curve for all of the players in this perfectly competitive market and so the new marginal revenue curve is gonna be right over there. Now, in this situation, what is the rational quantity for firm A to produce? Well, once again, as long as marginal revenue is higher than marginal cost, it makes sense for them to produce more and more and more, up until the point that they are equal. So now, firm A would want to produce less because the market price that it just has to take is less. But notice what happens as more and more entrants got into the market. The market price, which also defines this horizontal marginal revenue curve, went lower and lower to the point where firm A now in this situation is making no economic profit. At this point, where not only as marginal revenue intersecting marginal cost, but that's exactly the point in which marginal cost is equaling average total cost. So one way to think about it is in a perfectly competitive firm, they're productively efficient. They are producing the quantity that minimizes their average total cost. We've already talked about that point where marginal cost and average total cost intersect. That's gonna be the minimum point for average total cost. And why is that? Well, while marginal cost is below average total cost, average total cost is gonna get lower and lower and lower, and then once marginal cost gets higher than average total cost, well, then the average total cost curve will starting going curving up. So we just saw a situation that even where we see economic profit in the short run, in the long run, entrants are going to go into that market and it's going to reduce the economic profit down to zero and at that point, the firm that has a zero economic profit, they're productively efficient. They are producing at the minimum point of the average total cost curve. And we've already talked before that this equilibrium point right over here in our market, because our demand and supply curves, the intersection point defines the price, our equilibrium price and quantities, we're also allocatably efficient. We've talked about things like dead weight lost in the past. That is not happening right over here. Our marginal benefit is equal to our marginal cost right at that equilibrium price and quantity. Now, some of you might be saying, well, what about the other situation? What about if for some reason we were in a, let's call the supply curve here. Let's say people overshot. Too many people joined into this market. So let's say we went to supply curve three, well what's going to happen? Let me label this. This is right over here, this is marginal revenue curve one, which is equal to price one. This is marginal revenue curve two, which is equal to price two. And then this would define, so this right over here would be price three, price three, which would define marginal revenue curve three. So marginal revenue curve three, which is equal to price three. Well if too many entrants joined into that market, now firm A has a more difficult scenario. They are, would produce at this quantity, we've talked about many times already, but at that quantity, each unit their average total cost is higher than that revenue they're getting. So they're going to be running at an economic loss in the short run. But what would happen in the long run? Well firm A in the long run would probably exit the market and other firms who are running at economic loss would exit the market, and so that would shift the supply curve back to the left. And so we would eventually get, once again, to that reality where firms have no economic profit in there and we have a market that is allocatably efficient, no dead weight loss, and firms are producing at the minimum point of their average total cost curve, which is known as productive efficiency or productively efficient. |
Khan_Academy_AP_Microeconomics | Profit_maximization_worked_example_Free_Response_Question_Microeconomics_Khan_Academy.txt | - [Instructor] We're told corn is used as food and as an input in the production of ethanol, an alternative fuel. Assume corn is produced in a perfectly competitive market. Draw correctly labeled side-by-side graphs for the corn market and a representative corn farmer. On your graphs show each of the following: The equilibrium price and quantity in the corn market, labeled P sub M and Q sub M, respectively. The profit-maximizing quantity of corn produced by the representative farmer earning zero economic profit, labeled Q sub F. So like always, pause this video and see if you can do this on your own before we work through it together. All right, now let's work through it together. So we're going to do correctly labeled side-by-side graphs. So let me do, ooh, this is going to be my horizontal axis for the market. And then this is going to be the horizontal axis for the farmer. And this is going to be quantity in the market, quantity. Quantity, and then this is going to be quantity for the farmer, quantity. And then this is going to be price in the market. And whatever the market price is, that's also going to be the price that the farmer has to take, because it says it's a perfectly competitive market. So the farmer's going to be a price taker here. So let me make these axes. So this is price right over here. And this is price over here. So first let's draw the corn market. So let me label this corn market. And we've done this multiple times already. Our demand curve might look something like this. This is our demand curve. As when price is high, low quantity demanded. When price is low, high quantity demanded. And supply goes the other way around. So our supply curve would look something like this. And then this point, this helps us figure out this is going to be our equilibrium price, so that's P sub M, P sub M. And then this is going to be our equilibrium quantity, so Q sub M. Now this graph over here, we are going to draw the farmer. So this is going to be the farmer, the farmer's firm right over here. So the farmer's going to be a price taker, so whatever the equilibrium price in the market, that is going to be the price that the farmer is going to have to take. That market price is going to be the farmer's marginal revenue. Now they say the profit-maximizing quantity of corn produced by the representative farmer earning zero economic profit, labeled Q sub F. So we're going to have some quantity right over here. It is the profit-maximizing quantity, but it's also zero economic profit. So the zero economic profit tells us that the price must be equal to the average total cost at that quantity. So I can make an average total cost curve that looks something like this. And I'm going to make its minimum point intersect that market price, because we know from previous videos that the profit-maximizing quantity happens where the marginal cost intersects the marginal revenue, which, in this case, would be the price that the farmer has to take from the market. And we know that the marginal cost curve intersects the average total cost curve at this minimum point right over here. So I could draw a marginal cost curve. It might look something like this. So that is our marginal cost curve. And notice, the marginal cost curve intersects the average total cost at that minimum point. We explained that in multiple videos already. And we've explained in a previous video that the profit-maximizing quantity is the quantity at which the marginal cost and the marginal revenue meet. And the price is the marginal revenue. Beyond that point, every incremental unit the corn farmer's going to take a loss. It's gonna take him more resources to produce that corn than they're going to able to get in the market. And we also mentioned that this has to be a situation of zero economic profit. So the average total cost has to be at that price, at that marginal revenue right at that point. So this right over here would be our Q sub capital F. And we're done. |
Khan_Academy_AP_Microeconomics | Four_factors_of_production_AP_Microeconomics_Khan_Academy.txt | - [Instructor] An idea that will keep coming up as you study economics is the idea of the four factors of production, which are usually listed as land, labor, capital, and entrepreneurship. And the idea here is if you want to produce anything, so let's just say this circle is the production process, and this arrow is the output, you need inputs. Now, you might have many, many, many inputs. You might need supplies, you might need a factory, you might need people to work in the factory, you need all of these different things. But the idea of the four factors of production is that these things can all be classified in one of these four groups, as either land, labor, capital, or entrepreneurship. Now, these words have meaning in everyday language. And so, some of it might jump out at you. Of course, if you need to build factory or if you need to farm, you need land to do so. And you can see that in this example here, where we see a farm. Clearly, the need a lotta land in order to have the farm. Even in a garment factory, this is a picture of a garment factory from maybe a hundred years ago, even there, they needed land on which to build the factory. So, this floor is sitting on land. And land doesn't just have to strictly mean land in an economics context. It can mean natural resources in general. This could be things like water or air or energy. So, in some contexts, instead of land, some people might say natural resources for this first factor of production. Now, another important factor of production, and arguably they're all important, is the idea of labor. To produce many or most things, someone has to work on it. So, someone had to plant these seeds, and they will have to harvest these crops. The labor is very clear here. You see people putting in work in order to produce the product right over there. Now, capital is an interesting one. It means one thing in everyday language, and it means something slightly more specific when we talk about it in an economics context. In an economic context, capital is something produced to produce other things. So, examples of capital would be tools that you use to produce other things. It could be a building that you need in order to produce other things. It could be the machinery in a factory. So, in these two pictures, there's many examples of capital. You could view this table and the tools that these folks are using, that is capital. You could use, you could view the whole building itself and all of the light fixtures and all of that as capital. So, all of this stuff is capital. The hangers that they're putting the coats on after they produce it, that is capital. In this farm example, the capital would be the buildings. These were constructed so that they could produce the food from the farm. This little, it looks like some type of machinery there, that is capital for the farm. It's being used to produce the output of the farm. Now, the place that that's different than everyday language, in everyday language, when people talk about capital, they'll often include financial capital, financial assets that could be used to get benefit in the future, things like money. But in an economic context, we are not considering financial assets, we're only thinking about things that were produced in order to produce other things. The fourth factor of production is entrepreneurship. Entrepreneurship, in our everyday language means putting things together so you're trying to create other things. When someone's an entrepreneur, you might imagine someone who's trying to start a business. In an economic context, it has a related idea. Entrepreneurship is putting together all of the other factors of production so that you can actually produce things. You can't just randomly build buildings and randomly plant seeds. Someone has to think about how do you put these things together so that you can produce things in a reasonable way? And obviously, you wanna produce as much as possible given the other factors that you are putting into the production. A related idea, and it sometimes is used interchangeably in an economics course, is technology. So, sometimes, you'll see the four factors of production as land, labor, capital, and entrepreneurship; and sometimes, you'll see it listed as land, labor, capital, and technology. But when you see this, when you see technology as a factor of production, don't think about it as technology in everyday language, where you think of computer chips or software. When people are talking about technology as a factor of production, they are really talking about entrepreneurship. They're talking about the know-how of putting together the other factors of production in order to produce that output. Finally, I wanna leave on one idea, the idea of the two types of things that could be produced from all of these factors of production. Broadly speaking, we could produce something that could be used to produce more things, and we already talk about it. We could be, in that situation, be producing capital goods. So, that could be that we are constructing a factory that itself maybe produces tools for other people to use in some other production process. The other option we have is to produce what are known as consumption goods. Consumption goods. Consumption goods are goods that are just used. It might make people happy, they might find pleasure in it, but it's not being used to produce other things. And because our production resources are scarce, there's a trade-off when a society or a factory or whoever decides how much capital to produce versus how much consumption goods. You need some consumption goods; otherwise, frankly, we wouldn't have clothing on. We wouldn't be eating nice meals. We wouldn't be able to enjoy our lives. But at the same time, you also need capital. If we did only consumption goods, at some point, we wouldn't have all the things we need to produce the consumption goods. So, it's a very interesting trade-off that we'll explore more in future videos. |
Khan_Academy_AP_Microeconomics | Income_elasticity_of_demand_AP_Microeconomics_Khan_Academy.txt | - [Instructor] In previous videos, we have talked about the idea of price elasticity, and it might have been price elasticity of demand or price elasticity of supply. But in both situations, we were talking about our percent change in quantity over our percent change in price. If we're talking about price elasticity of demand, it would be the percent change in quantity demanded over the percent change in price. And if we're talking about price elasticity of supply, it would be our percent change in quantity supplied over percent change in price. And as we talked about in many videos, this is a way of measuring how sensitive is quantity demanded or supplied to a change in price. What we're going to see in this video is that this is not the only type of elasticity that economists will look at. There are many types of elasticity, where they want to see "How sensitive is one thing to another?" For example, you could look at the percent change, percent change in labor supply, so you could say quantity of labor, that would be our labor supply, divided by our percent change in wages. I'll just write it out, wages. And you could view that as a percent change in the price of labor. You might say, "Hey, this is just a price elasticity "of supply being particular to the labor market." But you can even see things, and we'll have a whole video about this, probably my next video that I will make, where you could have the percent change in, let's say quantity demanded of one good divided by, so let me call it good one divided by the percent change in price of not that good, then we would have price elasticity, but of good two. And so this is actually, thinking about how good one is a substitute for the other, and we'll go into a lot more depth there. But the focus of this video, as you can imagine because it was already written down in a clean font right over here is Income Elasticity. And here, we're gonna think about the income elasticity of demand. And you could imagine what that would be. This is going to be our percent change in quantity demanded, demanded, divided by, instead of thinking about the percent change in price of that good or the service, we're going to think about the percent change, change, in income of the people who might be in the market for that good. Normally you would expect that when our percent change in income goes up, that the same thing would happen to our percent change in quantity demanded. For example, let's say we're talking about the market for vacations, well, as my income, as most people's incomes go up, they might be able to afford larger or better vacations. And that would be a normal good. This is a situation of a normal good. Normal good, just as what you would expect. But you could actually have the other way around. You could imagine a situation where even though you have an increase in your percent change in income, that does not lead to an increase in your percent change in quantity demanded. In fact, it could lead to a decrease in the percent change in quantity demanded. Or another way of thinking about it, your quantity demanded could actually go down, so you would have a negative, a negative percent change right over here. Now could you image any situations like that? Well, imagine if we're talking about the market for car mechanic services. As people have more income, they might be able to afford better cars that are more reliable, that break down less, and then they would have to go to the car mechanic less. And so that situation, where our demand would actually go down when our income goes up, or our percent change will become negative when we have a positive percent change income, that would be, that is known as an inferior good. Inferior good. There's two big things to take away. One, you don't just have to think about price elasticity of supply or demand, there are other types of elasticities. But just to hit the point home on income elasticity, let's look at a few examples. We're told: suppose that when people's income increases by 20%, they buy 10% less fast food. In this situation, what type of good would fast food be? Pause this video and think about it. Well, their income is increasing but their demand is decreasing. That's the situation we just talked about. This is an inferior good. Inferior good. And for kicks, what is the income elasticity of demand right over here, calculate that. Just remember, our income elasticity of demand is just going to be our percent change in quantity demanded divided by our percent change, instead of price, we're going to say in income. I'll just write percent change of income, which is going to be what? Well, we know our percent change in income. It went up by 20% in this example, and what happened to our quantity demanded? Well, it went down by 10%, so negative 10%. And so here you have an income elasticity of demand of negative one half, or negative 0.5. Let's do another example. Suppose we knew that when people's income increased by 5% in a country, the demand for healthcare increased by 10%. What kind of good do people consider healthcare: Normal or inferior? First, calculate the income elasticity of demand for this example, and then answer these questions. All right, so first we are, our income elasticity of demand. Let's see, when our income increases by 5%, so we have a 5% increase in income, our demand for healthcare increases by 10%. Our demand for healthcare increases by 10%, so we get a positive income elasticity of demand. And so in general, if this thing is positive, you're dealing with a normal good. As income goes up, then you similarly see quantity demanded going up. This is a normal good. |
Khan_Academy_AP_Microeconomics | Comparative_advantage_worked_example_Basic_economics_concepts_AP_Macroeconomics_Khan_Academy.txt | - [Instructor] The countries of Kalos and Johto can produce two goods. Shiny charms and berries. Yep, you got to love these worlds created in these economics questions. The table below describe the production possibilities of each country in a day. So here it tells us that Kalos, if it puts all of its energy behind charms, it can produce 10 charms in a day. But if it put all of its energy behind berries, it can produce 20 berries in a day. And then Johto, all of its energy behind charms, 25, all of its energy behind berries, 75. Given these numbers are based on both countries having the same labor and capital inputs, who has the absolute advantage in charms? So pause the video and see if you can figure this out. All right, so let's just remind ourselves. Absolute advantage is just who is more efficient? Who, given the same inputs, can produce more? And they told us that these countries, they have the same labor and capital inputs, so this is really just a question of who can produce more charms in a day? And you can see very clearly that Johto can produce more charms in a day. And so I would say Johto, because they produce, let me write that a little bit neater, they produce more charms per day. Charms per day. With same inputs. Same inputs. So they are more efficient. More efficient. So they have the absolute advantage. Now this is an interesting thing, because our intuition might say well whoever has the absolute advantage, maybe they're the ones that should be producing charms. But this is what's interesting when we study comparative advantage. That is not always the case. And I suspect that this question will lead us there. All right, next question. They say calculate the opportunity cost in Kalos of charms. So the opportunity cost, in Kalos, of charms. So when Kalos decides to produce 10 charms, they're trading off 20 berries. Or another way of thinking about it, it costs them 20 berries to produce 10 charms. So we could say it costs 20 berries for 10 charms, which is equal to two berries, two berries per charm, in Kalos. So there you have it. The opportunity cost, they trade off two berries per charm. And actually, let me make it a little column here. The opportunity cost. So this is two berries per charm. And I have a feeling, and if you're taking an exam, say an AP exam, it's not a bad idea to just fill this thing out, so what is the opportunity cost, they haven't asked us that yet, but I'm just gonna do it really fast. What is the opportunity cost of charms in Johto? Well, they are trading off, to produce 25 charms, they trade off 75 berries. So this would be 75 divided by 25, this would be three berries per charm. 75 berries for 25 charms is three berries per charm. And if you want to know the opportunity cost of berries, well you can just take the reciprocal of each of these. So in Kalos, the opportunity cost is one half charms, charms per berry. And then in Johto, it is one third charms per berry. That if they wanted to produce 25 berries, if they wanted to produce 75 berries, they would trade off 25 charms. So it would cost them 25 charms to produce 75 berries, or one third of a charm per berry. So I'm just doing a little bit of extra. But then it's gonna be useful, because in the next question, they actually are asking us, who, we'll scroll up a little bit. They're saying who has the comparative advantage in berries, explain. So berries, whoever has the lower opportunity cost has the comparative advantage. So we see here that Johto has the lower opportunity cost in berries. One third is lower than one half. It's a lower opportunity cost of producing a berry. So Johto has one third charms per berry opportunity cost, opportunity cost. Which is lower than Kalos', Kalos' one half charms per berry opportunity cost. So Johto has comparative advantage. So Johto has comparative, comparative advantage in berries. And I apologize a little bit for my penmanship, I'm trying to save time by writing a little bit fast, but hopefully me saying it out loud at the same time is making it somewhat legible. All right. So the next question. If these countries were to specialize in trade, who would produce which good, explain. Well whoever have the comparative advantage of each will produce that one. So Kalos has comparative advantage, Kalos has lower opportunity cost in, in let's see, they have the lower opportunity cost when you compare them to, oh let me see, let me put it this way. For charms, let me write I this way, Kalos has a lower opportunity cost for charms. Kalos has advantage in charms. And then we already said Johto has advantage in berries. And so, Kalos, I keep saying it weird, Kalos produces charms, Johto produces berries, produces berries. And once again, this goes back to something we touched on at the beginning of the video. Even though Johto has the absolute advantage, in fact they have the absolute advantage in either, Johto is not, even though they can produce charms way more efficiently than Kalos, Johto is actually in this, if you buy all the arguments of comparative advantages, Johto should actually produce the berries, while Kalos should produce the charms, because they have a lower opportunity cost in terms of berries. Now let's answer this last question right over here. What would be a trading price that Johto and Kalos would agree on to trade charms for? Now you might be saying, well what's a price, I'm used to saying that in terms of just maybe dollars or some type of currency, how do I answer a price right over here? Well, the key is that we can give a price in terms of opportunity cost. So they want a price of charms. So it really could be in terms of berries. So let's see. Let's look at each of their cost of charms. So, Kalos' opportunity costs of a charm is two berries per charm, Johto's in three berries per charm. So let me rewrite that over here. So Kalos, Kalos opportunity cost of charms is two berries per charm. And then Johto opportunity cost of charms is three berries per charm. And here we're going to appreciate why comparative advantage works. We said that Kalos would be the one that would focus on the charms. And so notice. If they can sell the charms to Johto for something that is higher than their opportunity cost, and lower than Johto's opportunity cost, then they both benefit. And so a good price, let's say you could go halfway between the two, but it really could be anything in between the two, let's say 2.5 berries per charm. They both benefit. So they would trade at this, trade at 2.5 berries per charm. Why does this make sense for Johto, even though they have the absolute advantage? Well if they produce nothing but charms, it would cost them, or no matter what they do, it'll cost them three berries per charm. But now they figured out a way, through trade, to get charms at two and a half berries per charm. And so this will be a better deal for Johto. And so one thing to appreciate when we talk about comparative advantage, some people think that it's about one country benefiting more than the other. But if we assume all of the assumptions about comparative advantage in our models, then it's actually about both countries that are trading benefit. They will both be better off. They will both get gains from trade, and both will be better off. |
Khan_Academy_AP_Microeconomics | Deriving_demand_curve_from_tweaking_marginal_utility_per_dollar_Khan_Academy.txt | A few videos ago we saw that we could maximize total utility given our $5 spending by calculating the marginal utility per dollar for each incremental dollar we could spend on each of these goods and then just for each dollar maximizing it. Our 1st dollar, we got 100 utility points per dollar for that first chocolate bar and that was more than any than the first fruits, so that's where we spent it. Then we got more for the 2nd chocolate bar for that 1st pound of fruit, so we spend it there. Then for the 3rd pound or for the 3rd bar of chocolate. Then it became equal to spend for the 1st pound of fruit so then we spent the next $2 on that 1st pound of fruit because the price of fruit were $2. What I want to do in this video is explore what happens when I change the price of the chocolate bars. What happens to our marginal utility per dollar over here? In particular, what happens to the quantity demanded? If you think about what we're doing it, we figured out with 1 price what was the quantity demanded, we demanded 3 bars. If we change the price and we get another quantity demanded, we're essentially starting to plot our our demand curve and we can actually derive our demand curve from this information right over here. Let's see how we could do that. Let's now assume that our chocolate bars are $2. Now we're going to calculate. The marginal utility per dollar, this applies to both of these columns. This is for what it was $1 per bar, this is now when it's $2 per bar. Well, for that first bar, I'm still getting 100 points of marginal utility, but now it's $2. So 100 divided 2 is going to give me 50 marginal utility points per dollar. Then for that next bar, I get 80 marginal utility points. I'm still enjoying it but enjoying it a little bit less but I'm paying $2 for it. I'm getting 80 divided by 2 is 40 points, you know, and I'm just giving these arbitrary units, 40 points per dollar. Then the 3rd bar is 30 points per dollar. Then the 4th bar is 20 points per dollar. Now, how would I spend my $5? Let me do this a little bit, let me do it over here. How would I spend my $5 now? My first dollar, where would I get the most marginal utility per dollar? Where would I get the most bang for my buck? My very 1st dollar, I can either buy half a bar here, I could buy half a here and I'm assuming that, for the sake of simplicity let's assume that I get the same marginal utility per dollar for the 1st half a bar and for the 1st bar. That is constant until I get to one entire bar. That's also true for, and even if I buy a fraction of the pound here. My 1st dollar, I can't use these numbers, this is when the bars were a dollar per bar, now they're $2 per bar. This is the reality. Now actually it makes sense for me to, at least, for that 1st dollar I can buy a half pound of my fruit at a marginal utility per dollar of 60. My 1st dollar will go towards .5 pounds of fruit and I'm getting a marginal utility per dollar of 60. Where is my 2nd dollar going to go? Well, I can still get another half pound at a marginal utility of 60. Remember, we have to ignore these right here for the sake of this argument or for the sake of this scenario right now. I could still get another half pound for marginal utility per dollar of 60, so now I buy another half pound of fruit and my marginal utility per dollar is 60. Now, where is my 3rd dollar, my 3rd dollar going to be spent? Well, I could spend it now at a rate of a dollar per half bar, or $2 per bar for chocolate or a dollar per half bar, $2 per bar for fruit over here. I'm actually neutral. I could spend it. Let me just, for the sake of fun, say list on half a bar of chocolate and my marginal utility per dollar is 50. Then my 4th dollar, once again, I could do a couple of different things here. I could buy another half bar because I can buy up to a whole bar at this marginal utility per dollar up to a whole bar. So why not do that? I'll buy another half chocolate bar, so now I have a whole chocolate bar. Once again, I'm able to continue buying that at 50 utility units per dollar. Then my 5th dollar over here, what would I do with that? Well, I don't want to buy any more chocolate bars because my marginal utility per dollar of the chocolate bar because I've exhausted what I can buy at this utility, this utility per dollar. My marginal utility per dollar has gone down now, but now I could still buy fruit at that same 50. Now, with that dollar, since the fruit is $2 per pound, I can buy another half pound of fruit at a marginal utility per dollar rate of 50. Now I buy another half pound of fruit at a marginal utility per dollar of 50. You can calculate the total marginal utility I got, this is the marginal utility per dollar and this is a dollar spent at that marginal utility per dollar. My total utility I should say, the marginal utility is the increment, but my total utility now is 60 + 60 is 120 plus 50 + 50 + 50. So it's 120 + 150 = 270 total utility. Even more interesting here, let's think about the quantity of chocolate bars that I have now bought once the price is gone up. I have now bought exactly 1 chocolate bar. You could say my 3rd and 4th dollars were spent on 1 bar right over here, I bought 1 bar. Let's think about it, all else equal. Remember, ceteris paribus. We haven't changed the price of fruit, we haven't changed consumer preferences which would have changed your marginal utility numbers right over here. All else equal. What happened just when we changed the price of chocolate bars? Let me write it down. Just think about chocolate. If we just think about chocolate bars. Let me write price and quantity. When the price was $1 the quantity demanded was 3 bars. That was the 1st video we saw on marginal utility. We demanded 3 bars. Now when the price has gone up to $2, the quantity demanded is exactly 1 bar. We could do everything in between, we could see what happens if the price was a dollar 50 or if the price was 50 cents, if we actually lower the price. We would see how the, and there might actually be a situation where you would have to have higher quantities here especially when you lower the price. But by doing that, assuming you have enough rows here and we might not have it, if you lower the price. Assuming you have the marginal utility at different quantities for the two goods, you can figure out exactly how much chocolates someone would buy given different changes in price. We at least have 2 points for the demand curve now. If we assume that this is price and this is quantity right over here, when the price was $1, the quantity demanded was 3, and when the price is $2, the quantity demanded is 1. There, we have 2 points for our demand curve. Our demand curve might look something like that. If it was linear, it would go straight. It would go something straight like that. But we at least have 2 points on the curve and we could keep trying different prices out using these information to figure out the exact shape of that curve. |
Khan_Academy_AP_Microeconomics | Marginal_revenue_below_average_total_cost_Microeconomics_Khan_Academy.txt | In the last video, we finished up asking ourselves, how much do we produce if the market price is at $0.45? And just going with the logic that we introduced in the last video, you want to produce as much as possible to spread out the fixed cost. But you don't want to produce so much that the marginal cost is higher than your marginal revenue. And your marginal revenue is your market price. Every unit, every incremental unit, you're going to get $0.45. So you want to look at the quantity where your marginal revenue, the $0.45, is equal to your marginal cost. So we could look at it over here. So if we look at our marginal revenue, let's say $0.45 is right over there. You want to look where the $0.45 is equal to your marginal cost. And it looks like it is right over there. Now we could even see it on our table. When does our marginal cost equal $0.45? It equals that when we produce 8,000 gallons of our juice. Now the reason why this is somewhat interesting is at that point the amount of revenue that we're getting per unit, our marginal revenue, is less than our total cost per unit. We're selling each unit at $0.45, but our total cost for each of those units is $0.48 on average. So this right over here is our total cost. So you might say, look, I'm making a loss on every unit. The total amount of revenue I'm getting is a smaller rectangle over here. It's the quantity times the marginal revenue per unit. So this is the amount of revenue that I'm getting. Let me color it in carefully. That is the amount of revenue that I'm getting. While my costs are this larger rectangle. My quantity times my average total cost per unit. And so what I end up with is if you take that revenue and you subtract out that quantity, you end up with a loss of exactly this much. You are operating, in this situation, at a loss. You're operating at a loss when you are producing 8,000 units and you're getting $0.45 per unit. So does it make sense for you to do this? And we can even figure out the loss. You are producing 8,000 units. You're selling them for $0.45 a unit. And it costs you $0.48 per unit to produce them on average when you put all the costs in, $0.48 cents per unit. So you are losing $0.03 per unit, I guess gallon. We're talking about orange juice here. And it's times 8,000 gallons, means that we are losing we are losing $240. 8,000 times $0.06 is 24,000 cents, which is the same thing as $240. So does it make sense for us to do this? Well, one way to think about it, let's say we didn't do it. Let's say we're like, hey, I'm not going to produce any gallons. Well then what's going to be our loss? Well, we're assuming that this is our fixed cost. We've already committed ourselves to this expenditure right over here. Whether we produce no drops of orange juice, we are still going to be spending $1,000/ So if we produce nothing, we are guaranteeing ourselves a weekly loss of $1,000. And so this is at least better than that. So by starting to produce some units, we are at least able to offset some of that loss. And we're spreading out that fixed costs over more and more and more gallons. And you might say, hey, well why don't I just keep producing more and more units? Why don't I go here and maybe I produce 9,000 units where the marginal cost all of a sudden is higher than our marginal revenue? And the reason why that won't make any sense to do is because if you produce that many units, then all of a sudden each of those incremental units that you're producing beyond the 8,000, you're losing money on those. That 8,000 in first unit, the marginal cost is going to be higher than the marginal revenue that you're bringing in on that unit. So you're going to be losing money. You're going to start having a lower profit than even the negative $240 loss. It'll start going at a negative 240 something, negative 250, and so forth and so one. So you still don't want to produce beyond that point. And we'll touch more deeply on it in future videos, but this is essentially what differentiates the short-term supply curve from the long-run supply curve. In the short-term, we're going to assume that we have these fixed costs. And so it's just going to make sense to produce equivalent to our marginal cost. But over the long-run, maybe our fixed items, our capital, our machinery wears off or maybe the contract for my employees wear off, and then we have a different cost structure over the long-term. But we'll think about that in another video. But the simple answer is, assuming these really are your fixed costs, you still want to produce as many units as possible so that your marginal cost is equal to your marginal revenue, which in this case is the market price. We are price takers. So it actually is a rational thing to produce 8,000 units and take a loss on that and take a $240 per week loss as opposed to just producing nothing and taking $1,000 per week loss. Now it might not be rational once these things have been worn out-- your robots and the employees' contracts. It might not be rational to continue them past their term. And we'll think about that more because obviously, we are running at a loss. This is not necessarily a good business to be in, but now that we've gotten into the business, we might as well stay in it in order to recoup some of our costs here or at least spread them out, or at least not have $1,000 per week loss. Anyway, see you in the next video. |
Khan_Academy_AP_Microeconomics | Normal_and_inferior_goods_Supply_demand_and_market_equilibrium_Microeconomics_Khan_Academy.txt | What I want to do in this video is think about the demand curve for two different products. So this is some laptop that's on the market. And this, let's just say, is the cheapest car that happens to be on the market this is actually a picture of a 1985 assuming this is the cheapest car on the market. So let's just think about their hypothetical demand curves right now So once again, on the vertical axis, we're going to put price, and on the horizontal axis, we put quantity, and then over here let me do it for the same thing So this is price, and this right over here is quantity. And both of them satisfy the law of demand if the price is really high the quantity demanded is going to be really low for the laptop and so it might be right over there and if the price is low the quantity demanded is going to increase. So, the demand curve might look something like that. And it doesn't have to be a curve, or doesn't have to be a line, it could be a curve or anything like that. So that is the current demand for the laptop All else equal, so we're not talking about shifting any of those other factors that we've been talking about in the last few videos. Now we can draw a similar demand curve for this very cheap automobile. If the price is high, very few people are going to want to buy it, and I'm not going to specify what the price is, but this is a general idea if the price is higher, fewer people are going to want to buy it If the price is lower, more people are going to want to buy it So this demand curve will also have the same shape from the top left to the bottom right it satisfies the law of demand. So once again, that is the current demand. Now let's think about how the demand for each of these goods might change depending on changes in income. So we're going to focus on the income factor the income effect, for this video and see how these 2 products might change. So let's just assume that income in the general population goes up. So for something like a laptop, wow, if more people are making more money especially in real terms they have more money to spend well for any given price point, at any given price point, there's going to be a higher quantity that's demanded. At any given price point, higher quantity demanded. And so if income goes up for this laptop, the demand will increase. And the way we show demand increasing is the whole curve shifts to the right so this right over here demand increased demand went up when income went up. And this makes complete sense and if income were to go down, demand would go down because people would have less money to buy something like a laptop. And this is the case for most goods we call things like this, when income goes up, demand goes up, whole curve shifts to the right income goes down, demand goes down, whole curve shifts to the left We call this a normal good. So this right over here is a normal good. Now let's think about what happens with the cheapest car on the market. And let's assume we're in a developed country where almost everyone has some form of a car. Now, what happens when income goes up? So people have more money but are they going to spend that money buying the cheapest car on the market? Well, in most cases, if income goes up generally, people say, well I have a little bit more money, maybe I'll buy a slightly nicer car. So, and maybe in particular the people who were going to buy this car at any given price point So this price point, the people who were going to buy the car will say Wait! I can now afford a better car! Why should I you know, this is not safe maybe or not as safe as the other cars, and I want to impress my friends from high school and all that, so something very strange might happen for this car, the demand for this car. It actually will decrease so the whole curve could shift to the left. So income is a very strange thing for this good because income increasing maybe people say, hey you know what, I could trade out of this good I could get a good that I'd rather have than just getting more of this thing right over here Demand went down. And goods like this, we call them inferior goods. And the general way to think about inferior goods are the goods that people will want to not own if they had more money they would want to buy, I guess, less inferior goods. Or another way to think about it is, if income were to go down, and more people are budget strapped and they can't afford the Mercedes-Benz or the BMW or even the mid-sized sedan anymore, so if income were to go down and things were getting tighter, more people would want this car more people would have to trade down to this because they're strapped, they're tightening their belts and so you'll have this strange situation where if income goes down, demand would go up for this thing So income goes down, demand goes up. Remember, we're talking about demand, we're talking about the entire shifting of the curve. At any given price point, the quantity demanded will go up. Because, this is, or we're assuming, is the cheapest car on the market. So, and likewise, if income were to go down for a normal good, it'll do what you'll expect, demand would go down. So this, an inferior good, does the opposite of a normal good when we're talking about the income effect, the inferior good will do the opposite of a normal good and that's because people want to trade out of it when their income goes up or they don't want to buy it or they want to buy something nicer. And when their income goes down, they'll say I have to buy this thing, so you know, let me just do it. |
Khan_Academy_AP_Microeconomics | Marginal_cost_average_variable_cost_and_average_total_cost_AP_Microeconomics_Khan_Academy.txt | - [Instructor] Let's say that we run ABC Watch Factory and we want to understand the economics of our business. So, what we have in this table is some data that we've already been able to estimate or measure based on how our business is running and then we're gonna be able to figure out some other things based on this data. This first column is fixed costs, our monthly fixed costs, so these are the things that we can't really change in the short run regardless of how many people we hire or how many units we produce, so that might be the rent on our facilities or the cost of renting the equipment and so, for us that's $5,000 a month. Then you have your labor units and for the sake for this model, we'll say that a labor unit is a full-time employee who's at the factory working every working day in a month and so, you can see, we can go from one person working full time every working day in a month, all the way up to six. Now, this is the variable cost and for simplicity, this is mainly driven by the labor units and a real-world example would be driven by the labor units, it would be driven by how much material we're using to produce the watches but we have our variable cost right over here. And then we have our total cost which is just simply the fixed costs plus the variable cost for any given level of labor units and then we know how many watches we can produce in a month based on our number of labor units or you could view it as based on our total costs or based on our fixed and variable cost. Now, what we have here are other things that we would wanna look at. If we really wanna understand how our factory works. So, this is the marginal product of labor, MPL for short, then you have your marginal cost, then you have your average variable cost, then you have your average fixed costs and then you have your average total costs, so like always, pause this video and try to fill what these values would be for even one row of this table and then I'll do it with you. Now let's do it together. Let's start with marginal product of labor. Let's remind ourselves what that is. That says for every incremental labor unit, how much more are we able to produce? And so, we'd have to start at the second one 'cause we have to think about it incremental labor unit. So, as we go from one to labor units, we were able to go from 10 to 25 total output, so we were able to produce 15 more watches. I could just type in 15 but it's even better to do it with a formula so I can just scroll it down the rest of the rows. So, in this formula, I wanna find the difference in my total output, so 25, that cell minus this cell, that that's saying hey look, I was able to grow 15 output or increase my output by 15 when I increase labor by two minus one. And then I got my marginal product of labor is 15 when I went from one employee to two and then I can just figure that out for the other rows, that's the value of using a spreadsheet. My marginal product of labor when I went from two employees to three employees is 20, so that means by adding that third employee, I'm able to produce 20 more watches per month and so, you might be noticing two interesting trends here. Initially my marginal product of labor seems to be increasing and then it seems to be decreasing. And that's consistent with the way a lot of businesses or factories work which is initially you're getting the benefits of specialization where if you only have one person working in your factory, they have to do everything, they have to polish the glass and bring in the boxes and talk to your suppliers and fit the gears on your watches and whatever and do the wiring while as you add more people, they can start to specialize. One person can specialize on assembly. Another person can specialize on bringing the boxes in and so, initially you have these benefits of specialization and so, people can focus on just one skill and do it well but then you start getting diminishing returns, the office starts getting crowded, people are waiting for different supplies, they have to get out of each other's way and so, then you see this diminishing return trend where the marginal product of labor starts going down for those incremental labor units. Next, we'll think about marginal cost and as we'll see, the marginal cost trend's going the other direction as the marginal product of labor. So, marginal cost is just for every, for a certain increment and output, how much is that costing us? So, for example, if we are going from 10 to 25 output, for that 15 increment and output, how much is that costing us and I would say costing us on average but I don't want you to get confused, we're not talking about average variable cost or average fixed cost or average total cost but that would be, let's see our costs went from 7,000 to 11,000, so we'll do 11,000 minus 7,000. That is our change in cost divided by our change in total output. So, that's going to be divided by the 25 minus the 10. And we could just scroll this down, we'll extend that formula and you can see this trend that is as the marginal product of labor is increasing, your marginal cost is decreasing and it makes sense, in some ways we're getting more efficient through the specialization and what else but then once you have diminishing returns, diminishing marginal returns, your marginal cost is going up. And now we can do the, I guess you could say the average cost. So, first average of variable cost. That's just taking your variable cost and dividing it by your total output. And so, for at least those first 25 units, they cost on average or just the variable component, you have to be careful is $240. If you talk about the fixed component, well, that's just gonna be our fixed cost divided by our total units and then our average total cost, that's gonna be our total cost divided by those 25 units and so, you can see, our average total cost for those first 25 units is $440 and then it can be broken up between how much of that $440 is variable versus fixed and then we can just extend these formulas down, the magic of spreadsheets and what's interesting here and it's not gonna be going to be so obvious just looking at this spreadsheet is something interesting is happening when marginal cost seems to intersect either your average variable cost or your average total cost that at some point you're average variable cost, you see that same trend, it's trending down and then it starts to trend up again. Average total cost is trending down but then it trends up again and as we'll see when we graph it, the point at which marginal cost intersects with the average variable cost, that's when you have that change in direction of average variable cost and then same thing is true of when marginal cost intersects with average total cost. That's when you have that change in direction. Average fixed cost just continues to go down because those fixed costs aren't going up as you have more and more output, so you have those same fixed costs, you could view it has spread amongst more and more output, so that's just going to keep asymptoting downward. In the next video, we'll actually graph that and see these trends visually. |
Khan_Academy_AP_Microeconomics | Economic_models_Basic_economics_concepts_AP_Macroeconomics_and_Microeconomics_Khan_Academy.txt | - When you think about what the field of economics is about, it is quite daunting. An economy is made up of millions or even billions of actors organized in incredibly complex ways. Write down, this is complex real-world and each of the actors, human beings or organizations, these are incredibly complex. A human brain. I can't predict what you're going to do the next second much less what you're going to do the next day or the next year and imagine trying to make insights about what millions or billions of people will do but the field of economics has borrowed an idea from other fields. So for example, in chemistry, chemists have tried to understand at a high level, well, how do molecules in a container behave? Let's say molecules of gas. Well, you could imagine if you have a container here with trillions upon trillions of molecules, this is incredibly complex but by making some simplifying assumptions about the type of interactions these particles will have or don't have, they can come up with models like the ideal gas law which you might be familiar with or not from your chemistry class that relates the pressure to the volume to the number of particles you have to the actual temperature and so this right over here where you're taking something that's hairy and complex and making simplifying assumptions to help you understand it, this thing right over here is a model and this is in other fields as well. Sometimes, it's not an equation. Sometimes, it might be a simpler organism. For example, in biology, human beings are incredibly complex organisms and not only are they incredibly complex but certain forms of experimentation would also feel fairly unethical to our modern moral ethos and so what do biologists do? Well, they make simplifying assumptions or they pare down, they say, okay, we can't do that study on human beings but maybe we can simplify the problem by looking at simpler organisms. Maybe you can look at an individual cell right over here. Maybe you can look at things like fruit flies which are famous in the study of genetics. Maybe you can even look at fairly complex organisms. Even a mouse is a very complex thing but it's still simpler than a human being and at least to our modern ethics, we're willing to do certain things to mice that we aren't willing to do to human beings and so that's why in a biological context, you will hear people talk about things like a mouse model where they will test a drug on a mouse or try to understand how something happens in a mouse and then say, well, that's a pretty good indication that might be happening to human beings. In fact, when they do drug trials in medicine, they often will do it on mice first and when they have good confidence that it works there and that it's fairly safe, only then will they start to do the experiments on human beings. Well, economists are doing the same thing. Even before the advent of computers and computer models, economists make simplifying assumptions, assumptions like all of the actors in an economy are rational which we already know is not exactly true. I'm not always rational and I definitely know people who aren't always rational. They're simplifying assumptions that all of the people in an economy have the same access to information or that they all even have perfect information which we also know isn't necessarily true in a real economy. So depending on the model, there are going to be these simplifying assumptions that take this large, complex real-world thing and try to break it down into simple equations or lines or charts. We have models early on in our economic study. We will see things like the production possibility frontier where it assumes that you're only trading off between two things and everything else is equal, this notion of ceteris, ceteris paribus which means all things equal. In a real-world, you're not gonna be able to say, hey, let's just pick between these two things and then hold everything else equal. There's hundreds or thousands or millions of variables are operating but if you wanna make a model, maybe we can make these assumptions. Same thing with famous price equilibria that we're going to study later on where you have supply and demand and then you have these notions of equilibrium prices and quantities. These also make similar types of assumptions about rational actors and perfect information and these economic models can be very useful and that's why most of your study in a first year economics course is of these models. Now, with that said, you should also take them with a grain of salt and you shouldn't just accept them as the absolute description of reality. In fact, that's when economic models can get dangerous. You always have to be conscientious of what are those assumptions you made? In fact, Nobel Prizes have been won in economics by revisiting some simplifying assumptions and coming up with new models. The other difficulty about economics is it's hard to test it in as absolute a way definitely as something like chemistry or physics but even in biology where you're dealing with similarly complex systems, a human body and an economy, these are both extremely complex systems. If I wanna see in medicine whether a certain medication works, I can do a clinical trial. I could take hundreds or thousands of people and give maybe half of them the drug and I could try to control for a bunch of different variables but in economics, you can't take a thousand different economies that look very similar in what you think matters and then apply some type of economic prescription to half of them and then see what happens, to see whether your model is exactly true or whether your prescription for what makes an economy grow faster actually works and so the big takeaway, models are valuable across the various sciences including in economics but economics straddles between a social science and the sciences like chemistry or physics because you can't run experiments in the same way and we often make simplifying assumptions that even though we know aren't exactly true, they're the only way that we're able to make sense of an incredibly complex real-world. |
Khan_Academy_AP_Microeconomics | More_on_total_revenue_and_elasticity_Elasticity_Microeconomics_Khan_Academy.txt | I want to do one more video on total revenue and price elasticity of demand. Just to make sure that you, the relationship between the two is an intuitive one. So let's draw an arbitrary demand curve. So this is my price axis. That is my quantity demanded axis. Quantity axis. And let me just draw an arbitrary demand curve right over here. So let's say that is my demand curve. And let's pick some price and quantities on this demand curve. So let's say that the price is up here. Let's call that P1. And then, the quantity demanded. Let's call that Q1. And we already know that the total revenue is the area of this rectangle right over here. This is the total revenue. It's just the price times the quantity. If I'm selling 2 burgers an hour and for $9 a burger, I'm going to make $18 per hour. That's going to be this area right over here. Now, let's assume in this part of the curve that the price elasticity of demand is greater than 1. So we are elastic. So let me write this. So the price, the elasticity of demand-- actually, I should say the absolute value of the elasticity of demand. It will be actually be a negative number. But the absolute value of the elasticity of demand is greater than 1 which means for a 1% drop in price you have more, you have a greater than 1% increase in quantity. And that comes straight out of the expression or our formula for what elasticity is. Remember, elasticity is our percent change in quantity over percent change in price. So if this, if the absolute value of this is greater than 1-- these move in opposite directions. That's why it would be negative. But if we say the absolute value of this is greater than 1, that means that this quantity is going to be larger than this quantity. So if we have a 1% drop in price, the change in our quantity is going to be greater than 1%. And so for point right over here, if we lower this by 1%, we're going to increase this by more than 1%. So any drop in our any reduction in our height will be more than made up for. And this is generally the case. Will be more than made up for by an increase in our width. So total revenue will increase. So when price drops, so 1% drop in price and a larger than 1% increase in quantity means that total revenue will go up. Now, if we go down here. If we go down to this part of the curve. And let's say that this-- let's call this-- let's call that P2. And let's call that quantity 2. And then, this area right over here would be total revenue 2. Let's call that total revenue 1 over there. Price times the quantity. Now, what's happening over here? We're going to assume that our price elasticity of demand, the absolute value of it over here, is less than 1. So the absolute value of our price elasticity of demand is less than 1 at this point in the curve. And all that is a fancy way of saying that for a 1% drop in price, we get less than a 1% drop. Sorry, less than a 1% increase. They move in opposite directions. 1% increase in quantity. So we're lowering the height. If we have a 1% drop, we're lowering that by 1%. But we're not getting a 1% increase in our width. So the width isn't going to be increasing that much. So in general, this is going to result in a lowering of this area. This area will get smaller. We're reducing our height more than we are expanding our width. So in this situation, total revenue would go down. And remember, this is an elastic situation. So when it is elastic, total revenue tends to go up. And when it is inelastic-- I want to say, when it's elastic a drop in price tends to make total revenue go up. And when it is inelastic, a drop in price tends to make total revenue go down. And then, you can imagine, right when you're it unit elasticity, someplace around there, a 1% a drop in price will result in exactly 1% increase in quantity demanded. And so they will trade off. You won't get a noticeable change in your revenue. And the reason why I say that is that actually some, many econ textbooks will tell you that you don't get a change in revenue. But if you actually will do a detailed look at that math-- let me write it over here. So the absolute value of the price elasticity of demand at that point is 1. Which tells us that a 1% drop in price will, or goes along with a 1% increase in quantity. But if you look at the math. So if the old area. So let's call this price 3. And let's call this quantity 3 right over here. And so total revenue 3-- let me do this in a new color-- which is this area right over there, is going to be equal to price 3 times the quantity 3. Now, if we increase price by, or if we decrease price by 1%, then this will become 0.99 times our price. And if we increase our quantity by 1%, then this will become 1.01 times our quantity. Now, let's think about what this number right over here is. And this is why I'm saying it's not exactly, the total revenues aren't going to be exactly unchanged. If you multiply 0.99 times 1.01, you don't you get exactly 1. You don't get exactly 1. Another way to think about it, 0.99 times 0.01 is going to be 1% less than 1.01. And 1% of 1.01 is slightly larger than 1. Or another way to think about it, this value is going to be 1% larger than 0.99. And 1% larger than, 1% of a 0.99 is less than 1. So it's not going to get a 1. And you can see it with your calculator. 0.99 times 1.01 gets you to very close to 1. So this is going to be equal to 0.9999 times P3 Q3, which is equal to 0.9999 times total revenue 3. But it is-- total revenue 3. But it is roughly unchanged. So we can-- that's the general rule of thumb. So when you are at unit elasticity, then, a decrease in price roughly says, no change, approximately no change in total revenue. So I just wanted to make sure that it makes sense. It really just comes from these areas. If you're reducing the height by a less than you're increasing the width, obviously, the area is going to increase. Or most of the cases, I should say. It depends on where you are. If you are, if you're compensating, whatever you reduce the height, you are compensating perfectly with the increase in width, then you're not going to have a change in revenue. And if you decrease the height by more, if you're taking more area from the top than you're adding on the width, then you're going to have a total decrease in total revenue. |
Khan_Academy_AP_Microeconomics | Comparative_advantage_and_absolute_advantage_Microeconomics_Khan_Academy.txt | What I want to do in this video is make sure we understand the difference between "comparative advantage" and "absolute advantage". What we saw in the last video is that Patty had a comparative advantage in plates relative to Charlie because her opportunity cost of producing one plate was lower than Charlie's opportunity cost of producing a plate. Hers was one-third of a cup, his was three cups. So, that's why it made sense for her to specialize in plates. Charlie on the other hand had a comparative advantage in cups; his opportunity cost for producing a cup was only a third of a plate, while Patty's was three plates. So that's why he specialized in cups. Now, we can't confuse this with absolute advantage. Absolute advantage in a given product just means that you are more productive at that thing given the same inputs. And so if I were to just give you this graph, and you didn't know how many workers Charlie or Patty had and how many inputs they're using to produce either thirty cups in a day or thirty plates in a day, you actually could not make any statement about absolute advantage. But if we assume that in all of these scenarios they have the same number of inputs, so if we think about plates . . . If we say they each have one employee, maybe it's themselves, and given that one input, or the same number of inputs, Patty is able to produce more plates than Charlie, then it is true that Patty would have an absolute advantage in plates. And if given the same number of inputs, Charlie is able to produce more cups than Patty, then he would have an absolute advantage in cups. But it is not because of that absolute advantage that he is specializing in it. In fact, we don't even know what their inputs were. It might be that he doesn't have an absolute advantage. Maybe Charlie needs a hundred people to produce his thirty cups, while Patty can produce ten cups with one person. So in that case, actually Patty would have an absolute advantage, but it just wouldn't be obvious from this right over here. But to make everything clear, I want to do a scenario where Charlie improved his productivity in some way and he actually has the absolute advantage in both products, and still show that as long as they have different comparative advantages, then it still makes sense for them to specialize. So let's do another scenario. So Charlie has improved dramatically. So let's draw our little graph here. That's our cups axis, this is still our plates axis. Cups and plates . . . and let's just put some more markers here... ten, twenty, thirty and forty. And ten, twenty, thirty and forty, and let's still put Patty, let's assume Patty hasn't changed, so this is her PPF, so that is Patty's PPF, just like that. But let's say that Charlie has improved dramatically. And so Charlie's PPF looks like this. So this is Charlie's PPF now looks like this. So in a given day he can produce - and let's just assume they're using the same number of inputs- so using the same number of inputs in a given day he can produce forty cups when Patty can only produce ten. So he has the absolute advantage in cups. Or, in the same given day using the same inputs, he could produce forty plates while Patty can only produce thirty. So now Charlie, all of a sudden, has an absolute advantage in both products. But we'll see it still makes sense for them to specialize because they have different comparative advantages; they have different opportunity costs. So let's figure this out. So we have all the same numbers for Patty - actually, let me copy and paste Patty's numbers right here. Actually we have access to her numbers right over here so I don't have to copy and paste it. But let's think of Charlie's new numbers now. So this is the PPF for Charlie. So this is our new PPF for Charlie. Maybe he did some investment or R&D to get this new, awesome, productive PPF. So he's expanded his PPF. So what is his opportunity costs? Say he's sitting here - so he's producing 40 cups - what would be his opportunity cost of producing 40 plates? Well to produce those forty plates, he would have to give up those forty cups. So his opportunity cost of forty plates is equal to forty cups. Or you divide both sides by forty: his opportunity costs for one plate is equal to one cup. And this makes math very easy: his opportunity cost for one cup is equal to one plate. Now given this new reality - so we've already established Charlie has an absolute advantage in both. Using the same inputs he can do more of either of them. And remember, when you're talking about absolute advantage you have to think about the amount of inputs you use. Who's more productive in that way? But let's think about comparative advantage. If we think about plates, who has a lower opportunity cost for producing a plate? Patty hasn't changed. Her opportunity cost for producing a plate is one-third of a cup. Charlie's opportunity cost for producing a plate has improved, but it's still worse than Patty's. He has to spend one cup to make a plate, she only has to give up one-third of a cup to make a plate. So Patty still has a comparative advantage in plates. And if we look at the opportunity cost in cups, the opportunity cost for Charlie to make 1 cup is 1 plate. So it's actually a little bit worse than it was before, but as we'll see it ends up being a good thing, he's just overall more productive. But his opportunity cost for one cup, he's giving up one plate now, when before he was producing one third of a plate. And that's because in the other scenario, he was more one-sided, I guess is one way to say it. But his opportunity cost for producing a cup is still cheaper than Patty's. Her opportunity cost of producing a cup is three plates: her opportunity cost. While his is only one plate. So he still has the comparative advantage in cups. So Charlie should still specialize in cups . . . and Patty should still specialize in plates. And to show that they can still get an outcome that is beyond even Charlie's Production Possibilities Frontier, let's think about how they could trade. So Charlie's going to specialize in cups; he's going to sit right over there producing forty cups a day. And Patty's going to specialize in plates, and she's going to sit right there - let me use a different color, I don't want to use this color - she's going to sit right there and produce thirty plates a day. So how could they trade for mutual benefit? Well any trade that is - assuming that they don't want to have only plates or they don't only want to have cups. Any trade that is cheaper than their opportunity cost will be a good one. So for example, Patty is sitting here producing only plates. Her opportunity cost for a cup is three plates. So she would be willing to trade anything less than three plates for a cup, assuming that she wants it. Because, if she had to make the cups herself, she would have to give up three plates. So let's say that Patty would be willing to trade one cup sorry, one plate - actually she'd be willing to trade two plates for one cup. She's be willing to trade that, because if she had to make the cups herself, she'd have to give up three plates for one cup. So she's willing to trade two plates for one cup. And let's see if Charlie would be willing to trade two plates for one cup. So he has all of these cups - how many cups does he have to give away for a plate? Well he has to give away one cup for a plate. Now he would have to give away one cup for two plates, or he would have to give up half a cup for a plate. Either way, this is better than his opportunity cost of trying to get that incremental plate. So he would be willing to do that too: two plates for one cup. He'd be willing to do one cup for two plates. And to see how that would improve, he could have forty cups or he could trade one of them away - Actually, let's do a scenario where he trades ten of the cups away. So now he only has twenty cups, but for those twenty cups he traded away - Actually, that's a bad example because Patty won't have enough cups. So let's say he trades away ten cups. Let's say he trades away ten cups for twenty plates. So Charlie trades 10 cups for 20 plates. So now he trades ten cups and he gets twenty plates. So now he'll end up at this scenario over here, which was beyond, which was unattainable, when he was working by himself, when he didn't specialize and get gains from trade. So this is a good scenario for him. He's able to get outcomes he otherwise would not have been able to get. He could, depending on how he trades, he could get outcomes, well up to a certain point, because Patty only has thirty cups. So at best he can take all of Patty's cups. So he can get something along that line over there. But if we look at the same scenario, Patty traded twenty plates for ten cups: where does that put her? So she traded twenty plates, so she's down ten plates but she got ten cups, so that put her right over here. Once again, beyond her Production Possibilities Frontier, so this would look like a pretty good situation for Patty as well. |
Khan_Academy_AP_Microeconomics | Factors_affecting_supply_Supply_demand_and_market_equilibrium_Microeconomics_Khan_Academy.txt | In the last video, we introduced ourselves to the law of supply. And it was a fairly common sense idea that if we hold all else equal, that if the price of something goes up, there's more incentive for more producers to produce it or a given producer to produce more of it. And we saw that. As the price goes up, we moved along the supply curve, and the quantity produced went up. Now what I want to talk about in this video is all of the things we held equal in the last video. And the first of these, I'll call this the price of inputs, or another way to think about it is the cost of production. So if the price of inputs, maybe the price of labor, the people who would have to pick the grapes, or our fuel that we need to transport the grapes, or the land, if any of that increased, that at a given price point, we would make less money. There's less incentive for us to do it, especially if this is true only for grapes. Maybe we'll say, OK, if it's now more expensive to get grape seeds, maybe I'll start planting something else, because I'm not getting as much profit per pound of grape. So if the price of my inputs, or if the price of my cost-- or if the size of my costs goes up, at any given price point, I'd want to produce less. So if my price of inputs go up, my supply, the supply, would go down. So if this becomes, at this price point, I'd make less money, so I would produce less or maybe I would produce other things. So the whole supply curve would shift to the left. And also even the minimum price I would need to supply any of it would also go up, when you shift the curve to the left, because now all of a sudden, it costs me more to produce even that first unit. And likewise, if my price of my inputs went down, now all of a sudden at any given price point, producing grapes would become more profitable and I would have more incentive to maybe produce grapes relative to other things and use more land for grapes than other things. And then you would have the whole curve shift to the right. Now let's think about related goods. So what happens with the price of related goods. And we have to put our-- when we think about this, we don't want to think of it from a demand point of view, because we're talking about supply. You want to think about it from the producer's point of view. So when we think about related goods here, we want to think about substitutes for production. So maybe I'm a farmer-- and I know very little bit about farming, so I don't even know if this is possible-- but maybe on my land, I'm saying, well, some of my land is going to be for grapes and some of it is going to be for blueberries. And so what would happen if the price of a related good, in particular blueberries, what would happen if the price of blueberries went up? Well, if the price of blueberries went up, then I would say, wow, maybe I can do better with blueberries. And I would allocate more of my land to blueberries than to grapes. And so once again, the price of related goods-- well, it depends which related goods-- but if the price of productive substitutes-- so price of other things I could produce, other things I can produce. If the price of other things I can produce goes up, then my supply of grapes, once again, would go down. And the important thing is, is in any of these circumstances-- literally, just think it through. Do not just look at what I'm writing here and just try to memorize it in some way, shape, or form. This is really just a way to think about things. Hey, obviously, if I can make more money off of blueberries now all of a sudden, I'm going to allocate more of my land to blueberries than to grapes. Supply of grapes will go down. Now, let's think about what happens with the number of suppliers. And this one is pretty common sense. The more people they are supplying, the higher the supply would be. So if the number of suppliers goes up-- and now you wouldn't imagine-- this is a curve maybe for the aggregate supply. So if the number of suppliers goes up, then the aggregate supply would go up at any given price point. If the number of suppliers were to go down, then the aggregate supply would go down at any given price point. So this one, hopefully, is somewhat obvious. Then we could think about things like technology. And so this is just maybe, there's some innovation, some new type of seed that with the same amount of work, the same amount of land, can produce that many more grapes. So if we have technological improvements-- I'm assuming we're not going to go into some type of dark ages. If we have technological improvements, then that will also make the supply go up. You can also think of it as it might make it cheaper to produce. So it's kind of the same thing here. The price of inputs might go down. So that would make your supply go up. Or you could just say, hey, look, there's just going to be more grapes popping off of these new types of vines that we got, so we're just going to produce more grapes. And then the last one, I'll cover-- and it's a little bit strange in the grape analogy-- is the expected future prices. So the expected future prices, price expectations. Now let's go away from the grapes, because grapes, they're perishable goods, they go bad. It's not like you can save goods to use them later. But if, let's say, you are an oil producer. And oil is something you can store and you can use it later. If you expected oil prices to be neutral today, and then tomorrow, all of a sudden, you are sure that oil prices are going to go up in the future-- you're sure that a year from now, oil prices are just going to go through the roof-- what's your incentive? Well, you should hoard all of your oil. Do not sell it today and wait to sell it in the future, if you're sure that's what's going to happen. If there's a change in expected future prices-- so if you go from neutral to expecting prices go up-- prices go up in the future, then you're going to hoard your goods. You can't hoard grapes, because the grapes will just go bad. You might be able to, I don't know, turn them into wine or something. But if we're talking about something like oil, you would say, hey, why should I pump all of the fixed amount of oil in the ground today to sell at today's lower prices? I'm going to lower the supply today, so I can sell it in the future. So if the expected future prices go from neutral to you expect future prices to go up dramatically, then current supply-- and that's, I'm just going to emphasize by writing the word current-- current supply will go down. So you can hoard it to sell it in the future. |
Khan_Academy_AP_Microeconomics | Input_approach_to_determining_comparative_advantage_AP_Macroeconomics_Khan_Academy.txt | - [Instructor] In other videos we have already looked at production possibility curves and output tables in order to calculate opportunity costs of producing a certain product in a certain country. And then we used that to think about comparative advantage. We're going to do something very similar in this video, but instead of thinking about, or instead of starting with output, we're gonna start with input. So right over here we have a table that shows us the worker hours per item per country. So, instead of this being an output table where we say in a given country, how much of, say, toy cars can a worker in country A produce per day? Here we're saying, how many hours does a worker in country A take to produce A toy car? In country A it is two hours. That labor, that two hours of labor, this is the input. So we're not counting the number of cars per day here. We're saying how many hours per car, A, we need to put in to produce it. Similarly, we have the input required in country A to produce a belt. One hour of worker time. In country B, four hours of worker time produces a toy car. And in country B, three hours of worker time produces a belt. So what we're gonna do next is convert this into the world that you might be more familiar with, of thinking in an output world. And to do that, we'll just assume that there are eight working hours per day in either country. And so from this, can we construct an output table? Let me put this right over here. Output table, where once again we're gonna think about the output in country A. We're gonna think about the output in country B. And this is going to be in how many units of that product can a worker produce per day in each of those countries? So once again, we're gonna have toy cars in this row, and we're going to have belts in this row. And let me just draw some lines so it's clear that we're dealing with a table here. So there we go. Then one more column. And so, see if you can fill these in. So how many toy cars per worker per day can we produce in country A? Then think about it for belts. Then think about both of them for country B. Pause the video and try to figure that out. Alright, now let's think about how many toy cars per worker per day. Let me make it very clear. We're thinking per worker per day here. Because if we can fill out this output table from this, I guess you could call this an input table, then we can think about opportunity cost in the traditional way. And then we could think about in which country do we have a comparative advantage? So, let's see. Toy cars in country A. If it takes two hours to produce one toy car in country A, and if you're working, if the average, or if the worker is working eight hours per day, well then, a worker can produce four cars. Four cars times two hours is eight hours. So, an average worker per day in country A can produce four toy cars. Let me write than in that red color. Four toy cars. I just took eight hours and I divided by the number of hours it takes to produce a toy car. Similarly for belts, if I have eight hours and it takes an hour for a worker to make one belt, then per worker per day, eight divided by one, I could produce eight belts. And we could do the same thing for country B, and I encourage you to pause the video if you haven't done so already and try to fill this column out. Well, in country B, if it takes four hours to produce a toy car per worker, that means you take eight hours divided by four hours that you could produce two toy cars in a day per worker. If it takes three hours to produce a belt, well then you take your eight hours, divide it by three hours per belt, and you're gonna be able to make 8/3 belts per worker per day. This is the same thing as 2 2/3 belts per worker per day. So as you can see, we can easily translate between the input world and the output world. And then we could use this to calculate opportunity cost. So let's do that. Let me write opportunity cost. And I'll make another table here. So country A, country B, and then I have the toy cars, and then I have the belts. Let me do the belts in that orange color. I have the belts, and then let me set up my table. We're almost there. At any point in time, pause this video and see if you can figure out the opportunity cost given the information that we already have. We took this table to figure out this table, and now we could take this table to figure out this one. Well, let's do this together now. So, toy cars. What's the opportunity cost in country A? Well, one way to think about it is in country A, the same energy to produce four toy cars, I'll call it four c, c for cars. We could also use that to produce eight belts. So, if I were to divide both sides by four, the energy to create one car is equal to the energy to create two belts. So my opportunity cost of a car is two belts. And if I start with this original equation and just divide both sides by eight, I would solve for the energy for a belt. And so that would be four over eight is 1/2 of the energy to make a car is equal to the energy to make a belt. And so the opportunity cost of a belt is 1/2 a car. 1/2 a car. And like always, this and this are reciprocals of each other. And we could do this same exercise for country B. And once again, I keep emphasizing, try to pause the video. If you do this on your own as opposed to just watching me do it, it'll stick a lot better in your brain. Alright, in country B, the same energy to make two cars, toy cars, with that same energy I could make 8/3 belts. 8/3 belts right over here. So the energy to make a car, divide both sides by two, is equal to, instead of one car I can make 4/3 of a belt. And so I'll just write this as 1 1/3 of a belt. And then if I start right over here and I multiply both sides by 3/8, actually, let me do that over here. So I have 3/8 times two c is equal to 8/3 b times 3/8. These cancel out. And over here I'm gonna have 6/8 c. 6/8 c is the same thing as 3/4 c is equal to b. So, instead of making one belt, I could take that same energy and make 3/4 of a toy car. 3/4 of a toy car. So given everything that we've just done, which country has the comparative advantage in toy cars? Well, to figure that out, we just look at the opportunity cost for toy cars and we compare them. In country A, the opportunity cost is two belts while in country B it's only 1 1/3 belts. So country B has the comparative advantage right over here. Comparative advantage in toy cars. And then in belts, 1/2 of a car is less than 3/4 of a car. In belts, we see that country A has the comparative advantage. And now what's always interesting about thinking about this is notice, country B has the comparative advantage in toy cars. It has less of an opportunity cost in toy cars. Even though country A has the absolute advantage, its workers are more efficient at producing toy cars. A worker can produce four cars in country A versus two in country B. But despite that, because of the opportunity cost, it would actually make sense for country B to focus on cars and for country A to focus on the belts. But the big picture here is we're thinking about comparative advantage. And instead of thinking about with an output lens from the beginning, we started with an input lens, converted that to an output lens, calculated opportunity cost, and then was able to figure out which countries had a comparative advantage in which products. |
Khan_Academy_AP_Microeconomics | Introduction_to_price_elasticity_of_supply_AP_Microeconomics_Khan_Academy.txt | - [Instructor] We've done many videos on the price elasticity of demand, now we're going to focus on the price elasticity of supply. And it's a very similar idea, it's just being applied to supply now, it's a measure of how sensitive our quantity supplied is to percent changes in price. And we will calculate it as our percent change in quantity supplied for a given, for a given percent change in price, percent change in price. Now, to make this a little bit more tangible, let's look at a simple market, let's say this is the market for apples, right over here, where our vertical axis is price, and this could be thousands of dollars per ton, and then our horizontal axis is quantities, and maybe this is in tons per day. And this supply schedule and this supply curve are essentially describing the same data. So, let's think about our price elasticity of supply as we go from point A, point A, to point B. Well, on the supply schedule, point A is this point right over here, our price is four, our quantity is one. And point B is right over here. So let us calculate from point A to point B our price elasticity of supply. So, first of all, what is going to be our percent change in price? Well, we're going from four to six, so it's an increase of two, so our percent change in price is going to be equal to two, is how much we increased from a base of four, times 100% and that of course is going to be equal to a 50% increase in price. And then what is going to be our percent change in quantity? Well, we're going from one to two, so we're starting at a base of one. We are increasing by one, and then multiply that times 100%, that gives us 100%. So, when we have a 50% increase in price, that resulted going from point A to point B, in a 100% increase in quantity supplied. So, 100% divided by 50%, that is going to give us, this is going to be equal to two. Now, what if we go from point B to point C? So, this is point C right over here. I encourage you, pause this video and see if you can calculate the price elasticity of supply when going from point B to point C. Well, we're going to do a similar calculation. Our percent change in price. We start at a base of six and we are increasing by two. So we're gonna multiply that times 100%. So that is approximately, this is 1/3 times 100%, so approximately 33.3%. And then what is our percent change in quantity supplied? Well, we are going to go from two to three. So we start at a base of two, we increase by one. So plus one, and multiply times 100%. And so that's going to be given, it's going to be equal to 50%. And so when we have a 1/3 increase, or 33.3% increase in our price, we have a 50% increase of our quantity supplied, when we go from point B to point C right over here, and then one way to think about it is 50% divided by 1/3 is the same thing as 50% times three, and so this is going to be equal to, this is going to be equal to 1.5. So, just as we saw when we calculated price elasticity of demand, either when you have a linear curve here, your price elasticity of supply can change. It is not the same thing as slope. Now another thing to keep in mind is the way that I calculated price elasticity of supply in this video, which is arguably the simplest way, you would not get the same value when you're calculating the magnitude going from A to B than if you went from B to A. There are slightly more advanced techniques, the mid-point technique, for example, that will give you the same answer regardless of which direction you go in, but that's beyond the scope of this first video. Now, just as we discussed in the demand case, there are cases that you would consider to be more inelastic supply and cases where you would consider to be more elastic supply. So, one way to think about it is, if the magnitude of your price elasticity of supply is less than one, and of course this is magnitude so it's going to be greater than or equal to zero, well, then you're talking about inelastic price elasticity of supply, inelastic. That's a situation in which our quantity supplied is not going to change so much depending on, is not going to be so sensitive to our change in price. Now, if our price elasticity of supply is greater than one, that's generally considered to be elastic, for a given percent change in price, you're getting a larger than that percent change in quantity supplied. |
Khan_Academy_AP_Microeconomics | Long_run_supply_when_industry_costs_are_increasing_or_decreasing_Microeconomics_Khan_Academy.txt | - [Instructor] What we have here we can view as the long run equilibrium or long run steady state for a perfectly competitive market. Let's say this is the market for apples and this is idealized perfectly competitive situation where you have many firms producing, they're non-differentiated, they have the same cost structure, there's no barriers to entry or exit. And on the left you can see that this equilibrium price which is set by the intersection of the supply and demand curves, that that's just going to be the price that the firms have to take and we've talked about that at length in other videos. That's going to define that the firm's marginal revenue, not just this firm, but all of the participants of the market. In other videos we've talked about the fact that the rational quantity for this firm to produce would be where marginal revenue intersects marginal cost. And it's also gonna be the point where you have zero economic profit, where at that quantity, let's say the quantity for the firm, your average total cost is equal to your marginal revenue. If marginal revenue were higher than average total cost at this quantity, well then you would have other entrants into the market because you're having positive economic profit. If marginal revenue is below average total cost at that quantity, well then firms are running economic losses and you will have people exiting the industry. And either of those situations would get us back to an equilibrium state that looks something like this. But now let's imagine a shock to the market somehow, let's say a new research study comes out that says that the apples that this market produces, that it's incredibly good for you, it'll make you live longer, it'll make you happier, it'll make you have more friends. Well then the demand for apples goes up, and so you have a new demand curve that looks something like this, D prime. Well in that situation, what's going to happen? Well now you have a new equilibrium price, you also have a new equilibrium quantity over here, let's call that P Prime. This is going to define a new marginal revenue curve, for the participants in the industry. So M, marginal revenue, prime. And now all of a sudden, the rational quantity for them to produce would be out here, at least for this firm to produce, so Q prime for, this firm is out here and you notice at that quantity, it is making economic profit. For every unit it gets that much, it costs that much on average for every unit, so it's making that much per unit, and then you multiply that times the number of units or the quantity. This whole area is going to be the economic profit that this firm is getting, and it's like that all of the firms, or most of the firms in this perfectly competitive market are going to be getting it 'cause they all have the same cost structure. But as we said before, when you have this positive economic profit, and there's no barriers to entry, in the long run, more firms will enter because there's economic profit to be had. And in previous videos, we've talked about a situation where as firms enter into a market, or exit a market, it doesn't change the cost structures of the individual firms. So let's imagine for a second that because of everyone entering into this market that seems to have economic profit for the firms that are participating into it, some of the inputs of say, growing apples, which is is what these firms do, start to go up in costs. So we're not talking about constant costs, perfectly competitive market, now we're not talking about an increasing cost, perfectly competitive market. Well then firm A and every firm's cost structure is going to change because as more firms come in, you're going to have to pay more for maybe apple seeds, pay more for maybe pesticides, or wax, or maybe you pay more for land on which to grow them. And so you would have a different marginal cost curve. We have the marginal cost curve now looks like this. So marginal cost curve prime. You would also have a new average total cost curve, maybe it looks something like this. So average total cost prime. And so, you can imagine that firms will jump into the market in order to capture or think that they might be able to get some economic profit, but they would only do so until the economic profit for all firms goes to zero. So what point will the economic profit go to zero? Well that's when the marginal revenue for the firms is equal to our marginal cost is equal to our average total cost. So it's that point right over there. So we would get to this point right over here, let's call that marginal revenue prime. And so, more and more firms would enter into the market up until the point that the equilibrium price gets us to P prime. And so the supply would increase, those folks wanna get that economic profit but it would increase until this point. So it'd shift a little bit to the right, and then we would get to S prime. As you can see, based on this we can now start to imagine a long run supply curve in this increasing cost, perfectly competitive market. We were over here, that was our equilibrium point before, now we are over here. And so our long run supply curve in this increasing cost environment, even though it's perfectly competitive, might look something like this. So in a constant cost world, this was a flat line. Now in an increasing cost world, as more and more people enter the market, the cost structure, the inputs into producing an apple go up, now long run supply is that. Remember, the long run is enough time to go by for people to enter and exit the market. Or enough time to go by so fixed costs aren't fixed anymore, that they can be shed or that they could be increased. Now you could do another thought exercise. Let's say we're dealing with a market where the more people that enter the market, the inputs actually get cheaper. And if that seems hard to believe, you can imagine, well, now people are able to produce seeds or wax at a new scale, so the inputs actually get cheaper. Well then you would see the opposite thing. Then you would see that as more entrants enter the market, this cost structure goes down, and so the supply can increase more and more and more and more, to a point that the equilibrium price is now lower than it was before, and then you would have a downward sloping long run supply curve. |
Khan_Academy_AP_Microeconomics | Economic_profit_vs_accounting_profit_Microeconomics_Khan_Academy.txt | Background voice: Let's say this past year I started a restaurant and I want to think about what type of a profit I've been making at that restaurant. We're going to think about it in 2 different ways. We're going to think about it in terms of an accounting profit, which is really the type of profit that most of us associate with a business or a firm. We're also going to think about it in terms of economic profit, which we'll see is a little bit different. Instead of telling us whether a business is producing income, it tells us whether it makes sense to even run the business in the way that we're actually running it. First, let's focus on the traditional way of calculating profit. Let's say my firm, my restaurant, (my firm in a restaurant) in year 1 it brings in, in revenue, it brings in $500,000. Revenue literally is the amount of money the customers pay me to eat at the restaurant. They are paying for their dinners. This is literally the money that's coming in the door. Sometimes people call it the top line, because it's literally the top line of our income statement. I just wrote it. It's the top line. Now we have to think about our expenses. Expenses. Now, when you're running a restaurant one of the obvious expenses is going to be the cost of food. Food, we're going to say cost us $100,000. $100,000. Then, you have the cost of labor. I have the wait staff. I have the chefs and the bus boy. On all of those people, in this past year, I spent $100,000. Then, I have, and I am going to assume that I don't own the building, that I rent the building. So, building rent. I'm assuming this is on the building, let's say that that was $200,000. Then finally, I really just rented everything. I also rented the equipment, all of the stoves, the fridges, all of that stuff. None of this is stuff that I own, so the equipment rent. Equipment rent, I spent another $50,000. How much profit do I have here? Those are all of my expenses. I didn't borrow any money, so I didn't have any interest expense or anything like that. How much profit do I have before paying tax, or essentially my pretax profit? The reason why we think of it in those terms is because the amount you pay in tax is usually derived from your pretax profit. That depends on where this business is, what country, what state, what type of business it is. The easy way to calculate pretax profit, pretax profit. This is pretax and we're thinking in terms of accounting profit right over here. We take how much money comes through the door and then we just have to subtract out all of the payments we essentially have to make to other people. What we have left is out pretax profit. 500,000 minus 450,000 gives us a pretax profit (I'll do it in that same bright yellow) of $50,000. I'm assuming that I'm the only owner of this business, so I can essentially take it all out for myself. Maybe help pay my own personal rent or whatever else, or I could take some of this or all of this and reinvest it back into the business. Maybe I start buying my equipment or I expand in some way. Who knows what I might do with that money. This is just traditional accounting profit. This is how profit is calculated. Although, this is a super simple example. In the future I would like to do more nuanced examples in the accounting world. This, you would refer to as just accounting profit. Accounting profit. When people in the everyday world talk about profit, this is normally what they're talking about. Now, when economist talk about profit, they're talking about something slightly different. The best way to realize that is to just calculate economic profit for this exact same business, or this firm, as a economist would call it. A firm really is a general idea for an organization that is trying to maximize profit. Once again, it's year 1. Actually let me just copy and paste it. It's year 1, that's our revenue. I'm going to copy and I'm going to paste it. This right over here. So far, so good. Looks pretty similar. Now, we're going to think about things in a slightly different way. Economist view cost in terms of opportunity cost. As we'll see, some of the opportunity cost you can measure in terms of dollars. Some are less explicit. I'm going to write here, just so we can get in the economist frame of mind, opportunity cost. Within opportunity cost there are going to be explicit opportunity cost and implicit opportunity cost. First, let's do the explicit. Explicit opportunity cost. Actually, all of these are explicit opportunity cost. Let me just copy and paste that. I will copy and paste. All of these are explicit opportunity cost. The reason why they are explicit is I'm actually making up ... I'm paying money for all of these things. Even the equipment and the rent of the apartment, I don't own it. I'm actually paying whoever does own it. These are direct outlays out of the business. I'm explicitly making these payments. The reason why we can think of them as opportunity cost, even though they're given in dollar terms, is that if I was spending $100,000 on food, that's $100,000 that I couldn't spend on something else. If I'm spending $100,000 on labor, that's $100,000 that I couldn't spend on something else. I'm just measuring the opportunity cost in terms of dollars, but dollars that I could have spent on other things. So far, it looks pretty much identical. I'm just viewing it with a slightly different lens. You're like, "Well, what's the big deal here?" We're going to see a little bit of divergence when we start thinking about the implicit cost that really weren't taken into account here, the implicit opportunity cost especially. Implicit cost. If I am running this business and let's say, in order to run it I actually had to focus on it full time. I couldn't have actually quit my job. Then, there's an implicit cost of … An implicit opportunity cost of the job that I gave up, or my wages foregone. Let me write this down, wages foregone. Let's say, and this will depend on who we're talking about. Let's say I was a doctor and I was making a nice steady, risk free $150,000 a year. I was giving up $150,000 a year. Now, we've listed all of the explicit and the implicit opportunity cost. Now we're ready to calculate our economic profit. Let me draw a line over here. Our economic profit is going to be our revenue that we're taking in, minus all of these expenses. That gives us a positive $50,000. Now, we have to subtract the wages foregone. Then, I get to negative $150,000. This is interesting. This is kind of a big discrepancy here. In accounting terms, I'm profitable. In economic terms, I'm not profitable. The important thing to realize is economic profit, when it's negative, isn't saying, or you say that you have $100,000 economic loss, or an economic profit of negative $100,000. This isn't saying that the business or the firm isn't spinning out money. What it is saying, is it probably doesn't make sense to run this business or at least to run this business in this way. If this was 0, that means, hey, it's probably making money, but you're kind of neutral whether it makes sense to run it this way or not. If it's positive, that means it definitely does make sense to run the firm in this way and that it is definitely doing better than all of the alternatives. This right over here is saying, look, you're making $50,000 a year, that's the 50,000 that you have to spend, if you're the owner, or reinvest in the firm. This is saying, essentially, look, you could have been making more money than that $150,000. Instead of making $50,000 doing this, you could have been making $100,000 more doing something else. You are essentially giving up, you are giving up $100,000 to do this restaurant. If you are a rational decision maker and you're really are about maximizing your profit, this actually might not make so much sense for you. |
Khan_Academy_AP_Microeconomics | Terms_of_Trade_and_the_Gains_from_Trade_AP_Macroeconomics_Khan_Academy.txt | - [Instructor] Let's imagine a very simple world, as we tend to do in economics, that has two countries that are each capable of producing either pants or shirts, or some combination. And so what we have here are the production possibility curves for each of those countries, and this is in per worker per day. So, for example, in country A, per worker per day, they could, if they put all of their energy into pants, they could produce 20. If they put all of their energy into shirts, they could produce 10. Or there could be some combination that would sit on this line. Now, to help us digest the production possibility curves for these two countries, let me construct an output table. So this will be, this column will be the output for country A. This column will be the output for country B. And we're gonna think about the maximum number of pants, maximum pants, the maximum output of pants per worker per day. The input is the worker per day. And then let's think about the maximum number of shirts. So pause this video, and see if you can fill this out. What are the max pants and shirts in country A and country B? Well, in country A, I already talked about it, the maximum pants is 20, 20 pants. And then the maximum shirts, if they didn't make any pants, are 10. And in country B, the maximum pants are 30, and the maximum shirts, it looks like that is about 45. Now, from either of these production possibility curves or from this output table, because we have a constant opportunity cost, these production possibility curves are straight lines with a fixed slope, we can calculate the opportunity costs. So let's do that next. So this is country A, and then this is country B. And let me calculate the opportunity cost of pants, and let's calculate the opportunity cost of shirts. So pause this video, and see if you can figure that out. What are the opportunity costs of pants and shirts in countries A and B? And fill out this table. Well, one way to think about it, in country A, I could put all of my energy into pants and produce 20 pants, or I could put all of my energy into shirts and produce 10 shirts. 10 shirts, s for shirts, p for pants. And so if I want the cost of pants, I could just divide both sides by 20, and I would get pants, the amount of energy per pant is equal to, well, 10 divided by 20 is 1/2 a shirt. So the energy for pant is 1/2 for, is the same as the energy for 1/2 a shirt. And so we could say the opportunity cost of producing a pant is 1/2 a shirt. If we want the opportunity cost for shirts, we could take the reciprocal of this number. We could say it's going to be two over one pant. Or we could start with this equation right over here, and instead of solving for p, we could solve for s. How much energy, in terms of pants, does it take for us to produce one shirt? So if you divide both sides of this equation by 10, you would get, you would get two p is equal to s. Or another way of thinking about it, the energy to create one shirt is equal to the energy to create two pants. So the opportunity cost of producing a shirt is two pants. With that same energy of the shirt, you could produce two pants. Now, let's also fill it out for country B. And if you haven't done so already, try to use the same method to fill this, the opportunity costs for pants and shirts for country B. Well, in country B, I could put all of my energy into pants and produce 30 pants or all of my energy into shirts and produce 45 shirts. So the opportunity cost per pant, if I divide both sides by 30, it'd be 45 over 30, which would be equal to, they're both divisible by 15, 3/2 of a shirt. The energy for one pair of pants is the same as the energy for 1 1/2 shirts, I guess I could say. So let me write it that way. So the opportunity cost of pants is, for each pair, I'm giving up 1 1/2 shirts. And then, in the opportunity cost for shirts, well, I could just solve for s here. If I divide both sides by 45, I get the same energy for one shirt would be 30/45 of a pair of pants, which is the same thing as 2/3 of a pair of pants. And so I could write that as 2/3 of a pair of pants, or, if I want, oh, let me just write it that way, 2/3 of a pair of pants. So given the opportunity costs, what should each of these countries focus on? Pause this video, and try to figure that out. Well, let's first compare their opportunity costs in pants. So let's first compare their opportunity cost in pants. It is clear that country A has a lower opportunity cost for producing a pair of pants. It's only giving up 1/2 a shirt while country B is giving up 1 1/2 shirts. So country A has the comparative advantage right over here, so comparative advantage, right over here, in pants. And so it should focus all of its energy, according to the theory of comparative advantage, it should focus all of its energy on pants. And likewise, if we look at, so here we compared this to this, and likewise, if we try to look at shirts, right over here, if we look at their opportunity cost, country B is only giving up 2/3 of a pair of pants while country A would be giving up two pairs of pants. So country B has the lower opportunity cost or the comparative advantage in shirts. So country B should put all of their focus here on shirts. Now, I know what you might be thinking. People can't just walk around wearing only shirts. People might get cold below their waist. Or people don't want to only wear pants. They might get cold above their waist. And so how can people in these countries get the other type of garment? Well, the obvious answer is, if they focus in this way, they can trade. And what would be an acceptable trading price, let's say, for pants? Let's focus on pants for a second. So if we're thinking about the market for pants, so if you're country A, what would you be willing to sell pants for in terms of shirts? Well, a good price, so to speak, would be something higher than your opportunity cost. So A willing to sell, sell pants at price, I'll put that in quotes 'cause we're really thinking of price in terms of another good, at price greater than their opportunity cost, greater than 1/2 of a shirt. And you could think of this willing to trade or sell. I'll put that in quotes. They're really trading in our everyday language, right over here. And likewise, what about country B? Well, B willing to buy pants, they need pants, otherwise they would just be walking around with only shirts on, willing to buy pants at a price, at a price less than their opportunity cost for pants. And so that would be less than 1.5 of a shirt. So what would be a price that is greater than 1/2 a shirt and less than 1 1/2 of a shirt? And really any price in between these two values would work. Well, a nice round number is, well, they could trade at one pair of pants for one shirt. So a clearing price, a price that would work could be one p, one pants, for one shirt. And now, let's appreciate the gains from trade that they would both have here. So let's imagine this world where country A is producing 20 pants per worker per day. But let's say they decide that they want, instead of those 20 pants, they would want to trade 15 of them away for shirts. And so they would get, at this price, they would get 15 shirts. So they're gonna give up 15 pants. They're giving up 15 pants, so they'll only have five pants right over here. But they're going to get 15 shirts. So they're gonna get 15 shirts. And they're going to end right over here. This is where country A is going to end up. And what's cool about this is we've gone beyond the production possibilities curve. So you see, very clearly, the gain from trade. Country A could not have gotten to this point on its own. This is above the production possibilities curve. Likewise, country B was over here, with 45 shirts. It gave up 15 of those shirts. It now has 30 shirts. But it now has 15 pants. At least some of the people in the country are going to be able to wear pants now. So it now has 15 pants. Once again, it, too, is in a point beyond its production possibilities curve. It would not have been able to get here without the trade. So they are both better off. So the key thing, the key takeaway from this video is we now appreciate why comparative advantage is valuable, once again, making all the assumptions for these simplified economic models, because we can calculate out opportunity cost from that comparative advantage. And then we could think about what's a good price that they'd be willing to trade at and see that when they trade, they both are able to get beyond their production possibilities curve. |
Khan_Academy_AP_Microeconomics | Optimal_decisionmaking_and_opportunity_costs_APR_Microeconomics_Khan_Academy.txt | - [Instructor] What we're going to do in this video is think about optimal decision making by rational agents. And it's just thinking about how would a logical, someone with a lot of reasoning ability, make optimal decisions, make the best decisions for themselves? Well they would look at the costs and benefits of a decision and they would try to do the action that maximizes the difference between benefits and costs. So they would wanna maximize benefits, benefits minus costs. And this is an important idea because I think all of us would like to be rational agents, logical agents, making optimal decisions. Now when we think about benefits and costs, benefits you might try to quantify it somehow, maybe in terms of dollars. What's the benefit of say going to a movie or having some ice cream? And costs, we tend to associate with a price, the cost of going to a movie say might be $10. Those types of things are known as explicit costs, when there's an explicit price associated with it. But there's also something known as implicit costs, and the most well-defined implicit cost is the idea of an opportunity cost. And the opportunity cost in economic terms is defined as the cost of the next best alternative. So if I'm going to go to a movie, there might be the explicit costs of paying for the movie, but then there's the implicit cost. Maybe if I'm going to a movie, that time I could've used for something else. Maybe I could've earned money somehow. And so a rational actor would consider both of those and then compare them to the benefit of going to the movie, and if that's what maximizes the difference between benefits and costs, well they might decide to do that. Now to make this a little bit more tangible, let's look at that exact example. So right over here, we are told, suppose you have the choice between going to a movie for three hours versus working for three hours. Movie tickets cost $10. If you work, you can earn $30 an hour mowing lawns, $12 an hour working in an ice cream shop, or $10 an hour weeding your aunt's garden. What is the opportunity cost of going to a movie for three hours? And what is the total cost? So pause this video and see if you can figure that out. All right well we just have to go back to the definition. The opportunity cost is the cost of the next best alternative. So what is the next best alternative to going to the movie? Well, I can make the most money if I am mowing lawns, if I am mowing lawns. And so my opportunity cost is going to be $30 an hour, and if I'm going to a movie, that would've been three hours that I would not have been able to mow lawns. And so my opportunity cost, I'll write OC, I'm not talking about Orange County, the opportunity cost right over here is going to be $30 an hour times three hours, so 30 times three, which is going to be a $90, $90 opportunity cost. Now some of you might be saying, well what about the $12 an hour for an ice cream shop or $10 an hour for weeding your aunt's garden? Well, those weren't the next best alternative. The next best alternative was mowing the lawn. Some people might be confused and say, okay I'm gonna add all of these together per hour and multiply by three, but you're not gonna be able to do all three of these things. You're going to have to pick one of them. And we're assuming that maybe there aren't any extra costs that are not, maybe you get extra tired from mowing lawns versus working in an ice cream shop, but we're trying to simplify things, so let's not get overly complicated right now. So at a very face level or at a high level, the next best alternative is making $30 an hour mowing lawns. So that would be the opportunity cost. Now what would be the total cost? Well the total cost would be the sum of the implicit costs, which is opportunity cost is an example of that, plus the explicit cost. So the implicit costs, we already talked about that, that is $90. That's the cost of not, that's the opportunity cost of not mowing lawns. And then to that you're gonna add the explicit cost of just the price of the movie ticket. For $10 you get to watch a movie for three hours. So there's a $10 explicit cost. So the total cost of going to the movie is $100. And so how would an optimal decision or how would a rational agent use this information to make an optimal decision? Well they would wanna compare that to the benefit of going to a movie. And so if they could quantify that benefit somehow and say oh, yeah, the benefit to me of going to a movie is $200 and that difference between $200 and $100, that's the best difference that I can get out of all of my choices between my benefits and my costs. Well then I'm going to go to that movie. So I will leave you there and in future videos, we'll dig a little bit deeper into this. |
Khan_Academy_AP_Microeconomics | Tariff_and_imports_worked_example_AP_Microeconomics_Khan_Academy.txt | - [Instructor] We're told sugar is freely traded in the world market. Assume that a country, Loriland, is a price taker in the world market for sugar. Some of the sugar consumed in Loriland is produced domestically while the rest is imported. The world price of sugar is $2 per pound. The graph below shows Loriland's sugar market, and P sub W represents the world price. So we see our domestic demand, we see our domestic supply, and then we see the world price. All right, now let's try to answer the questions that they have given us. At the world price of $2 per pound, how much sugar is Loriland importing? So pause this video, and see if you can figure that out on your own. All right, now let's do it together. So at first, you look at your domestic demand and see, well, what would the domestic demand be at the world price? So that would be where these two lines intersect. So the domestic demand at the world price would be 14 million pounds. You might be tempted to put 14 million pounds here, but that would be the total domestic demand at that world price. But some of that is domestically produced, and some of it is imported. How much is domestically produced? Well, at a price of $2, the domestic producers are up for producing two million of those 14 million pounds. So this is domestic production. And so between these two points, that length, that represents how much is actually imported. So to go from two million pounds to 14 million pounds, 14 minus two is 12 million pounds. That is imported, 12 million pounds. Part b, suppose that Loriland imposes a per-unit tariff on sugar imports, and the new domestic price including the tariff is $4. Identify the new level of domestic production. So once again, pause the video, and try to figure that out. All right, so they say the new domestic price including the tariff is $4. So we are now in this situation. This is the new price. Now, they say what is the new level of domestic production? So the domestic supply, at that price, the domestic suppliers look like they are willing to supply six million pounds. So that is our new level of domestic production, six million pounds. All right, part ii, calculate the domestic consumer surplus for Loriland. You must show your work. Pause the video, and see if you can figure that out. Well, the domestic consumer surplus for Loriland in this scenario, where this is the price, well, then we are going to, let me scroll down a little bit so we can see the entire consumer surplus. That is going to be the area above this horizontal line at the price and below our domestic demand curve. So this right over here is the consumer surplus in that scenario. And so that is going to be this width, which is the quantity demanded, which is 10 million pounds, times the height, which is, you're going from four to $9, so it has a height of $5, and then times 1/2. If I just multiplied the quantity times the height, I'd be figuring out the area of this entire rectangle, so it's a little bit of a geometry review. To get half of that, we would multiply it by half. So this is going to be, it's going to be 10 million pounds. I'll do it right over here, so I have some space. 10 million pounds times, we have a difference of $5 per pound, $5 per pound. And then, of course, we want to multiply it times 1/2, so this one right over here. So let's see, the pounds cancel out. And so, if you multiply it out, this is going to be $50 million times 1/2. So it's going to be $25 million. All right, part iii, scroll down a little bit. Calculate the total tariff revenue collected by the government. This also says you must show your work. Once again, pause the video, and see if you can work through that. So the tariff revenue collected by the government, well, we went from a world price of $2 per pound to a domestic price of $4 per pound, so it was a $2 per pound tariff. And the government is collecting that $2 per pound on the imports. So in this situation, this is the domestic supply. We've already talked about that. And so this amount right over here, are the imports. So if you multiply this amount, which went from six million pounds to 10 million pounds, so this is going to be four million pounds, times the tariff, which is $2 per pound, per pound, you're going to get this area, which would be the government revenue. So this is going to be four million pounds times $2 per pound, times $2 per pound. Pound cancels out, and this is an area of a rectangle here. And so this is going to be equal to eight million, $8 million. All right, now let's do part c. Given the world price of $2, what per-unit tariff maximizes the sum of Loriland's domestic consumer surplus and producer surplus? Pause the video once more, and see if you can figure this out. All right, so you might be tempted to try out a bunch of tariffs and figure out if you can get a higher total surplus, but the important thing to realize is any tariff is going to reduce your total economic surplus. So you can immediately go and say that, hey, the ideal per-unit tariff is going to be $0 per unit. And if you want to see visually why that is, we talk about it in other videos. Remember, in the first situation, where we're just at the world price without any tariffs, the total economic surplus, this is the domestic producer surplus, which isn't that much, but you have a huge consumer surplus. You have all of this area as well. So those two triangles make the total economic surplus. Now, a tariff is going to raise this level. And as you raise this level, as you saw in the case of part b, well, you're going to shrink this upper triangle. You will grow this bottom triangle, but you're still going to be smaller than your two triangles that you had before. Because look at that second scenario, the scenario in part b. Your consumer surplus has now shrunk to this right over here. Your producer surplus has grown to that over there. But you haven't grown the total surplus. In fact, now for the consumers and the producers, you've lost this entire triangle. Some of it is captured by government revenue, but you also have deadweight loss. You also just have this section and this section as just deadweight loss. And so any tariff is going to reduce the, is going to reduce your possible economic surplus. So a $0 tariff, theoretically, would maximize your total economic surplus. |
Khan_Academy_AP_Microeconomics | Accounting_profit_vs_economic_profit_AP_Microeconomics_Khan_Academy.txt | - [Instructor] Let's continue thinking about how rational agents make decisions. So here we're told that Sally runs a business that only sells hamburgers in a building she owns. Every month, they sell 5,000 hamburgers at $5 per hamburger. She spends $2 per hamburger on supplies, bread, meat, lettuce, et cetera. She also pays Mike and Raj each $2,500 per month to work at the restaurant. Finally, utilities cost $500 per month. Sally works full-time at the restaurant and keeps the accounting profits for herself. What is the accounting profit of the business? So pause this video and see if you can figure that out. All right now let's think about this together. We're gonna think about it in terms of some of the types of costs we've thought about in the past. So when we think about benefits, and here we could think about it to a firm, although it's fully owned by Sally, the benefit to a firm of doing business is its revenue. So the total revenue that she collects, and everything we're gonna be doing is going to be per month, so her total revenue is going to be her price times quantity so it's going to be 5,000 hamburgers at $5 per hamburger. So that is going to be $25,000. Once again, all of this is going to be per month. So once again, we can view this as the total benefit that the firm, that her business, is getting. And now let's think about the costs. So first we could think about the cost of her supplies. It's oftentimes referred to costs of goods sold, but I'll just write supplies here. So costs, costs colon, so let's put supplies, supplies. That would be 5,000 hamburgers times $2 per hamburger, so that's a $10,000 cost, $10,000. Then she has the cost of her employees. So employees, I'll just write it, I'll abbreviate it like that. What's that going to be? Well she has two folks at $2,500 per month each. So that's going to be $5,000, two times $2,500, $5,000. And then last but not least, she has her utilities. So utilities, let's do util for short. That's going to be $500 per month. And so from this we can calculate the accounting profit. So we get the accounting profit, accounting profit, is going to be 25,000 minus 15,500. That's going to be $9,500 per month. And she gets to keep all of this, and so this seems like a pretty good amount of money to be earning. She's earning six figures a year. But the question is, is it rational for her to do this? Well some of you might be correctly thinking, well in order to determine whether it's rational for her to continue running this business, we have to know what the implicit costs are. Here we've only just looked at the explicit costs, and the most important of the implicit costs is the opportunity cost. And to factor that, we have to know, well maybe what she could've rented her building out, if she wasn't running this burger business, and maybe what she could do with her time if she wasn't working at the business full-time. So we need a little bit more information and let's see if we can get that. So now we are told that Sally could rent out her building for $5,000 per month. She can also make $6,000 per month as an accountant. Based on this, what is the economic profit of her business? So pause this video and see if you can figure this out. Well one way to think about it is, we can start with our accounting profit and then subtract out all the implicit costs, especially these opportunity costs right over here. So her opportunity cost, opportunity costs, are going to be per month, well if she doesn't run this business, she could rent out her building for $5,000 per month and then if she wasn't doing this full-time, she could make $6,000 per month as an accountant, $6,000 right over there. And so her opportunity costs are a total of $11,000. And so now her economic profit would be her total benefit minus her explicit costs minus her implicit costs. Her economic profit, economic profit, is going to be, well we could start at the 9,500 and subtract the 11,000, it is negative $1,500. It's important to realize, because economic profit always factors in the explicit costs and then other potential implicit costs, economic profit will never be higher than accounting profit. And assuming there are some implicit costs, it'll always be lower than accounting profit. So now based on all of what we've explored, is it rational for Sally to continue running her burger business? Well based on the information we've been given, it doesn't seem rational for her to continue running her burger business. She makes $9,500 in accounting profit from the business, but she's incurring $11,000 of opportunity cost to do so. And that's what makes her economic profit negative. This is not rational. Now if we had more information, maybe she hates being an accountant. Maybe there's a benefit for her working at the burger business. She has more flexibility with her time, she likes being self-employed, she doesn't have to listen to her manager tell her want to do. If that were the case, then it would change the calculations some because there would be an extra benefit from her running her burger joint. But we don't know, and based on the information we have, it doesn't seem rational for her to continue. |
Khan_Academy_AP_Microeconomics | Monopolies_vs_perfect_competition_Microeconomics_Khan_Academy.txt | - [Instructor] In this video, we're going to dig a little bit into the idea of what it means to be a monopoly, and so to help us appreciate that, let's think about the spectrum on which firms can be. So this is going to be my spectrum right over here. Now at the left end, we can imagine this idealized perfect competition, perfect competition, and we've talked about that in the other videos, but just as a review, this is where you have many firms. This is where they are selling an undifferentiated product or service, undifferentiated, undifferentiated product. The firms over here, well, they have no barriers to entry or exit, so no barriers to entry or exit. These firms that we've talked about in other videos, they need to be price takers. Why do they need to be price takers? Well, whatever the market price is, since no one cares which of these firms, which of these many firms they get the product from, none of those firms can really set their own price. If they were to go above the market price, well then no one will buy from them, and so they will just be price, price takers, and other things that we assume about perfect competition is that all of the actors in the market, both the buyers, the many buyers and the many sellers, they all know what the transactions are going on for. They know who's selling to whom for what amount. Now the other extreme, this is where we have the monopoly, monopoly. Here, instead of many firms selling or many firms producing, you have exactly one firm producing. Instead of an undifferentiated product, well, it's differentiated because it's the only firm. Instead of no barriers to entry or exit, here we have the exact opposite, so you could say insurmountable, insurmountable, mountable, I'll just abbreviate it, barriers, especially to enter, and instead of being a price taker, you are a price setter, price setter. You're the only player. You're the only actor who is selling anything, So you can decide what price to sell it at. Now, perfect competition as I talked about, it's a bit of a theoretical idea. It's hard to say any market that is absolutely perfect, but we can imagine markets that are on this spectrum, some closer to perfect competition, some closer to a monopoly. Things that I can imagine that are closer to perfect competition might be, let's say, agriculture or a certain type of agriculture. Let's say you are buying pistachios, and you might be, most people are indifferent as to where their pistachios come from, although some people might beg to differ that certain types of pistachios are better than others, but for the most part, that'd be closer to perfect competition. There will be just a price in the market for pistachios. If someone wants to grow pistachios, I'm not familiar with what it takes to grow pistachios, and I apologize to any offense to any pistachio growers out there, but maybe they can just get enough land, and there's very close to low barriers to entry, and they can start producing pistachios. As I mentioned, many would perceive it as undifferentiated, and there might be many firms in, say, the pistachio market. I actually don't know if that's the case, but let's just assume if that were the case it would be closer to a perfect competition. Now a monopoly, you can imagine things like things that take a lot of infrastructure in order to do that service. So I can imagine things like, over here, close to monopoly or at monopoly. You can imagine things like utilities providers, utilities, where it's hard for multiple people to run power lines to the various houses. You can imagine things like this. Telecom, telecom providers might be close, although in most geographies, you have more than one telecom providers, although in some parts of the world, you're getting pretty close to one because, once again, there's very, very, very high barriers to entry in either one of those. You gotta launch satellites and put cable under the ground and dig up roads and whatever until you get closer and closer to this notion of maybe there's one firm. If you're in a situation like telecom in a lot of the places where you have only a handful of firms, that's known as an oligopoly, but let's just think about the extreme, when you're in a monopoly situation, and so the next few videos, we're gonna dive a little bit deeper into what it means to be a monopoly, and what is the rational quantity for a profit-maximizing monopolistic firm to actually produce, and what would be their economic profit? |
Khan_Academy_AP_Microeconomics | Change_in_demand_versus_change_in_quantity_demanded_AP_Macroeconomics_Khan_Academy.txt | - [Instructor] What we're going to do in this video is a deep dive into the difference between demand and quantity demanded. In particular, we're gonna focus on change in demand versus change in quantity demanded. And so just as context, I have price versus quantity here for brand X of cars in a certain market and you see the demand curve for brand X of cars and we see the it follows the classic law of demand. At a high price, there is a low quantity demanded, and so this is already trying to draw the distinction. A quantity demanded, so I'll call this qd1, is associated with a particular point on the demand curve, it's not the whole curve itself. When people talk about demand, they're talking about the whole curve. But just following on of what I just said, following the law of demand at a low price, this is associated with, if we go to the demand curve, a high quantity demanded, quantity demanded two. And so to be very particular about this, quantity demanded is associated with a particular point on the demand curve while the demand curve is the set of all of these points that show how price and quantity are associated. So with that out of the way, to make things more tangible, let's go through a bunch of different circumstances and think about whether they would result in a change in demand versus a change in quantity demanded. So in this first scenario we say car dealerships slash prices by 10%. Would that result in a change in demand, which would involve shifting our demand curve, or would it involve a shift along the curve, a change in quantity demanded? Pause the video and try to figure it out. Well, there's a couple of ways to think about it. In order to shift the demand curve itself the one way I think about it is if you were to pick a given price, if you were to pick a given price, does what's described here in any way shift the quantity that would demanded at that price? Well, no, this is not shifting how much consumers would want to buy at that price. This is just shifting the price itself. So this is going to be a shift along the demand curve. So this would be a scenario where maybe the equilibrium price, and we'll talk more about that in future videos, maybe the equilibrium price and quantity demanded are associated with that point right over here before car dealers slashed their prices. So let's call this quantity demanded, let's call that quantity demanded three. But then when they slashed their prices, the prices go down, and so we end up with this point on our demand curve, and so this would be quantity demanded, quantity demanded four. So this would be a change in quantity demanded right over here, so change, I'll do delta for change in, change in quantity demanded. And in this case, the quantity demanded would go up. What about the price of gasoline increases? Pause this video. Think about what would happen. Would that change the quantity demanded for the cars or would it shift the entire demand curve? Well, I'll do the same exercise. Pick a given price. Let's say we're at this price right over here and this is the current quantity demanded, now if all of a sudden, actually for any price that I pick, if the price of gasoline increases, consumers will just have less money in their pocket, the cost of maintaining and using a car at any price would go up, and so they might be willing to buy less cars because the operating cost has gone up. And this would be true at any price, at any price. And so one way to think about it is the entire demand curve, the way I've just phrased it, you could view for the entire demand curve would shift. So if we call this D1 here, now this would be D2. So here we would say change in demand, and in this case, our change in demand, it would shift, it would go down. You could view it as shifting to the left. Actually, let me write that as shifting to the left because that's what it looks like on this graph. Let's do this third example, prices of public transportation goes down, what would happen? Is this a change in quantity demanded or would it be a shift in the demand curve? Well, once again, for any, for any given price that we are talking about, whether we're talking about here or whether we're talking about here, the substitute or one of the substitutes, which is public transportation, is now looking more favorable. So you could imagine people at a given price will just not demand, the market will just not demand as much of a quantity. And so this, once again, would be a change in the demand curve. When something is true for any given price along the curve, then you know that you're going to be shifting the curve. So our change in demand, and once again, we're going to shift to the left, so it's similar to bullet point two. Now, let's see, here we say the state lowers vehicle registration fees. Pause the video and think about that. Well, once again, regardless of where we might be sitting along the demand curve now, if registration fees has gone down, now the total cost of ownership of a car has gone down, and so for any given price, people might be able to demand a little bit more car. And so here we would have a shift of the demand curve to the right. Shift of the demand curve to the right. We could call this D3 right over here. So we have a change in the entire demand curve, not just quantity demanded, and we are going to the right. Let's do this, what is this, the fifth example. A recession leads to falling household incomes. Pause this video and think about it. Well, falling household incomes is actually analogous in some ways to the price of gasoline increases because people are just going to have less incomes regardless of what point we are on the curve. People are just going to be able to buy less. So that's going to shift the demand curve, the entire demand curve to the left. So it's a shift in demand or a change in demand once again going to the left. Last but not least, consumers expect new car prices to rise next year. What is that going to do to either, is that going to be a change in demand or a change in quantity demanded? Pause the video and think about it. Well, once again, this is something that applies regardless of where we happen to sit at a given moment on the curve, whatever the equilibrium price is, and we'll talk more about that in other videos. This is generally going to apply to any point that we are on the curve. If people expect prices to increase, if all of a sudden there's a bulletin that says, Hey, car prices are going to double next year, well then you can imagine wherever we are on the curve, people are going to say, Oh, if car prices are going to double next year, I better buy more car right now, so our entire demand curve is going to shift to the right, so it's a change in the entire demand curve, and it is going to go to the right. So the big picture here, if we're talking about a change in, well you could say a change in a particular price, someone raises the price or lowers the price, well that's going to change the quantity demanded. And later when we draw the supply curve and we see where they intersect and you have an equilibrium price, when one or both of the curves shift, their intersecting point changes and so then you will, you could have a shift in the curves, which will then result in a change in the quantity demanded. But if we're talking about things that are generally true regardless of where we are on the curve, that will just affect people's general demand for something, that is going to shift the entire curve, it's going to be a change in demand versus a change in quantity demanded. |
Khan_Academy_AP_Microeconomics | Review_of_revenue_and_cost_graphs_for_a_monopoly_Microeconomics_Khan_Academy.txt | - [Instructor] What I want to do in this video is review a little bit of what we've learned about monopolies and, in the process, get a better understanding for some of the graphical representations, which we have talked about in the past, but I wanna put it all together in this video here. So let's say that the industry that we are in, the demand curve looks something like that. That is demand and I'm going to assume that it is a linear demand curve. This axis right here is dollars per unit. In the context of demand, that's price, and this is quantity over here. This little graph here, we still have quantity in the horizontal axis, but the vertical axis isn't just dollars per unit, it's absolute level of dollars. Over here we can actually plot total revenue as a function of quantity, total revenue. Remember, we're assuming we're the only producer here. We have a monopoly, we have a monopoly in this market. So if we pick a quantity, if we don't produce anything, we're not going to generate any revenue, so our total revenue will be zero. If we produce a bunch, but we don't charge anything for it, and that's this point right over here, our total revenue will also be zero. We've done this in other videos, but then as we increase quantity from this point, our total revenue will keep going up and up and up. There'll be some maximum point and then it'll start going down again, so our total revenue would look something like this. Total revenue would look something like that, total revenue. And from the total revenue, we can think about what the marginal revenue would look like. Remember, the marginal revenue just says if I increase my quantity by a little bit, how much am I increasing my total revenue? So that's essentially the slope, the slope of the total revenue curve at any given point, or you can think of it as the slope of the tangent line. We've seen before, when you start here, you have a very high, positive slope and we've seen in other videos it actually ends up being the exact same value as where the demand curve intersects the vertical axis right over there, but then it keeps going lower, the slope becomes a little less deep, less deep, less deep. It's still positive, less deep, less deep, and then it becomes zero right over there and then it starts going negative. It becomes zero right at that quantity. The slope of this keeps going down and down and down, it's positive, then it becomes zero, and then it actually becomes negative and you see that here. Now it starts downward sloping even more steep, even more steep, and even more steep. That's the revenue side of things. Let me label this, this is our marginal revenue curve, slope of the total revenue. If we're gonna maximize profit, we need to think about what our costs look like, so let me draw our total cost curve. And I will do it in magenta. Let's say our total costs look something like this. Total cost looks something like that. Out here, where we have very few units, where we have zero units, all of our costs are fixed costs. And then as we produce more and more units, the variable costs start piling on over there. Even from this diagram, you can actually start to visually see economic profit. Economic profit, and when we're talking costs and profit in an economics class, like this is kind of one, I guess, remember, you should view it in terms of economic profit and when we're talking about total cost, we're talking about opportunity cost. So this is total opportunity cost, both the implicit, both the explicit, the ones that you're actually paying money for explicitly and the implicit opportunity costs. Total opportunity costs, that's total opportunity cost and the difference between your total revenue, so for a given quantity, the difference between your total revenue and your total opportunity cost, that gives you your economic profit. For this quantity right over here, your economic profit would be represented by the height of this little bar between these two curves. But what we see what's going on is, as we increase the quantity over here, these curves are getting further and further apart. That's because the green curve, the total revenue, it's slope is larger than this purple curve, which is total opportunity cost, or you could say it total cost. So we could go even further along, just the distance between the two curves gets bigger, bigger, looks like it maxes out right around here someplace and then the two things start getting closer and closer together. This purple curve's slope is now larger than the orange curve's slope, so then they start getting closer and closer together. If you were to just look at this graph, whatever the maximum distance between these two things are, it looks like it's about there, right over here, that would be your maximum economic profit. But we know we can also visualize it on this curve over here. And we can do that by plotting our marginal cost. And remember, marginal costs, this is marginal revenue, is the slope of your total revenue curve. Marginal cost is the slope, the instantaneous slope at any point of your total cost curve. So I will do that, let's do that in yellow. Right over here, you have a zero slope, or pretty close to zero, at least the way I drew it over there. So your marginal cost is going to be pretty close to zero right over there. And then we see this slope keeps increasing and increasing and increasing and so our marginal cost will keep increasing, increasing, and increasing, so it will look something like that. That is our marginal cost curve. If we pick a quantity and if we find that the marginal cost over here, I don't know, let's say that it's five dollars per unit, that literally means that the slope, that that same quantity, the slope of our total cost curve, that the slope over there would have to be five. That's what that is telling us. This is plotting the slope of this curve right over here. And if we want to maximize profit, we already talked about how you would do it visually on this curve, we can do it over here. Well, right over here, as we produce, if we start from producing nothing to producing something, for each incremental unit, the incremental revenue we get on that is much higher than the incremental cost. So, hey, we should produce it because we're gonna get profit there. We could keep producing because we're gonna get profit on each of these incremental units, so we'll keep doing it, we'll keep doing it, we'll keep doing it, until the marginal revenue is equal to the marginal cost. At that point, it doesn't make sense for us to produce anymore. If we produce an extra unit past that point, on that unit our cost will be higher than our revenue, so what will eat into our economic profit. So this right over here is where we max the quantity, which we maximize profit. When we see it, we see it right over there. The way I drew it, luckily, it looks like that is the maximum point between those two curves as well, and it makes sense. Before this point, when marginal revenue is higher than marginal cost, that means that the slope of the total revenue curve is larger than the slope of the total cost curve, so they're getting further and further apart. After this point, and right at that point their slopes are the same, so the slopes are going to be the same right over there, and then after that point, the slope of the marginal cost curve, sorry, the marginal cost is high, which tells us the slope of the total cost curve is higher, than the slope of the total revenue curve. And so they're gonna get closer and closer together and this distance gets squinched apart. That is where you maximize profit. And if you wanted to visualize the actual profit, on this graph over here, we cannot obviously visualize it here as the distance between these two curves. If you want to visualize it over here, we would have to plot our average total cost curve. And essentially, what you're doing is you're just taking this total cost curve and you're not just taking the slope at any point, that's the marginal cost, instead you're just dividing it by the quantities. So if you take this total cost curve, you take this value and you divide it by very, very, very low quantity, you're going to get a very, very, very, very large number. You can imagine, as you're spreading your fixed costs amongst a very small quantity, so you're gonna get a very large number. Then, as you produce more and more and more, your average total costs go down, but then your variable costs start picking up and your average total costs might look something like that. Average total costs. And so, if you wanna know your profit that you have maximized from this graph right over here, you say this is the quantity that maximizes my profit, marginal revenue is equal to marginal costs, the price that I can get in the market for that quantity, you go up to your demand curve and it gives you, this is the price that you will get for that quantity and so that is, on a per unit basis, that is the revenue that you will get. You can view price is equal to, price is the same thing as revenue, revenue per unit. So on a per unit basis, this is the revenue you're getting, and on a per unit basis, this is your average cost, this is average total costs. This is taking all your costs and dividing it by units. On an average, per unit basis, this is going to be your economic profit. On a per unit basis, and if you wanted to find your actual economic profit, you would have to multiply it by the total number of units. So you would, essentially, have the area of this rectangle right over here. This is your per unit average economic profit and so your total economic profit is going to be quantity times profit per unit and so this right over here is economic profit, or maybe I should call it the total economic profit. Let me write it out, total economic profit. And the area of that rectangle should be the same thing as the height of this right over here. The only reason, and we can maintain this is a sustainable scenario because we have a monopoly. No one else can enter. If this was not a monopoly, if there were no barriers to entry, then other people say hey, there's economic profit there, that means that there's an incentive for me to put those same resources together and try to compete because I'm going to get better returns than my opportunity costs than my alternatives is a good way to think about it. |
Khan_Academy_AP_Microeconomics | Cross_elasticity_of_demand_Elasticity_Microeconomics_Khan_Academy.txt | So far, we've been focused on the elasticity of demand for only one good. We've thought about how changes in the price of that good affect changes in its quantity. Now what we're going to explore is how we can go across goods. So we're going to talk about the cross elasticity of demand. And there's multiple different scenarios we could think about, but it's really thinking about how a price change in one good might affect the quantity demanded in another good. And to see an example of this, think about two airlines-- two competing airlines-- maybe it's the same exact route going at the exact same time, maybe between New York and London. So airline one, right over here-- airline two, very competitive, price right over here is $1,000 for a round trip. Quantity demanded is 200 tickets, let's say, in a given week. Airline two, price is $1,000 for the round trip, and the quantity demanded is 200 tickets as well. Now let's think about what will happen. What will happen if airline one raises its price from $1,000 to $1,100? In fact, we could even do something less dramatic than that, to $1,050-- so a relatively small increase in price. And remember, when we think about the percentage price increase, when we're thinking about elasticities in general, we don't just say, OK, $50 on top of $1,000, that's a 5% price increase. That's what we would do in everyday thinking. If you said you went from $1,000 to $1,050, you would say that's a $50 increase on a base of $1,000 or that is a 5% increase. But when you think about elasticities, because we want to have the same percent change between-- if you go from $1,000 to $1,050, or if you go from $1,050 down to 1,000-- we actually use the average point as a base. So the percent change in this scenario-- let me write it right over here. So our percent change-- and I'll write it in quotes, because it's a little bit different than what you do in traditional mathematics when you think about percent changes-- is you had a 50 change in price. Your price went up by 50, and on our base we will use 1,025, which is the average of 1,000 and 1,050. And so that gives us a change of 50 divided by 1,025 is equal to, let's say, roughly 4.9%. So this is approximately 4.9%, we'll say, "increase" in price, although we're going to put that increase in quotes, because we're using it on the average. And we do that so that if we said it was 1,050 to 1,000, it would still be a 4.9% decrease using this same idea-- using the midpoint as the base. Now, when that happens-- Everyone today, they use these travel sites where you can compare prices-- If these really are the exact same route, going from the exact same airport to the exact same other airport in London, leaving at the exact same time, everyone is going to gravitate to this one now, because it's only $1,000-- even just to save $50. Why would they ride on this airline? So this quantity demand is going to go to 0. And this quantity demanded is going to go to 400. And we're not going to think about the actual capacity of the planes and all that. We're going to have a very simple model here. So what was the percent change in quantity for airline two right over here? Well, once again, our change in quantity is 200, not 400. We went from 200 to 400. So we gained 200. And our base, we want to use the average of 200 and 400, which is 300. And so this is approximately 67%. So we have, all of a sudden, our cross elasticity of demand for airline two's tickets, relative to a1's price. And we get the percent change in the quantity demanded for a2's tickets, which is 67% over the percent change, not in a2's price change, but in a1's price change. That's why we call it cross elasticity. We're going from one good to another. So let's just say, for simplicity, roughly 5%. And so you do the math. So if you have 67% divided by 5%, you get to roughly 13.4. So this is approximately 13.4. So you have a very high cross elasticity of demand. In fact, if you even increase this, maybe by $5, you might have had the same effect. And so you would have had a very large number here. And that situation right here, for this cross elasticity of demand-- it's because these things are near perfect substitutes. The way that we set up this problem, we said, well, people don't care which one they take. They're just going to go for the cheapest one. And so when you have near substitutes, or nearly perfect substitutes, for each other, like this example right here, the cross elasticity of demand approaches infinity. It gets higher and higher and higher. In theory, if these are really, really, really identical, even if you raise this a penny, people will say, well, why would I waste a penny? I would just use airline two. And so this number would be even lower right over here. And so this thing might approach infinity. And notice this was a positive. When we just did regular price elasticity of demand, the only way that you would increase quantity for a traditional goods was by lowering price. But here, we raise price on a substitute competitive product, and we raise the demand for airline two's product, which actually made a lot of sense. So it wasn't a negative relationship. It's actually a positive value right over here. But you could have things in other-- you could have that negative relationship using cross elasticity of demand. This is an example of a substitute. We could think about the example of a complement. So what if we're talking about e-books? So let's say I have some type of an e-book, and the current quantity demanded in a given week is 1,000. And let's say that the price of an e-reader that you would need for my e-book is $100. But let's say that price of the e-reader goes down from $100 to $80. So you had a $20 decrease in price. Well, what's going to happen to my e-book, assuming its price does not change? Well, then the quantity demanded for my e-book will go up. So let's say the quantity demanded for my e-book goes up by 100, because more people are going to be able to afford this, or they're going to have money left over when they buy this to buy more e-books. And so I don't even know what the price for my e-book is, but at a given price point, the quantity demanded will go up. And so this goes to 1,100. And so I'll leave it to you to calculate this price elasticity of demand. But you will see that you will actually get a negative value, like we're used to seeing for regular price elasticity of demand. And when you do calculate it, remember, you want to do your percent price change in e-book quantity over percent change in e-reader price. And the other thing you have to remember, you don't just take negative 20 over 100. You take negative 20 over the average of these two, when you're thinking of it in the elasticity context. So this right over here-- actually, maybe we'll just work it through. Pause it, and try to do it yourself. So this value right over here is negative 20 over 90-- the average of those two-- and this value right over here is going to be plus 100 over the average of these two. So the average of those two is 1,050. And so this is 100 divided by 1,050, which gets you to about 0.95. So about 9 and 1/2% change in quantity demanded for my book. And then this denominator right here is negative 20 divided by 90. So you get a drop of 22%. And so if you divide the numerator by the denominator, you get 0.952 divided by negative 0.22222-- I'll just put couple of 2's there-- and you get a negative 0.43. So this is equal to negative 0.43. And this makes sense. If you lower the price of an e-reader-- this complement product, a product that goes along with my e-book-- it increases the demand. So just like you get with price elasticity of demand, you get a negative value over here. And what about completely two unrelated products? So let's say that I have basketballs, and the price of basketballs goes from, let's say, $20 to $30. What's going to happen to my e-book? Well, my e-book's not going to change. It's going to stay at $1,000. So my percent change in the quantity demanded of my e-book is going to be 0 in this example. So we're going to have 0, when we want to do this cross elasticity of demand, over my percent change in basketballs, which would be 30 over 25. So whatever that is-- 30 over 25 would be 10 over 25-- which is a 40% increase. So that would be 0 over 40%, which equals 0. So for unrelated products, products where the price of change in one of them does not affect the quantity demanded in the other, it makes complete sense that you have a 0 cross elasticity of demand. If they're complements, you would have a negative cross elasticity of demand. And if they're substitutes, you would have a positive one. And the closer the substitutes they are, the more positive your cross elasticity of demand is going to be. |
Khan_Academy_AP_Microeconomics | Profit_maximization_AP_Microeconomics_Khan_Academy.txt | - [Instructor] We've spent several videos talking about the costs of a firm. And in particular, we've thought about how marginal cost is driven by quantity and how average total cost is driven by quantity, and we think about other average costs as well. Now, in this video, we're going to extend that analysis by starting to think about profit. Now, profit, you are probably already familiar with the term. But one way to think about it, very generally, it's how much a firm brings in, you could consider that its revenue, minus its costs, minus its costs. And a rational firm will want to maximize its profit. And so to understand how a firm might go about maximizing its profit or what quantity it would need to produce to maximize its profit based on this, on its cost structure, we have to introduce revenue into this model here. And in particular, we are going to introduce the idea of marginal revenue. And we're going to assume that this firm is in a very competitive market, and so it is a price-taker. So regardless of how much this firm produces, the incremental revenue per unit of what it produces, maybe this is a doughnut company, the incremental amount per doughnut is going to stay the same regardless of how much this firm in particular produces. So let's say that the marginal revenue in this industry, in this market, is right over here. So one way to think about it is this would be the unit price in that market. So let me put that right over there, marginal revenue. Once again, for every incremental unit, how much revenue you're going to get, so it would just be the price of that unit. So how much would a rational firm produce in order to maximize its profit? If the marginal revenue is higher than the marginal cost, well, that means every incremental unit it produces, it's going to bring in some net money into the door. So it's rational for it to do it. So it would keep producing, keep producing, keep producing, keep producing. Now, it gets interesting as the marginal cost starts to approach the marginal revenue. As long as the marginal revenue is higher than the marginal cost, it's rational for the firm to produce. But right at that unit where the marginal cost is equal to the marginal revenue, well, there, on that incremental unit, the firm just breaks even at least on the margin. It might be able to utilize some of its fixed costs a little bit. But then, after that point, it makes no sense at all for it to keep producing. Why is that? Well, if the marginal cost is higher than the marginal revenue, that would be like saying, hey, I'm gonna sell a doughnut for $1 even though that incremental doughnut costs me $1.10 to produce. Well, no rational person, if they want to maximize their profit, would do that. So a rational firm that's trying to maximize its profit will produce the quantity where marginal cost intersects marginal revenue. It will produce this quantity right over there. Now, a natural question might be how much profit will it make from producing that quantity? Well, all you have to do is think about, this is the marginal revenue that it gets, and another way you could think about it, because this is constant, it's also going to be the average revenue that it gets per unit. And this right over here, is the average total cost per unit. And so what you could do is, this is how much it's getting on average per unit, and then multiply that times the number of units. And what you get is the area of this rectangle. So for those of you who are more visually inclined, one way to think about it is a profit-maximizing firm, a rational profit-maximizing firm, would want to maximize this area. Think about what would happen if they only produced this much. Well, then they're giving up a ton of area. Then the rectangle would only be this big. This would be the profit that the firm is going to be making from those units. And then if it decides, for some irrational reason, to produce more than this quantity that we settled on before, let's say this right over here, notice even though that the base of this rectangle is longer, the height is less, and this would actually have a lower area. And the reason why I feel very confident that this will have a lower area is because, in this situation, the firm is losing money on all of these incremental units where the marginal cost is higher than the marginal revenue. So big takeaway, a rational firm that's trying to maximize its profit will produce the quantity where marginal cost and marginal revenue are equal to each other. |
Khan_Academy_AP_Microeconomics | AP_Microeconomics_FRQ_on_perfect_competition_APR_Microeconomics_Khan_Academy.txt | - [Instructor] This is the type of question that you might see on an AP Economics exam, and it's talking about perfectly competitive markets. So it says, a typical profit-maximizing firm in a perfectly competitive constant-cost industry is earning a positive economic profit. So the first question they ask us is, is the market price greater than, less than, or equal to the firm's price? Explain. So pause this video and see if you can answer this on your own before we do it together. All right now let's do it together. So remember, we are talking about a perfectly competitive market. So in a perfectly competitive market, all of the players in that market have to be price takers. They have no pricing power. So the market price has to be equal to the firm's price. So market, market price equal, equal to firm price, firm price because in perfectly, perfectly competitive market, market, firms are price takers, firms are price takers. They have no pricing power. All right, part b. Draw correctly labeled side-by-side graphs for both the market and a typical firm and show each of the following. And they ask us to do a bunch of stuff here. So once again, pause this video and actually get out paper. This will be very valuable for you to have a go at this. All right, so let's see we wanna do these side-by-side graphs, and we wanna think about the market and the firm. And we've done this in multiple videos before. So let's think about what they're talking about is, so this is the market right over here. That's the market. And this is, on this axis is going to be price. On this axis is going to be quantity. And then let me do a similar thing for a firm here. So that would be the firm's price axis, price. And then this would be quantity for the firm, quantity. Let me make it clear, this is the market, and then this right over here is the firm. And let's see, they say, market price and quantity. So the equilibrium price and quantity in the market. So we could draw the supply curve for the market. It might look something like this, upward-sloping, we've seen that multiple times. We could do the demand curve for the market. It would look something like that. And then we have the equilibrium price in the market, which they want us to use P sub m, so P sub m. And then we have the equilibrium quantity in the market, which they want us to use Q sub m. So we've done this first part. All right now let's see what else they want. The firm's quantity, labeled Q sub f, the firm's average revenue curve, labeled AR, the firm's average total cost curve, labeled ATC, the area representing total cost shaded completely. So in order to do this first part, the quantity that it would be rational for this profit-seeking firm or the profit-maximizing firm to produce, to think about that, we'd actually also have to think about the firm's average revenue. And the average revenue, which is going to be the same thing as the demand curve for that firm, is going to be based on this market price. Remember, the firm, in this perfectly competitive market, has to be a price taker. So this horizontal line right over there, that is the firm's average revenue, AR, which is equal to its marginal revenue, which is equal to its demand curve, which is equal to this market price. And the quantity that it's rational for this firm to produce is where this marginal revenue curve, which is also the average revenue curve in this case, intersects our marginal cost curve. So the marginal cost curve might look something, something like this. So marginal cost. And so this right over here is our Q sub f. So we've done this part and this part. The firm's average total cost curve, well the average total cost at this quantity needs to be below the marginal revenue and the average revenue at that quantity because we know that the firm is earning a positive economic profit. So we are dealing with a situation that likely looks like this. So the average total cost might look something like this. And I drew it that way to ensure that at this quantity, Q sub f, our marginal revenue and our average revenue is above our average total cost. That tells us that we're earning economic profit in this situation. So I've done part iv. The area representing total cost shaded completely, well the area representing total cost would be the cost per unit, the average cost per unit, which is that much, times the total number of units. And the total number of units is going to be this length, which is equal to Q sub f. And so your total cost is going to be this shaded area. If they were asking us our total economic profit, then we would be talking about this area up here, but they're not. They're talking about our total cost, which is this area right over there. So we have done those parts. Now let's go to part c. If one firm in the market were to raise its price, what would happen to its total revenue? Explain. Pause this video, see if you can answer that. Well remember, we're dealing with a perfectly competitive market, a perfectly competitive industry. There's no differentiation between anyone's products. So if all of a sudden, someone were to stick their head out and try to raise price, no one would buy their product anymore because people can get identical products from other people for a lower price. And so, its total revenue, its total revenue would go to zero since product is undifferentiated, undifferentiated, and consumers could buy from others at lower price, at lower price. And this is another way to think about it. They have to be price takers in a perfectly competitive market. |
Khan_Academy_AP_Microeconomics | How_price_controls_reallocate_surplus_AP_Microeconomics_Khan_Academy.txt | - [Instructor] What we're going to talk about in this video is the effect of price controls on changing how the surplus, the total surplus is reallocated between consumers and producers. And we already touch on this in other videos. The video on rent control. The video on minimum wages. And so this is to make sure that we are taking away some of the big ideas. So, right over here I have my classic demand and supply curves. Of course, the rental market. At a high price the quantity demanded is low. The quantity that people would be willing to supply is quite high. And at a low price, the quantity that people would be willing to supply is low, while the quantity demanded would be quite high. And we've seen from many videos so far in our journey through economics that we have our equilibrium price and our equilibrium quantity where these two curves, or lines intersect. So, I'll call this Q sub zero and this is price sub zero. But let's say for whatever reason city officials in this city where this rental market that this chart describes the rental market for they decide that P sub zero is too high that their owners are complaining that rent are too high in the market. And so the city decides to put in a price control. And in this case, they try to implement a price ceiling. So they say, look, the price of rent per square foot per month cannot go above this level right over here. So this is the price ceiling. Price ceiling, right over there. Now, what's going to happen here? Well, if this is the price ceiling, then right over here, this is the total amount of square footage the quantity of I guess, square footage that is being willing that people are willing to supply that. The landlords, or building owners are willing to supply. But at this price, you have a much higher quantity that is being demanded. This is right over here is the quantity demanded. And when the quantity demanded at a price is higher than the quantity supplied well, then you have a shortage. Put this right over here. Is describing a shortage. And we talk about that in other videos. But let's think about what's happening to the total surplus. So when we let the market just get to an equilibrium price and quantity the total surplus, actually let me just draw separately the consumer and the producer surplus. So this was the consumer surplus. Right over here. Before the government intervention. And then, this is the producer surplus. And we've talked about this is other videos. But now what happens when we have this price control? Well, this is the quantity supply. Now, all of a sudden, the total surplus shrinks. The total surplus is now being depicted by this white trapezoid. And also, think about how things have shifted. So one thing that you see clearly, what is the producer surplus now? The producer surplus is only this little yellow triangle, or this little. I don't know my colors. This little blue triangle at the bottom. So you see very clearly that all producers suffer here. All producers all producers suffer. Now you might say, well, of course this is a rent control. But surely, the consumers will benefit here. Well, it isn't the case, it is the case that some consumers, the ones that are able to get into a unit, they might benefit. So this right over here, some demanders, or you can say some consumers consumers benefit. But, not all of the consumers benefit. In fact, you have a shortage now. Before you had more people who were able to get housing. Now these folks are not going to get housing. And, as in all of economics you should take a grain of salt in any type of oversimplified model like this. Or simplified, this is actually a very useful model for thinking about certain things. Because, even these consumers that are benefiting according to this model for these consumers it looks like their surplus has grown, or if you're this kind of marginal renter right over here, let me draw it over here. If you are the marginal renter that was before, getting this much benefit. Now you're able to get that plus all of that. But think about the things that this model is not capturing. What's the incentive for the landlord now? Would they want to invest in the building as much? Would they would they maybe let the building kind of suffer a little bit? And there's also particular quirks for how rent control is implemented that might also change behavior. So always keep in mind what's not captured by the model. But in broad brush terms you put in a price control, in this case, you put in a price ceiling you're going to create a shortage. All the producers are going to suffer. Some of the consumers benefit, according to this model. But not all of them. Because not all of them are now going to be able to get a place to rent. Now let's move over to another market. Let's say the corn market. And let's say once again we have our equilibrium price and our equilibrium quantity. So price of equilibrium. And this is our quantity sub equilibrium. But let's say in this situation the government let's say the farmer, the corn farmers are able to organize. And they're able to lobby the government and say, hey, we really suffer when there's low corn prices. So where we want to institute a price control we want you to institute a price control. We want a price war. So the government says, okay corn farmers you seem to be pretty serious about it so we are going to institute a price floor. So the price cannot go below this level. So this is a price floor. Well, what's going to happen now? Well let's think about the quantity demanded, and the quantity supplied. So that, right over there, is the quantity demanded. You see where the price intersects the demand curve. And this right over here is a quantity supply where we intersect in the supply curve. Quantity supply. So in this situation your demand is less the quantity demanded is less than the quantity supplied. So in this case, you have a surplus. The farmers would want to produce more than people would want at that price. And also think about what happens to this total surplus. And in particular, the consumer and the producer surpluses. So in the old world this was the consumer surplus. And, this right over here is the producer surplus. In the new world, the total surplus shrinks the sum of the two. We're now talking about this, the area of this trapezoid. Right over here, this sideways trapezoid. And you see that all consumers suffer because now the consumers surplus has been shrunk right over the consumer surpluses right over here. There's consumers who are now not even consuming corn. And even the ones that are consuming corn before, let's say this marginal consumer right over here was getting all of this benefit. Now, they are getting a smaller benefit. So we could say, all all consumers suffer. Now what about the producers? Well, the producers who are able to sell their corn definitely get a benefit. So if you were this marginal producer right over here your surplus right over here would have been like that. But now you get even a higher price for it. And so, one way to think about it is, some producers, some producers for sure, benefit according to this model. So let me write that. So, some producers producers benefit. But it's important to realize, once again, that not all of the producers benefit. Because once again, we have a surplus. If it's implemented this way well, you might have a lot of farmers who aren't even able to sell their corn at that price. So it's an interesting thing to think about. You should always take models with a grain of salt. But it is a pretty interesting framework where governments often will try to do some type of knee jerk solution to try to make something look good, or feel good to their constituents. But the end effect is that people might suffer more than they expect. They might cause a shortage when you put a price ceiling. Or, it might cause a surplus when you have a price floor |
Khan_Academy_AP_Microeconomics | Fixed_Variable_and_Marginal_Cost.txt | We here at the Khan Academy are working on some type of a software project. And we need to think about what's the optimal number of programmers we should hire, at least think about how much productivity we're getting per programmer when we're working on this software project. And so what I've done over here-- this is a spreadsheet so I'm not going to be able to write. I'll only be able to type. This is Microsoft Excel right over here. In this column, I have the different numbers of programmers. And then let's say based on other studies or industry studies or our past experience, this tells us how many lines of programming code we can get per month. And obviously, lines of code isn't maybe the best way to measure things because someone can write good lines or bad lines of code. But let's just say this is one way of measuring productivity for software engineers. So the first thing I'm going to think about is what are my fixed costs? So what am I going to spend no matter how many software engineers I hire for this project? And for the sake of this video, my fixed costs will be the office space and the electricity and let's assume I just have an office that can accommodate any number of these programmers. So that's a fixed cost. That's not going to change depending on the number of programmers I have. And then the other fixed cost, let's say I have a product manager for this project. And I'm going to pay her salary to essentially help spec out what this software should actually do. So her salary is, let's say, $10,000 a month and then another $5,000 a month in office space for everybody. So it's going to come out to $15,000 a month. And that's not going to change regardless of however many programmers I have. So I'm going to go into Excel and go to this little bottom right right over here. And I'm going to drag that down. And so it's going to be $15,000 in fixed costs no matter how many programmers I end up hiring. Now the variable cost. Well, let's just say that the full compensation for a programmer is $10,000 a month. So if you include the cost of their salary, if you include the cost of their health insurance, you include the extra goodies that they will eat from the company kitchen, whatever it might be. So it's going to be $10,000 a month. So my variable my total variable costs are going to be $10,000 times the number of programmers. So here I'm going to write equals, and I'm going to write, it's going to be $10,000 per programmer times-- that little snowflake, I pressed Shift + 8 to get that snowflake-- times, and I could say times whatever is in that cell. So you see it's cell D7. And actually let me scroll this over so that you can see the cells. So that is cell D7. And let me press Enter. So it's $10,000 times D7, which is this one right over here. And I just selected that. And I can press Enter. And right now, that's nothing. Let me scroll over so we can see everything a little bit better. Let me scroll over a little bit. There you go. I'm having trouble. OK. There you go. Now, what are going to be my total costs? My total costs are my fixed costs plus my variable costs. So that's going to be equal to-- and I'm just using my arrow keys right now-- it's going to be equal to F7, right? That's cell F7 plus this one, plus my variable costs. My total costs are my fixed costs plus my variable costs. And so it's $15,000. And actually, I can make this true for every row over here. And this is one of the really useful things about a spreadsheet, is I defined this cell as being $10,000 times whatever this cell is right over here. And so what I can just do is I can just take that, drag that all the way down. And for every one of these, it's going to take $10,000 times the cell that's essentially three to its left. So now it's going to be 10,000 times D8. This will be 10,000 times D9. Let's get that. So that we can see it right there. 10,000 times D9. And you could, if you click on there, you can actually see what the formula is. 10,000 times D9. So by dragging that, I was able to get the right formula all the way down. Now the total cost for every row here is going to be two to the left plus one to the left. And so if I drag that down, it'll do that for every row over here. So now this is 25, is the 15 plus the 10. 105 is 15 plus 90,000. Our total costs are fixed costs plus variable costs. Now let's think about the average fixed cost. And the average fixed cost, we're going to think about it in fixed cost per line of code produced. And over here, line of code produced is 0. So we're going to divide by 0, which is undefined. So we could leave that blank. But we could fill this one in. So our total fixed costs-- this is going to be our total fixed costs, which is cell F8. I just used the arrow keys to select F8. Divided by our total lines of code per month. So divided by our total lines of code per month. And so that gets me $3.75 fixed costs per line of code. And then I can do the same thing that I've been doing before. I can drag this down. And then we see what the fixed cost is. So if at any given point if I take the fixed cost, $15,000, divided by the lines of code, I get $1.38. And this actually makes sense because the more programmers I add on to this project, the more lines of code I get, I'm using the same fixed costs I'm. Using the same project manager. I'm using the same office space. So the cost of that project manager and that office space gets spread out along more and more code. So the fixed cost per line of code goes down as we add more and more programmers. Now what is the average of variable cost? So once again, the variable cost is going to be whatever the variable cost is per lines of code per month. So when we're talking about average, we're talking about average cost per line of code. So this is per line of code. Let me write it over here. Per line of code. And I can even say per month. Per line of code per month. Actually, I wanted that spread out more. But the way I've set it up. So let me scroll down. Oops. I'm having issues here. All right. Well these are all average lines of code per month. And so let's think about what happens with our variable costs. So I'm also going to start here because I don't want to divide by 0. So in this month, our total variable costs were $10,000 and our total lines of code are going to be $4,000. G8 divided by E8. And so average variable cost per line of code is 2.50 . And then what happens? So let's do that for every row over here. So when we do it for every row, something interesting happens. Our average fixed cost went down because we're taking the same cost and we're spreading it out amongst more code. But our average variable cost went up. As we added more programmers, per line of code, it actually costs us a little bit more on average per line of code. And that's actually, if you look here, as we add the incremental lines of code we get per programmer is actually going down. That first programmer by themselves, she can write 4,000 lines of code. But then that second programmer you're not getting to 8,000, you're getting to 3,000. And probably because they have to coordinate with each other. They have to plan a little bit more. It's not all in one person's head. Then, when you're at the third one, you're not even adding 3,000 lines of code. You're only adding 2,000 lines of code. And this is actually a real phenomenon that actually happens in companies. The more people you add on to a project, obviously they can maybe do more work, but there's also more coordination. There's going to be more meetings. There's going to be more interruptions. And so each person's individual productivity is going to go down. And this isn't to say that this third coder is somehow worse than the first coder. On average, all of them are now only going to produce 3,000 lines of code a month, when maybe individually they could have each produced 4,000 lines of code, but they have to spend some of their energies now coordinating it. And so that's why our average variable cost per line of code is going up. As we add more and more people, incrementally it's becoming more and more expensive on average to write that line of code. And now if we look at average total cost, that's going to be-- and this is once again, this is per line of code-- is going to be our total cost, H8 divided by the total lines of code per month. So if we just hire one engineer, we're going to have $6.25 spent per line of code. And this is actually just the sum of these two right over here. And then, let me set that formula for every row. And so we see something interesting. When we start to hire a few engineers, we're able to spread out our fixed costs, even though our average variable cost per line of code are going up, our fixed costs are going down. So it's actually we're getting a little bit of a benefit because we're spreading our fixed cost per line of code. But then it starts to get expensive again because, as we said, the more people you have working on the project, they're going to have spend more time coordinating with each other and maybe even undoing each other's work or redoing each other's work as opposed to just writing the actual software. And now let's think about the marginal cost. The marginal cost, the best way to think about it is, what is the incremental cost of that next set of line of code? So one way to think about it-- so this is going to be how much more you're spending divided by how much more code you're getting. So for example, how much more-- and this is going to be, once again, per line of code. So when we go from zero programmers to one programmer, we're going from $15,000 of total cost to $25,000 of total cost. We do it in parentheses. If we're going from 15 to 25, that means we're increasing our expenditure by 25 minus 15. And so that's why I'm doing H8 minus H7. So that's how much more we're spending in expenditures. And then, how much more code are we getting? Well we're getting 4,000 minus 0 lines of code. And the reason why I'm doing the formula this way is so that when we drag it down on all the rows, the formula will be right because it's relatively taking the right cells into account. And the reason why I'm saying it's average is because this is saying, what's the incremental cost per line of code for this first 4,000 lines of code? And then we can go from there. And we can drag it down. And now this tells us the incremental cost per code for the next 3,000 lines of code. And once again, it got a little bit more expensive because we're getting a little bit less efficient as we add more and more people. And there's something very interesting that happens here. And you might have even noticed it in these numbers over here. We actually get a negative marginal cost. And this isn't meaning that when we add more lines of code somehow we're getting money. It's actually saying that as we spend more money, we're actually killing lines of code. Because at some point, if you have too many people on this one project team, they actually start killing each other's productivity. And you can even see it right here in the numbers. When we had seven people, we were able to write 11,400 lines of code. But then the eighth person, because of coordination-- it's not that this eighth person is incompetent, it's just when you have eight people on a team, everyone's productivity goes down so that you're only able to produce 11,200 lines of code. And that's why you had this negative marginal cost. Now, when you get to eight people, all of a sudden by spending more dollars, you're actually destroying some of what you were actually trying to produce. So what I wanted to do here is just to really get you behind the numbers and really maybe give you a little sense of how you can actually do this with a spreadsheet. And get you thinking a little bit about how a firm's cost structure might actually work. |
Khan_Academy_AP_Microeconomics | Economies_and_diseconomies_of_scale_AP_Microeconomics_Khan_Academy.txt | - [Lecturer] In the last video, we were able to construct here in red this long-run average total cost curve based on connecting the minimum points or the bottoms of the U's of our various short-run average total cost curves. Each of those short-run average total cost curves were based on a certain amount of fixed cost in the short run, but in the long run, you can change your fixed costs. And here are fixed costs with the number of trucks. And so, we can vary it to optimize for a certain amount of quantity. Now when we did that, you could see a little trend here especially as we go up to, the way I drew it, it wouldn't necessarily be up to 200 or whatever you're producing, but the way I drew it, you see that this part right over here, it looks like our long-run average total cost curve is declining down. So, one way to think about it is, we are getting more and more efficient at producing our tacos in the long run as we product more of them until we get to 200 tacos. And so, at this part of our curve, we are experiencing economies of scale. And we've talked about where economies of scale can come from. It can come from specialization of labor or even machines. Specialization. So, as you get more and more scale, you can have different parts of your process specializing in making the taco shells or grating the cheese or cooking the meat, whatever it is. So, there is a specialization. You could get better at sourcing. So, as you get more scale, you might be able to order more of your supplies at a time, so you get better deals. You might be able to even, who knows, at some point, start a farm yourself and then cut out the middleman, and so forth and so on. Now as we get past that point, we see that our long-run average total cost curve, at least in this example, started to trend up. And so, this part of the curve, you could say that we are experiencing diseconomies of scale. Diseconomies of scale. So, what would cause diseconomies of scale? Well, these would most typically happen because what are known as coordination issues. As an organization grows, you have more people, more resources that you have to coordinate. And so, that complexity can sometimes make an organization more inefficient. There's other diseconomies of scale. At some very large scale, you might be depleting all of the low-hanging fruit of your inputs, and so, you have to pay more for some of your inputs. Maybe you've already depleted the people who are willing to work for less, so you have to raise wages, or you've depleted a lot of the resources you need, so you have to find new, more expensive resources. Now in this curve, it's not as obvious, but you can also have a notion of constant returns to scale. So, if we had a long-run average total cost curve that looked something like this, let me draw it over here, then in this section right over here, as the average total cost, the long-run average total cost is going down, that would be economies of scale. This section over here, as the long-run average total cost is going up, that would be our diseconomies of scale. But this section over here where it is constant, you might guess what that is called. That is called constant economies of scale or constant returns to scale, sometimes known as efficient scale. |
Khan_Academy_AP_Microeconomics | Marginal_revenue_and_marginal_cost_in_imperfect_competition_AP_Microeconomics_Khan_Academy.txt | - [Instructor] In this video, we're going to think about marginal revenue and marginal cost for a firm in an imperfectly competitive market. But before we do that, I just want to be able to review and compare to what we already know about a firm in a perfectly competitive market. So right over here, we're analyzing the firm's economics. This shows the marginal cost as a function of quantity, and we've talked about this before. Oftentimes, it will trend down initially, as you have better specialization and some efficiencies, and then it might start trending up, as there are just coordination costs or other costs that make the marginal cost go up. And we have talked about this notion that, in a perfectly competitive market, the firm is a price-taker. There's going to be some market price, let's call this P sub m, some price in the market for the good that they are producing, and there's many producers who are producing this good. And they're undifferentiated, and there's no barriers to entry. And so they just have to be price-takers there. No matter how many units they produce, they're just going to be able to get that same market price. So a firm in a perfectly competitive market, that market price defines their marginal revenue curve. Their marginal revenue curve will essentially just be a horizontal line like this, and we've already studied this in previous videos. And we talked about that here, if this firm was trying to maximize its profit and if it was rational, it would produce the quantity where marginal cost is equal to marginal revenue. So it would produce this quantity right over here. But now let's think about how things are a bit different for a firm in an imperfectly competitive market. In a previous video, we talked about how, in an imperfectly competitive market, there's some differentiation amongst the various players who are competing, and so their market price is a function of quantity. If they just produce a bunch of their product, the price that they get in the market is likely to go down. So they will have their own firm-specific demand curve. Maybe it looks something like this. So that is their demand curve. And we also saw in that video that that demand curve, essentially the price that they could get at any quantity, that that's not going to be the same as the marginal revenue curve. If the demand curve is downward-sloping like that, the marginal revenue curve is likely to be even more downward-sloping. So it's going to look something like this. That would be the marginal revenue curve. Now in this situation, what would be rational for the firm to do? Well, once again, it would want to produce the quantity where the marginal cost is equal to the marginal revenue. So they would want to produce this quantity right over here. But you see something interesting here. If they produce at this quantity, notice the price that they can get in the market is much higher than that. The price that they get in the market is higher than the marginal cost and the marginal revenue at that point. And because we see a situation where price is greater than your marginal cost, versus in a perfectly competitive market where you see that price is equal to marginal cost, that that is the optimal quantity, but because you have this gap, that people are willing to pay more than that marginal cost. But you still aren't going to be able to produce any more because it doesn't make sense from a marginal revenue point of view. This gap, the difference between the price and the marginal cost at this rational quantity for this firm in an imperfectly competitive market to produce, economists would refer to this as an inefficiency, inefficiency. Folks are willing to pay more than that marginal cost, but you still have no motivation to produce more. Because if you produce more, even though the price is higher than the marginal cost, your marginal revenue is going to be below the marginal cost, and so you would be taking a hit in aggregate on those extra units. |
Khan_Academy_AP_Microeconomics | Normative_and_positive_statements_Basic_economics_concepts_AP_Macroeconomics_Khan_Academy.txt | - [Instructor] What we're going to do in this video is discuss the difference between normative statements and positive statements, and you'll see these words used usually in an economic context, sometimes a philosophical one. A normative statement is one that really is a matter of opinion, maybe a matter of ethics, something that someone thinks is how the world should be. While a positive statement is something that, it doesn't necessarily have to be true but it's something that can be tested. So what we're going to do in this video is look at a bunch of statements around economics and think about whether they would be classified as normative statements, things that are opinions, that are a matter of ethics or morals, or whether they are positive statements, things that can be tested. So let's look at our first statement. This says, "Paying people who aren't working, "even though they could work, is wrong and unfair." So regardless of whether or not you agree with this statement, is it a normative statement or a positive statement? Well the fact that someone's saying it's wrong and it's unfair, this is pretty clearly a matter of opinion so this would be a normative statement. You can't test whether this is wrong or unfair, you would just have to believe that it is wrong and unfair. Now let's look at another statement. Programs like welfare reduce the incentive for people to work. Is this a normative statement or a positive statement? Well it might feel a little normative, it might feel like this is an opinion, but it actually can be tested. You could institute some welfare program on some small scale and compare it to a comparable place where there isn't a welfare program and see what it does for incentives to work, you survey people, you see how many people work in one situation or another. It might be a false statement, it might be a true statement, but either way, it actually can be tested, so this would be a positive statement so I'll put it in this category right over here, this is a positive, positive statement. Alright, let's look at another one. This say, "Raising taxes on the wealthy to pay "for government programs grows the economy." Is that a normative statement or a positive statement? Well once again, this can be tested. It might be true, it might be false, maybe your test is even inconclusive, but it can be tested, you could try to run a simulation, you could look at case studies of countries that did do this and see what happens to their economy versus ones that didn't do it. And so this is, even though it looks like something that someone who favors raising taxes on the wealthy maybe out of fairness arguments, something that they would say, this statement itself is not normative, the statement can be tested, so this is a positive statement. A good giveaway for normative statement, if it said something like it is fair to raise taxes on the wealthy to pay for government programs, that would have been a normative statement or we should do this, that would have been a normative statement but here, this is something that's testable. Now the next statement, "Raising taxes on the wealthy "slows economic growth." Is that a normative statement or a positive statement? Well, once again, this might feel like someone who is against raising taxes, who think it's unfair to raise taxes on the wealthy, something that they would say but the statement itself can actually be tested. So this is also a positive statement, even though in some ways it's the opposite statement as the one that we just did. Because once again, we could look at countries that did this and countries that didn't do this, we could run a computer simulation to try to understand whether this statement is true. Let's do one last statement. This says, "The government should raise taxes "on the wealthy to pay for helping the poor." Is this normative or positive? Well in this situation, the word should is a pretty big giveaway, should, or it's fair or unfair, this is someone's opinion, it's not something that's testable, you can't test whether this statement is right or wrong, it's based on, do you believe ethically, morally, that this is true? And so this is a normative statement, so I'll put it in the normative column. So big picture, these words normative and positive, these are fancy words but all they mean is normative is a matter of morals or opinion and can't really be tested while a positive statement, whether they're right or wrong or whether you agree or disagree with them, these are things that in theory could be tested. |
Khan_Academy_AP_Microeconomics | Price_elasticity_of_supply_determinants_AP_Microeconomics_Khan_Academy.txt | - [Instructor] In several videos we have talked already about the price elasticity of supply. In this video, we're going to dig a little bit deeper, and we're going to think about what factors might make a supply curve, or supply schedule, or portions of it, to be more elastic or inelastic. So we'll think about the determinants of the price elasticity of supply. And to help us think about that, I've drawn two different supply curves. And so, remember, price elasticity is thinking about how sensitive is the quantity, or the price elasticity of supply, we're thinking about how sensitive is a percent change in quantity supplied to a percent change in price. So, for example, if we were to go from this price, to this price, it's a pretty significant percent change in price. It looks like about a 50% change in price. But if we look at this supply curve, where it's getting pretty vertical, it's not quite vertical yet, but it's getting pretty steep, our percent change in quantity is going from here to here, so it's not as dramatic. So these parts of the curve that are relatively steep, as they're approaching more and more vertical, we would consider those to be relatively inelastic. If you had a perfectly vertical, or portions of this curve that were perfectly vertical, at those portions you would say that you have perfect inelasticity. And then, if on the other hand, in this example, if we were to change our price from say, this to this, that's a relatively small percent change in price, but notice it could result in a fairly large percent change in quantity supplied, we're going from this to this. So, when we are more horizontal, or the flatter our supply curve is, that would be, we're talking about relative elasticity, or we are much more elastic here than we are there. So, elastic. And if you were perfectly horizontal, then you would have a perfectly elastic part of our supply curve, or our supply schedule. Now what would cause, what are the factors that would make us get more inelastic, or more elastic? Well let's think about a situation, let's say we are in this world. Let's say we are in the world, starting at this price, where this is our quantity. Imagine that the factories that produce whatever good we're talking about here, they're already at full production. And to make more production would take a lot of time, or there might not even be the resources to have more production. Maybe the people who make the factories are doing other things right now. Maybe you have other resources, other inputs, into your production that there's just not more of. Maybe oil, as much oil in the world is being produced, and that's an input into our production. And so in that world, even if the price is, even if the price goes dramatically up, people might want to produce a lot more quantity, but they just can't, they don't have more factories, or they don't more inputs to produce them. So, the quantity might go up a little bit, they might run the factory, they might run the factories in overtime, or the people who are trained to do that skill, and maybe there just aren't a lot more people who can do that skill, they're working overtime, but they can't produce a lot more. And so, a couple of things that we can glean from that example is, this inelasticity of supply tends to happen in the short run, when in the short run, you're not gonna be able to build a new factory in the short run. You're going to be able to find more sources of let's say oil in the short run. You're not going to be able to train a lot more people to produce a lot more, in the short run. So if we're talking about a supply curve that's describing more of a short run time frame, then it tends to be more inelastic, and then the other thing is, and we already talked about this is, there's not available resources, or not a lot of available resources, not a lot of available, available resources. Once again, your factory's operating full-out. You've hired everyone who can produce that thing you're producing, or some natural resource that you need as an input, to make this good, well, the world is already producing as much as they could. Well, let's then go to the other situation. What would cause elasticity? Well, this could be a world where in the long run, it might be easier to get more resources. In the long run, you can build more factories. You can find and hire and train more people. So, the longer run that we are talking about, that tends to lead to a more elastic supply curve, in the longer run, and then another notion here, is that you aren't capacity constrained, or maybe I should say, aren't resource constrained. Aren't resource constrained. So, maybe our factories are running nowhere near capacity. So even a small increase of price, where it's like, "Hey, I'm gonna just make my factories run "a little bit more." Or there might be a lot of people who can help produce who have the skills to do, to produce that good, so as the price goes up, more and more people are going to start producing that good. Or it might be using resources where there's no shortage of it. Where, as the price even goes up incrementally, people will just use more and more of that resource to provide more and more of that good. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 32_Polymers_I_Intro_to_SolidState_Chemistry.txt | we will be talking about a new topic today. And that's polymers. And we're going to start by talking about-- what I want to do is cover two ways that we humans make polymers. And the first way has to do with something called the radical. So that's where I want to start our story. And what is a radical? Well, a radical is a molecule that has one or more unpaired electrons. So one or more unpaired electrons. But as you can imagine, if I've got an unpaired electron, it's kind of like a broken bond. It's kind of like a broken bond. That's a radical. And you could also imagine that a broken bond wants to not be broken. So it's going to try to find something to bond to. That's why we've got radicals all in us all the time. Sometimes they're important, but sometimes they can cause a lot of harm. And that's why if you eat blueberries, then you get these things called antioxidants. And antioxidants, that's just one of many, many types of antioxidants. What do they do? They go and they give these radicals what they need, what they're looking for, an electron. They give them an electron. And they don't mind. They don't become radical themselves. They just simply stop the radical from being radical. Now, why does this matter? Well, we're going to see what that means for polymers. But let's make a radical. So imagine-- and I'm going to go Lewis on us, because I want to see the electrons. I'm going to go Lewis. So imagine I've got methane. Well, methane looks like this, nice and happy. Oh, the octet. But now I introduced a chlorine atom. So I've got now chlorine. And chlorine is hungry. Chlorine wants to fill its octet by itself. And so what does it do? It's like, well, I'm powerful. I'm strong. I'm just going to grab from methane. It sees methane, and it's like, OK, you give me one of your electrons. How about that is a deal? And here we go. And here we go. And it takes it in the form of the hydrogen. And so we got this like this, and like this. Oh, the chlorine got really happy, but look at what happened to the methane. It lost an electron and a hydrogen with it, and it has an unpaired electron. That electron is not happy. I mean, it just wants to bond to something. That's all. You can't blame it. Don't we all want that? The problem is that this really needs it badly. And so this is called-- so what we did is we took chlorine, and we combined it with methane. And we made what's called a methyl radical. So this is called-- that's called a free radical. Or in this case, it's methyl, because that's what the chemistry is. It's a methyl radical. As we'll see, you can make radicals out of lots of stuff. Just take a hydrogen and an electron that have a free unpaired electron left, and it's a radical. I could start with ethane and make an ethyl radical, C2H5 with a dot. That dot is the unpaired electron. And so on, and so on. And so if you have free radicals and they come along, and there's some other molecules-- it's not going to be this one, but maybe there's something like an antioxidant in those blueberries. It's going to come along. I've got electrons I could give you. Here, have one. I see how needy you are. I see how badly you want one. You can take one of mine. And I'll be fine. I'll be fine. Or maybe we'll form a bond with that. Those are antioxidants. But that's not the topic. The topic is we're making this new material called a polymer. So how does that work? Well, to understand that, we've got to make a radical out of a special case. We've got to make a radical out of a double-bonded molecule. So now I'm going from methane to the double-bonded ethane. Now, hang on. Let me put this up here. And in the case of a double bond, something very important happens. Something very important happens. And the reason is that I've got two bonds. I've got two bonds, not one. Well, that's what double bond-- that is what double bond means. But I can now-- because I've got a double bond, I can do something very important. I can do something very important. When I turn that molecule into a radical, something special happens. So what I'm going to do is I'm not going to call it chlorine. I'm going to just call it R dot. And because now we're making a polymer, and this is called-- well, it's got different names. It's called radical, or sometimes you'll see it as addition polymerization. And this is the first kind of polymer I want to talk about. And so I'm going-- you saw how it worked with chlorine. I'm just going to say, I've got something called R. And it's called a radical initiator. And I'm going to draw the dot there, just like I had the dot on chlorine. This is going to be the radical initiator. Radical initiator. And I'm going to now bring this to my double-bonded molecule. That's the key. The key is in the double bond. So watch what happens. So first, here's what happens. I've got R with its dot plus-- and again, I'm going to go Lewis. I'm going to have all those electrons in the bonds spelled out for you with dots. So R plus C. Two pairs of electrons in there. Do you see that? There's my double bond. And here are the hydrogens. I'm not going to draw them every time. Those sticks have hydrogens on the ends. But now watch what happens. So now, this radical, like the chlorine, it sees this electron, and it says, aha, I can take you. I can take you. And I will be happy if I take this. And this double-bonded molecule can't really stop it from happening. So the radical now can bond to the carbon over here. It's got these two. Those are those two hydrogens that were there. But now watch what happens. Now I've got here, here, here. And then I've got this. And I've got this. But now this is a radical, because this has an unpaired electron. This has an unpaired electron. And so often what we'll do is we'll write this. We'll often just write it. I'm just going to rewrite it, just because then it gets us in the mood for what's going to happen next as a single bond with the two hydrogens and the radical sitting there. I just took this unpaired electron and put it out on the end there. That's a radical again. And I haven't broken the bond. That's the key. The double bond allowed a radical to come in, take an electron, make a new radical out of the whole thing, and remain stable. That's what the double bond gave me. So if I go to step two, what happens? If I go to step two, well, now, I've got this R that's sitting here. And I've got these here. And I've got-- I'll put them in explicitly, and there. Obviously, this is in a bucket of these C2H4 molecules. They're all over the place. And now this is going to see-- so we're going to do that, and this is going to see another one. Let's put that one in here. Here we go. A double bond, nice and happy, floating along. And this radical version of it sees it. And it says, nope, I need that electron. Thank you very much. And so you get this. Single bond, single bond. And we'll just put it out here. And so now, how many did I have? 2, 4, and here's another one. And there it is. And I've got another radical. I've got another radical. The thing stays radical until it finds another R dot. So this just goes on and on and on. This is a new radical. New radical. And each time it finds a double-bonded C2H4 molecule, it can take it, add it to the chain, add it to the chain. Sometimes this is called chain polymerization, addition polymerization, radical polymerization. You see it's all the same. You see how it works, because I'm making a chain, and I keep on adding the same building block, the same molecule. That's really important. The same molecule, and then et cetera, et cetera. Well, how long is et cetera? Well, typical polymers go on for a long time. We're talking a minimum of 100 times. 100, you might be able to draw that all the way down, maybe down to there. But a lot of polymers go to 100,000 or a million chain links, a million. That makes this class of materials extremely unique. And we're going to talk about them in the next-- over the next-- well, during this week. What I want to do is give you a couple of reasons why they're unique, and then we're going to go into my "Why This Matters." What we started with was a monomer. In this case, we started with ethylene or ethene, and so here was our monomer. And what we ended up with is a polymer. And in this case, it's-- oh, boy-- I'll draw it straight like I did there. OK, here we go. Here we go. This is the polymer. But see, I don't want to write 100,000 carbon letters on my page. You can you can't blame polymer chemists for that. And so what we do is we have a notation, where if this was the monomer, one. Poly, many. Mer, C2H4. Then what you do is you take the monomer and you write it in a special way to show that it's become a polymer. And you put the monomer inside. Oh, let's just write those hydrogens explicitly. Why not? There they are. They were there, too. But now what you do is you put parentheses. And to show that it's a polymer, this bond comes out the edge. That shows I had this monomer. Now, notice, this is the same as a monomer, but a single bond. Why? Because that double bond, to keep on making it radical, I change them all into single bonds. Each time I made the chain longer. And then we put an n, because it could repeat n times. And that's a polymer. So the chain bonds are sticking out of the edges like that of the parentheses. Polymer chemists aren't as-- the crystallographers, you don't want to upset. Polymer chemists are fine. So if you put brackets there, yeah, they're OK with it. It's not so bad. Parentheses, brackets. The main thing is those bonds that are polymerizing, those bonds that are forming the chain have to come out of the parentheses, and then the little n there is how many times. So a couple of things we can talk about. So polymer, monomer. Well, one thing, a couple of things. So the molecular weight is going to be high. It's going to be typically, and again, thousands of grams per mole, because it's a mole of the molecule. This is a molecule. This is a molecule. It's just a very, very long chain molecule. And so polymer people and bio people, they like using Daltons. 1 Dalton equals 1 gram per mole. Why switch to Daltons? I don't know! It's just a unit thing. So don't get confused. Dalton is a gram per mole. A kiloDalton is 1,000 grams per mole. We can say like a Da. Maybe they just like the symbol, Da. Or kDa for kiloDaltons. It's just grams per mole, thousands of grams. But now one long chain, if I know the molecular weight in grams per mole, then I kind of also could know how long it is. Back and forth, it's just a molecule. It's just that this is a very, very big molecule. And so sometimes you'll see it called a macromolecule. Macro. It's huge! It's a long chain. You just think about how the length is on the order of-- I'll spell it out-- microns. Just think about it. We've been talking about a lot of different materials in this class. We've been talking about a lot of different solid materials. Perfect crystals. Crystals with defects. Amorphous materials. Molecular solids. This is another class of solid, but you can really start to think about why this is so different. Because if this thing is 100,000 or a million units long, that strand is a very long piece of spaghetti. And that means that different things can happen. And that's one of the reasons why polymers are so interesting and can be so useful. And so if you think about taking it longer, this isn't even-- we're not even close to how long-- this is still just a little longer, but what I'm showing you is something important. If I've got spaghetti that that's long, you better believe it can tangle up. And so when you think about one of these strands or maybe a couple of them, they're going to usually, often, not always-- you can make crystals out of polymers, but a lot of times they're going to tangle, because they're just so long. And the degree of the length, by the way, that's another thing, degree of polymerization. Ah, why was my O so big there? I don't know. It was a moment. I'm feeling it with the O. And that's the number of mer-- yeah, it's a polymer-- mer units. And that's per polymer. And this is average, because as you can imagine, it's very-- I'm going out to a million. Did I hit a million or did I get 990,000? Or did I get a million and 10? You don't control the length exactly. And so the degree of polymerization is the number that is the average. It's the average. It's the same as the molecular weight. It's an average, because the strands have a distribution. And we'll talk about that more on Wednesday. But now I just want to get down some polymer insights. So we got things like molecular weight, degree of polymerization. Now, these are very long. And so you can imagine that it's kind of like glasses when we did glasses. These are really long chains. You can imagine as a liquid that might be pretty viscous. That might be pretty viscous. You could also imagine that when you solidify it, it might be pretty disordered, or maybe crystalline. It might have some similarities with glass. Again, we'll talk about that more Wednesday. But the one thing I want to highlight now is that what can happen because it's so long is that you can literally get what? You can literally get like one polymer chain that finds itself wanting to form a crystal literally like in the middle of itself. And then it's like, OK, I can stack and form a nice crystal, but now I can't again. And now, oh, look, I think I can pack in nicely as a crystal. And now I can't again. And it just goes, I could-- I'm really enjoying this. I could just keep going. That's one polymer strand! And it would keep going. It literally would, because it's like 100,000 units long. So you can imagine that polymers, just one strand of a polymer can be incredibly complicated, because it might have some regions that are crystalline and some that are amorphous. And by the way, we know that the crystalline region will be very different. This is going to have the lowest volume per mole. So this will have-- maybe this will be the hardest. Maybe it will be the most brittle. So does it crystallize? Does it not? That's going to be something that's very important. How much of it crystallizes? A micron, by the way, the length is also important. That's like visible light, wavelength. So optical properties will depend on these things. And so what you have is this new form of matter that we have in the last 50 years engineered and understood more and more how to control. And so we're talking about this addition polymerization. This list goes on and on and on now today. So what is this list? Well, these are all-- I want you to notice here-- monomer. This is the polymer name. This is the monomer. Here's the polymer. They should have put parentheses there. We'll excuse that, because you can feel that those are going outside of the parentheses or brackets. And then here's the use. So what did we just do? We just did polyethylene. Polyethylene is right there. That's the polymer polyethylene. There is the monomer. There is the polymer. Now, notice something about every single one of these monomers. There it is. They've all got that double bond, that double-bonded carbon. So when I put a radical initiator in a soup of those monomers, I can form polymer chains. I can form polymer chains. And those are the polymers you make. And notice they all start-- these are the same unit repeating over and over and over again. And notice the flexibility here. So in polyethylene, which is one of the simplest cases and is the most common polymer, these are just a few uses. If I just take one of those hydrogens off of one of the two carbons-- there's polyethylene up there-- if I just take one of the hydrogens off, and instead I put a benzene ring. So now I've got this. Oh, boy. Did I draw that? Yeah, it's OK. Here's my hydrogens coming off of the benzene. There you go. There you go. And there, and there, and oh, good. I've drawn polystyrene, haven't I? No, I have not! Because you've got to come out! That's a polymer. Now it's a polymer. I came out. Well, that's pretty simple. All I did is I just took the polyethylene, and I put a benzene ring there. And I get Styrofoam if I put air through it as I make it processing. Or I can make all sorts of other things out of it. That's polystyrene. That's polystyrene. This is why polymers have become such an incredibly important and enormous part of our lives, because you can see if I just change the monomer, I just change one thing in one molecule, and then get a bucket of those. I might completely change the solid that comes out of it. But all sorts of things are going to change. You can imagine that just the way the chain folds up on itself like that, maybe the way the chain comes and talks to another chain, or maybe a part of itself. When I say talk and I'm talking about chemistry, I mean bond. Is it Van der Waals? The surface area is enormous here. The surface area of one mol-- we did this, right? We looked at boiling points. As you get a little longer, a little longer, to C12. This is C hundreds of thousands. So the amount of bonding, London dispersion, is enormous. That's one of the reasons these can be very strong. But the tunability is also enormous, because I can just swap in a different group here, and polymerize that. This is why this was so exciting when it was-- polymers were first made really in the '40s and '50s. They really started coming into the market in the '70s en masse. And now I want to tell you why this matters. We're going to start with a commercial. This is an ad on television by Pepsi in 1978. And I thought this would be a nice way to start my "Why This Matters." It's only 30 seconds. Uh-oh. [VIDEO PLAYBACK] Pepsi Cola's new 2 liter plastic bottle. It's tough. Really tough. And besides being touch, Pepsi Cola's new 2 liter plastic bottle is 25% lighter than glass. It's tough and light. [END PLAYBACK] Tough and light. Now, apart from some other potential differences you may have noticed with today's times, that was the key. That was one of the keys. So they could make-- Pepsi and Coke, Coca-Cola, could make bottles that were lighter than glass, but not break. That was a very big deal. That was a very big deal. And they really started-- in a lot of ways, they started the revolution of plastics and polymers in commercial products. Now, if you fast forward to today, I just want to show you how bottles are made. That's a 2 liter bottle. Here is a 2 liter bottle making plant. And I think it's always interesting to see how what you're buying is actually made. So here's just a little short video. [VIDEO PLAYBACK] Harden almost instantly thanks to a built-in cooling system. These preforms are now on their way to becoming single-serving juice bottles. This is another plastic injection molding machine. It uses the same method to make preforms for a different model, 1 and 1/2 to 2 liter bottles. The preform's next stop is a machine called a reheat stretch blow molder. In a matter of seconds, it heats each preform just enough to make the plastic malleable, then inserts a rod to stretch the preform lengthwise, while at the same time blowing in air at extremely high pressure. This forces the preform into a bottle-shaped mold. Cold water circulates within the mold to cool. [END PLAYBACK] So the reason I wanted to show you this is it just gives you a sense of the pace, the pace. Now, the pace is connected to the material and the chemistry. And it's things that you already know. When we did stress strain curves, one of the whole regions of it is called plastic deformation. These are plastics. So one of the big advantages of these things is you can make like a small mold like you do first, and then that can be doled out to different processes. And in a matter of less than a second with a little heat, a little pressure, it can be made into any bottle you want, and then the next one, and the next one. Well, let's see. We buy-- I'm going to write a couple of things here-- let's see. We buy as humans 1 million of those bottles, plastic bottles-- anybody want to give me a unit? Day. It's per minute. [GASPING] We buy a million globally per minute. And the thing is, if you're talking about plastic bottles, 91% of them are not recycled. 91% of them are not recycled. And so my "Why This Matters," it's going to turn a little dark right now, because the thing is, if you compare with what nature does, these are nature's polymers. And I will talk about them on Friday a little bit more. And you can say, well, nature's had millions of years to work on this, sure. Nature is the world's greatest polymer engineer. We are all polymers. We are all polymers of different types. Amino acids, sugars, cellulose, keratin. Nature does this in so many ways. And of course, by definition, these are biodegradable. Now, that's what nature makes. Here's what we make. Now, this is a beach. That's a coastline. And there's a lot of places where you can get some data. I like Our World in Data. You guys can take a look at that and many other resources. Here's a beach. Here is a guy. It's hard to see, but he's actually in a boat. That's a boat. So that's all water. And of course, there's a lot of information lately on things like how much of this stuff animals are eating. And there's also a lot of pictures that are sometimes difficult to see, where animals get stuck in a piece of plastic, for example. And that can lead to death, or deformity, as in this case. And the thing is that we have to think very carefully about what we're doing. And we have not done that with regard to polymers. We have not done that at all. And just to give you a couple more numbers, here's a chart. This is how polymers are used today. Packaging is 40%. 40% of plastic produced today is in packaging. And the thing is-- sorry. 40% of the plastics are used only once, and then thrown away. We all know that. We drink a bottle of water out of plastic, or all the way back to the '70s, the Pepsi and the Coke, and we throw it away. But the thing is that most of that, a lot of that, most of that winds up in the oceans. And it takes about a half a thousand years, 500 years, to decompose. So let's see. A couple more numbers. So first of all, only 7% of all plastic is recycled. Now, Coca-Cola and Pepsi could use a lot more recycled plastic. One of the main reasons they don't is the containers wouldn't be see through. And we can't have that, right? As consumers, we've got to see it. It's got to be this beautiful, clear container. But as I said, most of what's not recycled winds up in the oceans. And let's see, in 2050-- here's one. In 2050, the plastic in the ocean will equal the weight of all fish. That includes whales and large sea animals, everyone. Now, the thing is that-- let's see. Many of these plastics, like polystyrene, there is an example, they decompose. And so when they're large, large, there is a chance to collect them. But the sunlight out in the ocean, they float up to the top or they get currents that mix them, and the sunlight, the UV energy from the sun, decomposes them. Eventually they get smaller and smaller and smaller. And eventually, they get really tiny. Those are called microplastics. You say, well, why is that bad? Because it's toxic, for one thing. Polystyrene is toxic. The styrene molecule is toxic. It's a carcinogen. It's a carcinogen. People who eat seafood in some studies are estimated to be eating 11,000 pieces of microplastic a year. That's what they're ingesting today. So this is the problem. We're making these single-use products, single-use, put it in recycling because I'm a good citizen, but 9 out of 10 times it doesn't matter, because it's not getting recycled. Oh, and then it goes in the ocean where it stays for 500 years. There's a plastic bottle. There's fishing lines. So this is the problem. We should not be using things on the minute timescale that then take a half thousand years to decompose and are extremely toxic when they do. This is clearly a problem. Now, there are a lot of people working on this and thinking about this. Not enough. And I want to tell you about a book that I read when it came out a long time ago. It's a wonderful book. And I got reintroduced to it last week. By David McKay, who unfortunately passed away recently, but he wrote this book called Sustainable Energy Without the Hot Air. And I highly recommend you read this if you haven't. It's free. It's a free download. WithoutHotAir.com. Because he just goes through sort of simply the math. And the reason I'm bringing this to your attention-- he doesn't really talk about plastics. He talks about energy-- is I love this discussion in chapter 19. Chapter 19 is titled "Every BIG helps." He says, if everyone does a little, we'll achieve only a little. That is true. We have this way of talking about these crises in little bits, because it makes us feel better, and it makes companies feel better. You talk about this with the plastics in straws has gotten a lot of press. And companies kind of do their part. Although, Starbucks is kind of funny, because they replaced all plastic straws with paper straws encased in plastic. That's actually more than the plastic straws to begin with. Single-use plastic again. But anyway, they're trying. They walk away. They say, we tried. No! Because that's doing a little. The plastic in the ocean is 0.03% from straws. There's so much more we have to do. We can't think little about problems like this. We can't. We can't do that. Now, there are some big projects. This is one that's gotten some attention. This is called Ocean Cleanup. Now, they've gotten a lot of attention. They've got these rigs out there that could collect some of this plastic in the ocean. And it's at least thinking big. There's a lot of bold claims that they're making. A lot of bold claims, a lot of hype. But at least it's an attempt to do something big. One of the limitations here though is this technology can't go below a centimeter. It can't go below that. It's not going to collect below a centimeter. But there are some studies that estimate that 90% of the plastic in the ocean already is below a centimeter, and 60%, 70% of that has already sunk. Talk about the need to think about this in a big way. This is a very big problem. And I hope that more and more people get excited about trying to tackle it in big ways, not little ways. That was my "Why This Matters," nice and happy. Nice and happy, why? Because we're going to solve these problems. That's why. We have to. We have no choice. I mentioned there were two ways to make plastics that I want to talk about. I want to talk about the second way now. So the first way is called radical polymerization, because I take one mer, and then I make it into a really long chain. The second way-- and that right there is nylon, by the way. Nylon can't be made that way. Nylon can't be made that way. And here's the reason. Because in the second way of making polymers, we take two different types of mers. And this way is called condensation polymerization. It's a different way of forming very, very long chains. So on Wednesday, we'll talk about what you can do with these chains, how you can modify the properties, kind like we do with glass. We talked about the fundamentals of glass, and then we talked about engineering it. On Wednesday, we'll talk about how to engineer the properties of these polymers. But right now, I want to just focus on how to make them. And so in condensation polymerization, the goal is also to make these super long strands, but you do it with two different starting molecules and no initiator. So how does that work? Well, I'll take a-- I'll get to nylon in a sec. But first, let's just take something called the dicarboxylic acid. And what I'm going to do is I'm going to leave a box inside. And I'm going to say that could be something. That could be something. But I know that on that something, I've got this. I've got a carbon, oxygen, and hydrogen, and on this side I've got the same thing. OH. So I've got the same thing on both sides. That's a dicarboxylic acid. It's carboxylic acid. Now, on this side, I'm going to have another mer, and it's going to be different. And this one, I'm also going to have a box. And over here. [SNEEZING] Gesundheit. I've got NH2. And over here I've got NH2. NH2. Emphasizing the N is what's connected to the box there. That's why I drew it that way. And that's called an amine. This is an amine. And this is called a diamine. So this is a diamine dicarboxylic acid. And what happens is when those two molecules-- so the box can be anything. The box can be whatever for now. If it's got those ends to it, if it's got the carboxylic acid and amine-- so on the one hand, the molecule's got these groups on the end, and on the other hand, it's got these, then what happens when I put them in solution together? Well, what happens is these two pieces react. These two pieces react. And so what you get is-- let's see. You've got C. Here's my box there. There's the O down there. And let's see. Now here's the carbon on this side. Oh, boy. I'm going to run out of room, and I don't want to. So I'm going to move over here. And recycle these boards so that I can draw it nice and big. So now what happens? So these are reacting. And so now what do I have? So I have C, double bond O. There's my carboxylic acid. There's the box from the left-hand molecule, the one on the left there. Here's another C, double bond O. But look at what's happened. I've formed a carbon-nitrogen link. We'll talk about that in a second. And then here's another box that came from the amine. And this is an NH. And then there'd be like another H there. And there'd be another H-- yes, another OH here. Yeah. Yeah. NH2. So I got that back. And what's happened? What's happened is I've made water. I've made water, plus I've connected these two molecules together. So if I want to be complete about it, I've got to add a water molecule, because that's what had happened in the middle here. The OH group saw the H on the NH2 group, and it said, hey, I want to make water. Can I do that? And then the carbon and the nitrogen were like, you know what? Let me check it out, internally. Let me check out my Lewis structure. And it's like, yeah, you can take the OH and the H and make water, because then we could connect and form this bond here. And that bond is called an amide link. That's that kind. And so I've formed a kind of bond between these two molecules that's a covalent bond. And this polymer is a class of polymers called polyamide. Because you can see now, OK, I've given off some water, condensation polarization, but I'm also now ready. I've got my carboxylic acid, and my NH, my amine groups there ready to go again. So if this side finds another one of these, it can attach and react. And if this side finds another one of those, it can attach and react. And it keeps going and going and going. In much the same way, it's just now instead of a radical, instead of initiating it by tearing an electron out, I simply have brought together two types of molecules that want to, in giving up water, react and form a bond. That's very powerful. It's very powerful, because I can do-- I can now, like I said, this can have whatever inside. So now it's a different way of thinking about it. Before we had the mer. And I could make the mer whatever I want. Just keep a double bond somewhere, and I can make a polymer. That's a lot of flexibility. Now, instead of keeping a double bond, just keep these amine and carboxylic acid groups there, and I can put anything inside the boxes that I want. Again, massive tunability. And one of the funnest things to do is the nylon rope pull. We couldn't do it. Apparently, it's a little dangerous. Nylon is when the box winds up having a carbon chain in it. Six carbon atoms. And when this has six carbon atoms and this has six carbon atoms, that's called nylon 6,6, which some of you may have seen. So you get nylon 6,6, or sometimes it's just called 66. It's six carbon atoms in the boxes. You can imagine you can make 6,8, 4,6. You can play around. You can put lots of things in those boxes. The nylon rope pull is such a cool thing. I had to show you a video. Here's what you've got. You've got one molecule in solution on the top. You can tell. It's the slightly lighter liquid. And the other one on the bottom. The amines on one side, these carboxylic acid on the other. This is nylon. So the box is six carbon atoms. Now, what's happening? At the interface, this reaction is occurring. So what I can do, and I'll play this. Or what this person can do is put a little piece of glass stir in there and pull nylon out. So watch how he does this. So you get it started. There is a piece of solid coming out of two liquids. What's happening? As he pulls it, more is reacting and forming more of these chains. And you can just keep going and going and going. What's so cool about this is you're literally creating this solid with functionality that's tunable out of these two liquid baths. It's all happening at the interface where the two molecules, the amine and the carboxylic acid can see each other. And there you can see it. I wish you didn't have the text there, but look at that. You're pulling a polymer right out of the interface between two liquids. That's how nylon is made. And again, this had an incredible impact on so many applications. I thought I'd show you this one. That's the first line in New York to buy stockings, nylon stockings. And it's going on and on and on. Of course, then there are a few other applications, too, like parachutes, and many, many, many more, because again, it's a whole other way of making polymers. Now, I'll talk more about this on Wednesday. But in your goody bag, you've got old school fun. And we'll talk about this in the context of the things we're going to learn more about polymers and polymer engineering. See you guys on Wednesday. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 22_Xray_Diffraction_Techniques_II_Intro_to_SolidState_Chemistry.txt | Happy Friday. How is everyone? Woo! Woo! Woo! Cool. Cool. I am excited. We're going to keep talking about XRD today. We're going to pick up where we left off. So I filled in some numbers. And then, we're going to fill out this chart and really get to a deep understanding of XRD. I'll also talk about Moseley's law and a little bit about another type of XRD that is done. Now, where were we? Oh, OK. We were here. So on Wednesday-- and before we fill this chart out, I want to go back and make sure we, kind of, get this. Right? We talked about the Bragg condition. And we talked about how the unique thing about x-rays is that their wavelengths are right in that, sort of, precious distance regime of the spacing of atoms in a crystal. And so if I shine them on the crystal, then they can diffract. And they can have constructive and destructive interference, but only if they line up. And that's what the Bragg diffraction tells us. It tells us the the Bragg condition for when they line up. And in particular, this is a pretty nice image. So you can see here's one x-ray coming in. And it's a wave. It's a wave. So it's going to be a wave that goes like that, and then it bounces off of this surface. Now a second one doesn't bounce off that top surface. It bounces off of the one beneath it. OK? And they're separated by a distance D. So this could be any set of planes. Right? So here are my planes now. So these black atoms are one plane. Right? But these yellow ones are another one. That's another plane that you could see. Remember, one of the assumptions of the Bragg condition is that these planes are like mirrors. So we lose a sense of where the atoms are exactly, and we just write a mirror there. And we say it's reflecting off the mirror. So where's the next one? And you can take the orange ones. That's another plane. The red ones and so forth. Now you have a wave on top. OK? You've got a wave on top. And then, you got a wave coming down and bouncing off the one beneath it and for those to constructively interfere so that you pick up a signal in the detector. There's only one way for that to happen. And that's if this distance plus that distance is equal to a multiple of the wavelength. Right? And that is the Bragg condition. It has to come all the way down to here, which makes that theta equal to that theta. You see that? Because they've gone up to this point, up to that line. They've gone the same distance. So they're waving together. All right? And it's that point where now this is the extra that the one that bounces off of this plane, that's the extra distance it goes. That's the Bragg condition. OK? That's the Bragg condition. And remember, in this class, OK, so it could be any multiple. But here, we're just going to assume n equals 1 because it's the easiest case. OK, good. That's the Bragg condition. Then, we did something beyond just that because that tells us what the wavelength has to be to get a spot, to get something the detector can see because it constructively interferes. Everything else is destructive. No signal. But we did one more thing. We also said that d for some set of planes. Remember, this HKL is just telling me which one of these I'm talking about. That's what HKL is. I'm going to pick one of these planes. And the distance between them is DHKL. OK, good. But we know from before, from our Miller times, we know that that is equal to the square root of a squared plus k squared plus l squared for the [INAUDIBLE] we're talking about. All right? So you see, it's kind of Bragg plus Miller. Bragg plus distance Miller. Right? Miller cubic. That's what we're really combining here. The distance given by this formula when we talk about Miller planes, that combined gave us this. Lambda over 2a squared equals sine squared theta HKL over h squared plus k squared plus elsewhere. And all I've done is substitute and square. OK? And we did one more thing on Wednesday. And we talked about constance. And we said, well, if something is equal to a constant, then it's a constant. All right? That's some good math. That's some good math. And so we said, well, that's the whole beauty of these characteristic x-rays because if you take a k alpha line from copper, it's always 1.54 angstroms. It's always lambda equals 1.54. That's what's so nice about it. Doesn't depend on anything else. Just the metal you put in there is the target. It's called a target because you're targeting it with electrons. So the copper target, k alpha, lambda constant. Lattice constant? That's built into the name. Constant. Constant equals the angle and the plane it's reflecting off. And so our job then in crystallography and doing x-ray diffraction, our job is to figure out how to keep this thing a constant. The same constant. All right? That's what constant means. OK? That's our job. And that's where we left off. And we say, OK, well let's take this to the next level. Let's do an actual experiment. Now how do you do experiments? Well you have equipment. And this is what a really cool XRD machine looks like. This is what happens when you don't have any money. [LAUGHTER] So this is what's in my lab. [LAUGHTER] It's true. You can come see it. But you see, the point is they knew the same thing. We don't need fancy equipment at MIT. No, it's the ideas that matter that you're putting into this thing. But that's kind of cool still. I like the way that looks. Same thing. Right? Source. Sample. Detector. That's hard to see. But source, sample, detector from the 60s-ish. OK. [LAUGHING] All right. So now it's easier. Of course, you can understand. I mean, some of these things have different ways of doing it. But you can understand changing the angle is going to be easier to move that sample. Right? And then, maybe the detector moves. But you might keep the source x-rays fixed. And what happens? Well, you do this, and you generate a spectrum. So you're changing the angle. You're keeping this fixed. And you're saying, what do I got? Well here, in this case, I'm telling you what I got. I got aluminum. I put aluminum in there. And I shot x-rays. If anybody gives you an XRD pattern, they got to tell you what kind of x-rays they used. Right? That's really important. At some point, you got to know this. This is pretty important. The lambda is from a copper target, and the k alpha lines is 1.54 angstroms. And this is the spectrum. And I'm being asked to get the crystal structure of aluminum and the lattice constant. And I can get that because, look, I can simply read off peaks. And those are the peaks, and I've kind of highlighted them there. 2 theta. More for historical reasons, you take 2 theta measurements, even though, in the Bragg condition, it's just theta. OK, we can divide by 2. Right? And so here are those first 4 peaks. And I've just listed them here. Peak 1, 2, 3, 4. Sin squared of half of that. Gesundheit. OK, there it is. And then, we have our recipe. I gave you this recipe as a way to keep this constant. All right? It's an easier way to do it if you scale and clear fractions. It's just easier. So that's why I've given you that recipe. So let's do it. So first we normalize. What does that mean? Well, you have the smallest value of science course data. I'm just going to divide everything by that. OK? So that's where we left off. One because that's what I'm normalizing by. And then, this one would be 1.33. And this one would be 2.67. And this one would be 3.67. And so that's good. But I don't want fractions. It's going to be easier for me to think about this as integers. And so we clear fractions. What does clear fractions mean? That! Clear the fractions. Get rid of them. Make it integers. How do you do that? Find what you multiply this by so that they're all integers. OK, I got an idea. How about 3 times 3. In this case, times 3 equals 3. OK, good. And in this case, well, I don't mean-- OK, anyway, times 3 equals 4. Times 3. Don't multiply it by a different number. I'm clearing the fractions for the whole table. I'm multiplying by the same number. OK, good. 8. And this is times 3 equals 11. So I've got 3, 4, 8, 11. And that's what I need to work with. That's what I need to work with because now, you see, it's now-- I hope it's kind of starting to make some sense-- if I can get this to be 8 squared plus k squared plus l squared, well then I've done my job. I've kept this constant. That's the trick. Right? I've done my job. I've just made the math easy. If I can get h squared plus k squared plus l squared to equal these things, then I'm keeping it constant. So let's see. What would that be? So this has to be h squared-- let me just write here-- plus k squared plus l squared. OK? I want those to be an h squared plus k. So if I do that, well, this looks a whole lot like 1 1 1. That works. And for 4, 2 0 0. OK, and for 8, 2 2 0. And finally, for 11, 3 1 1. Aw, it's a beautiful thing. I've gone from angles to planes. These are planes. H, k and l mean one of these. It means one of those. That's what it means, a set of those. And I have found now, literally, which ones give me each peak. This peak is the 1 1 1 plain. It's x-rays bouncing off of that, reflecting off of that, and the one below it, and the one below it, and seeing where it gives you constructive interference, and gives you the Bragg condition. And therefore, a peak. OK. All right. Now hold on. Hold on. Now, OK, have we determined the crystal structure? Have we determined the crystal structure? Can anybody? How do we get the crystal structure from this, from these peaks? How do we get? Well, let's see. OK. Let's see. Could it be bcc? What else did we learn on Wednesday? Right? What else did we learn on Wednesday? We learned about selection rules. You got to remember the selection rules. Those factor in. So let's write those down here. Let's remind ourselves of, OK, that's important. Right? Now I mentioned how this came about. I don't need you to know how to derive the selection rules. But I did talk about why they come about because, for some of these cuts, you could have a plane in between that destroys the signal, depending on which crystal structure it is and which plane you're talking about. And when it comes down to it, for a simple cubic, it's anything goes. Anything. Oh, polonium. And for bcc, it adds to an even number. And for FCC, no mixing. No mixing of odd and even. Those are the selection rules. You don't need to know how to divide them, but I do want you to know what they are. So now we can see, well OK, hold on. If this were a bcc crystal and they had to add to even 1 plus 1 plus 1 is 3. Bam! It's not bcc. Right? Yeah? OK. Yeah, but now, hang on. If it's no mixing on even, did I handle that one OK? Yeah! Odd! Even! Even. Odd. This looks like an FCC crystal to me. Right? This looks like an FCC crystal. And I can go even further. I can calculate the lattice constant. I can calculate the lattice constant cause it's right here. It's right here. Right? So that's the lattice constant. That's a number that's a constant lambda. So I could take any one of those planes, any one of those, any one of these four peaks. And if I did this right, they better give me the same lattice constant. And that's actually the last step in the recipe. Did I put the recipe? Yes! All right. This is the recipe I gave you on Wednesday for you all to become crystallographers. X-ray defractioners. That's OK. We'll make it work. You're X-ray diffractioners. And the thing is, you got to remember how powerful this is. Yeah, how powerful this is because we've learned about perfect crystals. And now, we've just gotten the single most important way to figure out which one we have. We shine x-rays on it, and we look for diffraction peaks. That's it. This is a very powerful tool. And it is used all over the place in materials chemistry, physics. You name the field, x-ray diffraction is a starting point for characterizing the crystal. And so, the last sanity check that you want to do here because it'll help you make sure you did these points correctly, is compute these values because, you know, this is it. Right? Sine squared over a squared plus k squared plus l squared better be a constant. If it's not, then you've done something wrong in this analysis. Right? You could just calculate the lattice constant for each peak, and they also better agree. So I won't do that, but it turns out we did it right. For each one of these, that is indeed a constant. We have achieved our goal. With great power comes great responsibility. How you use XRD. How you use XRD. Well, let's see. Let's use it to figure out what phase of material we have. Why does this matter, right? Well I have a why this matters, but there's multiple reasons for why this matters. As I think we've talked about, the properties of a material depend on the chemistry in the material, but they also depend on the structure of the material. And we're making solids, so those structures are crystals, until next week when we destroy the crystals by putting defects in them! But for now, they're these perfect crystals. And they have this, kind of, repeating pattern. Worry about hcp. bcc, ah we're comfortable. fcc, comfortable. bcc. Now, this is called a phase diagram. This is called a phase diagram. Phase diagrams will be on the quiz next week. I'm kidding. They're not going to be on the quiz because this isn't a class about phase diagrams. But I want you to know what a phase diagram is. It is a map of materials. It is like a treasure map, literally because you find I need iron to have a different property. Oh! If I go to this pressure and that temperature, aha! It can have a different property. Why? Because it has a different crystal structure. So the crystal structure of something-- in this case, it's iron-- depends on the pressure and temperature. And the properties depend on the crystal structure. So we've got a way now to change materials. Same element. I'm just changing its structure. I love this one. This is water. Now don't worry about how complex this is. Actually, the whole point is it's complex. This is the phase diagram of water. Now this doesn't even have them all. There are 17 current phases of ice. 17 different crystalline structures of ice. That is incredible. Right? You can actually imagine why, from your knowledge in this class already, those are hydrogen bonds. They're kind of strong, but they're not too strong. There's a lot of ways that those water molecules could form a solid. And they do. 17 to date and counting. Let's see. And this tells you which one you have. Now Earth's operating conditions are right around here. And that's a really good thing because there is a liquid phase in there. Ice 1 is here. ih is ice one. And it's a really good thing because of the fish because if you weren't ice 1, then when it freezes, it wouldn't have a different density. And the ice wouldn't float to the top. And the fish could still live, which they do in the winter because you don't freeze from the bottom. You freeze from the top. That's because it's ice 1 and not ice 8. It would be terrible. But yet, we know about these. And how do we know? We know because of crystallography. We know because of XRD. We can figure out. Now if you're out and maybe it's a Friday night, so maybe this is a good idea-- it's a good time to mention it-- you might get ice in your water out at a restaurant. How do you know which kind of ice it is? Talk about XRD. It's the only real way to figure out which one of these you really have. OK. So for example, if I were to think of other types of problems you might get, well, here's a good one. OK, here's a good one. So this is an X-ray diffraction spectrum. And I give it to you, and I don't even tell you what it is. Why is that? Here's a pattern that I put something in there, and I measure these pics. And I tell you the source. It's the k alpha line of copper. So that's the wavelength of the source. First thing is what lattice, what symmetry, does this correspond to? Is it simple cubic, body centered cubic, or face centered cubic? Can we get that? We could get that because of the exact same thing that we just talked about. Right? This can not be. If it were simple cubic, you'd see the peaks. Mm! But know that some peaks are forbidden, so they don't show up. Right? And you only get peaks that look like an FCC, according to our selection rules. No mixing odd even at work. So this is an FCC crystal. So I've already learned that. And then to say, well, what's the-- OK, use the 1 1 1 peak to calculate the lattice constant. OK. Well, we can do that. Right? So the 1 1 1 peak is even labeled there. It's 38 degrees. 38 degrees. So to theta is 38 degrees. So that means that theta-- OK, I can do this one-- is 19 degrees. OK. Oh, oh, oh, from Bragg. Lambda equals to d sine theta. But look, label, label the plane. This corresponds to a plane. So some distance between some planes corresponds to some angle that gave me constructive interference for those planes. Let's pass it around so you guys can look at these planes. They're beautiful things. Now that means that if I look at the first peek, I know what la-- well I labeled them here. In the last one, I didn't. And you want to find those. Here, I'm giving you the planes. So if I do that, then d 1 1 1 equals lambda over 2 sine theta 1 1 1, which equals 2.37 angstroms. OK. Have I answered the question? I haven't answered the question yet because, you see, the question says what's the lattice constant. But I'm very close. The lattice constant is related because of Miller. The Miller plains equation for lattice. Lattice constant. Distance between any planes. Oh, this is all just coming together. And so d 1 1 1 is equal to a over 3. And so a equals 4.1 angstroms. Oh, you could go even further. You could figure out, now that you know it's fcc-- it's a single atom base, it's fcc crystal, you could actually figure out the atomic radius. I got a, and I know the close pack direction. Right? I could actually figure out the atomic radius. And I could go into my periodic table and figure out what I have just from the atomic radii listed in the periodic table and the crystal structures. FCC. Automic radius. Which one is it? It's gold. It's gold. So this is the kind of thing that you guys now have the power to do. You have the power to do this. You can go back and forth between-- based on your knowledge of crystals and now your knowledge of x-rays and how x-rays interact with crystals you can do this. Oh, that went too far. OK, so now I'm going to get to the why this matters. So there's two different sources, two different targets. Remember, we can call them sources or targets. And notice they both have this continuous radiation. And I will talk about that. The [INAUDIBLE] will end with a discussion a little bit on what you can do with that. But for now, let's keep focused on these lines, these discrete characteristic lines. So there's the k alpha from molybdenum. There's the k alpha for copper. OK, now Moseley, Henry Moseley, was a brilliant scientist. And he was working with Rutherford and others. And he was really interested in looking at the trends. So he took all of these elements. So from calcium all the way to zinc. 20 to 30. And he said I want to look at the k alpha lines of all of these. I want to look at the K alpha lines of all of these. And what he found was extraordinary. He did more. 38 in total. But I'm just going to show you his data for these. OK? And what he did was absolutely profound because what he noticed-- there are the lines. These are actually his measurements. OK? These are his k alpha lines. And what he noticed in going from calcium down to-- ohh-- brass? OK, we'll talk about that in a minute. It's supposed to be zinc, isn't it? Why is it zinc? Well, why is it not zinc? Let's think about that later. For now he noticed that this has a square root relationship to the energy. So Moseley came up with-- he was working on this in 1912. OK? You got to remember 1913. 1912 was when Rutherford did the gold foil experiments. It had been 44 years since Mendeleev put his periodic table to paper and published that. So for 44 years, we had a periodic table. But see, the thing is, there was a huge problem with the periodic table because Mendeleev had this, sort of, brilliant realization that periodicity-- periodicity-- was related to, both, the atomic mass-- and remember, we talked about this-- and the properties, the chemical properties. That chemical properties. That allowed him to create in ordering of the elements. In ordering of the elements. That is still the ordering, essentially, that we have today. But the problem is why did they have that ordering? I didn't really tell you why, sometimes, he was like, well, the properties win. OK, maybe the mass-- no, no. Properties win. Properties need to be aligned in this column, so I'm I'm going to move those over. Like, that's what he did. But he didn't know why, except that it made sense to him. Moseley's experiments told us. They gave us the why. Aw, it was so important. And Moseley's law-- Moseley's law-- was, essentially, him thinking about the Bohr model for these characteristic x-rays. So he said that h nu-- so that's the frequency for some k alpha line-- is equal to 13.6 ev. All that looks familiar. Times z minus 1 squared. And then, he did the difference in energy, just like we've done now a number of times in this class. That's what he did. And this is 3/4. Why this? Well, let's talk about that in a second. So 13.6 ev times 3/4 times z minus 1 squared. Two things about this, right? One is why is it 1 over 1 squared minus 1 over 2 squared? Well that's because they knew, or at least they were pretty confident, that you had this positive charge in the middle here. That's the protons and the nucleus. And then, you had these electrons. Right? So you had, like, the 1s electrons. And then, you had another shell out here. And so they knew, OK, 2s. Maybe that's combined with 2p, so it goes on. So what happens in the x-ray experiment? Well what happens is you shoot an electron in. Rank and just crank the voltage up so high that an electron could come and knock that out. Right? And so that's what did it. And so now one of these can cascade down there and give off a k alpha photon. Right? This is nothing new. We've talked about this. But you see it here in the equation. You see it in two ways. First, we're going from 2 to 1. 1 squared minus 1/2 squared. 3/4. Right? But second-- and this was critical-- the z minus. The z minus told us that all these positive charges in here, all these positive charges, are screened perfectly by one electron, this one that was left. And it works. He assumed perfect [INAUDIBLE]. So what do all these see? They see z minus 1. They see z minus 1 if z is related to the atomic number. Now here's where this was so powerful because when you-- and if you can't read this, don't worry about that. I just want to show you that it is a perfectly straight line. When you plot the K alpha transitions, these are different elements. These are his different elements. And if you plot the square root of the frequency versus element, it is a perfect line that holds. And so what Moseley wrote in the paper is we have your proof in 1930 that there is, in the atom, a fundamental quantity. A fundamental quantity, which increases by regular steps as one passes from one element to the next. This quantity can only be-- only be-- the charge on the central positive nucleus, of the existence of which we already have definite proof. Now we know about that. We know from experiments that Rutherford did that that nucleus had the positive charge. But they didn't know that it was connected to the position in the periodic table. In fact, years later, people talking about those experiences-- I mean, people didn't even take Rutherford's experiments as seriously until Moseley's work came along. 44 years had come since Mendeleev. But Moseley gave it the foundation that it needed. Periodicity is because of atomic number. Periodicity is atomic number. That was not known. That z gives you the periodicity. Right? And that gives you, also, the number of protons. This was a time when they didn't know what was going on. Why did the mass change so much the neutron wasn't discovered till 1932, 20 years later? But this gave the grounding that was needed to the chemistry of the periodic table. It was a very important discovery. Very, very important discovery. So that's my why this matters. And it's really tragic because he died, tragically, in World War I at age 27. And the Nobel Prize was not given in 1916 for either physics or chemistry. He died in 1915. And most people, at the time, believe that he would have won it at age 28. That's how important that discovery was. OK, let's go back to brass. I got to go to brass. What's going on here? What is going on with brass? Right? Any ideas? Why didn't he just put zinc there? It looked so good until brass! What is going on? Well, ha, here, I'll give you a hint. OK, the melting temperature. Let's see. Ah, brass. OK, zinc melts around 420 C. Brass, which equals zinc and copper, melts at around 900 C or more. Greater than 900 C. That's a hint because if we go back to the video-- let's go back. Oh, the sound is on. Those are the electrons! A small portion of the time. Let me set that up because I'm doing in the middle. The electrons are coming off of a cathode because Rankin has cranked the voltage way up, and pumped all the air out. So those electrons are coming off with thousands and thousands of KEV. OK? They're coming out. And that's what's on the right here. There they are. The cathode is shaped to focus the electrons onto a small portion of the target, which increases the intensity of the x-rays produced. The target is, typically, a piece of tungsten or molybdenum, which may be embedded in a stationary water or oil coiled rod, or on a rotating disk. The high speed electrons collide with the target and rapidly lose energy. There are two x-ray producing interactions between the incident electrons and the atom. Ah! Look at the power! You can feel the power coming into that tiny little piece of metal. Now some of the time, you get the transition that we talked about. You knock an electron out, and you emit a photon-- a k alpha photon-- or maybe an l beta photon. Right? And sometimes you slow the electron down so you get the Brahms fraulein. But there's so much energy being pumped into this material. And a lot of times, those electrons just collide and heat up the metal. That metal is getting extremely hot. And in these experiments, either you cooled it, or you might rotate it quickly so that it gets to cool down. It gets a little break. Or you do both. Or you just throw up your hands and you say, you know what, I'm just going to go for an aloy that's got some zinc in it, but won't melt every time. Yeah, that was a good idea. So you put brass in, OK, you're going to get the copper lines. But you'll get the zinc lines. You'll get the zinc lines. And that was why he used brass. And that's why you can see you get extra lines in the case of brass. Al right. Good! All right. Now I am very excited about Moseley. You know, it allowed us to actually understand this not just because. Look at this. Mendeleev was like, you know what, cobalt and nickel can't be swapped. Hey, all you people who just think about mass, you're not going to put them in the order you think. I'm changing that. I'm changing it because properties matter. Moseley said, no, you're changing it because it's the right order because of the blue numbers. You're changing it because there is an actual count here that matters. And that is why your periodic table is what it is. Absolutely critical. It put the lantinides in the right place. It allowed the prediction of elements-- like [INAUDIBLE] and a number of others that hadn't been discovered yet-- and it set it all on this much more solid grounding. So that is the contribution of Moseley. I can't help it! I'm very excited so I brought t-shirts. I knew I was talking about Moseley. Now-- Woo! I'm going. OK. All right. There. And in the middle. And over there. [CHEERING] And over there. [CHEERING] And over there. This way! And, oh! That's a bad arm. [CHEERING] And up there. [CHEERING] Hold on. Hold on now. OK, there. There. And this is all for Moseley. Ah! This is all for Mosley! [CHEERING] All right. Now, OK. [SIDE CONVERSATION] Now apart from me showing off-- I don't have much of an arm-- one thing I'd like to say is, if you did already get a t-shirt, please, just hand it off to a friend or somebody. You know, let's share. Sharing is always caring. Now here's another type of x-ray I want to talk about. And z, actually, this is much less used today. I hope you've gotten a sense of how powerful these characteristic x-rays are. You know exactly what the wavelength is. But in the very earliest XRD experiments, it was the Bremsstrahlung radiation that was used. And that was by Van Laway. So I want to just tell you about that. And so here's an example of an Mo target. Now let's remind ourselves what we're looking at. These are x-rays. These are x-ray intensities versus wavelength. And this corresponds as 5, 10, 15, 20. That corresponds to the [INAUDIBLE].. Electrons. That's thousands of electron volts of the incoming electron. That electron that came in here and knocked this out from the cathode. Right? OK. Now that electron-- if it comes in, and we talked about this already, but let me remind you-- if it comes in at, kind of, a low energy, well, you're still going to get this continuous spectrum. This continuous spectrum. Oh! Why does it look like that? Beyond the scope of what we need to know. But in case you're wondering, it has to do with very complicated effects that happen when x-rays come off and then are reabsorbed by the same material. You can imagine that that could lead to all sorts of interesting cascades of absorption and emission. That's what leads to this shape. But we don't need to know those details. What we need to know is that as you increase the power of those incident electrons, you're going to get more and more intensity of x-rays. And this goes down. The minimum wavelength or maximum energy of the continuous spectrum-- the maximum energy goes up because the maximum energy you can get in the continuous spectrum is equal to the incident electron itself. So it's set. Right? Remember the Duane something limit? Duane Hunt? OK, we talked about that already, how to get that value. Yeah, but then you get to this certain point! And look, all of a sudden, the characteristic lines appear. And those are the K alpha lines for molybdenum. Why all of a sudden? Well because you got over the amount of energy needed to kick out a 1s electron from molybdenum. That's exactly why. Before that, you had enough energy to produce x-rays, but not enough to kick out the 1s electron. It's kicking out the 1s electron that gives you characteristic peaks. OK, good. This is all reminder stuff, but I want to get back on that page. But see now, OK, this continuous part is also used. And it was used, like I said, by Von Laue. And Von Laue was, again, a really, really smart cookie. And he used the Bragg condition to prove that the x-rays and the space light of structure of crystals can be interfered with each other to make this diffraction. But what he did was different. OK, so the Nobel Prize was given for this epoch making discovery. And I'm actually very happy that the people that worked with him also shared the Nobel Prize. Walter and Paul. What they did was different though. They took this continuous spectrum. And instead of fixing the angle and the lambda-- I'm sorry, instead of varying the angle but having a fixed lambda, they did the opposite because the continuous spectrum has all lambdas. Right? And so you can imagine how this works. Right? You can imagine how this works. So in the Laue version of XRD, you shoot the x-rays, which have all the wavelengths in them, at a crystal that's fixed. I'm not changing the angle. OK? I'm not changing the angle. And what you get are still diffraction peaks. You get spots. So In Laue XRD, you have a single crystal. It's got to be the same everywhere, in this case. You have a single crystal. Theta is fixed. You don't move it around. But lambda varies continuously. OK. And the pattern that you get reflects the symmetry. Pattern reflects the crystal symmetry. How is that possible? Well, you can actually see this by thinking about the-- do I have another picture? I just have his one. You can see this by thinking about that model, which is somewhere out there in the sea. Where is the model? I don't see it. Ah! All the way in the back waving around. Let's see if I can draw this. Let's see. Ah, I'm not going to try. If I shoot x-rays that have a fixed lambda in, but I vary the angle, then as we've been talking about, there's going to be certain angles where you get constructive interference. Now if I keep the thing fixed and I shoot all lambdas into it, well, take a look at the crystal. Imagine that I'm holding that structure right now. OK? Well I'm going to see this set of planes here. So I'm to see this set of planes. Maybe they're [INAUDIBLE] planes. I don't know. Right? And I'm going to bounce off of that and constructively interfere, maybe, if I've got the right lambda. But see, if I'm looking at the crystal and it's fixed, I'm going to see also there's another set of planes like that. Do you see? There's another set of planes like that. And if I'm a different wavelength-- not the same one, but a different one-- I might constructively interfere with those. But you can see that the angle that I make will be different. And so I'll put a spot in a different place. You see? So I've got all the wavelengths there. And as they constructively interfere with different planes that they see, they make different spots. They make different spots. And that's what those are. In those spots, it's more complicated to get crystal structure from those spots. It's more complicated. You can get symmetry fairly easily. But getting the structure itself is a little bit more complicated. It can be tricky. And for that, another reason most work today is done with characteristic peaks, XRD. But this happened before. And so I wanted to show it to you. And this resolved, actually, in one of Laue's very famous papers. It resolved the structure of zinc sulfide. It was very important work. And those are the spots. Look at that pattern there. See, the thing that you could get is the symmetry. Oh, by the way, by the way, I'm not going to test you on Laue. I'm only talking about it for five minutes here. So in this class, I want you to know about using characteristic peaks, and all the things we've been talking about. Laue is just on the side for knowledge. We won't test on this. But I want to end by telling you a story. This tells you something about crystal symmetry. What is symmetry? Well it just means, if I rotate it, is it the same? Or if I translate it, is it the same? Give me just one minute because this is actually a really interesting story. That's translational symmetry. We looked at this picture already. There's no rotational symmetry. To get back to this picture, I'd have to rotate the whole thing by 360 degrees. But see, there are crystals that actually have only rotational symmetry and no translational symmetry. And Dan Shechtman was the one who discovered these in the 1980s. Nobody believed him. His advisor told him to go back and read the book. And you know who wrote the book? Linus Pauling. Linus Pauling wrote the book on chemical bonds. Literally, the book was called The Chemical Bond. And Linus Pauling took the lead. He won two Nobel prizes, not one. He took the lead in attacking Shechtman. Shechtman had discovered quasi crystals. Shechtman had discovered quasi crystals, but nobody-- he was fired his papers. His papers weren't published. And for 10 years, nobody believed him. Linus Pauling said famously, "There are no such things as quasi crystals, only quasi scientists." That's how bad it was. Oh! But Shechtman had his day. He won the Nobel Prize. And he had his day. And he was proven right. And these quasi crystals are absolutely fascinating. Rotational, but no translational symmetry. Have a great weekend! |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 9_Lewis_Structures_I_Intro_to_SolidState_Chemistry.txt | OK, this is the thing. We've done a lot of work on ionic bonds. And I drew the ionic bond energy again, on Monday. And remember, in an ironic bond-- so let's take sodium chloride-- we have, oh, we're going to go Lewis, right? And people say, well, why did I draw that dot there. I could have drawn two dots there and that dot there. Sure. Yeah. But they're kind of looking at each other. So, you know. [LAUGHTER] And remember in an iconic bond chlorine is like, give me! I want! And that's the kind of relationship they have, right? And so this went into something like this, where sodium was like, OK, fine. Take it. I'm a plus now. And chlorine was like, thank you very much. I'm fully octeted, right? Like that. And that's the ionic bond. So this is ionic. And we learned this. OK, but see, there is another kind of bond that can happen. It's not all about taking and one per-- no. It can also be about sharing. About sharing. So if I have hydrogen instead, let's suppose I have hydrogen like up there. And I've got hydrogen now. And here is hydrogen. Well, see when they see-- geshundheit-- each other, a very different thing happens. Because one of the hydrogens doesn't say, gimme. And the other one doesn't say, sure. Well, but something else happens. Because they get closer. They get closer, and closer. Oh! Let's look at this in terms of like, OK, so here's a plus and minus. Oh, yeah. It's not an orbit, it's an orbital! And here's another plus, and here's another minus. And these electrons-- you know, the thing is, the electrons are like, I don't like you other electron. You're repelling me. And the protons are like, I don't like you other proton. You're repelling me. But the electrons do like protons. And at a certain point, if you bring these together, one of them is like, hey, I got an idea! What if I'm attracted to you sometimes, over here, and I'm this electron. I'm like, I want to be attracted to you. And I'll tell you what, you can be attracted to my proton, and we'll share. We'll share. And so, when these come together, what you get is the protons are a certain distance apart, that's set by how far you can push them in without them feeling that repulsion. And the electrons-- as long as they don't get like, you know super on top of each other, because they're going to repel each other-- they're happier. They're happier because they got two positive charges that they can kind of go around and be near. That leads to happiness-- lower energy bond! But you can see right now, this is a very different kind of bond. Right? This is not ionic. This is covalent. This is covalent. When share electrons it's a covalent bond. OK? And that is something we write differently. We write it with a dash. Well, or you could've also written it-- or maybe you should say, well, let's just put those electrons in between them like that. Right? And it's showing that those electrons didn't go to one or the other, they're in between. They are what are making up the bond together, sharing those protons across the atoms. OK. So that's nice. That's hydrogen. What happens when we get to something a little bit more complicated, right? So let's go to water. Now, if I have something like water-- so now, I'm going to go through some examples. Let's suppose I have water. Well, I've got this here, and I've got this here. And now, remember, we're going to think about things in terms of Lewis dots, which I introduced to you on Monday. Those are the valence electrons in an atom, which are what do all of the chemical bonding. So those are the ones we're going to carry. Now, how many dots do I have for oxygen. I forget. Six! Six. And if you didn't know, you know where to look. Because you're always carrying your periodic table with you-- always! And so I might write six like that. OK. What lines do I draw where? How are these shared? So how are the electrons shared in covalent bonds? Covalent bonds. And that is where Lewis will help. That is where Lewis will help. How do I take these valence configurations of electrons and make a molecule with these kinds of sharing bonds in it? And there's actually a fairly simple recipe that I want to teach you today, because that's how we're going to do it. And the recipe is written here. And don't worry, there's a lot of text here, but I'm going to go through it for three different examples. And if you haven't done Lewis-- I know some of you have in high school, some of you have not. That's OK. If you haven't, practice is really key. All right? We'll do three molecules, maybe four, today. And I encourage you to do a bunch more as practice. And this is the recipe for drawing Lewis structures. And Lewis structures is what tells us how electrons are bonded in these molecules. And so, let's follow this for the one I have up here. Let's follow this for water. OK? So we're going to go through this recipe. Now, the first-- so, one through six. I will write one through six for the example. I will parallel that recipe. So first, connect atom central. Central atom is often the least electronegative-- already I'm stuck! Because I didn't tell you anything about electronegativity. That's coming a little later. Today, I will tell you what electronegativity is. For now, trust me. So the first step is connect atoms. Central atom is often the least electronegative. OK. Well, so let's see. So for water, so step one, the H atoms are going to be terminal. That means that they're on the outside. Terminal. OK? Outside. They're also called terminal. And so, the arrangement should be something like H, O, and H. OK, good. So that's going to be my arrangement. OK. Step two-- determine the total number of valence electrons. OK, I can do that. So two. Well, I've got one, two , three-- OK. I've got eight valence electrons. Good. OK. Step three is where it gets really fun. Place bonding pair of electrons. Go all in, just like I did for-- So there's a bonding pair right there. There they are. Two electrons in a bonding pair. So place bonding pair between adjacent atoms. OK. I can do that. So let's see. Here we go. H, O, H. Right? OK. So there's my bonding pair. I put two electrons. But now, you see, OK, I only used up four electrons there. OK? Now, starting with that, now add enough electrons to each one to form an octet. Start with terminal atoms, add enough electrons to form an octet. OK? So let's see. But hydrogen is special, because hydrogen's octet is just two. Hydrogen doesn't want eight. Hydrogen just wants two. So we called it out here, in the recipe. Two for hydrogen. OK. So add enough electrons to form an octet. Well see, each H atom already has-- Remember they're sharing. They're sharing. OK? So each H atom already has the two electrons that it wants. And so, for four I've got that-- oh, I started with the terminal. Atoms. OK. And so for four, I've got each H is OK. It has two. All right? It has two electrons, which is what it wants in its valence. Next one. If-- oh! Now, it gets-- If there's electrons left over put them on the central atom. OK. So I went from the terminal in. I went from the outside in. H has two. H has two. That's good. Oh! But how do I know if I have any left over? Well, I did that in step two. I had a total number of electrons. Now, how many have I used? I've used two here, and I've used two here. I have four left over, because I started with eight. Right? And I was just told from step five to put those on the central atom. OK. So step five now, is going to make this look interesting. So H, O, H-- and now, I'm not going to put them in a bond. Right? I'm just going to put them on the atom. And those are called non-bonding. Well, OK. That's a genius name. Nonbinding. Because they're not bonding. [LAUGHS] That's a good name for those electrons. You can also call those lone pairs, because they're lone. They're not involved in a bond. All right? So those are my four extra electrons. And then finally, if the central atom is less than an octet, used the lone pairs from terminal atoms-- Oh, this looks complicated. I don't want to deal with it. Why? Because I've got an octet here already. So six is not needed. Because everything since the oxygen-- since O is happy. It's got its octet. So that is a very slow-- we went slow. We'll go faster now. Right? So that is a very slow introduction to how to do a Lewis dot diagram for a molecule. In this case, H2O. Now, you may notice I drew it linear. Some of us know that is not the shape of this molecule. But shapes come later. Shapes come next week, right? Right now we're not thinking about shapes. Let's do another example. So my next example is O, Cl. I'm taking this and I'm putting a charge on it. So here's my next example. Well, anyway, that's my example. I'm going to go through the numbers again. We're going to go a little faster. So I want to know how this looks in a Lewis. Structure well, OK. So there's only two atoms, so-- Maybe I should have called this example B, because now I have point 1, all right? So there's only two atoms. So there's no central atom. OK, that sounds like that might be easy. And then step two is, how many valence electrons are there? Well, there's six, plus seven, plus one, right? So oxygen is six. Chlorine is seven, plus one, equals 14 electrons. Where did the one come from? That's because it's got a negative charge here. It's a negatively charged molecule. OK. Good. Now-- OK, now it gets interesting. Place bonding pair of electrons between adjacent atoms. So my adjacent atoms are these two atoms, which I could write like this, or I could also have written it like this. OK? Those are equivalent. Now step four. OK. Now, starting with the terminal atoms, add enough electrons to each one to form an octet. So let's go through this. So now step four. OK. Now, this can count the two electrons. This can count those two electrons. You see how this works? Because we're sharing now. No one took. We're sharing. So they can each count those, because of what I drew up there in terms of how a convenient bond works. But now, they need another six to form an octet. So how many? I used two. How many do I have left? I've got 12 left. It's perfect. Right? It's perfect. So I've got exactly as many as I want. And I can write O has all of those. Cl has all of these. And if we want to be careful about it, we will emphasize that this molecule is not neutral. It got an extra electron. That's why we put the minus sign up there. Right? It's charged. And this molecule, OK. Each atom-- let's see, OK. I used the extra electrons. If electrons are left over-- No electrons left over. If central atoms-- no, they're already octets. So five and six are not needed. Oh, but you know the last example is going to be more interesting. So let's do that. Last example. O-- C, example C. O-- so I've got H, OK. I'm just going to write the atoms here, because I'm starting where I don't even know what it's going to look like. I don't even want to hint at it. Those are my atoms. I've got two hydrogens, an oxygen, and a chlorine. OK. I'm sorry, a carbon. Oh, yeah. I meant this to be a carbon. There we go. Two hydrogens, an oxygen, and carbon. OK. Now, as we will see when I tell you what electronegativity is, carbon has the lowest electronegativity. OK? And so, I'm going to put that in the center. That's going to tell me that if I answer point 1, it might look something like this. Oh, now it's taking some form, right? OK? So the hydrogens are going to go like that. Their terminally oxygen is outside, carbons in the middle. And how many electrons do I have? OK, so now, let's see. Answering step two, I've got 12 valence electrons. OK. Because I've got six, four, five, right? So 12. OK? Good. And now, I'm going to put the bonding pair in there. So let's do that. So step three-- OK, so now I've got C, H, H, O. So I've done step three. OK. So I've got two, four, six. I've used up six electrons in this sharing covalent bond situation, and I started with 12. So that means I've got six left. OK. So I'm going to go to step four. Now, starting with the terminal atoms, add enough to each one to form an octet. Now hydrogen is good, right? Because hydrogen, remember, it's octet is a doublet. It's two. So that's good. But what about oxygen? OK. It looks like oxygen needs some. So now, we're going to write this structure like this. I've got to add six more electrons to oxygen in order to fulfill its octet, right? And oxygen is happy. All right? Yeah, it is. Just like that. But if electrons are left over-- but are there any electrons left over? No. So I can't place electrons on the center atom, yet now we invoke number six. If central less than an octet-- it is two, two, two-- then use lone pairs from terminal atoms to form multiple bonds to the central atom. Oh, yeah. Now we're there. Now we're there. So now we jump to step six. Because look. Let's continue. Because look, if I add all these up and I go to carbon, it's not happy. But let's make it happy, because that's what Lewis lets us do. And I'm going to go down to here-- six continued. And all I need to do to do that is to follow the instructions, which is, taking one lone pair and put it here. So I'm going to draw these explicitly now as dots, right? So I took one of those lone pairs and I moved it here. And I created a stronger sharing bond. I wrote down the guidelines of Lewis, remember, on Monday. And the third point was octet! Right? Atoms want to achieve their nearest octet. And this this tells us then, by following these steps to implement Lewis' dream and vision, we get nothing less than the nature of bonding itself in molecules. That's cool. And so you could write this with sticks if you want. C, H-- that's a double bond with those electrons up there. It's the same thing. And now, if you count-- happy, happy, happy, happy. Everybody's happy. Caring is sharing. They're all sharing. So this is the general idea of Lewis. Following the Lewis prescription tells us about covalent bonds, and it tells us how many electrons are going to be shared. Now, it also tells us something else that's very important, and it's through this concept called formal charge. So that's the next thing that we need to learn. Because now, I go to another molecule. In this case, it's formaldehyde. OK. CH2O. But look, now, neither of these is done. These are not done. But it's like my starting place. OK. Carbon central here. It's kind of central-ish there. Can I start with either of these? And the answer is yes, you can. And you can draw-- and let's do that-- you can actually draw correct Lewis structures. Let me use the board over here. All right? So I could draw this like this. H-- Let me see here. H, C, double bond, O, hydrogen. And I could put that there and that there. Or I could draw it like this. Do I have it this way? Yes. All right? Like this. And carbon double bond O. And when I look at this, and I look at the electron count, it all works out. All right? I got to add my lone pairs to the oxygen. Don't let me forget those. It all works out. The total number of valence electrons is 12 in both cases, and everybody's obeying the octet rule. But the concept of formal charge allows us to distinguish between which one is likely to be more stable. And so, what formal charge is-- and I've got it written out here-- OK? So formal charge-- non-binding electrons count for the given atom. Bonding electrons are divided equally. And so formal charge is the total valence of an atom-- so I can take atom here and say, what's it's total valence? Carbon, four. Oxygen, six. All right? So that's the total balance of the free atom. And then I subtract from that how many nonbinding electrons it has, plus the bonding ones over two. Right? And so, if I do that-- let's write that. So if I think about that-- gesundheit-- from my pictures, the formal charge is equal to the number of valence electrons of the free atom. That means the atom, before it ever participated in a bond, minus the number of dots, minus the number of lines. Because that is really the same definition, right? That's the same definition. So let's go up to this one. So now, if I look at carbon in this configuration, and I calculate the formal charge of carbon, it's going to be 4-- the valence electron of the free atom-- minus 2-- because it's got the two dots there, right? Minus 3-- 1, 2, 3. Or the total number of electrons involved in bonds divided by two-- 6 over 2. 3, right? So those are equivalent. But that means that the formal charge on this is-- just spell it out here for now-- formal charge is equal to -1. And if I do the same thing here, then it's +1. But you see, if I look at this structure, and I think about the formal charge on the carbon, atom-- Again, I'm basically counting a change from the free atom. OK? And so carbon had four. What did this new bonding environment do to the valence chemistry? And in this case, the formal charge on carbon is going to be 4-- starting with the number of free valence electrons-- minus no dots, but it's got one, two, three, four bonds. So that's zero. And the formal charge on the oxygen is going to be 6 minus 1, 2, 3, 4, 5, 6. So that's also zero. And you get a sense-- how much did I push this atom away from where it normally thinks about its charge balance? And you get a sense for why this might be true. Lewis structure with a set of formal charges closest to 0 is usually the most stable. And that's our guideline. And so, this one not very happy. That one happy. And so you can calculate the formal charge on any atom in any Lewis structure-- gesundheit-- by following this very simple procedure. You take any atom and you look at how you've drawn it's Lewis structure, right? And you just count. So for example, if I take CO2, and I think, well, I can do the same thing here. I've got two Lewis structures and they look like they both satisfy the recipe. The recipe is satisfied. But which one is more stable? Well, if I look at this, it's the same idea. And I won't go through it in detail, but if I look at this then-- C, O, O, with lone pairs on the Os, like that. All right? This is going to give me formal charges of 0, 0, and 0. Whereas, if I do the one on the right and I put a triple bond-- so I put too many electrons sharing-- then yeah, I satisfied the octets, but no, I'm not as stable, because the formal charges aren't as close to zero. So this gives me to my brief, why this matters for take. Why would I care which one of these CO2 takes? All right? OK. Well, it turns out that as long as we're not at absolute zero, this molecule is moving. And so, obviously, that's related to this. It's not obvious. But cars in the US are now the number one emitter of CO2. OK? They're the number one cause of CO2 emissions. And here's a nice little corner somewhere. And so, this is coming out now from the transportation sector. More of the CO2 comes from that than any other sector. Why does it matter? Well, why does this Lewis structure matter? Well, it has to do with how it moves. Because, you see, I showed you this before. And I didn't complete it because we were talking about electron transitions, and then I gave you the example of ozone being really important for absorbing in the UV. And I showed you the chemistry of ozone degradation with CFCs. You see, look at this absorption out here. There's CO2, right? And so, there's the sunlight above the atmosphere. Here it is on Earth. And you could see CO2 there. But see, the reason CO2 absorbs there has to do with how it moves. It has to do with its vibrations. Now, that's not something that you need to know for like a test or something, but I wanted to tell you about it because it is directly related to what we just did. Those vibrations-- and by the way, that first mode is the one out here doing all the IR absorption. Right? This one. And you can see right away, if this thing is wiggling, it's going to wiggle very differently. Right? It's going to wiggle very differently whether it has two double bonds on either side of the carbon, or a triple and a single. Right? And so that, alone, tells you something really important about how it interacts in our atmosphere with radiant heat, right? With IR radiation. OK. That's my 'why this matters.' Now, I've been talking about all these things in absolutes. But you see, the world is not so absolute. And in fact, we're missing a really important part of the picture here. So this is H2, right? There is an iconic bond where, remember, it's like, I'm picky! OK. And then it's like a Coulomb interaction. And here it's a sharing. Each electron sees both of those protons. But there's a whole bunch of room in between, right? There's a whole bunch of room in between. And so what we need next, as we talk about covalent bonds, is we need a way to think about how ionic they are. Right? Because there's only a certain kind of covalent bond that's purely covalent. As it's written here-- non-polar covalent. So let's talk about that next. And what we're going to do is use this term that I alluded to in the beginning, which is point number one in your recipe, and that is electronegativity. OK? So I'm going to get it done. Almost. Practice makes perfect. OK. All right. So there is a symbol kai, that we use for this concept of electronegativity. Electronegativity. OK? Now, this is the tendency of an atom to attract shared pair of electrons to itself. OK? Let's write "in a bond," just to be sure. All right? In a bond. So if an electron in a bond, the question is, how much was it able to pull those bonding electrons to it? All right? And you can already think about this in terms of concepts that we've learned already. Right? Like the size of an atom. How far out electrons are? Right? The radius of an atom. Whether there's shielding going on. Right? This kind of leads you-- how many protons there are. All right? This leads you already to be able to think about this. Which atom is going to want a pair of electrons in a bond, right? Which one is going to want it more, and how much? But it was Pauling who said, no, I want to go further that. I want to write down-- even if it's empirical-- I want to write down some way, some number. I want to quantify this. And so he came up with a scale for electronegativity. The term and concept of electronegativity goes back long before Pauling, to, I think, [INAUDIBLE].. But Pauling is the one who said, I want to quantify this. And the way we're going to do is we're going to measure bonding energies between all sorts of different atoms. And what he found is that, if a bonds to a, and b bonds to b, a to b is not just a simple sum of the two. And that clued him in for how to think about this partially covalent or polar covalent nature. I'll tell you why that word "polar" is there in a few minutes. And so Pauling developed this scale-- the electronegativity scale-- to tell us this. And he arbitrarily set fluorine to four and hydrogen at first 2.1, now it's 2.2. But basically, this is just some of the elements of the periodic table, and these are their electronegativities. OK? And you don't need to worry about how this scale was developed quantitatively. There are actually many electronegativity scales. OK? What I want you to know is what it means conceptually, and then how to use it to think about whether a bond is going to be ionic, or covalent, or somewhere in between. And so, you can see already here, OK, the extremes of this are fluorine and caesium. These are the most electronegative, least electronegative, or if you want to be positive about it, most electropositive. When you're talking to these elements you want them to feel good. They're not the least. They're very important. So they're electropositive. Right? And now you know conceptually. But look, now we also can think about differences. Because, if the difference between this concept, from one atom to another is zero, well then, that must be a pure covalent bond. Because neither one could bring in the electron pair more than the other, right? So like chlorine two-- the chlorine dimer-- then the change in electronegativity between one chlorine and another is zero, it's a pure covalent. I mean, just think about that conceptually, right? So that's what electronegativity means. So if I have two that are the same, neither one can draw in. They have the same kai. Neither one can draw the pair towards them more. Right? So the same is true for any dimer like this, where it's the same atom. The H2 dimer that we started with, also has delta kai of 0. Those are called pure covalent, because it's just pure sharing. Nobody took more charge than the other. But see, if I go to something like sodium chloride, well now, from this table I can calculate the difference in electronegativity between sodium and chlorine. And it's 2.23, which is pretty high. And it's ionic. So if that difference is high, then it means one of the atoms grabbed the electrons. And if it's kind of 0, it's pure covalent. And if it's somewhere in between, it's this polar covalent. So, for example, HCL. So these are all with the same atom-- chlorine. But here delta kai is equal to 0.96. So this is called polar covalent. OK? So this is also partially ionic, right? These mean the same things. There's ionic character-- is another way that we say it. This bond is not purely covalent because one of them took a little more charge, because it's electronegativity was higher. And therefore, there's a little bit of that kind of ionic thing going on, because you got a little plus over here, and a little minus over there. Right? And that is something that can be looked at for a whole bunch of different bonds. And so here, this is what Pauling was trying to fit. So he came up with these empirical fits. Right? OK, things kind of tend to lie on this line, which is the plot of the electronegativity difference between two different atoms the percent ionic character. Is it really fully ionic? Is it is it fully covalent? And you can see, over here, it's kind of not interesting, right? It's just anything with itself is zero. And over here, you've got some of the same ionic bonds that we've already talked about, right? And then you've got all this stuff in between. And what this says is, as you've now seen, right? [INAUDIBLE] is a suggestion of a rule, and then it gets broken 20% of the time. This is also a way of qualitatively classifying bonds and thinking about the ionic character in a bond, but it is not always quantitative. But in general, what you'll see is that if delta kai is greater than 2, then it tends to be an ionic bond. And if delta kai is less than, let's say, 1.6, it tends to be covalent or polar covalent. And in-between it depends. How do I know? Right? I'll show you an example. Where's my example? I've got an example here. Sodium bromide. It's not on there. But if I take sodium bromide and I take HF, they both have delta kai equal 1.9. Now, some textbooks will simply say 1.7. You will see that in some textbooks, because they just want it to be all or nothing. It's always ionic, or it's-- But the fact of the matter is that in this intermediate regime you get variations that depend on other things. And so, for example, HF is a gas at room temperature. So these have the same electronegativity difference, but this really behaves like a polar covalent molecule. And this really behaves like an ionic solid-- an ionic bond. How do you know? It's the properties. Right? Remember, I showed you the properties of ionic solids on Monday. And so, I can look at those and check them off-- solid at room temperature, et cetera. And I can see, well, does this fit the bill? Does it look like an iconic solid? It absolutely does not. So even though this is the same, you got to be careful in this intermediate regime. Now, you can also go even further, and you can get quantitative. Because, you see, what we're really talking about here is something called a dipole. That is why it's called polar covalent. OK? It's called polar covalent because we made a dipole. And as far as I know, from my electricity and magnetism training, the dipole moment-- mu, it's written as mu-- is the charge times the distance. Where Q is that the charge. In this case, it's literally the charge that was pulled. It's exactly that. There's the chart, right? Tendency to pull a shared electron. I pulled it over here, so I created a little more negative charge on me, and a little positive charge on there. And that meant that I've got a partial charge now, separated by a distance. Right? And so you can actually use this to get quantitative. So let's do sodium chloride, right? So for sodium chloride the dipole moment is nine to debye. And this is just one debye. This is charge times distance, so one debye is equal to 3.3. 10 to the minus 30th coulomb meters. That's the units of a dipole moment-- charge times distance. So if I take sodium chloride and I tell you that its dipole moment is nine debye, then I also have to tell you that the distance between them, r, is equal to 2.36 angstroms. That is this distance here. I've made a bond between these two things, and that bond is some distance apart. Well, that's exactly the distance I need to think about the dipole, right? So remember, a dipole is because I've got positive charge and negative charge. And in fact, you might see it written like this. You might see there's a little positive charge here, and a little negative charge there, and the dipole moment goes like that. All right? There's a dipole. And now, those charges at this distance in the bond allow me to actually be quantitative, because Q is equal to mu over r. And if I plug that in, then it's 1.3 times 10 to the minus 19th coulombs. So for sodium chloride-- because I was given the dipole moment in that molecule, and I can look up the bonds distance-- I can actually tell you how much charge are on those atoms if I know that information. But I also know what the charge of a full electron is. This is almost there. It's something like 80% of an electron. And so that's a pretty strong ionic bond. But it's not 100%, right? So this is the message I want to tell you. In reality, we've been talking about absolutes. I fully took a charge. Right? No! You didn't. You took 80%. And there's a little bit still over there. Right? But we do like to still categorize materials. And that's what this does really nicely, is it says, well, this is going to behave like an iconic solid. Meaning, it's going to have those properties that I showed you on Monday. Right? Even though it's not fully ionic, it's pretty darn ionic. Right? And now, even though these are not fully covalent, meaning, pure, non-polar covalent-- non-polar, no dipole in the bond. But these have some dipole, but that's OK. They're still acting like covalent molecules. Right? And so there's this spectrum in between. And you can use these concepts to think about the bonding, and to think about in particular this-- that so many bonds are actually like this. Even though we categorize them like this, and we think about them like that, they actually look like this. And electronegativity and dipoles are what help us characterize these differences. Now, there is more to Lewis that I will talk about on Friday. But there's something called resonant structures, and that's a very important part of Lewis. I'm going to talk about it Friday, and it will not be on the exam on Monday. OK? But on Friday, we'll talk about resonance structures, we'll do a few more Lewis examples, and then I'll talk about exam one. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | Additional_Lecture_1_Phases_Intro_to_SolidState_Chemistry_2019.txt | [SQUEAKING][RUSTLING][CLICKING] JEFFREY GROSSMAN: All right, how are you guys doing? [CHEERING] Thank you. You know why? This is such an amazing day. All right, first of all, we're going to talk about phases. And there are two lectures that I have introduced this semester that are new, that have never been seen before, until right now. One of them is this one. Right, Dane, thank you. And then, check this out, this morning over the wires comes this news. And it's the Nobel Prize in chemistry this year was given to 3 of sort of the founders of lithium ion batteries. And the other new lecture that I'm giving this semester, later on, is on the chemistry of batteries. It's like coincidence? [LAUGHTER] I don't think so. Did someone in Stockholm know? I don't know. But it's really, really cool. And this is awesome. And there's so much great stuff online now about lithium ion batteries, because of this. I mean there was before, but there is a lot of press on this now. It's very interesting to read about. OK. So now, what is happening? I say the word boiling all the time. Right. I've been saying it all the time, especially the last week. And I mean at home, too, I'm saying boiling a lot. And it's because it's a proxy for bond strength, intramolecular bond strength and boiling, I've been kind of going back and forth. Right. But so, today, we're going to go, what is happening when you boil? All right. Well, you know that you're changing a phase, as you can see from this highly accurate drawing of water molecules in the gas phase, water molecules in the liquid phase. But what's really under the hood? And of course, the story starts with Otto. Now who was Otto? Otto Von Guericke. Otto Von Guericke in the 1600s. He was like-- you know what-- Aristotle. Aristotle, he was all about like nature abhors a vacuum. Does it really? I don't know. I don't-- you know, does nature really hate? Like, let's try some stuff out. Let's try to make a vacuum, and see what happens. So he figured out a way to take these two hemispheres, these two big bowls, and kind of pull all the air out. There he is. He looks very happy there. Right. And he's pulling-- because he's pumping the air out inside of here, and look he did. He went around the countryside in Germany, and he would attach-- gesundheit-- just attach eight horses to one side, and eight horses to another, 16 horses pulling. Eight that way, eight that way, and they couldn't pull it apart. This is what he did. It doesn't look like he's got a lot of crowd, but maybe he did better. I don't see any houses either. I don't know where he is. But that's the experiment that he did all throughout the countryside. And he called it the power of the vacuum. Nature abhors a vacuum, the power of the vacuum. Power of the vacuum. It's not the power of the vacuum. Oh, Otto, it's not the power of the vacuum, because vacuum has no power, and nature doesn't hate it. But pressure-- so atmosphere has power. Technically, it has force. Atmosphere has pressure. What he showed in that experiment, and if there's one experiment I want you to remember from this class, it's not this one. [LAUGHTER] But this is a cool one. But what he showed wasn't the power vacuum, he showed the power of one atmosphere of pressure. He showed the power of pressure. And so we start. Pressure is force over area. Typical units would be like newtons per meter squared, right. And if you're going newtons per meter squared, then this is called the pascal. And for reference, if you have one pascal, one pascal, it's about the pressure of a dollar bill, $1 resting on a table. I don't know. Why do I use that one? Because I remember it, $1 resting on a table. And so that's the force per area of the bill resting on a table. So if you had 100,000 of them, right, if you had 100,000 of them, and you stacked them up, then that's one atmosphere. That's just a different unit. It's about 100,000 pascals. So the power that he was exhibiting in this experiment is the power of the atmosphere. We've got-- we've got atmosphere all around us, pushing on us. It's pushing on us. And so, we got like a car worth of atmosphere, literally, on us, but it's pushing on us in all different directions. And we're pushing back. Luckily, we've got stuff inside of us, so we don't like implode. Right. That force is strong, and that's a demonstration of how strong, how much one atmosphere is. Right. It's force per area, the power of one atmosphere. Well, that's also directly related to boiling. And how do we get from here to boiling? Well, so we need a couple of concepts, right. So first, we need the concept of evaporation. And I'm going to use this-- I'm going to use this example of like water molecules, right. So this is a-- this is a glass of water. Right. And so, if I have a glass of water, I'm going to draw them like these little things-- yeah, and now molecules kind of near the surface-- well, it could leave. And that is-- that's a molecule, not a v, right, and that's evaporation, evaporation. Evaporation. But, you know, the other way you could go is, you could say, well, what if there was this liquid here, and you've got these molecules, kind of like that. And you had one here. And it kind of comes back. Well, that's condensation. You guys kind of know this, right? Condensation. Evaporation and condensation. Oh, but this is not what's interesting. I mean it's interesting, but it's-- but it's more interesting now if you close it off. And so, let's do that with these super amazing animations here-- they're there. Look at that. OK. So now what I've done, I've got a liquid and a solid. And in both cases, there's a liquid. You could have solid. Molecules can come off. We'll get to that. How? Why? They can come off. But now I've closed it. Now if I close the container, what would happen if I start with nothing? So now I start with a liquid, and I close it in a container. You see-- look at the molecules. They're kind of coming off. And maybe they'll hit the walls or bounce around, maybe at some point, they'll come back into the liquid. Right. OK. So evaporation went that way. Condensation went that way. Now-- ah-- now, if I were to plot, if I were to close this and plot, for example, a rate. And I say, well, I want to know how quickly evaporation is happening. It's just going to keep happening. I mean I'm starting with, you see in the videos-- these videos are real-- you see it doesn't stop happening. right. It keeps happening. So if I have no vapor in there to begin with, well, then, OK, evaporation is going to go like this. All right, maybe this is time, it is time. And then-- so that's evaporation. But then, at a certain time, at a certain point, I'm going to have enough gas molecules, gas water molecules in there. So it starts out at 0, and they're going to build up in there, and at a certain point, you got that. Right. And so, this would be condensation. OK. We're starting with the basics, condensation. Right. And then at a certain point, they reach what's called dynamic equilibrium, dynamic eqm, eqm is equilibrium. So the dynamic equilibrium is really important, because what happens there is nothing is stopping. It's not the thing stopped. You can see it in the videos. I found it online, so it must be right. So like, you know, it's continually going. It's a dynamic process. It's a dynamic equilibrium, but they are equal. That point that happens in these containers, you can measure the pressure. That is very important. Right. Because it's exerting a pressure on the container. It's exerting a pressure on the liquid. Pressure. Atmosphere. Pressure, inside of a closed container, vapor pressure. P vapor. That happens when you hit that dynamic equilibrium. Now why am I telling you all this? Well, the vapor pressure is going to be how we understand phase change. And so-- and so-- you kind of get a sense already, kind of get a sense already, that this is going to be related-- this is related, you know, this depends on the bonding. Right. So I just told you that water was really special. So that means that how it-- it's the bonding between the water molecules and how often can it get out of the liquid, it's related to the bonding. Right. So if you take molecules that we talked about on Monday-- these are just from Monday-- propane and butane, now you understand, right? Why is there this difference in the vapor pressure of protane-- propane, and the vapor pressure of butane? Well, it's how they bond together. And you know exactly why. Because these are only able to do [INAUDIBLE].. And so, this has a-- has a higher surface area. We talk about this in terms of the boiling point, remember, as a proxy for the intermolecular bonding strength. And now I'm giving you the vapor pressure. So this is the vapor pressure of these things is different. Those are measured-- those are given a kilopascal. So if it's water-- so the water vapor pressure-- right-- let's see, make sure I get this value-- yes-- P-- I'll just put a Pv for water, is about 2.3 kilopascal. So it's a lot lower than those. Right. Those are-- now, in atmospheres, and this is going to be important, 0.02 atmospheres. Just switching around units, that's all. It's same thing. Gesundheit. Right. So now, if a liquid is- you can think about this as, well, this is a proxy already. I can understand this conceptually. Right. It's a proxy for the intermolecular bonding. If you take something like glycerol-- well, glycerol is-- let's draw glycerol just for fun. Carbon, carbon, Oh-- let's see, it's got another carbon, OH, it's got another OH, look at all that, OHs. It's got some hydrogens down here. It's got another hydrogen in there, and it's got two more hydrogens out here. No, I am not paying attention to VSEPR right now. I'm just getting it on the board. But look-- but look at that. Right away, you're like, oh, look at those hydrogen bond opportunities. Yeah. This could bond more strongly than water to itself, right. It's got all that hydrogen bonding. It's got even more than water. And so the vapor pressure of this is 0.01 pascal. So much lower. And as a general rule, we like to compare everything to water. So if something has a vapor pressure that is higher than water, so it's called volatile, Pv greater than Pv water, it is volatile. It's a volatile liquid, like butane and propane. If it's less than Pv water, then it's called-- here it comes-- all those chemists-- non-volatile. Look at that. Look at that beautiful naming originality. OK. But this is the thing, right, I've given you this concept of vapor pressure. I've talked about it a little bit. You can relate it to something you just learned on Monday. But I still have not come back to boiling. And that really is it. It's that boiling is actually understood by vapor pressure. That is how we understand boiling. And so you can think about it-- let's just give it to you conceptually. Right. So if you had-- let's go back to those buckets there. Think of it-- we'll go back to like evaporation and condensation. But now, instead of being in a closed container, right, now I'm outside. So if I had, let's say I had this container here. And I've got all these waters in it. Right. But now I'm outside. So I've got the power of one atmosphere, right. One atmosphere. Well, you know, now you see, like, OK, if a molecule is able to evaporate from the liquid, if a molecule is able to leave the liquid bonding environment, breaking the bond, that's why we can relate it to intermolecular forces, right. It's breaking those bonds in the liquid, and becoming a gas phase molecule. It's changing its phase. Well, you know, then the question is, is it like going to-- what's it going to do? You know, is it going to kind of hang out here? And are the forces of the atmosphere, is the atmospheric pressure going to kind of hold it around there? Or, is it going to be able to leave? Well, this is a very simple conceptual picture. Right. If you thought that the forces of the atmosphere were going to always hold it there, then you'd say the atmosphere is a container. And it's not. But, actually, evaporation only happens because a breeze can come along and blow this away-- right-- out in the air. So those forces are super important, because as soon as the vapor pressure is greater than the atmospheric pressure, well, now you can get a sense of what-- the vapor pressure is greater than the pressure outside over that surface of the air here, then the molecule is just going to fly right out and power through it. That is the definition of boiling. Boiling is a pressure effect. It's that the vapor pressure of this material, which is related to how strongly it's bonded in the liquid compared to the gas phase, that that goes higher than the surrounding pressure. That's the definition of boiling. So you can see why water doesn't boil. Like having a bucket of water, and I watch it, it's not going to boil. 0.02 atmospheres is not greater than one. Right. But so I haven't gotten there yet. I haven't made it boil. But I want to make a boil. And that is where we go next, because that is really what temperature is. See, the vapor pressure is really a link between temperature and boiling. It's the length that you need to understand between temperature and boiling. The reason why-- when you increase the temperature of the water, the reason why it boils is because you change its vapor pressure. That's why it boils. That's what boiling is. OK. It's when you get it above the atmospheric pressure. No. OK. So the-- so we got-- OK-- so are we-- where-- there we go-- we'll come to that in a minute. So if I were to plot now, the vapor pressure-- let's do this carefully here. OK. I'm going to make a plot of the vapor pressure, Pv, right, versus temperature. So I just told you there's a dependence on the temperature. Right. So versus temperature. Let's go to-- oh, I don't know-- let's go to room temperature. So around here, and I just told you that at 0.022 for 25 C. OK, good. So 0.02 for 25. 0.02 for 25 C. So that's the vapor pressure. That's the temperature. But I also told you that boiling-- if I'm at-- so this is 0-- but I also told you that boiling-- so this would be like one atmosphere. Let's suppose the units are atmospheres. Right. OK. So I told you that boiling happens-- now we know that water-- we know what the answer is, right, up here. That's got to be boiling for 100 C, right. That's where the water goes above or meets the atmospheric pressure, and then goes above it. So once the vapor pressure-- so if this is the vapor pressure of water, this is what we have so far. But we need more than that. So you can say, well, why is it-- so we know this-- why is it that if I go to Denver, the-- so like this is in Boston-- this is in Boston. This is Boston. But if I live in Denver-- how many people live in Denver-- well, I mean you're not living there now, but-- but, welcome to Boston. Oh, ho, ho. And if I go like this-- now see the thing is, now you boil at-- this is-- this is in Denver. So you guys don't have as much air in Denver. The pressure in Denver is 0.83, 0.83 atmospheres. But I just said that, you know, you boil when you have the vapor pressure is higher than the atmospheric pressure. So that means, whatever this curve is, it means that I'm going to boil at a different temperature. And it turns out that's 100, and if you're in Denver, it takes longer to boil potatoes, because the boiling temperature is 95. That's why. But it's all because of this. It's all because of how evaporation happens as it relates to the atmospheric pressure. Yeah, but-- the thing is, though, what we need-- and by the way, you know, you can go the other way, too. Right. You go up. If you go below sea level, then the atmospheric pressure is higher. So if you go to the Dead Sea, your water boils at 101.4. If you go to the top of Mount Everest, it's 70. You never boil potatoes there. Well, you can, but they'll never cook. So how do you-- how does this get connected? That's really the question. What is-- so what is this? Right. Where does that-- where does that dependence come from? And that's where we need something called kinetic theory. Now I'm not going to teach you kinetic theory. And I'm not going to say that you responsible for knowing it. But I'm showing you what it means. What kinetic theory does, which you would learn in, say like a thermal class, right. Well, kinetic theory does is it gives us this relationship between how many molecules have a given energy, in this case, it's a kinetic energy, right, and the temperature. That's what can in theory does for us. So let's see, if I had-- if I had to write this in a kind of just simple dependence way, what it's telling us, in, general is-- so this would be kinetic theory-- kinetic theory of gases, if you want. That's what we're talking about there. OK. This would give us the following. This is what you get is that the log of the fractions-- fraction of molecules with some energy, E, OK, with energy-- I'll write it out just so there's not confusion-- energy, E, the log of that, the natural log of that, is going to go as that energy divided by temperature. It gives you a connection between what I'm plotting here really. That's where that plot comes from. OK. The fraction of molecules with kinetic energy, E, goes as that energy divided by temperature. The log of that. That's what kinetic theory gives me. That fills it in, except that I-- it still isn't vapor pressure. Right. This still isn't vapor pressure. And so for that, you need one more piece. And that is given to us by Clausius-Clapeyron. So the Clausius-Clapeyron equation-- ah-- Clapeyron-- does not-- I'm not going to derive it. You don't need to know how to derive it. But it goes from the kinetic theory to vapor pressures. So this one tells us that the log of the vapor pressure equals minus-- and here we go-- this is the next really important concept-- the heat of vaporization-- and we'll get to this in a second-- divided by RT plus a constant. So that's the ideal gas constant. That's the temperature. And this is the enthalpy or the heat of vaporization. So it's the energy to vaporize a liquid. Good. Now, often, this is given in per mole. So it would be like the energy of vaporizing a mole of a liquid, but it's an energy. It's an amount of energy needed to turn it, whatever you got to put into it, to turn it into a gas from a liquid. That's that enthalpy of vaporization. And R is the ideal gas constant, in this case. You know, so if it's R-- if R is in the equation, it's got to be per mole, right, because it's R. Remember, we've already talked about that. Since it's R, it's the ideal gas constant. So that's, now, OK, so that is how you get between-- that is how you get this relationship. You get it, and, again, this is an important equation. You get it from the kinetic theory gases, but this is what the outcome is for us. This is what matters, is the Clausius-Clapeyron equation. And so, and so now, you can actually now, if somehow you knew how much energy it took to vaporize a mole of this stuff, well, now you've got a curve. That's all I need to know, right, is how much energy is-- so now, I can say, well, let's see, that's OK-- I got-- I got, maybe butane is going to go like this. And then glycerol might go like that. Maybe this is propanol. Right. Propane, glycerol, and so on. Yeah. But I'm still missing this concept. So I can now get dependence, but how do I get delta H? That's the next piece of the puzzle. Right. How do I get delta H? OK. So, you know, so far I've got this really cool result, right. I've got that P-- so far I've got that-- I'm going back to bowling-- I'm going back to boiling. Right. For boiling, I know that-- so the ion-- so P vapor is related to the IMFs, right. I know that P vapor is related to the temperature. And I know that the atmospheric pressure is important. Right. Atmospheric pressure is related to boiling. Yeah. OK. That's redundant, because you've got to get higher than the atmospheric pressure to boiling. That's how we defined it. So, but, again, this specific dependence is what I wanted on temperature, and I have it, except that I now need to know what this delta H is. OK. So here we go-- why is this related to pizza? Everything's related to pizza at the end of the day. But see, now we need to talk about delta H, and what it means. And you know, the way-- the easiest way to show what it means is to plot how this relates to phase change. How heat, right, that's how we're changing phase here with temperatures. We're changing the temperature. We're adding heat into the system. What does it do? What does it do? Well, you know, so one type of plot that you can make for that would be temperature versus thermal, thermal energy input. Now a lot of times what you'll see in textbooks is they call that q. It's an energy term. So it's like in joules, right. Let's say. But it's how much thermal energy you put it in, then you can draw it out like this. Right. So if you draw it out, then you would have your phases. So this would be like a gas, a liquid, a solid, and these would be phase changes. This, and that. So I'm adding thermal energy into the system. It's raising the temperature of the system. That's why these are going up. But the solid is getting hotter, and then, all of a sudden, it melts. OK. Good. So it melted. It melts. And then it's a liquid, and, all of a sudden, it vaporizes. Right. Vaporize. But you can go the other way. If you go the other way, it's condensation out here. All right. Or freezing in here. Melting, freezing, vaporization, condensation. But I like to draw it this way, because this way we're going to see very nicely what delta H is. So here's-- now I'm going to draw it with temperature on the x-axis. So temperature-- and this is going to be H. And I'm drawing this as the thermal energy-- let's see, I'm going to write it in the same terms-- right. So this is thermal energy. Now in this case, thermal energy, we use this thing called enthalpy, enthalpy. But you don't need to worry too much about it, for this class, you can think about it as the thermal energy content of the material. So if I have a change in that, wells, it's because I've lost or gained thermal energy. Right. If you take a thermal class, you'll learn a lot more about enthalpy. And that this is true only if you have constant pressure conditions, but that's not relevant here. I just want you to know, this is like this. OK. But this is the total thermal energy in your material. And by the way, you know, q, heat, the word heat is q. That is the definite-- heat means thermal energy transfer. Thermal energy transfer, a flow of heat. That makes sense. But you cannot say that this has 10 joules of heat. No, don't say that. You will sound like you're from another school. [LAUGHTER] You don't have-- you have that much enthalpy. Yeah. But not that much. Heat is a transfer of thermal energy, please. OK. Now here we go. All right. So let's suppose I were to draw this. Well, now I'm going to have the same-- I'm going to have phases. Right. And so here we go, here we go, here we go. And this is my gas, and this is my liquid, and this is my solid. There's a couple of cool things, right. So now-- so that's the thermal energy, H. But so now, if I were to take part of the solid like that, and just look at it, while it's a solid. I'm adding thermal energy to it. I'm increasing its temperature. We got a way to relate those things. It's called the heat capacity. All right. It's called the heat capacity. So if I look at that, OK, I can do that. Let's go here. All right. If I look at that, then I can relate those by-- well, I'm using q, because it's a change, right, that's the change in the enthalpy. All right. That's the energy transfer. It goes as the heat capacity times the change in the temperature. That s just means it's the heat capacity of the solid. Literally, that's the definition of heat capacity. Right. This would be like units of joules per kelvin. OK. So and, example of heat capacity, I love pizza, and so an example of heat capacity is the following. You burn, hopefully, you don't. But if you take it out of the oven, and you take a bite, right away, you will burn the roof of your mouth on the cheese. On the cheese, because cheese has a lot of water in it compared to the crust. And water has a really high heat capacity. What is heat capacity? It's the capacity of it to have thermal energy. And the water has a really high one, and so, when you take this out, everything's at the same temperature for just an instant. They're all at the same temperature. If you eat it right here, please don't, you burn your mouth on the crust the same way as on the cheese. But since you're waiting, now the crust gives out-- it doesn't have much heat capacity. So it all goes away. All that thermal energy leaves very quickly from the crust, but not the cheese, because it's got so much more. And if we want to get a little more serious about it, we can talk about the oceans. And so, I put this in here. Because we talk a lot about climate change, what you're talking about are the temperatures of the atmosphere. We talk about that a lot. But what gets sort of shoved aside is the temperature of the ocean, and that is rising, and it's not rising that quickly, luckily, but that's because the ocean has such a high heat capacity. That's where most of this thermal energy is going. It's going into the ocean. Right. And we need to start talking about that a whole lot more, because it's a lot higher than it used to be. This is the historical reference, pretty flat. A volcano erupts, 500 years ago, you get a blip this big. Now we're here. Right. So the amount of thermal energy this ocean is holding for us is very high. OK. Now so that's high capacity. That's kind of an important part, but, look, the other thing we have here-- ah, I keep hitting the wrong-- is this. And that's what I want to talk about. This is that delta H. Now in this case, it's the change in energy of fusion, because you're fusing to make a solid. In this case, it's called delta H of vaporization. You could call it condensation vaporization. That's the thing we needed. And now you see it so beautifully. All right. You see it so beautifully right there, because, now, you see-- so what is it that causes me to need to put a lot of energy into a material to change its bonds so that it can vaporize? It's all about the bonding strength. It's all about the-- everything comes to the bonding strength. Right. And so I have these kind of two things, right. Within one phase, within one phase, I can relate delta H to delta T. And that relationship is here. I'm in the solid phase. I can relate them. That's actually called sensible heat. Sensible heat. It's the energy that you would give off when you cool, or it's the energy that you'd absorb when you heat. But now I've got this other thing, which is at the phase change, at the phase change. Well, now, the temperature doesn't change, but I'm putting all of this energy in, or getting it all out. That's what delta H is. All right. So delta T is 0, because the temperature is constant. And delta H is sort of, generally, called the latent heat, latent heat. If you're talking about the phase change between a solid and liquid, it's the heat of fusion, heat of melting. If you're talking about between the liquid and gas vaporization, or condensation. So that's that energy change. And it's related to, well, what we've been talking about, but here it isn't a nice diagram. Right. Now, where do we go from here? Well, there's something so cool about this graph that I can show you really easily. I can also tell you how to do this, which is that things can-- these are the phase change transition temperatures, but things don't need to transition at those. In fact, oftentimes, you might get something like that. Not as much in the gas, but liquid to solid, you certainly can get it. This is called super cooled. Why? Because, for some reason, [INAUDIBLE],, for some reason, it's remained a liquid below its phase change temperature. Here's water, super cooled, poured into a glass. It's one of the most fun things you can do in life. There it is. It's a liquid. As soon as it hits the glass, it's like, wait a second, I want to be a solid. I'm over here. I'm over here. I need to be a solid. And you can do this. You can make-- I can tell you how to make it. You can make super cooled water at home. It's really fun. But that was what you're doing there, now you know, is you're bringing this down below the phase change. It's remaining liquid. Now all of a sudden, it hits the glass, and, but what am I doing? I need to solidify. All right. OK. This is the reason why all this matters is heat is a big deal. 90%, you now, this is-- you can study this. There are wonderful charts to study. But depending on what you want to do, heat is almost always involved. Heat being a transfer of thermal energy. And so 90% of the current energy budget, basically, goes through the middle in some way. That's a lot. But then, if you look at things like this, which you shouldn't, because you can't read it, but if you blow up-- these are all the inputs on the left, all the ways we make energy. Here's how we use it. And here's the result. Here's where I want to get the punch line is, rejected energy. Whatever these units, doesn't matter, it's 60%. What is rejected energy mean? Wasted. All this precious fossil fuel stuff we're burning and all the solar energy we're collecting, anything we're doing here, 60% of it goes into heat, wasted heat. And so the reason why this matters is because there are these materials that we can use to try to capture some of that. Right. There are materials we can use to try to capture some of that, and it all comes down to chemistry. These are called phase change materials. And if you plot the melting temperature, that's this-- that's this melting temperature, right, Tm, versus how much of its delta H, how much delta H you have, you get all sorts of classes and materials. But, you know, so you might want a certain melting temperature from wherever you're operating, but then you might not get a high enough delta H, or you might want really high delta Hs, but then the thing doesn't melt until 700c, which is way too high. So we got to fill this out. We need materials here. We need materials here. That's a call to chemistry. That's a call to chemistry. So then, it all has to do with the phase change. It's a weird name. These are called phase change materials, but that's true of all materials. But, here, when you're talking about storing thermal energy, we use the term phase change materials. OK. Now the last concept, and I'll just touch on, and then spend a little time on Friday on, is the concept of a map. These are maps. These are maps. But you can go even bigger. And back in the day, back in the day, PVT was a big deal. You went to a party, you were going to talk about PVT. You were. And you know, and so, everybody had their own way of looking at it. Gay-Lussac, P versus T. But look, if I do P versus T, and hold volume constant, right, I might get like a line there, like P-- if this is pressure, all right, and this is temperature, you might get something like P versus T is a straight line, if you hold other stuff constant. Right. That's good. But that's not what I'm interested in today. I'm interested in where the phase boundaries are. Right. This is a materials map, and it gives me the boundary, that's what this is all about. That's what this was all about. Because, look, that's why I went through all this. You boil. What is boil? Phase change. Phase change. You boil when Pv is greater than P at [INAUDIBLE].. So that is the phase boundary. It's when you have both the gas and the liquid coexisting. Where is it? Ah, there it is, because I can get my pressure of vaporization, if I only know this, or vice versa. Right. And then I get that. It's up here. That's the phase boundary. Oh, yeah, that's all cool, but this is where it's at. I want to know where stuff changes phase. All right. And, of course, you can go further. And I've got to show you something here. There's a couple more things about-- these are called phase diagrams, right. These are maps of materials, and they are beautiful things, and incredibly important. And there's a couple last points, one of which I want to share now, and then the other I'll share on Friday. But the one I want to share now is the triple point. So look at this. Here's the gas liquid boundary line. That's what we've been talking about. You also got the solid liquid boundary line. That's here. You got gas liquid here, delta H- here's delta H. You also have sublimation. So you go straight from a solid to a gas, and you have deposition. You go straight from a gas-- but look at this. There's one point where they all co-exist. One, and only one where all three phases exist at the same time. That's pressure. That's temperature. And what's so cool about this is there's only one for this material, and so it, actually, is a really nice way to have a repeatable, reproducible point, which I'll get to in a second. But first you got to see this. This is a triple point. And what they're doing is they're lowering the pressure and the temperature. They're changing both. You could just do this with one or the other. Yeah. There it is. OK. How did they get it to boil? Lower the pressure. You now know why. You lower the atmosphere inside the container. You lowered that, so you went like this. You went down. And all of a sudden, you converted some of the liquid to gas, because you went down in pressure. And then you said, well, OK, but wait a second, I'm only seeing these two phases. So let me do this, and go over that way, getting closer, closer, OK, what's happening now? Then do this, right? And you're trying to find this special point here. And they're going close to it. They're going close to it. And then a couple minutes go by, that, look, it's freezing. It's freezing. It's-- but wait a second, I thought I said triple point. And I did. It's boiling and freezing at the same time. It is boiling and freezing at the same time. Yes, that is actually what all materials can do. That is incredible. Somebody just said what. Thank you for making my day. That makes my day. That is a triple point. We got a one or two more things to talk about phase diagrams. We'll do that on Friday. Have a great rest of your day. This is where we left off. And so, I want to start right here. And because a triple point is just so incredibly cool, I've found another random video, and I thought I'd show that. So this is where we left off. And here it is. And what they're doing, they might be controlling the temperature, they're definitely controlling the pressure. All right. They have a liquid in here. Now watch what happens. We know now from the phase diagram that you can find those boundaries on the diagram. So they're lowering the pressure, and at some point, yeah, nothing's going happen. There. OK. Now it's playing. And so at some point, this is going to start to freeze, because I'm crossing over the boundary. All right. I'm crossing over the boundary. And so when you cross over, when you get to a phase boundary, you've got two phases coexisting. Oh, and there it starts freezing, so you've got the liquid and the solid phase coexisting, but there's-- but now can it coexist with, also, the gas. And at one point of pressure and temperature, it can. There it is. It's kind of boiled a little bit. I like this one, because it kind of goes big boil. So here it is. It's trying to freeze, and now it's boiling. And now, watch, it'll refreeze. All right. And then it's going to boil again. It's all three phases at exactly the same time. So you can Google it. There it is. It boiled, again. And then it's going to freeze, again. It can do all three at once. That is cool. All right. And how do we get there? How do we get there? Well, what I thought-- on Wednesday was the first time I told you there's two new lectures this fall. That was the first time I gave that one. That was one of them. The other one will be on the chemistry of batteries. And because it's new, I, you know, I want to make sure that you feel your oneness with the concepts that we learned. And so I have made some summary slides for you, that I'm not going to take time to go through. But that tells you how we got to here. That's the end of the lecture on Wednesday. And I want to make sure that how we got there, and what the key points were. There are five key points. And I've created a slide here, again, this is Wednesday, so I'm not going to go through this in detail, but I'll leave this in your notes. One point was boiling and vapor pressure, and what that means. The second point is the vapor pressure versus temperature, which we get from kinetic theory and Clausius-Clapeyron gives us the relationship between that vapor pressure and the temperature through this thing called the vaporization. The thermal-- this, remember, this change in the thermal energy. Right, this delta H, this change in enthalpy. OK. And by the way, you can-- these are these like vapor pressure versus temperature curves we drew. You can take two of them, for example, that gets rid of the constant in that equation, two pressures, two vapor pressures, two temperatures. You can calculate that vaporization energy. That's that interrelationship we talked about. The third point is the thermal energy that delta H is related to the phase change. It is the energy of the phase change. Right. So we talked about that. And then, I couldn't find a good-- so I just drew this myself here with fancy PowerPoint-- and what you can see here is a heating curve. And you can draw it in two different ways. You can draw it as the temperature versus enthalpy, or you could draw it as the energy input versus temperature. These are equivalent. It's just not mm, and mm. Right. And then, in this, you can find all so much information, like there's that delta H between the phases, right. Here is the liquid phase, the gas, the solid. And then, if you're in a single phase, we talked about that change in enthalpy related to the heat capacity. If you're at the phase change, the temperature doesn't change, but you're putting a lot of energy into it. And that, finally, got us to phase diagrams. This is where we left off. So I'm leaving this all here for you to look at if you want. It's kind of a summary of Wednesday's lecture, the key points. This is that two-phase curve. Right. And this is-- on these curves, remember, the phase diagram is a map. It's this beautiful map of the material. Right. It's a map of the chemistry of the thermodynamics of the phases of whether it's a gas, liquid, or solid. And these coexistence curves are where two phases coexist. Right. So you see at these temperatures and pressures, and that's why we started the lecture talking about vapor pressure, because the vapor pressure is such a nice way to get into the idea of how a material changes phase from liquid to gas, which is what we did. And then, we also talked about how that triple point, that's the video that we left off with. And then there's one last thing I didn't mention, which is that there's this critical point here. What's happening out there? What's happening out there is really interesting, too. There is a temperature, right, above which-- so there is-- if you keep going up in temperature, there is a critical point where this boundary just disappears. It stops. The liquid, there's the liquid, and the gas, and now there's no line. What happened? What happened is, you've reached a temperature where it, basically, can't exist as a liquid anymore. If you keep going, the thermal energy is just too high. But the pressure is too high, too. So this is a different phase. And this phase it out here is called supercritical. This is called supercritical, where the material has properties of both. That's really cool. Right. So this material has maybe properties like it can flow like a gas, all right, but it's got a higher density than a gas, but it's still lighter than a liquid. And it's that same material. It's that same material, whatever it is, water. Right. That's called supercritical. So these lines can end like that. And that's called the critical point. Now, there, you know, there are many reasons why these maps are important. I thought I'd give you one which is strawberries. Because, if you go and you buy-- some of us like to buy like dried fruit-- but if I take strawberries and I leave them out on the counter, they're not going to look like this. You know, how do you know that? Well, you know that from the phase diagram, because if I draw my temperature and pressure, and I've got some phase diagram that looks like this, right. And so now, you know solid, liquid, gas. Well, you know, I could-- if I just-- if I just tried to dry these things out, that's this. That's that. But so, I'm-- maybe, maybe, I'm going to, or maybe I want to do this. Oh, that's even worse. What if I go like this. I'm going to heat the strawberries up. I'm making jam. No. That's not jam. Right. That's nice kind of maybe still solid, it's not like mushy and gooey. How do I get it? Well, you use the phase diagram, of course. You go-- because this kind of phase change, going across that phase boundary, it creates a lot of disruption in the material. It creates a lot of disruption. You saw it, right, this boiling. So if you do that for the water inside of the strawberries, you're going to destroy all the framework of the good stuff, right. So what you do is you freeze it. You go this way, and then you go that way. If you do that, now the solid as it sublimates is much less of a kind of violent, disturbing phenomenon to the surrounding matrix of strawberry. And so you get these freeze dried fruit that way. It's much less disturbing, destructive way to get rid of the water. That's called freeze dried. You can also go the other-- if I go out here, I've got all sorts of uses to use this supercritical region. I've got all sorts of ways where that becomes useful. That's how coffee is decaffeinated, because it's a different phase of this material. So the phase maps give us this information. They tell us how we can engineer properties and materials by engineering their phases. Right. And if you pull out another thing, you know-- a lot of the textbooks and what we did is we look at the phase diagram down here. But if you pull out, you also can see how complex materials are. All right. This is water. We've been focusing on this part of phase diagram, now you plot it. Water has 17 phases, 17, not 3. And just two years ago, the 17th was discovered, and it's a very, very light form of ice. All right. So you can get all sorts of really important information about materials by looking at their phase diagrams. You can compare different materials, like this is the phase diagram of CO2 versus water. And you can learn about sort of where their critical points are. You can see all the slope here in water is-- most material slope like that. Water slopes like that. And that's because of hydrogen bonds. Because the hydrogen bonds make it so that the solid is less dense than the liquid, which means, if I squeeze on it, I can-- then I go to the liquid. That's really weird. If I squeeze on this, it's more soft. But here, it goes-- it can cross a phase boundary. That's very unique to water, and the fish are very happy about that, because ice floats. OK. Now, OK, now, so that's the kind of last part of the phase diagram stuff. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 20_Xray_Emission_and_Absorption_Intro_to_SolidState_Chemistry.txt | How's everyone doing? [APPLAUSE] Parents only, how are you doing? [APPLAUSE] How is it that the parents were louder than the kids? Kids, how are you guys doing? [APPLAUSE] All right. Welcome, parents. Welcome, parents. I'm really excited to have you here in class. This is 3091, Introduction to Solid State Chemistry. And today we're going to have a little bit of fun. As you can see, we got some activities. Now, the way I like to frame this-- are there any other seats? No, not so much. But make yourself comfortable. The way I like to frame this is you've got a lot of activities going on already. This is the parent weekend. So if we look at-- why is that happening? If we look at the-- oh, that's not good. OK, there we go. If we look at the MIT family weekend, there is activities. And so what I thought is that what we could do in this class-- we've got to learn a little bit. We're going to learn about x-rays today. But what I thought we could do is also add some ideas for activities that you all could do with your kids. But it doesn't even matter. It could be with any kids. But just more activities to do to give you more options. And so in particular, I've got five suggestions for you that we'll talk about today. The first one is staying warm. It's gotten really cold, so we're going to talk about staying warm. And then the next thing we'll do is we'll appreciate the fall and all the colors. And we'll cook. Cooking is such a wonderful thing to do. I hope you guys can do it together. We'll talk about that. And we'll talk about phones and how they crack their screens. And we have some sort of larger versions here that we'll talk about. And of course, it wouldn't be complete if we didn't blow bubbles, because that's a really fun activity no matter what age you are. So that's our plan. That's our plan today. Now let's get started. So let's start with the first activity, and it's staying warm. So I love candles, and I think a lot of us have seen candles. The difference is between everyone else's and your kids who are in this class who see candles is they don't see candles. They see C25 H52 combusting. And your kids now, when they see a candle-- if you guys go out to dinner and there's a candle on the table, the first thing-- I guarantee you the first thing they're going to do is assess the volume of the room, figure out how much oxygen in grams there is in that room, and they're going to decide which one is the limiting reagent, the candle or you. And they'll make a choice. They may get you out of the room because you don't know. You've got to know that. And of course, they'll have their periodic tables with them at the dinner. Don't be surprised when they take it out. Now, your students-- oh, yeah, that's perfect. Your students also know-- your kids, my students, your kids-- also know that a lot of the earliest chemistry, going all the way back to the times of Democritus thousands of years ago, the way it started was because people were breaking things and smashing things and lighting stuff on fire. And we've got the whole periodic table in this class, but we still know that the most fun element of all to light on fire is the very first. It's hydrogen. And so I thought I would show you what that's like. OK. So here's hydrogen. And like I said, this is also-- it's a way to think about-- oh, that's a beautiful thing. Is that a beautiful thing? I've got to do it again. And as your kids know, I can do this a lot. But it's just-- and my hand doesn't-- I don't feel it because of the heat capacity. But that's another class. But what we're interested in is combustion. So there's that one there, too. So this is all about the context of staying warm, of course. You don't need to light hydrogen on fire to stay warm. But I also have suggested discussion topics. Because I don't want you just to do these activities. I want you to be engaged. It might have been a while since you guys have connected-- I don't know-- in person. And so here, maybe you don't know what to say at first. And so for example, you could talk about balancing chemical reactions. You could talk about the concept of a mole. And you could also talk about limiting reagents, which are really, really important in life. So that's my first suggested activity. Like I said, we'll have five new activities. That's the first one. But like I also said, we've got to learn something today. And students, this is not on the exam, as a reminder. But we do need to learn what X-rays are. Because next week, we're going to be shining them on stuff. So today, we're going to learn about how X-rays are generated. And it all starts with the very first selfie. This is it. Does anybody know who that is? That is Anna Rontgen. That is the wife of Wilhelm Ronten. That's the very first X-ray ever taken. That's the very first X-ray ever taken. And Wilhelm Rontgen was playing around with cathode ray tubes. And so he-- I love this quote of him. He said, I did not think. I investigated. And that's-- now, we think. Of course we think. But what we do in this class is we investigate. We're explorers. Were explorers of chemistry and how chemistry connects to what we see in life and what we're doing every day. And so how did Rontgen do this? OK. Well, literally, it was a dark and stormy night, literally. And he was in the basement in his lab. That's him. And it's going to be stormy tomorrow, so you guys could even try to think about how it felt. And so what he did is he-- everybody was playing with these cathode ray tubes. We learned a lot about cathode ray tubes. So here, you put some kind of cathode here. And then there is an anode here, and you hook it up to a voltage. And what happens is electrons fly. This is a cathode ray tube, so it's the CRT. Electrons will fly off of the cathode and go here. Now, if they go through the anode, and then you put a screen here, those are the very first cathode ray tube experiments. And we talked about how this led to the discovery of the electron itself. But what Rontgen did is he said, now, hold on. What if I changed-- I put a voltage here. What if I amp that up? What if I just keep going? Oh, by the way, there is a vacuum inside of here. There has to be a vacuum. And if I am the voltage up-- you know what? Actually, I'm going to put a metal here for the anode. I'm going to make that some kind of metal, and I'm going to hit the metal with the electrons. Now, this is exactly what happened. What happened is the room glowed. The room glowed. So Rontgen had-- showing you there, the lights are off. The lights are off, but the room was lit. And he said, oh my goodness. What is going on here? What is going on? And what he decided-- he literally discovered a new type of ray, a new type of light ray. And because he didn't know what to call it-- he didn't have a name on that dark and stormy night-- he wrote down the letter x, which he regretted later. He really did. Because everybody started calling it-- he didn't know what to call it, so he called it an X-ray, which is what happens when you do this experiment with a very high voltage. You get rays that came off of this and illuminated the room. And he said, I don't know what to call it. I'll put the letter x for now. And that stuck. Now, these X-rays are unique. They're unique because they have very high energy. They have very high energy. So the X-ray energies-- the energies are in the range of 100 eV to 100 KeV. That's a big energy. But we know how to go back and forth between energy and wavelengths. So we know that that means that the wavelength range is between 0.01 and 10 nanometers. And that's what's really important. You came up with a source that has wavelengths in this very, very special regime. This is a very special regime because it's the distance between atoms. Now, there's two types of x-rays. But first, let's-- at the time, you might think, people-- why wasn't he wearing anything, like at the dentist? Why wasn't he wearing anything? They didn't know that radiation could be dangerous. And so if you look at the time, there were ads like this for a system to remove the hair. And it was advertised as effective, safe, and painless. This is-- those are X-rays, one of the first uses of X-rays. This is a token to get into a bath that's called a radium bath. It's a water bath where you have radium that you either drink or is in the water itself. And look at what was happening in Paris. Radium cure, a fad of Paris society. Patients play bridge and take tea for two hours daily-- daily-- in a drawing room. This is what it says. December 9th. The afternoon radium cue is the latest craze in Paris society. The popularity of the treatment, new to Paris, has devolved quite suddenly and is due, no doubt, to the fact that it is exceedingly pleasant. Oh, France. Right? They didn't know. They didn't know. And it was the same with these X-rays. What is this thing? Well, it turned out it was a very, very important type of light. And because of its importance, Ronten won the first ever Nobel Prize given in physics. There is an element, number 111, named Roentgenium. And 60 years later, the DNA was discovered-- the structure of the DNA, the double helix was discovered because of X-rays. So the impact of X-rays is tremendous. Now, the next activity, as I mentioned, has to do with enjoying the fall. Now, the thing is that New England has beautiful colors. And so if you haven't taken a stroll around the parks, or maybe you go up to a nice hike-- you'll see these beautiful colors changing in the leaves. But the thing is that in our class, your kids, they don't see that anymore. No. They don't. Because when they see colors, they think about electron levels. That's what they think about. They can't help it. They cannot help it. So when they see orange or red, all they're thinking about is, oh, my goodness, there is an electron level here, and an electron getting excited, and it's coming back down and emitting a photon, and how those levels are changing as you go from one leaf to another, and how chemistry is at the heart of it all. Chemistry is at the heart of it all. And so that's what they're thinking of. And we can use chemistry. I don't know if we have the next one ready here. We can use chemistry not only to change the levels of electrons that are in this material, which then changes the color of light that comes off of it-- it's all about the electrons-- but we can use chemistry to control how quickly that happens. Nature does it on this yearly time scale. We can do it on any time scale because of chemical kinetics. So we're going to get that experiment out here in a second. Meanwhile-- oh, there it is. Can we give the helpers a huge hand? Because they are-- thank you, guys. [APPLAUSE] All right. Now what do I do? And that's it? This is pre-mixed? Oh. OK. Now, this is an example of how we can change-- we can control the color of a material using chemistry. We can control when-- do I pour it all? Do I pour it all? OK. Why not? Yeah. OK. Now I want you to focus on the larger bottle. Nothing is happening. Keep watching it. You've got to keep watching it. See? That's an example. But your students, your kids are seeing this and they're saying, no, it's all about electron levels. I've got to go and write down those electron levels. I don't care if it's a Bohr model or a semiconductor bandgap. I've got to figure out how that happened. How did the color change? I changed the electron levels in the liquid. That's what I did. And I controlled the kinetics. So my suggested topics for this one of discussion-- chemical kinetics. Now, we're going to talk about chemical kinetics later in the fall, but you guys can get a head start this weekend. Because that's what happened there. We controlled when it-- it didn't happen right away. It happened later. Why? Acids and bases, also coming later, get a head start. And then how to paint with electrons, because that is what we're doing. We're changing how electrons interact with the material to change the color. We are painting. And that's what nature is doing with its leaves. Now, you might be on this walk, and you might be strolling around, and you might not go into the countryside. You might just be down Mass Ave. Now, if you're walking down Mass Ave from here, and you go not towards Boston but the other way, you get to another school. Now, I've got to-- I have talked to your kids about this already. What I want to do right now is address the parents. You have activities you're going to, right? And some of them may involve-- I don't know, a dinner or a party, or a lot of other parents. Some of those parents may have kids that go to the other school. Now, here's what I want to tell you. It might not be their fault. It might not be their fault. I'm just saying we do the best we can as parents. We do. We do everything we can. We did everything right. And then you realize sometimes, kids have to make their own mistakes. So I just-- and the last thing I'll say about this-- look, be nice, because they're probably going through a really hard time. So that's what I want to say about the stroll. You might wind up down there. I don't know. OK, back to our X-rays. Where were we? So x-rays are coming off of this metal. It's super high voltage. It's a a dark and stormy night. What happened next is that we had to figure out what were the properties of this light. And what we really mean are things like intensity and wavelength. Now, I wrote down ranges there. But basically, what you have with X-rays are two types. You have type 1, which is called continuous. And it arises continuous-- continuous. And it actually-- this is a pretty good name, because it's a continuous energy up to a point. It's a continuous spectrum. So what do I mean by that? Well, what I mean by that-- to understand that, we have to know how we get it. There are two types of X-rays. In this first type, what we have is we have some atoms with a bunch of protons in there. This is like that metal atom. And I've shot really high-energy electrons at it. But that atom has electrons sitting around it. We know that. 1, 2, 3, 4, 5, 6, 7, 8. OK, and then again. And these are the quantum levels. This would be n equals 3, n equals 2. And in here, we've got n equals 1. But now, here's the thing. I'm shooting high-energy electrons at this atom. So one of the first things that can happen is nothing at all. So if I shoot it, say, up here, way away-- if it's all the way up here going way away from the atom and doesn't feel it, then it might just come through. It might have some energy-- call it e0-- and it just came all the way through. It didn't interact. Well, that's kind of boring. What if I interacted-- what if I'm out here, and I interact-- I got through some of those outer electrons. Immediately, what do I see? If I'm a negative charge, I see this positive charge, and I'm like, ah, that looks good. It's an attraction. And so the electron curves like that. Now, that means that it accelerated. This is not a electron stuck in an orbit. No. This one was free, and it remained free. But because it got inside of the electron cloud of the atom, it saw a a positive charge. And that bends it, and that accelerates it. And when something that's charged accelerates, it gives off a photon. It gives off a photon. So that might be some kind of energy that it has left over that we call e1. And this would be like the photon 1. And we know that it's lost energy to the photon. It's decelerated. It's decelerated. So we know that e0 is greater than e1. But I could have shot another one through, and this one got even luckier. These are my electrons, incoming electrons. This one got even closer to the positive charge, felt an even stronger attraction, and decelerated even more. And so it also emits a photon. And so we know that e2 must be smaller than e1. It's left with less energy. Now, what you're doing is you're decelerating electrons by having them pass through the cloud of charge of an atom. And that deceleration emits photons. And you can see right away what I'm doing is I'm breaking the electrons. Well, that's exactly what this kind of radiation is called. This radiation is, in fact, called Bremsstrahlung. And that literally means braking radiation. Braking radiation. It means breaking. Well, if I take this very simple picture, and I try to make a plot-- like I said, if I want to understand my light, I've got to know about how it behaves as a function of, say, wavelength or frequency. And so if you plotted the wavelength with the intensity, well, you can see right here I'm going to get-- OK, so I'm going to get some photon that comes out that has some lambda, and then another photon that comes out that has another lambda. But you can see that that electron could have been braked at any loss of energy. And so what you get is something that is very continuous up to a point. This is the shape that it looks like. So I can get-- this is lambda. So if lambda is out there, this would be like lower energy photons. And up here, it's higher, higher-energy photons. Now, these are all x-rays. They're in this very high-energy range. But you've got this continuous spectrum. Because the electron could have been slowed down in any way except higher than this point. Why? Well, that's just determined by how much energy that initial electron got. Because you can imagine this point is a special point. This represents a fully stopped electron, fully braked, fully stopped electron. So that incoming electron has transferred all of its energy to the photon. And because this is higher energy, that's as high as it gets. That makes sense, right? OK. We have a name for that too. That's the Duane Hunt limit. And basically, it says that the energy of-- the maximum energy, e max of the photon that can come out is equal to the electron times the voltage, the charge of the electron times the voltage across, ev across the tube, and that that is equal to hc over lambda min. This is just saying max energy equals hc over min wavelength, energy wavelength. So that's one type. And if I had-- but now you can see if I shot it, now I'm bumping up the voltage even more, and I'm shooting it with higher-energy electrons. Well, you can imagine that if the electrons I shoot in have a higher energy, then that point is going to go in and in like that. So this would be Bremsstrahlung radiation, and this would be increasing incident-- inc? That's not good. Incident electron energy. And I will put a box around this so that it's increasing, that arrow going like that. So if I shoot it with higher and higher-energy electrons, I'm still going to have a minimum lambda, but it's going to get smaller and smaller. Continuous x-rays. OK. That's type 1. Now, speaking of breaking-- now, speaking of breaking-- OK. Now, here is my third activity, and it's cooking. And I love cooking, and I hope you do too. It's a really fun thing to do. And I think that I love cooking with cast iron. So maybe you've got a piece of cast iron in your dorm that you can pull out, and pull out a recipe, and start making something. But what I want you to think about-- and this is the thing-- is your kids, our students don't see it as cast iron anymore. They see it as iron with interstitially doped carbon. That's what they see it as. And so I want you to see it that way too. And then that gets you thinking. How is this cast iron so strong? And oh, wait, the early chemists-- I heard they smashed and they lit things on fire. How could we break this cast iron? Could it break? Could it break? Should we be worried about it? And so what I think you should do-- oh, yeah, there it is. Let's take this. You take the-- I don't have a pan, but I have this. Is this filled? OK. And you take out a hammer because you have a hammer in the closet. And you go like this. And nothing happens. And then you do it again. You say, well, you know what? Maybe we need to go a little bigger. And you go into your closet, and you find a sledgehammer. And you say, you know what? I think if we just use this handy sledgehammer-- hopefully not in your dorm-- then maybe we'll break it. And so you keep hitting-- not over the electrical. And you hit it again. And I won't use-- I won't go extreme here, but I will tell you that I've done it before. And nothing happens. Nothing happens. There's not even a scratch on it. But these are your kids who go to MIT, not another school. And they know-- and they know that intermolecular forces have power. They've got a lot of power. And they know about bonding. And they know about the hydrogen bond, and the power of the hydrogen bond, and how strong it is, and why it is. And then they start thinking about water. I wonder if we could leverage the power of the hydrogen bond. Now, do we have liquid nitrogen? We do. OK. So they put it together that we might be able to leverage that power of the hydrogen. They knew. They saw the gecko. They say, the London dispersion lets a gecko climb a wall. Well, then hydrogen bonds might be what it takes to break the cast iron. All that you need is to pull the liquid nitrogen out of your dorm closet, and-- yeah, OK. You want to go ahead? Yeah. And fill it with water because you need to take advantage of the power of the hydrogen bond. And then let it sit, of course with the blast shield that you also have handy. OK. Well, let's see how this goes. And we might take a-- we'll see what happens. Again, this is a bonding moment. This is a hydrogen bonding moment. But this is where the two of you-- the three of you, the four of you-- it could be a huge group. This is where you talk about things like the crystal structure of iron. This is where you talk about things like intermolecular forces. And we've learned them all. We learned about them all in this class. Where you talk about the power bonds and the power of the bond between you as a family. This is it. Just do it on this side of the blast shield. OK. Now we'll let that sit. We'll see how that goes. And OK, yeah. That's good. Now, where were we? Oh, right. Oh, there's another type of X-ray. There's another type of X-ray. And the other type of X-ray-- that's the only board left. The other type of X-ray is from a different phenomenon that also has to do with-- it also has to do with the atom and the electron. Because the one type is because the electrons are coming through this cloud, and they get a little attracted, and they lose energy, and they give that off to an X-ray photon. But see, the other kind comes from a direct excitation, the same kind that we learned about when we looked at the Bohr atom. And so this is drawn here. Now, there's a very important thing that happens. As we've learned in this class, when you go from one area of expertise to another-- like crystallographers don't like commas. They don't like commas. Don't upset a crystallographer. It makes no sense. It's not worth it. This is the same thing with X-ray specialists. Don't use n equals 1, n equals 2, n equals 3. Use KLM, because they prefer that. So the KLMs there are simply the quantum numbers, the principle quantum numbers. So 1 is k, 2 is l, and 3 is m. And all we're doing is the same kind of excitations as drawn there very well. So you see the incident electron comes in, and now it's got enough energy to excite one of those core electrons. It's got enough energy to excite it. And so what it does is it trades energy with that core electron. The core electron goes up, and then it comes back down. Now, if we had, say, an m level-- so this is n equals 1, 2, 3. So if we had-- now I can draw it. If I had some electron in my k shell, my n equals 1 shell, and that electron got bumped up to an l shell, well, now you know-- from our work with the Bohr atom, you know that when this goes back down, it's going to emit a photon. But this is very different now. This is very different. This photon has only that energy, nothing in between. It's not continuous. I'm getting used to wearing this. It feels good. It feels pretty good. So I could have had an electron-- instead of going to the l level, maybe it went all the way up to the m level. And then from there, it cascades back down, and it emits a photon. And the way this works in terms of labeling is because it goes down to the k level, these are k photons. And we go in order. So this is the least energy k photon. It ended in k. So that's a k alpha or k-- I'll just write it out to be clear-- k beta. Oh, I hear sizzling. Oh, that's because you're pouring. That's good. More liquid nitrogen is never a bad thing. That's great. No, no, I'm good. You guys are doing such a good-- now, OK. So if it's k beta-- because it went to k, it ended at k. It gave off a photon. The photon is either alpha, beta, gamma, depending on which one. But now here's the thing. You know if you go back to that plot of energy-- of lambda versus intensity, well, you know that there's nothing in between. There are no electronic states in here. So you can't excite in between. This is forbidden. It's forbidden. So unlike the braking kind of radiation, in this case, it's very, very discrete. And so now, if we plot what you get from this, well, you're going to get a k alpha peak that's very, very discrete, and you're going to get a k beta peak that's only at the one energy that is the difference between these levels, k alpha k beta. So that's another kind. This is a type 2. This is second type of X-ray. OK. It sounded interesting. We'll let it keep going. We'll let it keep going. It's good because it gives you a time. And I hope-- you guys can just leave it. I hope that during this time that you're here doing these activities, don't have your phones. Put them down. Listen to the sounds of cast iron popping. You can't appreciate it if you're on your phone tweeting it or DMing it. You can't do that. It's not going to happen. It won't be as much of a bond. Did I get it right? No? OK. So we got our second type of X-ray, and it's discreet. And we're going to put them together in a minute. But first, we've got to talk about glass. Now-- oh, more. OK. Good. Good. Keep going. Oh, glass. That's perfect. The thing is that-- well, a lot of times this happens to your phone. And we know about crystals in this glass. And this is the one-- if you were willing to pay hundreds of dollars more for your iWatch, you can get a quartz screen. This is quartz. This is what we know now. We don't look at the Apple ad. We look at this in this class. I'm good. We're good. Yeah, thank you. And so we can hand this out because we like to touch and feel the chemistry in this class. That is quartz. That is quartz. That's not glass, by the way. And that's another thing we're going to learn. Glass means amorphous. It means not crystalline. It means not crystalline. That's the thing on the right. Now, quartz has a lot of really great properties. It's a very strong material. But it's a lot more expensive, which is why this happens. And you've got all this glass, this amorphous SIO2 that is-- oh, that was a good one. Yeah, I see water coming out. But there are ways to try to make that stronger, to try to make the amorphous SIO2 stronger. And there's no better way to think about-- and I don't want you to break your phone. But if you think-- speaking of phones-- if you think about glass, you're probably thinking about baseball. And if you're thinking about baseball, you're thinking about the Red Sox, of course. Right? Am I right? And so I brought glass, and I want to think about how glass breaks. That's what we're going to study in a couple of weeks in this class. But I thought let's bring this to life. But I need some volunteers, and I'd like some parents to come down. I need some baseball arms, some people who can throw this at the glass. It's OK. Come on. I need some volunteers. Come on down. There you go. You don't look quite like a parent, but it's OK. All right. OK. Now, OK. We need to get-- let's get you a little further back. And we're going to have you put this on. This is for aim. And what I want you to try to do is throw this as hard as you can at that piece of glass on the left. Can you do that? All right. Really let it go. Now, hold on. Hold on. Let's move you right over here. The one on the left right there as hard as you can. OK, that's all right. Try it again. We get multiple tries here. We'll take turns. We'll take turns. OK. Try again. OK. That was good. I think we need to try a little harder. One more time, and then we'll get-- what parent? That's the annealed one, isn't it? Last year it didn't break, either. Thank you very much. Who's next? I need a volunteer. Come on down. Oh, you look too happy not to try. You look too happy not to try. Can you hand those to this gentleman right here? All right. Come on down. Now, we're going after that piece of glass on the left. And we're going to see if it's a good piece of glass for your phone. OK? All right. Let's let it rip. Oh, yeah. That's all right. Try again. Try again. Multiple tries. You've got this. Oh. That's very strong glass. Try one more time. Really? OK. Let's try the one on the right there. Let's see if that one breaks. We might have them reversed. We don't have them reversed? No? All right. Let's try that one. OK. All right. Thank you very much. [APPLAUSE] Now, does anyone else want to try? Apparently we have very unbreakable glass. Maybe we've invented something new here at MIT. The sledgehammer? Oh, that's a great idea. No. But there's a reason for it. Because, see, what we're going to learn is that baseballs aren't the only way-- even though this is Boston, the Red Sox-- it's not the only way to break glass. Now, I don't know what's going on with this one. This is tempered. That means that it's been treated. It's been treated so that it's got super strong mechanical strength this way. You saw that it didn't break. Of course, neither did that. But when a glass is tempered, as we'll realize, it has to do with putting certain chemistry inside the glass, ions, that are bigger, that causes a strain in the material, that tighten the surface. And so there's all this stress built up, and it's all at the very edge. So this part is super, super strong. But this part, just a simple little tap-- I didn't even-- I barely even touched it. And so that is-- now, so the moral of that story is glass might be stronger than you think, I guess, in this one. And also, don't tap the edges of glass, especially if it's tempered. Because that's what makes it so tough on this on the surface. But the strain builds up. And if you relieve it with just a tiny, little tap, boom. Now, where were we? Discussion topics-- crystal versus glass, ordered versus disordered, substitutional dopants. That's how to temper glass. That's what's in your iPhone glass. And of course, the Red Sox, which are on tonight. And you should watch game three. Now, the thing about these peaks is that these are called characteristic. And what's nice about them is, in fact, that no matter what, they only depend-- I don't care how much energy I send in. These peaks only depend on the metal atom. So Ronten put in a metal, and he'd always get the same peaks if I give it enough energy. Maybe I don't have enough energy to excite anything from down here. Then you don't get these peaks. You still get Bremsstrahlung, but you don't get the characteristic peaks. They're called characteristic because they're characteristic of the metal. Now they're dependent on the chemistry. If I put in different metal in, I'm going to get different Peaks And so with these two types of X-rays, I've got a new kind of light that I can shine on materials. I've got a new kind of light that I can shine. And the light-- when I plot it all together, it might look something like this where maybe these are the peaks-- uh oh. OK. Maybe these are the peaks due to going down to K, and these are the ones due to going down to L. So those are K, and these are L. and the rest is continuous, and they're on top of each other. And I get it all. But with that new kind of light, I can do new things. And that's very important. And that's what we're going to do after the exam, which is on Monday. The exam. Don't forget about the exam. You're doing all these activities. You've got to think about the exam. Why is this important? This is important because we will be shining x-rays as a source of light onto crystals. And there's a very important thing that happens when you shine light onto something. And you can see it here. Light is a wave. Light is a particle. Light is a wave. Light is a particle. Have that discussion too. But here we're looking at it as a wave, and waves diffract. That means that they can add or cancel each other, right like water waves. But it depends on the size of the thing they're interacting with. If the size of the thing they're interacting with is the same as the wavelength of light, then you can get a diffraction pattern. And so when we shine X-rays, which have wavelengths that are the same as the spacing between planes in a crystal, well, then it interacts. And it can diffract. And that allows us to determine the structure of the crystal. That's called X-ray diffraction. And this is how it all started. It started with the Bragg father and son pair, who actually said in their paper the crystal is, in fact, acting as a diffraction grating. And from that, you can literally tell us which kind of crystal it is. I want to point something out here. That's a father and son. They both won the Nobel Prize together for this discovery. Now, I'm not putting pressure on you guys. I'm just saying it's not all about the kids. The parents got to step up too. So that's Bragg defraction. And that is one of the most-- X-ray diffraction-- what we're going to do with these powerful sources of light, this is one of the most important ways to characterize crystals that we have. And just as a reminder, I mentioned the exam. Don't forget-- this is what we're talking about. These planes-- notice the negative. Remember the 1, 1 bar? Don't forget about negative directions. These are the things that we're diffracting off of, literally the crystal planes. So just a reminder not to forget about that. But I can't end by talking about exam one, so I have to end with something happy. And bubbles always make us happy, always. I love blowing bubbles. I always will. I always have. But we don't just blow bubbles in this class. We don't just blow bubbles. We really blow bubbles. And so what we're going to do is we're going to show you how we like to blow bubbles. You want to do this? You go ahead. Now, what we like to do is we like to really blow bubbles in this class. And so we're going to use exothermic reactions to blow bubbles. And I think that's really a good way-- you go ahead. You go ahead. You're the one. All right. That's the only way to blow bubbles. That's combining an exothermic reaction with bubbles. That's happiness right there in a bottle. That is happiness. That is happiness. [APPLAUSE] Oh, yes, and look at this-- success. Success. The power of the hydrogen bond. OK. I hope you guys have a fantastic weekend, and I hope you enjoy yourselves and follow some of these discussions. [APPLAUSE] |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 18_Introduction_to_Crystallography_Intro_to_SolidState_Chemistry.txt | Big picture-- I showed you this big picture before. We're now going to add to it. Look at what we've done. Ha! Electronic structure all the way up through [INAUDIBLE],, [? Louis, ?] molecular orbital theory, how to handle multi-electrons, all these different kinds of bonding-- check. We've covered every single one of these, and this is solid-state chemistry. OK, but now here we are, right? Now OK, remember we also talked about this on the very first lecture. When you classify solids, there are two things that are really important. One is the bonding type. How do the atoms in the solid want to talk to each other? How do they interact? And that is something that we have covered in a lot of ways. But the next is how they come together, the actual arrangement of those atoms. And that is what we're going to talk about today and on Wednesday. And then after exam two as we learn about X-rays, that's a way to characterize the arrangement so that you can see how they're ordered, OK? We're going to use X-rays and do something called X-ray diffraction. But for this week I want to just classify it. I want to talk about the tools that we use to understand how a solid forms, how to describe the formation of a solid, how the atoms can be described. And OK, so, of course, a solid is that which is dimensionally stable, simple definition, right? But we can break down how these atoms arrange into two basic categories-- atomic arrangement, the order, and what we call it, right? In the first category, it's regular. So what that means is that everywhere I look, I have the same thing repeating. And you know what? It repeats over a long, long way-- like really long, OK? So that kind of regular, long range order is a crystal. That's a crystalline solid. And that is going to be the topic of this week, crystalline solids. Now, there are also solids that you can make where there isn't order. Or maybe there's some but not much. So, things are kind of jumbled up. They're not regularly repeating. They might be random, the atomic arrangement. Or maybe there's repetition, but it's very short-range. So things look ordered, but only for a few bond lengths, not 10 to the 23rd bond lengths, right? Those are called amorphous solids, or also glass. Now, as I mentioned, this is the topic of this week, and this will be covered on exam two. This is the topic after exam two next week, OK? So we'll talk about both of these types of solids as we go. All right, now, OK, it turns out that nature has been thinking about order for a long, long time. And there is a reason for this. Because nature wants to be efficient. This plant is like, well, I'm only going to grow this big, and I got to put as many seeds as I can on the surface of this thing. How do I fit them-- how do I pack them in? Well, I could fit them randomly, but it doesn't seem like I could get as many seeds on there. What if I packed them in in some regular pattern? What if I packed them in in some regular way? Well, you can pack a lot more in. And so nature has already been thinking about symmetry and order and even long-range order for a long time. People have been thinking about this, too. Robert Hooke, the same person from Hooke's law, in the 1600s he was all about cannonballs. And he was like, how can I stack cannonballs? And he tried this, right? He's like, well, maybe if I just put them like this, is that a good way to stack them? I don't know. And then he sort of put them all together. But now, he tried this, and then he tried another arrangement, and he'd be like, well, but how-- first of all, are they stable? Do they fall over easily? Because you don't want cannonballs falling all over the place. But then also, did you pack as much in this way as another way? So he was thinking about it. And he figured out that you could pack more cannonballs in the same area or volume if you sort of switched it up, right? And then he thought, well, maybe other things have packing, too. And he thought about all sorts of things. He looked at salt crystals, and he said, they must also pack in some way, because cannonballs do. So people have been thinking about this for a long time, OK? Now, people are also inspired by regular order, right? So if you look at, for example, an Escher painting, you can see that there is a repetitive pattern here. This is a crystal of ducks, right? OK, so now, when you look at patterns like this, you have to-- well, you don't have to, but we have to in this class. We've got to figure out what is the repeating unit. What is it that is repeating everywhere? So that's the first question we need to ask. And you can see if you look at this picture, you can draw a box like that. There's a box around a light colored, dark colored duck. And now you see if you take this box and you move it over, you translate it over there, you get exactly the same thing. If you take that box and you put it up there, you get exactly the same thing. So that is a repeating unit. Oh, we have a definition. It's a unit cell. It's called a unit cell. And it's the repeating unit in a crystal. It's a repeating unit. Now, it's not always obvious. So, there's one way that I can repeat and take this and make the picture by repeating it everywhere. But check this out-- I could have done that. If you stare at that, you can see that that is also a repeating unit, right? That is also repeating unit. If I take this and I move it down here, I get exactly the same thing. And I move it over there, and I can tile all of space. I can tile all of space. And something is really important here. I can take these repeating units and stamp them. Oh, I like that word. And I can tile all of space and recreate this picture without any voids, right? I didn't miss anything. I filled all of space. I filled all of space. And that is what a crystal system is, right? A crystal system-- let's write it here-- crystal system. Crystal system-- that is a way of enumerating-- enumerates ways that space can be filled, OK? Enumerates ways that space can be filled with no voids. That's a crystal system. OK, well, let's go back to the canons for a minute, because we can think about this first in 3D before we-- we're just warming up before we get to 3D, OK? Oh, there it is. That's a unit cell. OK, now the thing is, so I've drawn a unit cell. And notice what I've done. I've taken this, and I've drawn it this way, OK? And those vectors define the unit cell, right? Those vectors define the unit cell. So those are my stamps, the vectors there. So these things here-- well, I went a little far there, but they should go to the center of the next cannonball. So now these are vectors. If I just take these, and I put them-- now, if I translate them over to there, then I get the next unit cell and the next one. And I go up, and I go over, and I get all the tiling of all space. And so those vectors are really important, and they have a name. Those are called the lattice vectors. Those are the lattice vectors. I really like to think of them as my stamps, or my stamp. It is a stamp. Mm! Mm! Mm! And no voids-- these reactors are my stamp. Now, I'm stamping all of space. I didn't really say what I'm putting there. I mean, here there's cannonballs. In Escher there's ducks. Doesn't matter right now, right? It's a stamp that fills all of space. That is how we're building up our knowledge of crystals. OK, well you can also ask the question, which is, how much could you fill in there? And that's a very important question that we want-- ah-- --that we want to know. OK, mm! OK, now, so I've got my lattice vectors. But I want to know now what am I-- because I could put there-- look at this. I could take a lattice vector like this. This is a square lattice. OK, if it's a square lattice, then those are the same length, right? A is like the length of the-- this is like the length of the lattice vector. Now, if I took this lattice, this stamp, and I say, OK, now I'm going to decide what to put there-- I'm going to put circles. OK, so I'm going to put a circle there, circle there. And what you do then is you put a circle everywhere that you stamp, right? OK, so I have now taken my stamp, and I've stamped everywhere. And I've put something at each place-- a circle. But I might ask a question. And this is a very important question, which is, if I have circles, what is the maximum that they can pack in for this lattice? How much can I pack? Well, you can just visualize that now. Let's grow them. A stamp means it's the same thing everywhere, right? So now if I make one of them bigger, well, I've made them all bigger, right? Because they're all the same everywhere. And if I make them all big enough, eventually they're going to touch like that. That's a special thing, because now I've grown the shape that I'm putting there to the point where it's the maximum packing, right? And so you can imagine this. The area of maximum packing is a very important parameter. And it's something that we'll talk about today with 3D crystals, which is the solids of elements in the periodic table. But if you say with this square analogy-- so my lattice is a square. My stamp is a square. My sides are equivalent. And what I put there is a circle. Then the area of maximum packing is something you can calculate, right? Because the circle itself-- in this case, if I-- [STUDENT SNEEZES] Gezuntheit. If I look at it, then-- oh, that's not a good circle. But anyway, you'll get the point here, which is that-- oh, boy-- which is that if this is the lattice vector is a, like I drew there and I'm looking at the maximum packing, then you know already that the radius of the circle is a half a. So radius equals 1/2 a. If the radius of the circle is 1/2 a-- if the radius of a circle is 1/2 a, then you also know what the area is. I'm in 2D right now. I'm in 2D. So if I've maximized the packing of these circles on the square lattice, then the area of the circle is pi times r squared. So 1/2 a squared equals pi a-squared over 4. But see, I also know how much I packed it in, right? And I'm doing this slowly on purpose, because we'll go faster when we do some of the other ones. But I want to do the first one slow, because then we'll all be able to see what we're talking about with the simplest case. The area of the square is a-squared, right? And so the area of max packing is equal to pi a squared over a-squared-- pi over 4. because the a-squared is cancelled. So it's 78%. This is what Hooke was-- but did I max it out, or could I do something different? Could I pick a different lattice? What if we made cannonballs into rectangles? Bad idea. No, he stuck with spheres, but how do you pack them in? How can you arrange them in a way to change the max packing? That's what nature does with atoms, as we'll see. So for example, if I had a 2D hexagonal lattice, instead of my stamp being a square-- so if it were a 2D hexagonal lattice-- I won't do this one out. But then the max packing goes all the way up to 91%. That's a lot more per area-- max packing. So this is something that we care a lot about. It's something that we care about because, when atoms see each other in a solid, what's the first thing they want to determine? The first thing they want to do is they want to go back in here and they want to be like, which one of those can we do? Can we find some way to bond? That's what atoms do. That's what everyone does. How can we bond together? What kind of relationship are we going to have? But they want to max that out, right? And so they want to try to pack in, given the bond that they're going to have. And that is what's going to dictate the kind of arrangement they can have in the crystal, OK? So OK, so here we are. And we're going from 2D to 3D. Now, the thing is, this is not going to be an in-depth dive into symmetry and space groups and group theory. Those are all great topics. But I'm just going to tell you-- and it's actually kind of cool. In 3D, there are only seven unique crystal systems. There are only seven ways that you can do this. You can only pack in and have no voids-- and have no voids-- you can only pack in these seven different systems where, as you can see, what's changing here? Well, it's Cartesian coordinates, so you've got the length of each lattice vector and their relative angles, right? That's all you're changing. Well, sometimes you change all the angles at once, OK? Sometimes you change only-- well, sometimes you change only a few of them. Sometimes you make them all the same. Sometimes you change a few. Sometimes you have all of them be even. And sometimes some of them are equal, some of the lattice lengths are equal but another one is different, all right? These are the seven ways that we can fill all of space. And there are only seven. In this class, we're only going to focus on one, and that's the cubic system. And the reason is that this gives us plenty to work with to understand crystallography and how chemistry relates to crystallography. And there is the benefit of the fact that a lot of elements take one of these three-- actually, only two. But they take cubic symmetry. What does that mean? Well, it means that all the lattice vectors are the same. Each one has a length a. They're orthogonal to each other, which means that the angle is 90 degrees. So we've got a cube. Now, it turns out that now we say, well, how do you pack? How can I pack things together in the cubic system? And there's three different ways to do it. So what this is is it's more symmetry and more group theory and math that goes into this that Bravais figured out. And so we call these Bravais lattices. Now, what Bravais figured out-- and again, you don't need to know all the math behind this, but I want you to know these three cubic lattices. What you figured out is that from the seven crystal systems, there are only 14 ways to pack. There are 14 ways to pack. That's it, no more. That's pretty cool. There's no 15. There's no 16. There's 14 exactly-- 14 ways that you can, by having a lattice, which means a stamp, there are only 14 ways you can do it for the seven crystal systems. And for the cubic system, there's only three. So I have three types of stamp. That's what it means, three types of stamps where I leave no voids, right? I leave no voids. So I can tile all of 3D space, right? OK, and it forms a cubic unit cell, cubic unit cell. These are the three ways-- a simple cubic, a body-centered cubic, and a face-centered cubic. These are the three types of lattices that have cubic symmetry. They are the only three. So in a Bravais lattice, the difference is in a Bravais lattice what Bravais did was he enumerated-- so here we enumerate the ways space can be filled. He enumerated the packing, all the ways that the space can be packed with, for example, packing within the unit cell. So there's three distinct ways to pack within a cubic unit cell, OK? That's what we're going to bring to life today and we'll talk more about on Wednesday as well. And what I want to do is I want to go through each one. So we're going to go through simple cubic, body-centered cubic, face-centered cubic. And then with each one, I'm going to talk about it, and then I'm going to show you a little video so that we can really see how the unit cell gets kind of sliced up. And there's like a razor that comes in and slices it. It's really cool. And so you can really see and feel what we're talking about. And speaking of seeing and feeling, so I have these. And I had wanted for these to be what go in your goody bag. But apparently like $100,000 is too much or whatever. [LAUGHTER] But so you have smaller versions of these. And I encourage you-- there's no better way to think about crystallography than to build stuff and see it. You've got to do that. It's really cool. And so this is the simple cubic. OK, so I'll pass that around. And this is the body-centered cubic. And here you have spheres, right? That's fine. But all we've done so far is set up a lattice. This could be anything. It's a repeat lattice. Well, that's what a lattice is. The lattice just defines that this is the same as this. Whatever is here is the same, is the same as this, is the same as this. That's what we're defining here, OK? OK, all right, now let's talk about these three. This is a picture from [? April. ?] These are the three systems I just talked about, the three Bravais lattices, three different ways to pack in a cubic symmetric cell. And notice there's three ways of looking at it, right? So here is sort of a ball-and-stick model, which makes it easy to see the bonding and the atoms. But the space-filling model is really important here, because the space-filling model lets us think about the maximum packing. That's really important. Remember the maximum packing, because remember these are bonding. Really? These are bonding together, right? So drawing them all the way up to where they touch, that's going to help us understand what their maximum packing can be. So let's look at that for the simple cubic, OK? So for the simple cubic, we'll do that here-- simple cubic. OK, simple cubic-- oh, let's just call it SC for short. Why not? Now, one of the things you want to think about is, how many nearest neighbors do I have? And I lost the-- so you can see. Look at look at the edge and think about how many nearest neighbors-- well, you know you got six, right? If you're in the corner there of the cube in your simple cubic, you've got six nearest neighbors. OK, that's important, right? Six nearest neighbors-- why is it important? Well, it's important because that's how many other atoms you could bond with in this solid, OK? But the other thing we want to know is the packing. Now, again, what you want to do to figure out the maximum packing is you want to think about it in the same way we did here. Think about putting something at each of the lattice points and then growing it out until they can't grow any more, right? And that's a good way to think about it. And in this case, we're going to take spheres. We're going to be taking spheres because that's how we're going to represent atoms. So we're going to be taking spheres. And we're going to grow them out and see what the packing is. Yeah, but I've got to do the same trick I did here. I've got to do the same trick I did here. I've got to write the radius of the atom in terms of the unit cell. So if the unit cell-- oh, boy. I'm going to be drawing a 3D structure. Danger! a, a, a-- that's a cube. All the sides are a. I made myself very happy. It's a cube. Now-- oh, but we have a cube. Now, OK, here we go. This one-- oh, but see, I want maximum packing. So I'm going to grow them out. And you can see what's going to happen as I do that. And so you can see that in the case of a simple cubic lattice, it's just like in the square lattice. It's just like in the square lattice, right? When I've grown out the sphere, the radius of that sphere, the radius is r equals-- now, oh, wait a second. OK, so I have 4/3 pi r-- OK, but that doesn't matter yet. Don't do volume. Think about the radius OK, the radius is just going to be 1/2 the lattice, right? So radius is 1/2 a. Volume is 4/3 pi 1/2 a cubed. Now, this is important. That is at the max packing. OK, so I've grown these volumes out, and I've seen what the packing can be. Now, we actually call this-- so now I've got the volume of an atom. I'm pretending that these are atoms now. No, I'm not pretending anymore. It's real. These spheres are now atoms. And OK, so that's the volume of the atom, right? And so we can write atomic packing fraction as the volume of the atom over the volume of the cell. That's exactly what I did up here, right? That's how we got the area of the square, the area of the circle, right? But there's one more thing that's really important, which is it has to be times the number of atoms in the cell. When I say cell, I mean unit cell. It has to be times the number of atoms in the unit cell. So it's however many atoms I have times the volume of the atom. OK, that's good. That's how much is inside of the cell divided by the volume of the cell, and that gives me the atomic packing fraction. So in the case of a simple cubic, the APF, the Atomic Packing Fraction, is equal to pi over 6, which is 52%. That's not very good, all right? That's not-- well, I mean, it's relative. We don't know yet. But it feels kind of low-- 52%. So with a simple cubic lattice, all I can do in my wildest dreams is cover 52% of that volume. That's not very-- that doesn't feel like a lot. And in fact, when you look at the periodic table, what you find is that there's only one element-- one-- that takes the simple cubic lattice. Anybody know what it is? Of course-- it's polonium. [LAUGHTER] Of course. That's a really good conversation starter. I'm just saying. That is a great way to meet people. Which one? It's polonium? Why? Relativity. In polonium, those electrons are so high energy-- literally. They're so high energy they're like close to the speed of light. They're relativistic. Their mass is heavier. All sorts of interesting things happen. Polonium is the only one that goes simple cubic. I promised a video. Oh, there's the-- oh. Ha. a-- a-- a. There is a diagonal. There is another diagonal. Oh! We're going to come back to this. Let's draw that. Because that might be important, right? So if I have a cube, then this diagonal, if that's a and that's a, then that is a root 2. And the body diagonal-- that's the body diagonal-- is a root 3-- equals a root 3, right? OK, we'll come back-- oh, I didn't need that, because this is just a over 2. But whoa, hold on. We're coming to it. But I promised a video first. So here's a video with the laser coming in. And you're going to really see-- this will come to life a little bit. [VIDEO PLAYBACK] - [INAUDIBLE] of a cube. The simple, or primitive cubic unit cell has particles at the corners only. In reality, the particles lie as close to each other as possible. Note that the particles touch along the cube edges but not along a diagonal in the face or along a diagonal through the body. By slicing away parts that belong to neighboring unit cells, we see that the actual unit cell consists of portions of the particles. When the cells pack next to each other in all three dimensions, we obtain the crystal. If we fade the others out, you can see the original group of eight particles within the array and the unit cell within that group. We find the number of particles in one unit cell by combining all the particles' portions. In the simple cubic unit cell, eight corners, each of which is 1/8 of a particle, combine to give one particle. A key feature of a crystal structure is its coordination number, the number of the nearest neighbors surrounding each particle. In a simple cubic array, any given particle has a neighboring particle above, below, to the right, to the left, in front, and in back of it, for a total of six nearest neighbors. The body-centered cubic unit cell has a-- [END PLAYBACK] OK, so you see-- that was so cool, right? Like unh, unh, and you're figuring out how many atoms are in the cell, right? Well, why didn't you just put the atom in the middle? You could do that, but then it's not at the point, right? So you've got to understand, the concept of the lattice is that whatever I put here, wherever my vectors go, that's what I put there and there and there. And so we often as a standard, we say, well, OK, what did I put there? I put an atom and then I put another atom. But then your unit cell is that cube. And so we cut the unit cell, and you're like, well, how many atoms do I have in it? Well, let's see. I got one here, but it's shared with all these other cubes, so there's an 1/8 of an atom there and 1/8 of an atom there. We add them all up, and you get one atom right back, which you knew-- which you knew. And by the way, this was not 1 over a. It was. I wrote it mistakenly 1 over a. The radius is 1/2 a over there, which is correct here. OK, good. Now, next crystal structure-- oh, look. How do you know what crystal structure-- there it is right there! It's the periodic table! To the rescue again-- he had it right there. [APPLAUSE] Thank you. [CHEERING AND APPLAUSE] That means T-shirts and T-shirts and T-shirts and T-shirts. [CHATTER] All right, I'm going all the way up there and up there. We'll bring more. You bring a periodic table like that-- what is a crystal symmetry? I know. Which one is simple cubic? Polonium. Where's-- it's only polonium. Oh, there it is-- polonium. Thank you for being representative of simple cubic crystals. But now we're talking about the next kind, which is body-centered cubic. And look, a lot more have a body-centered cubic. And here, OK, so if it's BCC, Body-Centered Cubic, well, now you know if I take these atoms at these corners, and I add now-- there's a lattice that takes me to the middle. Remember, my lattice is giving me this symmetry. My lattice is giving me this. So it means whatever I put here, I put in all the corners and I put in the middle, right? That's BCC. That's what BCC means. Well, and yeah, if you grow that-- OK, so first of all, eight nearest neighbors-- eight nearest neighbors-- well, you can see that, because I've got the six, but then I've got these ones here, right? I've got actually it's not the six. I've got the one along the diagonal, which is kind of the point I wanted to make next. Oh, this is BCC. This is BCC, right? Now, how do you see that it's BCC? Because you stare at it and you stare at it and you build it. You get together with friends. You get together with friends, and you build even more. And you see how much you can build, and then you look, and you see, look here it is. There is my square, right? OK, and there's another square. And that's my cube. And you go like this. And then you're like, oh, look, and there's an atom in the middle. That's BCC. And they're all the same. That's BCC, right? But see, because this is defined by the lattice, I could have made my square anywhere. This didn't have to be the middle atom. I could move it over, and then this would be the middle atom or this, right? They're all equivalent. It's a lattice. Now, if you look at this and you take that square or that cube, then what you'll find is that if you go along the body diagonal-- see, now I'm going to take each of these, and I'm going to grow them. Where do the atoms touch? If I grow the atoms in a BCC crystal, they touch along the body diagonal first. Now, that's really important. Let's pass this one around, too. That's really important, OK? Because now we have this concept that is also very important, which is the close-packed direction-- close-packed direction. And for the BCC structure, it's the body diagonal. Over here it's the cube edge. For a simple cubic, it's the cube edge. Cube edge is the close-packed-- OK, how is that going to work? I'll do this. No, that's going to be confusing. Cube edge close-packed-- oh, that's really small. But you get my point, close-packed direction. Close-packed direction, body diagonal-- what that means is that-- and you can see it right here. If you go all the way back to this-- there it is. As I grow these-- these are the same. They're colored blue just for clarity, just you can see. But these are the same because it's a BCC lattice. And as I grow them, these all have the same radius. These are the ones that touch first along the diagonal, not those, all right? Not those. And that feeds into what the radius is of the atom. I do the same trick, and the APF is going to be 4/3 pi. And now I've got my a root 3 over 4. Why? Because a is the length of the cube. And you can see here I've got four radii, right? Two diameters of the sphere go down the body diagonal. So r cubed, 4/3 pi r cubed, is going to be the volume of a maximally-packed sphere in a BCC lattice. So this has to be divided by the whole volume, which is a-cubed. But I'm missing something, because there's not just one. There's not just one. There's another one. There's actually two atoms in a unit cell. There's two of these. You can see that one in the middle and then that eighth, eighth, eighth, eighth. You've got two. So I've got to multiply it by 2, and it gives you 68%, is the APF. And that's why so many more structures, especially metals, are happy to be BCC. Let's watch the video again, or a different video, this one on BBC. [VIDEO PLAYBACK] - --particle at each corner and one in the center, which is colored pink to make it easier to see. With full-size spheres, you can see that the particles don't touch along the edges of the cube. Close-packing - But each corner particle does touch the one in the center. The actual unit cell consists of portions of the corner particles and the whole one in the center. Eight eighths give one particle. And the one in the center gives another, for a total of two particles. In this tiny portion of a body-centered cubic array, you can see that any given particle has four nearest neighbors above and four below, for a total of eight nearest neighbors. [END PLAYBACK] OK, now it's really important, right? Just so you see conceptually, visually how we got this, the close-packed direction for BCC is four times the radius. I want to write that, four times the radius. And it's that thing that we're doing now a couple of times, where you've got the radius of this sphere that you're packing in with a given symmetry. But I've got to write that in terms of the lattice vectors, which are in this case all a, right? So it's really easy. But they're all a, so I've got to write it in terms of the length of the lattice. That's how you get the atomic packing, OK? Good, now the last one is FCC. Look at how many elements take the FCC form. How do you know? You bring out the periodic table, and you look it up. Now, many of these elements can take different crystal symmetries. The ones that are listed are the ones that are often-- well, always the most stable one, right. They're the most stable one. But sometimes they can take many different symmetries if you do something like add temperature or pressure or other things to the system. What's in the periodic table is the most stable one. Now, the FCC is the last one that we're going to care about. And it is the face-centered cubic. Now, OK, so on this one, this is the face, right? So we've got these atoms that are sharing across the face. And then we've got these atoms here at the corners. There's nothing in the middle now, but they're on the faces-- sorry, the corners and the faces, all right? And so if you look at this, what you can see from looking at this-- and you've got to look at it. Where's my cube? Where's my unit cell? You now know it's called the unit cell, right? Oh, and I see. And then there's the face atom, right? There's a square there. Then there's the atom in the face. And then there's nothing in the middle. It's not BCC, right? But if I look at another edge of that, there's the atom in that face and so on and so on, right? Now, what's really cool about FCC is that it's got a lot of neighbors. And it's got more packing. So for FCC, this is the close-packed direction. Close-packed direction is the face, is the-- did I write down? I didn't write down. OK, that's fine. It's the face diagonal, which is a root 2 in length. And so the nearest neighbors is 12. And the atomic packing fraction for FCC is 4/3 pi times-- and now here we go, right? When I expand them, it's along the face that they're going to touch, right? So that's the close-packed direction. Face diagonal, that's this one here, right? So it's a root 2 over 4, right? Because again, I've got a radius and another and another and another. There's two diameters worth that go across that face. And so that's going to be cubed, OK, divided by a-cubed. Now, is there something missing? Yes. The number of atoms. If you look at a FCC crystal, and you look at the corners, each one of those shares with eight other unit cells, so there's 1/8. That gives you the one, like in our simple cubic. But now you've got all these ones sharing across faces. You've got six of those, each one sharing 1/2 atom. So that's three more. So there's 4. And this gives you a very nice packing of 74%-- 74%. Let's see what our video guy says about FCC. [VIDEO PLAYBACK] - The face-centered cubic unit cell has a particle at each corner and in each face, which are colored yellow here, but none in the center. The corner particles don't touch each other. But each corner does touch a particle in the face. And those in the faces touch each other as well. The actual unit cell consists of portions of particles at the corners and in the faces. Eight-eighths at the corners gives one particle. And half a particle in each of six faces gives three more, for a total of four particles. In this tiny portion of a face-centered cubic array, notice that a given particle has four nearest neighbors around it, four more above, and four more below, for a total of 12 nearest neighbors. Stacking spheres shows how the three cubic unit cells arise. Now we talk about packing. - Arranged a layer of spheres in horizontal and vertical rows. Note the large, diamond-shaped space among the particles. Placing the next layer directly over the first gives a structure based on the simple cubic unit cell. Those larger spaces mean an inefficient use of space. In fact, only 52% of the available volume is actually occupied by spheres. Because of this inefficiency, the simple cubic unit cell is seen rarely in nature. Polonium! - A more efficient stacking occurs if we place the second layer over the spaces formed by the first layer and the third layer over the spaces formed by the second. That simple change leads to 68% of the available volume occupied by the spheres and a structure based on the body-centered cubic unit cell. Many metals, including all the alkali metals, adopt this arrangement. For the most efficient stacking-- Now watch this. - --shift every other row in the first layer so the large, diamond-shaped spaces become smaller triangular spaces, and place the second layer over them. Then the third layer goes over the holes visible through the first and second layers. In this arrangement, called cubic closest packing, spheres occupy 74% of the volume. Note that it is based-- [END PLAYBACK] --on the FCC crystal, Bravais lattice. That's what he meant to say there. And you really see it come to life, right? I wish I could do this somehow with the 3D models. But you stare at the 3D models and you see that-- you know, what's interesting is the way you put it-- you slide-- this is what Hooke did with the cannonballs. You slide that base layer, and then you can stack them differently. But it's still a cubic system, FCC, all right? We'll pick up on this on Wednesday. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 17_Metals_Intro_to_SolidState_Chemistry.txt | Let's get started. How is everyone? [CLASS RESPONDS] Whoo. You know why you're so excited? Because it's a goody bag day. And I'm not going to deny you a weekend of goodie bag action, which is why you're getting it today. Even though the topic in the goody bag is crystallography, we will not start talking about crystallography until Monday. But I didn't want to wait. I know you didn't want to wait to have these models in your hands because it's an enhancement of your weekend activities. So the goody bag, again, just sort of like last time, I'm giving it out to you today, but the topic will start on Monday. Today, instead, we're going to talk about metals and metallic bonding, and the properties of metals that are related to the kind of bond that you get in metals. And you know, we kind of motivated band theory of solids with conductivity and we talked about conductivity. There is a chart of conductivity and there's some elements in there, some elements and some materials that are more complicated, right, like glass isn't necessarily one element or silica. There's diamond, silicon, germanium, copper. And we talked about insulators, semiconductors. And I've mentioned metals and that's what I want to focus on today. We did this in the context of the bands, right? So remember, here you've got your conduction band, and here you've got your valence band, and the valence band is filled. And so if there's a large gap, that's the gap, if it's large like, you know, over 3 and 1/2 EV then it's an insulator. And then the one in between we talked about a lot on Wednesday that, you know, if this is your gap, and it's say between 0 to 3.5 EV, then that's a semiconductor. And so, you know, that's what we talked about. That's in your goody bag. We talked a lot about semiconductors. And then we talked about how, well, if you have some kind of band that is filled in the middle there. So it doesn't have to be filled in the middle. It can be filled here. It can be filled up here. But the point is that the filling of electrons does not stop where there is a forbidden zone of states, right, where there is no states. That's what this is. That's what a gap is. It's a gap of electron freedom, right. Where there's metal, there is no such thing because you fill up to a point where then right above that infinitesimally tiny bit of energy away, there is a free state right? There's a free state, so you're in the middle of a band and that's a metal. Well, I mean, this is you know-- so we should say that the gap here has to be greater than zero, here the gap would be equal to zero. There is no gap. You can't define a gap in a metal. And we talked about semi-conductor problems. I looked up, just for fun, you know, I mean, why not? Because what else do you do late at night? You look up old exam problems and you play with them. And you know, OK, so from last exam, all right, so this is for the semiconductors, we had-- oh, here's a good question. 2.8 centimeters cubed of pure germanium. Now I want to highlight this because there's one more thing I want to say about semiconductors before I move on. Now, it's doped. And that's what we spent a lot of time talking about on Wednesday. It's doped with 50 micrograms, OK, of selenium. That was the problem. And the first part of the problem is, how many carriers are produced? And in the same question, it's like, are they n-type or are they p-type? What type of carriers? Those are things we can answer based on what we learned on Wednesday. And so, you know, oh, then you have this. And you say, aha, I know where to start because everything starts here, right? And so pure germanium is doped. Well, first of all, where am I? This is my map, right? Germanium is there. Now, it's doped with selenium. Is selenium to the right or to the left of the period? It's to the right. Right? And so right away you know, if you put a selenium in there instead of germanium, so if you take a germanium atom and you put a selenium atom in, it's going to-- if it's a doping, it's going to be in the bonding network that germanium was, but add electrons to it, if it's to the right. Right? And so you know that it's going to be n-type material. And if you do the math, the number of carriers is equal to 7.6 times 10 to the 20th per centimeter cubed. Now, here's something important. That is twice as many selenium atoms, twice as many because it's two over from germanium. So it had two extra electrons for each dopant atom, right? So the way you solve these problems, you figure out how many selenium atoms are there per centimeter cubed and how many electrons for each of those acted as a free carrier dope. As a dopant, remember, dopant simply give you carriers either of their-- if it's n-type, or down here, if it's p-type. OK? And this is n-type. N-type. OK. So that's nice-- now the reason I want to do this is because, again, it is-- you can never have enough fun doing problems like this and looking up old exams and stuff. But I also want to go back to this picture that I drew for you on Wednesday, OK? So this picture was of the number of carriers, and remember these are electrons that carry current, right? And we sort of drew something that looked like this. Now remember, this is, right, these are thermally activated and this is thermally activated, and these are from dopants. Now, if I put dopants in, and then I can get you a constant number of carriers because it's just defined by how much of the impurity I've mixed in. How much impurity and I can control that really carefully, which means I can control the number of carriers, and then eventually, thermally excited carriers will just start to dominate. But this ability to dope with chemistry is what enabled the semiconductor revolution. Now, you say, OK, what's going on over here? What's going on in this part here? All right. Well, you know what's going on there because we drew this picture on Wednesday. Because if I have an n-type, if that's my VBN and that's my CBN, the way this works is you introduce a donor level. Right? That's how doping works, right? The electrons from selenium go into here and then you have enough thermal energy to get them up to there and that's where they're free carriers. That's where they can carry electricity. And so at a certain temperature below which, right, at a certain temperature, if you're lower than that, well there's not even enough thermal energy to get from here to here. All right. Not even enough to do that. So it's got to go down eventually. That's what that part is. And then the one other thing I want to say about this plot is that we have names for this. These are called intrinsic carriers, intrinsic, and these are called extrinsic. Why? I mean, that kind of makes sense. That's actually a pretty good way to name them. Because, you know, intrinsic means I didn't need to do anything. It's pure germanium, pure germanium, intrinsic. I didn't do anything to the material. But it's still-- you can get carriers from thermal activation above the gap. Whereas if it's extrinsic, it means I did something external. I added dopants. Those are extrinsically doped. So those are just important distinguishing factors. Now, I'm going to come back to this plot later. I want to talk about metals. Metals are all the way over here. Those are semiconductors and doping them or thermal activation gets me carriers, which gets me this connectivity range, but metals are much, much, much higher. Metals are much, much, much higher. Why? What is going on in metals that's so special? OK. Well, the way we understand everything is from the bonds. So what is happening in a metal? There's aluminum. There's aluminum. There's aluminum. And there is aluminum. For us, this is all we see now. We don't see any of this anymore. We only see atoms and electrons because those are the building blocks, right? Those are the building blocks. And the thing is that, if we write down all the orbitals of an aluminum atom, there's the core, there's a shell, there's another shell, there's these ones out here, way out here, right, in the case of aluminum, three of them that are really, really, kind of weakly bonded. Right? They're way out there. They're loose. They have the ability to come off easily. We talked about this in the context of other properties, right? You know, the polarizability, the electronegativity, right? All of this feeds in. Everything is determined by where these electrons are. And then also, as I mentioned, by how the atom's arranged. That comes next week. But if they're really, really kind of almost free, then you can kind of picture it like this. Let's dial it back to sodium. And you can kind of picture a metal as truly free electrons, right. They are truly free. They're not in-- they're in the middle of that band and they've got all this freedom because the states are free all around them, and they're basically not really that attached to any one atom or another. And this is really the picture of the metallic bond. So let's use-- let's go over here. They're so free when these atoms come together with those very, very loosely bound outer electrons, those electrons are so free we actually call it a sea. We call it the electron sea. It's a simplified way to think about it, but it's a good picture to start with. So in metals, you've got this electron sea. Electron sea, we'll call it a model because it's a way of thinking. And in this model, you've got a positive charge, nuclei, so you've got positively charged. Why? Because they lost an electron to the sea. They've lost it. Those outer ones have gone. Well, they're not gone because there are others around too, but they're all just kind of roaming free, and so you're left with these positive charges, right? And these are the cations. Positively charged nuclei are cations. Right. And they're in a sea of electrons and so this is because the valence electrons is not tightly bound. This is what I've been saying, but this is why you get it. Cations, so positive charge nuclei, these are cat islands in a sea of electrons. So if you think about a metal in this way, you understand a lot of the properties of metals. This is a very useful way to think about metals. And let's go back to our pictures of bonding. Let me use this board here. I'll put it right under here. And we're going to finish what we started. We're going to finish what we started. Because, remember, we talked about covalent. We talked about ionic. And then we did all these intermolecular forces and the bonding that comes about from those, but we didn't do metallic. So if I do covalent over here, and I do ionic over here. Well, OK, covalent we know, right. You have a maybe HH. That's a covalent bond. That's a non-polar kind of covalent bond, caring is sharing. Or we could have HF, right, where it's sort of like, you know, it's a little less sharing. But you know, it's still a covalent bond. It's a polar covalent bond, right? I mean, now in the ionic sense, you have sodium chloride. We did that example where you literally-- chlorine is like give me and sodium's like OK and that's it. That's the relationship. Right. But see now here, let's take sodium and let's just put a bunch of them around. Well, but see, if I've got an electron sea, then my electrons are simply wandering around in the sea. So every ion has this kind of ocean of electrons around it. They're not attached. It would be like if you say, how are you going to parent? Well, I'm going to parent them until, you know, you can only go into the backyard and play. That's It, right? You are confined to be in the backyard. And now you say, well, OK, but now if I'm a proton and I'm parenting an electron, I say, you know what? You can be in any yard. It doesn't matter. Don't even come home. [LAUGHTER] I don't care. Another kid's going to come into my backyard at some point, so it's all good. That is the relationship of a metal inside of a metal. Maybe here it's like the two houses switch kids. I don't know what's going on. No. One of them just took the kid. [LAUGHTER] But anyway. All right. I'm going too far. But here's the thing. I want to give you that sense that they're just roaming around in a sea. They're roaming around. They're roaming around. Now, here's the point. I've got a sea of electrons-- oh. Oh, huh, we can finish our table. I'm going to finish that table. We started the table. Let's finish the table. Because now, if I have metallic, all right, and I have this kind of, you know, this kind of set up here, right. That's the picture. That's the model. And here I have cation-- remember our table-- slash delocalize, delocalize electrons. And the bonding strength for this table in kilojoules per mole is 75 to 1,000. It's a big range. We'll talk about that. And there is an example, iron. We finished our table. That's it. That's the last entry. Now, remember, this part of the table was the basis of attraction. Remember that? And we filled this out for all the other types of bonds, the basis of attraction. I want to point out that every single basis we have covered involves the same thing, understanding what electrons are doing. What is the electronic structure? That's the fundamental principle of this class, right? That's the key component, the key ingredient, that's in all of our recipes. What are those electrons doing? That leads us to every single one of those kinds of bonds, including metallic. All right. So now, you can imagine how this controls the properties, right? But first you say, well, if I had a group one metal, if I had a group one metal, then I've got two-- I've got one, right, like sodium. But if I go over to magnesium, you can see that I've got a higher charge because both of those electrons are likely to leave and wander around in this sea of electrons. So that would be what magnesium looks like in terms of this electron sea model. But the thing is that this sea, this picture, is what helps us understand the properties of metals and that's what I want to talk about. That's what I want to talk about. And these are some of the properties, right? You can go through metals and there's a whole range of properties, but these are some of the ones that are pretty consistent. They are reflective. They're shiny. There are literally settings in like photo apps. Metallic, make it look metallic. Right? Because metals are shiny. They have luster. Why? They have high electrical conductivity. That's the thing we've talked about a lot already. They have a high thermal conductivity. I'll mention that. High heat capacity. They're malleable and ductile. All of these and many more properties of metals can be explained from this one simple picture of the electron sea. This one simple picture. OK. So let's start. So if I think about-- let's look at these top three, OK? So I'll start, why is a metal shiny? Why is a metal shiny? Well, it goes back to the thing we've been talking about all along, you know, or certainly in a lot of parts, right? The bohr model, and atoms, and electrons getting excited. The semiconductors and LEDs, right? It has to do with how photons interact with the material. But see, now I've got these electrons that are sitting out there at any energy level ready for the photon. There's no gap. Right? So I've got electrons sitting around in the sea and they're ready for anything, any kind of energy that hits them, well, they've got room to be excited. Right? They've got room to be excited. And it's that, combined with other things, but it's that, that is a big part of why metals have their shininess. Those electrons can be excited in almost any way, right, because there's an infinite number of states right there. There's no gap. Right? So the way they can respond to energy from photons is extremely diverse. And you can imagine, if I'm a photon and I excite an electron, it doesn't matter what the wavelength is. That electron can fall back down and re-emit the same energy photon. OK. So that's luster either way. But now you know and you really should do this. And look at. I just said, you know, you have this. I've got electrons. I shine photons on these. They might come up and go back down. There's all this room here for electrons to be excited. There's no gap. Whereas in diamond, in diamond I've got this huge gap, right? Remember, diamond is an insulator. Diamond has a gap of 5.5 electron volts, roughly. Now you know why diamonds are transparent. It's of course, the electrons. Of course. Because if I shine photons on this, and those photons are indivisible, you know that even if I go all the way up to purple photons, they're not that high. So any time I try to-- so I'm shining photons on my diamond and those electrons are like, can I go here? No. Can I go here? No. Can I go here? No. You have to shine a very high energy photon to excite an electron in diamond. That's why it's transparent, right? That's the reason. When you go to the store to buy diamonds, you should ask them about the electrons. That's what's making it all work. That's what explains the properties. But back to metals. Now, OK, so that's luster. What about electrical conductivity? Well, it's the same thing. It comes from the sea. It comes from the sea. If I take a wire of copper and I look at this as a sea of electrons, so I'm not drawing the ions, they're in there. I'm just drawing the fact that I've got this sort of ocean of electrons that are fairly free, right? Well, what is happening when I try to move them, it is with some kind of electromagnetic force, right? There's some kind of force. And what that means is that I have a charge that this doesn't like. So like it's a repulsive charge over here. It's going to push one of these into the other one. But then that's going to feel-- that's going to feel-- a force is going to move and then that's going to push this one, and that's going to push that one, and so forth. That's how electricity works, right? So the actual movement of the electrons could be slow in the material, but the current, right, you can push a lot of current through because these ones over here feel pretty quickly the forces you applied over there. But they only feel it, they only feel it if you can get this freedom. If those electrons are not free, they're going to feel this repulsion there and be like, you know what? I'm stuck in my state. I'm stuck in this full valence band. I got nowhere to go. Sorry. You know, put your energy into the atom or something else. I don't know, but I'm not moving. Ah, but if it's a sea, then you can push them and they can push each other. Right? That's the same thing. It's a conceptual picture, but it's the same thing as almost an infinite number of free states. That's what that means, right? That's what that means. Now, if you've got copper, we see copper now as something like argon with those d electrons filled, 3d 10, and an s electron here. And so if it's a copper wire, those d electrons it's all filled up in there and they act as a way of shielding, it's a beautiful shielding, so that this is even more free and less interested in staying around. So copper's such a great conductor and that whole column is such a great conductor and this is, by the way-- oh, such a good thing to do on a Friday night. I'm so glad it's Friday. Look at this, ptable.com. Ha. You could spend hours on this. I do spend hours on this. [LAUGHTER] Property. Electrical conductivity And then it shows you dark green is high-- dark green is high-- and then no coloring is low. And then you can just click around. You just click around. I could do it now, but I would never leave. But you go there and you can click on different properties and see the trends in the periodic table, and it's a beautiful thing. And you now can understand them because of the ways in which electrons behave. So you can see-- look at this. There's copper, right? So there's copper, and below that, we've got silver and gold. And this is a really high set of elements. The electrical conductivity in these three elements is super high, and this is why. Because in each case, you're going down but you're filling the d's. Oh, and then you're even filling f's. Screening. Further away. Really free s-electrons. High conductivity. Electron sea, right? Very easy to push them around. Now. OK. This also goes back to this picture, because, remember, I snuck it in because it was on the graph, and I didn't take it out. But there was a picture of silicon, where I showed silicon increasing its intrinsic carrier concentration with temperature, and on top of that was a metal. I think was tungsten. Now if you draw the metal, it would be way up. It would be way up right here because this is higher, but it went this way. But now you know why. It's totally different. The reason is totally different than why this changes, but you can understand it for the same root, which is the behavior of the electrons. So, in a metal, in a semiconductor, this temperature dependence-- here, I've taken the temperature dependence away because I've doped it. But here, that's the thermal activation into a place of freedom, into a place where they can be free in the conduction band. Here, they're already totally free. So what's happening? Well, when I increase the temperature, I'm increasing the vibrations. You can imagine if these electrons are trying to do their thing, and they're in a sea of electrons, the last thing I want to do if I'm moving along is to see some big ion moving around. And if it moves more, then it's going to make me more upset and get in my way. That's temperature. That's why this goes down and not up. For metals, the more thermal energy you put in, the more those ions-- those positive charges-- they're positive-- are going to interfere with the electrons being free. Oh, we are understanding things. We understood why these have-- now, you might say, why are these different? That's complicated. Relativity. Not part of this class. But if you want to understand why these are different from a-- they're all really high-- why are they different, it goes beyond the scope of this class, but it's also a very interesting story that has to do with, on the one hand, relativistic effects. Now there's another thing, which is that if I go back to that table and I did this-- I really did this a lot-- I clicked back and forth between electrical-- now look up here-- thermal, electrical, thermal, electrical. And you can click-- you can go, mm, mm, mm, mm, and you can watch the table, and do that for like five minutes or so. And what you'll find is almost nothing changes. You can't do that with very many properties. But for metals, and those two properties, they're almost identical. Again, it goes back to the sea. It goes back to the sea, because if you have electrons that are free in a material, those can carry not just electricity, but those can be the dominant carriers of thermal energy as well. And I'm not going to go into those details. I just want to mention it. So this is actually called the Wiedemann-Franz law, and it basically says for metals, that the thermal-- let's see. OK, I'll just write it here. Thermal divided by electrical conductivity for both-- conductivities-- OK, I'm putting it in parentheses. It should be thermal conductivity divided by electoral conductivity goes as some constant times the temperature. So if you look at that-- so you can look at it by toggling back and forth and see how similar these are. You can also just plot stuff or find plots of stuff. Here's one. Electroconductivity-- this is at some fixed temperature-- orders of magnitude variation, orders of magnitude variation, tons and tons of different materials. This is the Wiedemann-Franz law. Now you know that law is simply a result of the fact that these free electrons can carry both electrical-- their charge-- as well as thermal energy, and that the way they carry those are dependent on the material, dependent on the number of electrons, and the way they bond and all that, but if they carry it in one way for one, it carries it the same way for the other. And look at all these things. Now notice here, this word that's used a lot-- alloy. Alloy. Alloy. These are all-- alloy. And I mentioned the word alloy on Wednesday, because I said if you wanted to make a good blue LED, you had gallium nitride but it was a little too high in its band gap, so they alloyed it with other-- then that meant mixing in. Well, that is done for metals. That is done for metals in a very wide range. Why? Because the metal properties that you have, you may like some of, but you may not like others. So like for copper, it's got this really high conductivity, but maybe you don't want it to be so malleable. What does that mean? We'll see in 10 minutes. Malleable, ductile. So what do you do? Well, you alloy it. And so here's an example. Brass. Well, we know brass, but now you know brass. Now you really know brass. You didn't know brass. You don't know anyone or anything until you see it as an atom and electrons. That is how you know someone. Copper? With zinc substituted in a two-to-one ratio-- copper to zinc-- the connectivity drops. Those electrons are not as free, but they're still pretty free. Maybe they're a little more tied up in the bonding, and maybe what you've done by alloying has changed its mechanical properties, because I kind of want something copper-like but I wanted to have different mechanical properties, so I alloyed it. And this is really how we think about materials design. I need something to have, let's say, a certain set of properties. Well, I don't have them all here, and I don't have them all here, but how can I use a combination? How can I use a combination to get what I want? Bronze is copper with tin-- four-to-one ratio of copper and tin. The conductivity drop even more. It's not so simple. It's not so simple as, oh, if it's four-to-one, then maybe it's like 20%. No, because it has to do with what happens to the freedom of these electrons. That's what it has to do with, and that's more complicated. But maybe I've changed the properties in another way that I really like. Well, those are substitutional. Substitutional alloys means I took one of the coppers out and I replaced it with something else. Here's brass with zinc in there. You can also have interstitial alloys, and interstitial alloys-- so if we-- we'll be coming back to this when we talk about crystals next week-- but alloys-- actually when we talk about defects. So you can have substitutional-- substitutional-- and you can have interstitial. This is just a preview. Interstitial. This is just a preview. We're not going to be testing on this or thinking about this much until we get to actually putting these things in to a crystal later as defects. But right now, I just want to make sure you see them for the first time. Substitution, interstitial. Why would you want one or the other? Well, it depends on what properties you're trying to change. In this case, I really like iron. It's really cheap. It's a cool material, but I need it to be stronger. And that's called steel. And if you just put a little bit of carbon-- really, not much, like a percent by weight-- it can dramatically change the mechanical properties of iron. You can make it a much stronger material. There is a massive, massive range of properties. And part of why metals are so tunable is because of the electron sea. I just came across this article actually, and I thought I'd show you. This was just literally yesterday. Was that yesterday? I thought it was an interesting example. Car Wars: Steel Getting Stronger, Lighter to Curb Aluminum Rise. Well, see the thing is that as you want cars to be lighter-- think fuel efficiency, or range, if it's electric-- you've got to use lighter metals. And aluminum is lighter, but aluminum is more than twice as much a steel. It's a lot lighter though. So there's a lot of interest in making steel-- that's just iron with carbon interstitial-- and making steel stronger. And you can see, "Newest alloy." That's what caught my attention. Newest alloy. They've come up with something to add to iron to make its mechanical properties really, really good at a very, very thin layer because they need steel to be even stronger in a much thinner layer. Otherwise, it's too heavy. And you've got to make it really thin and really strong. Those are going to be alloys. OK. Back to our property list. Now, malleability and ductility. Why are metals malleable and ductile? Well, I thought this was a good place for a video because I really like it. I think this is a really simple explanation of malleability and ductility as it relates to the electron sea. So let's watch this. This looks like a minute long. [VIDEO PLAYBACK] - [INAUDIBLE] shatters. And this is because the applied force pushes light ions close together. They violently repel each other, breaking the crystal apart. That's an ionic crystal. - In contrast, if you hit a metal with a hammer, it doesn't break. It just dents. Metals are able to deform form in response to an applied force. The mobile sea of electrons shields the cations from each other, preventing violent repulsion and allowing the metal to change shape. The most malleable metal is gold. A similar property to the malleability of metals is their ability to be pulled into long thin wires. We call this ductility. Ionic compounds are not ductile for the same reason they are not malleable in general. If an ionic compound is forced into a long cylinder, it breaks apart because of the repulsion of like ions. In contrast, a metal can be pulled into a long cylindrical shape because the cations can line up, shielded from each other, as the fluid-like like sea of electrons flows around them. That's my favorite part. - The most ductile metal-- [END PLAYBACK] Fluid-like sea. Oh. The most ductile metal. What is it? We may never know. We may-- it's platinum. But see, now, OK. OK. Now that's a-- I just gave it away. OK. Ductile, malleable, sea of electrons. Sea of electrons. Because did you see them flowing around? You're moving the atom-- that was like a dance thing. That can be like a new dance move that you try out tonight, the sea of electrons dance move. I'm just saying. I'm not going to do it again. But the sea of electrons is able to move around the ions as they're distorting because the metal bond is not so directional. The metal bond is not so directional. Because think about it. What is holding the metal together? It's these interactions between the house and the kids that are roaming all over the place. Is the kid over there or over there? It doesn't matter. Just give me a little bit of negative charge so I can feel an attraction. That's the metal-- that's why it is so able to deform in the way it does. Because those atoms are moving around, and the electrons are just kind of there as this happy sea, and the bonding remains. It doesn't break. That's the key. Otherwise, this thing would just crack. So wire-pulling-- check this one out. You can take a 10-centimeter by one-centimeter platinum rod, and you can spin that into a wire, and this is done-- this is a very thin case for effect. But you can make a 0.0006-millimeter diameter wire, and you get 2,777 kilometers of wire. Now that's because of ductility. The only reason I can get from point A to point B there is because I'm not breaking this apart, and that's how thin I can get-- 27-millimeter diameter when I pull it. And I thought that was a really good example of why this matters, because maybe you've noticed, but wires are really important to us. We use them a lot. And so, I you just saw in a cartoon with electrons. Here is a cartoon that I thought was interesting that shows a wire-pulling setup. So, now this is-- Die, reducing the diameter. That's a die. Further reduction can take place using extra dies. The drawn wire is finally wound on a spool. This transformation elongates the wire. One meter of 5.5-millimeter wire rod can be drawn to 30 meters with one millimeter diameter or 484 meters with 0.25 millimeters diameter. OK. Now, that is a cartoon, and I think it's always good. When you see cartoons like this, you don't know. Go look up what the real deal is. What does it really look like to draw wires? And I found it pretty cool. So this is an actual wire-drawing plant, and this is what it looks like. Now, there's no audio here, but I just wanted to show you the magnitude. Those are huge towers that are melting metals to give you this first rod, and then it goes through one after the other. And look at it being pulled-- being pulled in at some diameter, and it comes out in a different diameter. But I wanted to show you the scope and size of this. This is just one of them. There's a person, and so you get a sense of the size, and then you can also see something-- when you go visit these factories, and I have these factories, there's always a lot of this happening. Why? Things are getting really hot. When you're spinning things like a wire at a very high velocity, things heat up, so you get water all over the place. So there's water being sprayed in factories all over the place, because you've got to keep stuff cool, or there's water baths that you run these things through. And then you've got to think about, well, does this thing interact with water? How does the water play a role? It all comes down to the electrons. And once they've actually drawn the wire out over this huge factory, they can actually weave it back together. So I found this. This is the same factory. Now these things here are doing the opposite. So these are huge drums, about the size of a person, and each of these got a wire that was drawn. And now, again, because of the ductility, you can also move the wires, play with the wires, and you can weave them together. You can weave them right back together into a thread that looks like that, which we all have seen I think many, many times. All of this processing, all of this manufacturability, has to do with the electron sea. That's why we have wires OK. Now, a couple more small points. So we're going to go back to this picture because this is how we started. And this is the 1-D-- remember that 1-D model that we used to introduce band theory-- and I want to go back to this, because what you can see is that if I think about this as bonding and anti-bonding orbitals all kind of mixed in here, then you can also get a sense, just like in the dimers that we make-- N2, O2, remember NO? Bond order. It counts how many electrons are in an anti-bonding orbital. Just because it's in an anti-bonding orbital doesn't mean it's not there in the system. It's the same with these metal bands. Some of them may be good for bonding. Some of them may not. But just like in the dimers, the bond order is related to the strength of that bond. Remember? And in NO, it's like a bond order of 2 and 1/2, and then if I took an electron out, it actually got stronger. The bonding got stronger. It's the same thing. You can think about this the same way in metals. You have a band. Some of these orbitals are not going to be bonding orbitals, and some of them are going to be much more about the bonding. And so that's going to relate to things like boiling point. Metals have boiling points too. So if you look now, go back to the periodic table, remember the highest electrical connectivity was here. Well, that had to do with that right there, with the position, with the filling of the orbitals to give me the freest electron I can imagine, after those d-orbitals are filled, that s. But if I'm thinking about whether I filled anti-bondings orbitals or not, because now I'm thinking about the strength of the bonds, you see a different trend, and you see something that you would expect. First of all, you see a huge variability, but you see that the ones that are strongest-- the highest boiling points-- are going to be the ones where you fill that band-- those bands-- roughly halfway. And this is also why, in a metallic bond, in a metallic bond, you've got this enormous range. So you've got-- it's the weakest for elements with near-empty, near empty, valence subshells, valence subshells, and I would say, like OK, caesium is an example, or near-full, like, for example, in mercury. Now mercury is actually a liquid at room temperature because of this. There it is. There it is. Why? Because I-- remember the beryllium dimer. I filled all these electrons in there, and they're causing this really weak bonding, this really weak bonding, because I had to put all of them into the anti-bonding orbitals. I couldn't do this trick that right over here, right one column next to it, these guys can have tricks, where they-- now this is bonding strength. This is not electroconductivity. But here, they're filling the d, and then the s is free. Not here. Here they're all filled, bonding and anti-bonding. No tricks. Now, if you look at the melting temperature-- that's one of the strongest elements would be tungsten. The strongest would be example tungsten, where you've got a range of melting-- the melting point of tungsten-- and this is not the same as the boiling point-- the melting point of tungsten is 3,680 degrees C. That's a liquid at room temperature. And the melting point of cesium is 28 degrees C. Look at that range. And it all has to do with how these electrons are filling the metallic bands. And more than that, because it also has to do with the way the atoms come together, and that is going to be the topic of next week. Now that would have been a perfect line to end on because I said the words, "next week." I just want to make one more point. I can't stop because there is one more thing I want to show very quickly in one minute. And that is-- going back to this picture, if you're thinking about filling, and I've got like my 3s-- remember, I've got like my 3s-- and let's put 2p here and then 3s and 3p. Now remember, for metals, for a lot of these metals, what ends up happening is what I showed you before on Wednesday, which is that you get actually an overlap of this, so you get a band that would be the 3s/3p band. Sodium, magnesium, aluminum-- these all come together like this to form a single band. Now remember, I don't need you to know how to do this. We would tell you if it does this or not. And we talked about that. I'm not going to ask you to know how these come together, do they form a continuous band or not? But in this case, they do, for that row. And so you know. I've got eight electrons, eight-electron capacity in that band per atom. That's the capacity of the band. So now I want to talk about, how is this band filled? Well, for sodium, I'm going to fill it 1/8th, because it's got just one electron that it's putting into that band. And for magnesium, I'm going to fill it with two electrons. And aluminum would be three. And so on. So it goes back to the same picture as before. We're filling the bands, but now we've got 10 to the 23rd and 24th atoms, and so we think about filling the bands as what's the occupancy per atom you can put in there? And so I could tell you if I had a mole of sodium, how many electrons I have in this band, right? It's a mole. And how did you fill it? And that ties into what I just talked about, which is how that relates to bonding strength. OK. Now we can stop. On Monday, we will start talking about crystals. Have a very good weekend. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 14_Intermolecular_Forces_Intro_to_SolidState_Chemistry.txt | Today, we are talking about bonding between molecules. Now, we have talked a lot about bonding within molecules. Right. And that got us to saying things like, well you can use electronegativity, You can use electronegativity, for example, to say, I got an ionic bond. All right. Or I got a non-polar covalent bond or I've got a polar covalent bond. But today, what we're going to do is talk about how these molecules-- how these molecules bond together. What sorts of interactions can they have? And those are really important. Right. So that's going to be the topic. Now, if we just get started with what we know what's up there. All right. Well, there's ionic. So listen, I'm going to fill this in. Bond, a model. You'll see what I mean. I'm going to draw a little cartoon. The attraction, how it's attracted, the energy range, and an example. Right. So for an ionic bond, we've got ionic-- ionic. OK. And here we go. Watch this. Plus and minus-- there's my model. OK. And the attraction here-- we'll get a little specific. It's a cation and an anion. And we know that that's going to go like Coulomb's Law. Right. So the attraction, the energy, not the force, the energy will go like 1 over R, roughly. Now, what is the energy? Well, the energy is pretty wide ranging. I'm going to do this in kilojoules per mole, since those are the units I have. I'll write that in a second. And an ionic bond can have lots and lots of range in here. And it typically can be very strong. Something we've already talked about. An example would be sodium chloride, right there. Let's put the units here. The energy units are in-- I'm going to write it here-- kilojoules per mole. Now, in a covalent bond-- so covalent wouldn't be here. OK. So in a covalent bond, remember, we've got something very different. We've got this carrying a sharing kind of thing, and we've got maybe two atoms. Right. Two atoms like this. And then the charge is sort of-- the negative charge is sort of floating around and sharing those two centers, like in H2 or Cl2. I'm going to use Cl, so let's keep using Cl, all right. That would also be an example of a covalent bond. And the energy range there is also quite large, 150 to 1100 are the numbers I got. The attraction is shared electrons-- shared pair of electrons. And I'm not putting an R dependence, because for covalent bond, it's complicated. It's complicated. There is no. It is. It's a complicated-- but it's a very nice, happy relationship. But it's a complicated R dependence. So you can't simplify it like you can with a 1 over R or like some of the other interactions we'll talk about today. OK. 1 over R, not for covalent. All right. Now that's where we've been. Mm. Mm. Here we go. Here we go. Look at this. I'm not even-- I'm just getting started. Because the thing is, now, we got to talking about how these molecules can talk to each other. And there's a variety of ways. What I want to focus on first is this thing here. That's that polar covalent bond that we've already talked about. So we've already got a grounding for it. Right. But I want to remind you of what happens. Right. So if I have, let's keep with HCl. If I have HCl, that's a nice version of a polar covalent bond. And remember what happens there is you've got a sharing but sort of unfair. Right. So one of the atoms took a little more of the charge. And so sometimes we represent that with like a more of the cloud, right, on one side than the other. But it's still a covalent bond. Yeah. But the thing is that-- and so what you're left with like in this case is some positive charge on the hydrogen atom and some negative charge on the chlorine atom. And we did this already. We used these symbols. And we also talked about how that leads you to a dipole moment. What is a dipole? Well, a dipole is just the description of what you have in this case, where you've got two charges of opposite signs separated by some distance. OK. Two charges. Those aren't full charges. Those are partial charges. Right. And they're separated by some distance, the distance between these two atoms. And you get a dipole moment. And it has a magnitude and a direction. And remember, just to remind you, I did share this with you in a previous lecture. But, you know, mu equals the charge times the distance between the charge. Right. So that would be like the charge would be the deltas here and the distance between them, right, would be this. Right. 'r'. Now, remember the units are in Debye. So for this molecule, for here, mu is around 1.8D. And 1D is equal to 3.3 times 10 to the minus 30th Coulomb meters. Those are the units. Charge times distance. Charge times distance. Right. OK. 3.3 times 10 to the minus 30th Coulomb meter is a Debye. And again, just to remind you, because we did talk about this when we talked about polar covalent bonds, the dipole moment of these molecules tends to range from 0-- mm-- right there. Right, no dipole, to 10 or so-- 11. Right. Debyes. Now, why is this important? Because if you now think, well, what would happen? What would happen if I brought another charge up to the dipole? What would happen? Right. Let's say that I bring a positive charge. Let's say that I'm a sodium and I've lost that outer electron, which we know is pretty easy to lose, if you're sodium. Right. So I'm positively in charged, and I'm coming along, and I see this dipole. Well, it turns out that, as you can imagine, I'm going to be attracted to that negative charge. Right. So this molecule's going to orient in such a way that I can actually bond to the molecule and be attracted to the dipole of the molecule. So that is actually another type of bond. Right. And we'll call that ion-dipole. It's an ion-dipole bond. OK. Let's do this so that I stay within the lines. All right. Now, OK. So what does that look like? Well, we just said it's a charge. I'm taking a single charge, and I'm bringing it to this thing that has a shift of the charge within it that led to these two partial charges. So, as you as you can imagine, it's going to want to orient to form a bond. All right, so let's draw it that way. All right. So maybe if I were to take HCl, maybe I would have sort of like, you know, a cloud like that. Right. Where it had it's a minus and it's plus there. OK. Why did I-- I don't know why used dash lines, but there you go. And so you can see that is-- what I wanted to put is the dashed line for the bond itself. Give myself a little space. Right. That's the bond we're talking about. That's the bond we're talking-- Now, it turns out that this is an important type of interaction between molecules and ions. It's an ion dipole interaction. And it has a range-- I wrote it down. It has a range of 40 to 600, huge range, 40 to 600. And the kind of attraction is really not describable in any other way than what we just described. It's an ion and a dipole. And it goes as 1 over r squared. If you look at the separation, the distance between these, right, that distance-- the dependence on the energy, on the bond energy, will go like that distance, like, 1 over that distance squared. Now, these dependencies you can get from fairly simple electrostatics. I won't derive any of it. I'm telling you what it is. OK. Good. So an ion double-- now an example of that would be the system that I just drew. Right. So Na+ plus and HCl. OK. Good. That's a different kind of bond. But see, now, I could also have two dipoles. And I got a picture for that. So here's a dipole. There's my HCl dipole, and there it is with the negative partial charge and a positive partial charge. And it's also color coded. Right. OK. So that's cool. And there they are just floating along. And I put an electric field on. And I put an electric field on. And you can see that with the electric field-- so there's my positive plate and my negative plate-- they're going to try to align. Right. These dipoles mean that they're going to align with the field. They're going to align with the field. So all of those positive edges are going to try to turn and face the negative one. And all those negative ones are going to turn and face the positive one. And that's exactly what we just did with this ion. We basically did the same thing here. That's what the ion did. OK. That's what the ion did. It said, hey, come and face me and let's form a bond. But these can also be attracted to each other. All right. And so I've got a picture there. So you can see, like, if I have a dipole in another dipole-- so this is not an external field, and it's not an ion. It's two dipoles, two HCl molecules. Right. Well, they each have a plus charge and a minus charge. They each have a plus charge and a minus charge and, you know, partial charge, right, polar covalent. And you can see that those can line up, actually, in different ways to form attraction. And they can also repel. Right. But overall, when they get together, or when two of them get together, they're going to find a way to attract, because they can-- because they can. And so that is another kind of inter-molecular attraction, and it's called a dipole-dipole. Dipole-dipole. But, let's draw this. And if we do that we have-- let's see. I'm going to try to draw this. So I've got one dipole. I'm going to stick to that picture-- oh, I'm going to stick to this picture, but it's the same as this picture, right, where I'm actually using the kind of shape here to show you that more charge has gone over to one side. OK. So let's go back to here. Right. And so if that happens, then I've got minus and plus, thank you. OK. And then I'm going to have another one that's going to want to line up like this. Oh. Did I do that right? Yeah. And this is the bond. This dashed line here is the kind of bond that we're talking about. Yeah, those are almost sort of asymmetric. OK. Now-- gesundheit. Now, that's a dipole-dipole. We don't really have a better way of calling that either. They're dipole charges interacting with dipole charges. And that's going to give you a 1 over r cubed dependence. And, you know, if you've taken some basic ENM or if you do, you'll learn this. Right. You'll learn how to derive these distance dependencies. So I've got two dipoles, right, because there are two molecules that have not-- is not a non-polar. It's a polar covalent bond. And those are attracted to each other in this way. And the range of that bonding strength for two dipoles is 5 to 25. So it's a lot less. Right. And well, if you had HCl and HCl, that would be an example-- now instead of an ion, I'm putting two polar molecules in. Right. So this could happen. This could happen from a molecule to a molecule. It could happen from an atom to a molecule. It could happen from a molecule to an atom. Right. So for example, you know, I could have-- if I have HCl, and more charge has gone to the chlorine side. Right. So this has my delta plus. This is what I wrote over on the other side. That's my delta minus. And if I have just a single atom-- I'm going to put a xenon atom here. Well, it's just sitting around happily, you know, minding it's own business. And what's going to happen to that? Right. What's going to happen to that? Well, see, if this comes up to an atom that didn't have a dipole, it still has feelings. It still has feelings. Why? Because electrons have feelings. That's why. Because it has electrons. So those electrons are in there, and it doesn't have a dipole. But that's OK, because those electrons still can feel. They can feel the dipole of this. They can feel the dipole of this molecule. And when they feel that dipole, they react to it. They react to it. So there's a whole other thing that can happen. By the way, it could happen-- it could happen with a sodium atom. Na+ also is a charge that this happy go lucky xenon, it doesn't know what's coming. But it's got electrons that feel charge. And so you bring in charge to it, and they're going to react to it. They're going to react to it. Whether it's a single charge or whether it's a dipole, these are charges that are going to change the charge around that atom. Now, there's another name for that. And that leads to another kind of bond. It actually leads to another kind of bond. Let's see. So I'm going to just-- let me draw the cartoon. I really want to make sure you understand this, because this sets up the next kind of bonding. And so let's suppose I take a sodium, and I'll use H2. H-- gesundheit. H. And mine could have been a single atom. It could be a non-polar molecule. Right. So this is what this looks like. And now this is going to-- these are kind of walking along happily, and then they see each other. And all of a sudden, this is going to induce a shift of the charge around that H2 molecule. So what you're going to see is now more-- which way is it going to go? Which should have the bigger side, the more charge? Right, yeah, here. Because that's positive, and electrons are negative. So now it's going to look like this. Now, that was not a dipole. It is not a molecule with a dipole. But it's an induced dipole, induced dipole, because of feelings. It comes down to feelings. That has a name. All right. That's ion. that one right there is ion-induced-- see if I can fit it all here-- induced dipole. Ion-induced dipole. And I'm going to try to draw this now. Let's see. OK. Hang on. OK. So the ion-induced dipole-- I'm going to have a plus charge here, and here's what I'm going to do. I'm going to draw the original shape as what it was. Oh, boy. Here we go. That's what it was. But then it got induced. OK. It got induced. And so then, once it's induced, you've got like a minus and a plus. Happens all the time. It happens all the time. Now, the way that it will-- we'll talk about this in a little bit of a different way, because this attraction is between an ion-- by the way, an ion just means an atom or something lost a charge or has a charge, plus or minus. Remember, cation anion. But the ion in this case is interacting with what is called a polarizable-- polarizable electron cloud. Now, the electron cloud is nothing new. We got an electron cloud nailed. We know about electron cloud. Right. Because those are orbitals. But what is this thing called polarizable? Right, polarizable. And that's a very important term. So I'm going to write down its definition. Because you can imagine if I have a way of being attracted to a non-polar molecule by shifting its charge, by shifting its cloud, then how easy was that? How much could I shift it? You can tell that that's going to affect the bond strength. By the way, which is-- let's see-- 3 to 15 and has a 1 over r to the fourth dependence. And let's give us-- let's give the example that I gave there. Na+ and H, H. OK, non-polar molecule. OK. It could have been an atom. But It could have been a non-polar molecule. Yeah. Now, polarizable-- polarizability, is a very important concept. Polarizability is the measure-- gesundheit-- Of how easy it is to temporarily distort the electron distribution. That is a very important concept for these types of bonds and for a lot of other properties and, of course, for life in general. How polarizable are my electrons? I need to know. Are they just willing to deform from their happy orbitals? Temporarily, because if they were going to deform permanently, then you'd have a dipole, a permanent dipole. These are induced dipoles. OK. That's important. OK. Now, OK, good. So that's an ion-induced dipole. Now you could also imagine, you know, this could have easily been a dipole over here. It didn't have to be an ion. It could have been a dipole. I could have had a dipole here that came up to a non-polar molecule or an atom, right, and it could have induced the same shift in the charge density, which then allows me to be attracted, because there's a little bit of a fluctuation of negative charge there. Right. So if I were to-- so the thing is, we've been talking about dimers. But you could talk about, what is the dipole of something more than a dimer. So this is polar. What about these other things? Right. So if I look at molecules, right, that are polar, it's not just about HCl, right, which is that first one up there. But that's kind of the obvious case. But what about CCl4? What if I take a carbon atom and I put 4 chlorines? Those are all polar bonds. And you can see-- well, it's hard to see-- but they've drawn in the dipole moment. You've lost some charge from the carbon. It's gone out to the chlorine atom. There's a dipole movement within each bond. But now here's where VSEPR comes in, because these are going to be tetrahedrally distributed and fully canceling out. So it's like you've got all these kind of electrostatic things going on that all cancel out. So this molecule is not a polar molecule. Right. On the other hand, molecules NH3 or CH3Cl-- right. Now, this is going to be polar, because these dipoles do not cancel out within the molecules. So you can kind of use this thinking. There's BF3, trigonal planar. Right. All those dipoles cancel. It's a non-polar molecule. But if you take a classic example-- which we'll come back-- of, say, water as a polar molecule, I could do the same thing. Right. I can take water, which is oxygen, hydrogen, hydrogen. And you see you've got delta plus there, delta plus there, and you've got some minus there. Right. Now, there's a dipole moment, so part of it cancels but not all of it. So there is a dipole moment for the net water molecule. The net dipole is what matters to determine if a molecule's polar or not. So there's a net dipole, net dipole when you sum them all up. And that's what's going to determine if this can come along to that xenon atom, right. And so here's my xenon just sitting there doing nothing. And now when water comes up to xenon, what's going to happen? Well, what's going to happen is I've got my water molecule, my water molecule, my delta minus, my delta plus, my delta plus. And because it's got a net dipole, it's going to form the atom. And so I'm going to have-- let's see, make sure I draw this right. So it's going to look kind of like this. Delta minus, delta plus, and that allows a water molecule to bond to a xenon atom. It is bonding with another bond, which is very similar to what I just wrote-- gesundheit-- But it is a dipole-induced, not ion. It is a dipole-induced dipole. You see that? Because now, I didn't have an ion. I had a water molecule that had a dipole moment. So now I've got my dipole-induced dipole, and that's going to look-- oh let's draw it. Let's see. So I got my dipole-induced dipole. So we're going to go, kind of, oh, boy, there we go. OK. Minus, plus, and then-- now that might be the dipole of, say, the water molecule or something that. And then I've got another thing that didn't have a dipole, but it got induced to have one. OK. Oh, boy. I went out of the lines. Ah. I was so careful. But you can't-- you can't hold chemistry. Uh-uh. No way. No way. Mm. OK. Delta plus, delta minus, it's induced. That's what that means, right. Now, it turns out that this has an even weaker-- so this would be a dipole. Let's just-- right, let's finish this up-- dipole interacting with the polarizable electron cloud. And the value of this is even lower range, 2 to 10, and the dependence is 1 over r to the sixth. And the example-- let's go back to HCl. HCl, there's my dipole coming in ClCl. I like that because it's got three Cls. Why not? That's a non-polar molecule. That's a dipole. That induces a dipole in this by shifting its polarized electrons around. Right. And then it allows there to be a little bit of attraction. You might be saying, but that's so little. We'll get to that in a second. That's not so little. That's actually not so little when you've got a lot of it. OK. OK. Dipole interaction-- now it turns out that even-- and this is what's mind blowing. You knew this was coming. This moment today was going to come. I can have an interaction without any dipole to start with. It's true. I can take two non-polar covalent bonds, two non-polar covalent molecules, H2 and H2, and they are attracted to each other. How is that possible? Well, it's what explains this. I mean, we know-- why does helium, argon, and xenon-- why do these have different boiling points? By the way, the boiling point of these-- all right, so these can be made into liquids and you can boil them, right-- is really different. And it's a measure of how strongly they're bonded together. Right. So this is a direct kind of way of thinking about, of measuring something that's related to the bond strength. So why do these atoms-- they're all just atoms-- you know, they're all just like this. They don't have this dipole. Right. They're non-polar. They're just atoms sitting there with their symmetric charge clouds. How is it that they can be attracted to each other in the first place? And how is it that they can have such a strong trend? And that has to do with another kind of force. And there is a picture. So here's two examples. There's two helium atoms. So that's the first one in this list, boiling point minus 269 degrees Celsius. It's pretty cold. And there's two non-polar molecules, H2. What happens is that you get fluctuations. You get fluctuations. And what I mean by that is that, just randomly, literally, randomly, right, because of collisions or thermal energy-- randomly, one of those could form a dipole. I didn't mean it. It wasn't my happy place, but it happened. Now, if it happens and your charge kind of goes to one side, even if it's just for a split second, that's a dipole that then, if you're near another atom, can induce a dipole. And we know that if we can induce a dipole, if we're here, if we're at this stage, we've already got that covered. We already know that I can induce another dipole if I'm charged. Right. And so what happens is you get this kind of fluctuation that leads literally to a bond. It leads to a bond. Right. And that is called London dispersion. That's called the London force, and the London dispersion is the name of a kind of bonding you have. Let me just make sure-- ha. OK. London dispersion. Notice there's room for one more. It's coming. There's more. We're not done. We are not done. I can literally take two non-polar atoms or molecules, and I can find a way for them to be attracted. And the way it happens is thermal fluctuations. This is such an important force, and it's such a strange one that I want to write down what it is. It's that non-polar-- we've been talking about polar and charges, but non-polar molecules are attracted by fluctuations. Fluctuations. I should say, by dipole fluctuations. That's really what's happening. Right. By dipole fluctuations. So one of them gets a little shift, and there's a dipole. And then that dipole, if there's another one around, can induce a shift, and then there can be a bond. And now, this gets back to something very important, which is that-- you can imagine, how easy was it for temperature-- how easy was it to create that shift? Right. Well that's going to have to do with your polarizability. Now, OK, so what is it that has to do with polarizability? Right. Well, if-- oh, I do have a board here. Ha. OK. You can already imagine, if I go back to this list here-- if I go from helium to argon, what have I done? I've gone down in the periodic table. And we've already talked about this. Those electrons that are going to do the shifting to cause the fluctuation-- those electrons are more loosely bound. Right. They're further out. They're more loosely bound. We've already talked about this. We've looked at this in a lot of ways. Right. Whether it's radius or ionization energy, those are-- if they're more loosely bound, then they're easier to shift around. I'm not taking them out. Right. I'm not taking them out. But I'm moving them. That's what London is. I got to fluctuate it. So you can imagine now, why, if it's heavier, and those electrons are further out, they're more polarizable. Right. So let's write that down. Right. So the London-- so we're going to get a stronger London-- two reasons. One, heavier atom, which really, in this case, right, it's heavier, but it really just means more polarizable. If it's more polarizable, that's the key. That's the chemistry key, more polarizable, more polarizable electron cloud. Write it all out. But the other thing that you can imagine with London-- right, so that explains that trend. The other thing that you can imagine with London is that these interactions, these fluctuations, they're happening all over the place. And the more contact I have-- the more places in a molecule I have to induce this, to have this fluctuation happen, the stronger these forces could be. And this is exactly what we see. Look at this trend. Methane, ethane, propane, and butane. All I've done is I've added another carbon atom. Oh, we're going to get to these. We'll talk a little bit about Orgel later. But for now, look at the boiling points of those. Again, boiling point being a proxy for the bonding strength. What bond could they have? None of those, only this. Ha. I got excited. Only in London, because look, they're non-polar molecules. They've got no charge on them. This is the only way they can talk to each other. But look at how much they talk to each other. Now, why? Because of the second thing, right. Which is that the greater the surface, the greater the surface. You can have more. The more surface you have-- the greater surface, more contact area, more contact area. And that's how this bond-- that's how this thing works. Because as long as I've got more surfaces see each other, then there can be more fluctuations, right, that are happening, that then induce the other one to have a fluctuation, that then create a little moment of bonding, that dramatically changes the properties. And look at this. This is a little bit small, but, you know, these are the-- so let's go all the way, right. CH4, OK. Let's go all the way, and look at this. It determines when you have a gas at room temperature or when you have a liquid at room temperature. Boy, is that important for how we use these molecules. Right. Boy, is that important. It's all London. there's no other option. Right. But that's what's dictating this trend. It's even more mind-blowing than that, because look at this. This is one molecule that's identical. This is C5. This is 5 atoms of carbon, n-Pentane. Same exact grams per mole, because it's the same chemical formula. But look here, 36.1 degrees C is the boiling point, and here it's 9.5. Why? It's all this. This is not the ground state. The most stable structure is here. But there is a version that's called neopentane. It's like a nickname for it. There's a version of it that you can make that has a different shape. And because this shape cannot come into contact with itself, look at this, you can see it from the red line. It's coming into contact with itself. In the liquid or gas, it can't have as much London forces, London potential. So it cannot bind as strongly, and therefore the boiling point is a lot lower. It's a lot lower. Now, there's a little bit of naming confusion. I want to clear this up. Oh, boy. I have nowhere to go here. So I'm going to get rid of London for a second. And I want to make something very clear that there's another word that's used, which is called Van der Waals. OK. Now, this ability of a fluctuation to happen, which then induces a fluctuation, that can happen no matter what molecule you have. It can happen. And it does. It's always there. All right. So all molecules, whether they are polar or non-polar-- polar or non-polar, all molecules are attracted by London plus any other attractive forces-- any other attractive forces present-- forces present. OK. So London is always there. Now, people interchangeably use the term Van der Waals. And what Van der Waals encompasses is more than London. Van der Waals, what you will sometimes see in textbooks, is the Van der Waals force. OK. So that is any or all added together. This is the combination of these weak forces of dipole-dipole-- dipole-dipole. Let's see, dipole-induced dipole, and London. Those taken together you will sometimes see referred to as Van der Waals forces, or weak forces. But we know more than that. We know the distinction. Right. We know that London started like this and then went like this. Ah! Really bad. Gesundheit. Right. We know that it started with non-polar, and that there's a difference. And so this is all about the polarizable electron cloud. Polarizable electron cloud, that's all you got. Right. And this can be really small, 0.5 to 40. Oh, yeah. Let's just keep with chlorine 2, chlorine 2. Two non-polar molecules, London dispersion, and this also has a dependence of-- in terms of the distance-- of 1 over r to the sixth. Now, in case you think that's small, I want to show you a little video of why this matters. So weak forces-- now, weak forces are really strong. We know this because of "Mission: Impossible". We know this also because of our own students. There was one of my former students, Rory, and he won all sorts of prizes, because he was trying to be like a gecko. And so he made gloves that were like gecko gloves, and much to the delight of the facilities people at MIT, he actually climbed one of the buildings using his gloves. In case you are wondering, it wasn't actually their delight. But really cool stuff. So let me show you. This is like a 30-second video. He found the answer in the sheer number and design of the hairs on the geckos feet. Geckos have millions of microscopic hairs on their toes. And of course, we can't see this with our naked eye, because each hair is only 1/10 the size of a human hair. And each of those hairs branch down to billions of little split ends. And they can make such close contact with the surface that weak intermolecular forces can begin to add up to something really strong. Turns out geckos exploit something called the Van der Waals force. No. London, London. If you think of an atom as a dancing couple, when you bring two atoms into very close contact, part of one atom can get attracted to part of the other. That very weak bond is the Van der Waals force, and it sticks atoms together. Proximity is the key. But bringing two materials that near each other is harder than you'd think. All right. Now, obviously, that got me very excited when I saw that, because that is how I see atoms, and that is how I see electrons. And I saw this, and I almost fell over. And of course, I played it, and I know all those moves. I would suggest you guys-- it's a Friday. There's some good moves in there. There's some good moves in there. And you can take this on your phone and just kind of work it out at the club. And speaking of the club, there's one more thing I've got to teach you. Because it's that last line. It's that last line. See, if you go along a trend like methane, silane, germane, right, these are non-polar molecules, and they're all attracted to each other by these Van der Waals. And you can see that the boiling point, remember, our proxy for the force, the bonding strength, is going up. That's what you'd expect. That's because these are getting larger, and there are going to be more polarizable electrons. But look at what happens as you go to other molecules. This is the expected trend until you get here. Right. NH3-- and that is because of one last type of bond. That is because of something called the hydrogen bond. It's a very unique kind of bond. And so a hydrogen bond is our last intermolecular bond. Hydrogen bond, right. And it's a very particular kind of bond. And it has to do is something very particular that hydrogen does. Look at those boiling points. Look at the change. I mean, H2O is H2O because of the hydrogen bonds. Right. Earth's operating conditions are luckily right here. But this is only good because of hydrogen bonds, and it's all the way up and not all the way down there. So hydrogen bonds are unique. This is the electronegativity scale, and there's something very important that happens for those three elements. Those are the three elements that are in this trend. Those are the three elements. See, nitrogen, chlorine, and oxygen, those are the three elements here. And what's important about them is that they have such a huge electronegativity. And so in a hydrogen bond, what's going to happen-- what's going to happen in a hydrogen is they're really going to pull a lot of charge from the hydrogen. So you have a very high positive charge on the hydrogen, but there's something else that happens. For hydrogen bonding to occur, you see, it's the H with say fluorine, and you have a delta plus and a delta minus, and this is high. This is very high. All right. But you also have a lone pair somewhere. You've also got a lone pair somewhere. And we know that we have that on the fluorine, right. We know that, and we've got those two. And the bond that a hydrogen bond represents is the fact that I've done two things with the same molecule. I've taken a lot of charge off of hydrogen and created a very strong delta plus on the hydrogen. But I've also got a lone pair somewhere that that can bond to. And when I have those two things, which I really have for these three elements, right. When I have those two things, I can make this very special kind of bond. Now, this is what it looks like. Right. So there's an H2O molecule, and I want to make a point here. There's the covalent bond between the oxygen and hydrogen. There's the delta plus. And there's that hydrogen bond right there. Mm. Mm. There. OK. Hydrogen bond that forms between the delta plus and the lone pair. Now, you can see why water is so special. You can see. Because unlike NH3 or HF, you can see where water had the biggest kick up from what would be expected. Because unlike NH3 or HF, you have exactly the right balance. I've created two of these very strong delta pluses, and I've got two lone pairs. It all all works out. There's no extra lone pairs, like here. And all these extra lone pairs, right, that don't-- you know, each molecule only has one hydrogen. Here, it's two and two. That's why water is so special. This bond is so important. I know you've probably heard about it for water, but here's the example I want to give you for the club, because some of you might be able to drink ethanol, only if you're at the right age, please, of course. This is what's in alcohol, right. This is the ethanol molecule. And it has a boiling point of 78.5 degrees. Again, that tells you something about the bond strength of this molecule to another molecule. But look at this. That's an oxygen with a hydrogen. This took a lot of charge from the hydrogen, right, so you've got a strong delta plus. But you've got those two lone pairs there ready to bond to the other ethanol molecule. But look what happens. Methoxymethane is a gas at room temp. It's boiled off. It's the exact same chemical formula. It's the exact same chemical formula. All I've done is shift the oxygen in, and so now there's no hydrogen that's had that big of a delta on it. So it cannot hydrogen bond. It can London. You know. But it can't hydrogen. It can't. Because the oxygen has the lone pairs. But there's no hydrogens that have that strong of a delta. So when you're at the club tonight, trying out those moves, go up to somebody and say, it's a really good thing that's not methoxymethane, isn't it. Really good thing. And tell them about hydrogen bonds. Have a great weekend. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | Goodie_Bag_9_Polymers_Intro_to_SolidState_Chemistry.txt | [SQUEAKING] [RUSTLING] [CLICKING] CAROLYN JONS: This is Goodie Bag number nine. In today's video, we will be exploring the topics of polymerization and cross-linking and learning how these processes can be used to make slime at home. You'll need gloves, Elmer's glue, borax powder, plastic cups, and spoons for stirring. One product made of polymers that many people are familiar with is Silly Putty. Silly Putty is made of a polymer called polydimethylsiloxane. Polydimethylsiloxane is made of repeating units called monomers that include a silicon, an oxygen, and two CH3 groups. Today we're going to make an alternative material that behaves a little bit like Silly Putty, but is instead made of a different polymer. This can be made by combining Elmer's glue and cross-linking with borax. As you can see, after I added borax to the Elmer's glue, the material became much more viscous. This is because of cross-linking. Cross-linking is forming a bond to link one polymer chain to another. In this experiment, the polyvinyl alcohol and Elmer's glue forms hydrogen bonds with the boric acid. These cross-links make it more difficult for the polymer chains to slide past one another, so the material becomes more viscous. By adding in more borax, we increase the number of cross-links, and this results in a material more resistant to flow. You may have noticed that when we pull quickly, the slime breaks in half. But when we pull slowly, the slime is able to stretch. This is because the slow pulling allows hydrogen bonds to break and then reform, allowing the chains to more easily slide past one another. In this Goodie Bag, we hope you had an opportunity to make a polymer of your own and witness how cross-linking can change material properties. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 33_Polymers_II_Intro_to_SolidState_Chemistry.txt | Let's get started. We're going to keep talking about polymers today. Polymers-- that's also going to be the topic of the quiz. And because of the exam and everything-- and so we're having our second lecture on polymers today. The quiz tomorrow will be at the end of recitation so that you can have a full second recitation on this topic. OK. I mentioned this on Monday, right? Monday we introduced polymers and I covered two different ways to make polymers, OK? Let's write those down. Let's write those down with chalk, which is somewhere. Here we go. On the one hand, we had what we called radical initiation. So we use a radical initiator. Remember this R dot? And we talked about radicals, OK? And how that can lead to this kind of chain reaction. Right? And so this is called chain polymerization. Sometimes it's called addition, polymerization. But it all involves a radical to start it and then a double bond. Remember, we talked about that double bond. The importance of the double bond. Right. So this would have some monomer. This would have some monomer. But it's got to have with a double bond. That's not an equal sign, that's a double bond. Now, the second way we talked about was called condensation polymerization. So condensation polymerization these are two ways we covered that allow you to make these really, really long chains. That's what a polymer is, right? Polymer-- "mer" being from say hundreds to thousands and even millions. And in condensation polymerization, you would have two different-- you could even have more. --monomers. And these two different monomers, well, let's write it explicitly. They react, OK? And they react because they've got these end groups that when they see each other, right, from one monomer to the next, they form a bond. And in the case that we did on Monday, they formed an amide bond. And they gave off water, right? So water left. So when you do condensation polymerization, you start with two different monomers. They've got the end groups on them that when they see each other, they want to react. You wind up with a polymer that weighs a little bit less than the two monomers, right? Because you've given off water. You could give off other things. There's a lot of examples. So in the reaction you give off maybe H2O, like we did. Could be HCL. Could be NH3. There are a number of ways. I don't want you to get the feeling that the only way you can do condensation polymerization is the nylon example, all right? There's a lot of ways you can do this. But one thing that is certainly true is that in forming that bond, you give something else up. And that's why sometimes people like calling this addition. Because in addition polymerization, right, you take a monomer and you just keep adding it. And the sum is n times the monomer, right? The sum of the weight. Whereas in this case, it's a little bit less because you did a reaction. OK. So that's all kind of getting us back in the mood. That's what we did on Monday. Now today, I want to talk about the properties of polymers and how we control them. How we can change them. Different ways that we can change them. I can't cover everything but I want to cover some key elements of engineering polymers. So we're going to talk about the properties and a couple of specific things that we use to control them. So let's start a list here in the middle. And the first thing is something we already did talk about Monday, which is just the monomer itself, right? So what is it that matters in a polymer? What is it that dictates what a polymer will be like, right? Well, one is the monomer. So that's the unit-- well, OK. We just said that you need two here. So maybe there's monomers. But that's the unit that you're repeating. Clearly that's going to be really important, right? What did you put there, right? Is it C2H4? Is it polyethylene or is it something else? Did you add something to it? The benzene ring, right? And so that's going to be really important. The other thing that we talked about-- and so we're going to create this list --is the molecular weight. Now remember, molecular weight. Oh we, like every little community, likes to come up with its own names. We know grams per mole. And then they say, oh that's a dalton. And so now we have to say, OK, a dalton is a gram per mole. Kilodalton would be 1,000 grams per mole. But we also said that this gives us the degree of polymerization. So that's really important, right? Because how much to the degree of polarization-- I mean, if I know what my mer is and I know the molecular weight, the grams per mole of a strand, then I know how many mers went into it. So I know the degree of polymerization. That's just n, right? It's some number. But this is important and I'll put it over here so I can keep going down on that board. When you have a synthesis of a polymer, doesn't matter which one of those you use, you're going out 100,000 units. Yeah, I can't say well, every single one is going to be 100,000. That is really, really hard to do. So instead you get a distribution. So when you say the molecular weight, what you mean is some average, right? So this would be like maybe the molecular weight, maybe this is in daltons, maybe it's in kilodaltons. And you say, aw, the molecular weight is 40 kilodaltons. No, that's the average. You might have ones that are way, way longer out here. Maybe orders of magnitude longer. This distribution can be orders of magnitude. But it still can be you know, you can try to make them longer so you have a distribution that's more out here. Or you make them shorter, right? So this is something that is important and that can be engineered. You can imagine if my chains are really, really long, then maybe it's a tougher material, right? Because there's just more covalently bonded backbone to break, right? You can imagine that that might be one thing. Oh, but the thing that we gotta talk about is the chemistry which dictates the bonding. OK, so the chemistry. So the interactions. And this takes us back to so many happy moments from this fall where we talked about intermolecular forces. All of those apply to polymers. They all apply. What are the IMFs? Well, if I gave you an example, like a question-- OK, maybe you could say, well, let's see-- OK, I've got polyethylene. Remember, this how we write it? And in this case, I've left off my Hs. But the little sticks there means there's an H above and below each one. I've got polystyrene. So that's polyethylene. Here's polystyrene. Remember, this one has a benzene ring. Oh, that's a terrible drawing of a benzene ring. But it's what it's supposed to be. They have to come out of the parentheses. If it doesn't come out of the parentheses is not a polymer because you're not indicating repetition, right? OK, that's supposed to be benzene. And that'll be polystyrene. And then how about another one? I'll put it here. How about polyvinyl alcohol? So now watch this. I'm going brackets because remember, it's OK. And so here OH, right? And each one of these would have some number of repeats. That's polyvinyl alcohol. I can say well, just look at those monomers. Just look at the mers, right? Remember, the monomers if these were made with chain polarization, the monomers would have a double bond, right? Remember that. But so this is the repeat unit in the polymer. That's what goes into the brackets or the parentheses. I can look at it and say, which one of these is going to be the strongest mechanically? And you know from your IMF days. You know which one it is because only one of them can form a stronger bond than London. They're all going to have London. Everyone has London. But this one has hydrogen as well. That's got hydrogen bonds. So the polyvinyl alcohol will be stronger, right? That will be the stronger one. And you know that bond is going to be in there. It's going to bond to other hydrogens from other chains or maybe from itself. Because remember this is what it looks like. But even much, much, much, much, much more. These are macromolecules. These are macromolecules. OK. Yes. But now I wanted to put these here so I had them. So I'm going to take one of these. Which one is it? It's polystyrene. It's this one with the very poorly drawn benzene ring. . And I didn't change anything about it but look at what it can do. That's polystyrene in three different forms. They're all cups but you know these cups well, right? One's Styrofoam. One, I don't know what you call that. But they're all plastic cups but they have very different properties. They have the exact same polymer. Polystyrene. How do we do that? That's not the IMFs. It doesn't even have to have anything to do with a molecular weight. It could. No, it's something else. And that is the density. Oh and so now we get to talk about density and crystallinity. And I just saved three letters. Makes me very happy. Now crystallinity density, memories. Good memories. Very good memories are happening right now from when we talked about amorphous materials and the glass transition, right? Because remember, I drew this on the board. Let's see. Let's get these to come down. What that actually looks like-- each strand in there, remember, goes like this. And then it's like, oh wait. I can make a crystal. And then it's like, nope. I'm amorphous again. And then it's like oh wait. Hold on. I can crystallize right here. And then it's amorphous again and it's so fun. And then you go like this and that. And that's like one one hundredth of a strand. And so you know the crystalline region-- what do I mean by "crystalline?" Well, the strand is stacking up in some ordered regular repeating way. What do I mean by "amorphous?" Well, I mean spaghetti. But a polymer, in its own strand, can crystallize to some extent or to more or less, right? And that's something we can control. We can engineer that. And that's what leads to these differences because you know that in a crystalline region, it's going to have a lower volume, right? It's going to take up less volume. That's what we did before with glasses. So let's put that on the board, right? So you know that from out good old days that if I melt a material, well, it could form a crystal. That's temperature. And this would be volume per mole. Right, this would be like the melting point. It might form a crystal. But with polymers, imagine I've got the spaghetti strands and now they're miles long. It's really hard to line it all up. And so a lot of it will not find that place. And instead it will be a glass, which you know is another way of saying so these are now TGs, right? TG. And so you know now all about this, right? And you know that cooling rate is one way to change the density. To change the volume per mole, right? So the density of the amorphous region is one parameter that could be glassy in different ways, like we did before. And then how much of it you can make into a crystal is another parameter. Both of which play into the overall density and crystallinity, right? Both of those are extremely important and those are going to depend on all sorts of parameters related to the processing and the temperature and so forth. Those are the things that can lead to changes like this from the same exact chemistry. So let's go from polystyrene back to polyethylene. So if you look at polyethylene-- so this is a list. I showed I think a version of this on Monday. It's hard to read. Don't worry about it. These are all monomers. And this is the polymer. They didn't put the brackets there but you can see-- oh maybe they did. Yeah sort of. You can see the line, right, coming out. That's the repeat line. And this is the monomer. Notice the double bonds ready for chain polymerization. And here are the properties and here are the uses. Look at this-- LDPE and HDPE. Now LDPE and HDPE. So that's low density and high density of the same exact polymer. So the interactions are the same. But all we've done is we've changed the density and the difference between the properties is tremendous. So when you go to the grocery store and they give you a plastic bag, which you'll wind up feeding to fishes in the ocean as you now know, it's very soft. It's the exact same material that's in over 30% of all toys on the planet. It's still polyethylene. All we've done is mess with that. We've messed with the density. So how do you mess with the density? How? Well, it turns out with polymers there are different ways you can do this. One we just talked about. So one, let's do a little-- I'm going to do this. So one would be let's say processing. I'll call it processing. That's right here. How quickly did I melt it? What were the conditions that I formed it in? But see you can imagine changing its ability to crystallize our pack by just changing something else about the polymer itself. And so another way to do that would be to change the physical structure of the chain. Physical structure. And there are two ways that you can do this. Oh boy, let's see. How am I going to continue this? OK let's do this. Bullet one. One way is called branching. And another is called tacticity. So what are these? So branching is exactly what it sounds like because the polymer I showed you here-- this is a mess and it's showing how it's a spaghetti chain. But notice that it's just a line everywhere. We would call this a straight line. This is a straight line, it just kind of wanders around. Yeah. OK. Because it's in contrast to-- if it's a straight line, then it's sort of locally straight. But what if it did this? And then it's like, well, which way do I go? You can be the exact same chemistry but because maybe one of those radicals in the soup while this was being made came in and got to move a hydrogen around in just the right way. It was able to start growing off of the side, right? That's a branch. So you can imagine that controlling this branching would be a really big deal in terms of the crystallinity right? So now if you go to pasta. It always seems to come back to pasta. Maybe it's just me. I don't think so though. I think there's something deep about the pasta in here. But see, this is a polymer. But you can imagine now if I make spaghetti and it looks like the left, it's all packed in. This is actually an interesting idea for a new pasta brand, right? And what if each spaghetti strand looked like that? And now you take those and you try to make your-- it would be actually pretty cool but it wouldn't pack as well. You can feel it with the pasta, right? You can feel it. It's going to take up more room in the bowl. I mean, it's the exact same thing with polymers. So if you want a polymer to be stronger, to have maybe more crystallinity or higher density, HDPE, then you don't want very much branching. So you better dial that in. When you synthesize it, you better think carefully about how you're making this to prevent those kinds of side chains from growing, OK? Well, tacticity is another thing. And that's not the same as branching. Tacticity is remember-- ah! Where did it go? I had it up there. There's the benzene ring. Are they all on the same side or are they on alternating sides? How did this thing go? Did two go on the same side and then one and is it random? I mean, you can imagine that if I have a way to break the symmetry of the chain like this, then you can imagine that if I break it a lot so I'm atactic, then that's going to be harder to line up as well. That's going to make it harder for the spaghetti to kind of fall right next to each other. But even if this spaghetti had little you can call these little, mini benzine branches. It's not branching. But you can call them little branches. But if they're all on the same side, then you can imagine they can still stack really well. So when you have side groups like this, this is another aspect of engineering the polymer that you got to think very carefully about. Because if it's isotactic or syndiotactic, then it's going to be-- so that's the physical structure of the chain. --then it's going to be able to pack more easily and it's going to be stronger, right? if it's atactic and we're talking about like a hundred degrees. Same polymer, same chemical unit make the molecular weights the same. All you've changed is whether there's symmetry or not. And you can change the melting point by a hundred degrees. Right, of the same material. So that's another way that we engineer the properties of polymers, OK? Same polymer garbage bags to Tupperware. Same chemistry. Isotactic to atactic. OK, now we get to something really fun, which is cross-links. And this is number five, OK? So we're talking about ways to control polymers. Cross-linking is one of the single most important ways to engineer the properties of polymers. And it basically is exactly what it says-- you're linking two different polymer chains together across. Cross linking, OK? Now, it turns out the one-- remember, I showed you the video of the nylon. It's called the nylon rope pole. If you want to see that again, Google it. It's so cool. You have two liquids with the two-- it's a condensation polymerization example. Two liquids with two different monomers and you pull a solid out of the interface. And that's this. That's exactly this. And those are the molecules, these are the monomers that were in the liquid. And there is nylon 6.6. Remember, it's because you have a chain there. And this is what you get. And it keeps repeating. And there's your polymer. The reason I'm saying that nylon actually cross-links is because it's something very important. It's not just that these have different interactions with each other. No. It's something a little bit more than that. So if I look at this and say, OK, here's a carbon and a nitrogen. I'm going to just try to draw a part of this. Carbon, nitrogen, there's a hydrogen, and oh boy, here we go. Here's an oxygen. OK. And it goes on. Oh, I'm not going to draw what's in there. It's 6.6. And then comes another. OK, nitrogen, hydrogen, and then it goes up to a carbon, oxygen, and then ah, 6.6. Or whatever it is, right? And then it goes to a carbon and oxygen. This should have been like that. And this goes to nitrogen and so on and so on. Oh remember, this is called the amide bond. That's cool, right? That's the bond that formed. That's the amide bond. That's the bond that formed this polyamide, right? We did this on Monday. But look at what happened. So if I have another one of these that comes up beneath it-- and I'm not going to draw the whole thing. I'm just going to draw that part that has the amide bond. You can see that if I have an oxygen here with the carbon and let's do oh boy, nitrogen, and like that OK? And so on, then if this lines up-- but it can line up. And then over here you've got a hydrogen. And so you can get this kind of really nice line up to form a hydrogen bond between these chains. You can. It can happen. And that can add, as we know, that can add strength to the material. So that's a cross link because it's a very specific linkage between these strands, right? It's not just sort of an average over different types of bonding. It's the specific link that's in the chain. Well, it turns out that that just seemed to happen. Because of the chemistry that was there it can happen. But look, we can also make it happen. We can engineer polymers and we do to have all sorts of cross-linking chemistry. And it's really one of the most single important ways to control the properties of polymers, right? So I want to talk about that a little bit. And I'm going to do it with maybe the most well-used, at least, example of what a cross-linker does. We're going to talk about rubber. Now, rubber is a natural polymer. You can get it out of trees, right? I think maybe I have a picture. Yeah, I do. Oh, poor Goodyear. But we'll talk about him in a minute. There's rubber coming right out of a tree. And when we see rubber in this class we think of isoprene. And isoprene is this molecule, OK? So here's isoprene. There's a CH3 and then there's a double bond to a CH2 and another double bond to another CH2. Now, the polymer that is coming from this rubber tree is what? It's polyisoprene, OK? So I've got two double bonds. We're going to talk about that in a second. But I only need one to polymerize this. I only need one, right? And so if I think about this, I can make a polymer out of it and the polymer would look like this. This would be polyisoprene, OK? So H, I'll draw the Hs in just so we keep track. Here's another C. And this has my-- oh boy. I said I'd draw them and now I have to. Once you commit, all right, there it is. H, right? And then you've got the double bond in there. And an H there. And then you've got this CH2 unit there. Oh, and that is polyisoprene. It's not-- it looks complicated. It's got a CH3 unit and some CH2 units. But here's the key-- it's got left over, a double bond. All right, that's the key. Now, you see, that double bond, I can do more chemistry with. I can do cross-link chemistry with. So I started with two and I'm left with one because I needed one to make the 100,000 long chains of these things, right so this is isoprene. And this is polyisoprene. Yeah, but the thing is that polyisoprene is not very useful. The Mayans used it. People tried to use polyisoprene directly from nature for centuries. The Mayans would because it's soft, you know? It's like a mold. It's this really cool mold. And probably, at some point, somebody stepped in it. And they're like, look, my footprint. And it stayed. And so then, what they did is they made boots out of it. And so they would put their whole foot inside of bucket of isoprene. and then it would sort of mold really well, right? And they'd make shoes that way. The problem is people tried to do this much later too. The problem with making a boot out of natural rubber is that the natural rubber doesn't hold together very well. So like seriously, on a hot day, the boot melts, which is not what you want. People made clothing out of it too. Jackets were made with polyisoprene inside of two fibers. But then it would melt and get like sticky, like, on a summer day or if it rained out. So you had some issues with rubber. But people really wanted to make something out of this stuff. It seemed really neat. It could kind of be strong but then very moldable. And so along comes Charles Goodyear. And he accidentally left rubber on his stove with sulfur. And literally, the way the story goes is his house started smelling like burnt rubber, which was probably the first time ever that smell was made. And what he found was that this rubber was completely different when it was left on the stove with sulfur. It all comes back to the double bond. It all comes back to the double bond. Why? Because now I've got a place where I can add a covalent link. And that's exactly what sulfur does, all right? So if I take this-- I'll try to draw it again. I'm not going to draw the Hs this time. So I've got this and here's that special double bond, right? And then I've got this and this. OK, that's natural rubber. But now see, with a double bond, I can-- remember, what are these? Well, these are four electrons? But if sulfur comes along, I might be able to take two of them and move them like this. If sulfur comes and offers something that the carbon wants. Hey, come and bond with me. It'll be more fun. Maybe we'll go to a lower energy state, right? And then the carbon's like, you know what? I get that. That sounds pretty cool. I'm going to give an electron from this double bond. I get to stay connected here with a single bond, right? And then I can be here and here and the sulfur can provide that other electron and form a nice single bond, right? Because there's that double bond, you can do the chemistry to create covalent strong bonds to sulfur. And then you get like, another sulfur and another sulfur and so on and these can be maybe 2, 3, 4 sulfur units long, depending on how you cook it. And then this will come. And this comes down and finds another one, finds another strand. And it finds the double bond that's in another polyisoprene strand. That's called cross-- that's well, accidental. He was a tinkerer. And he played with lots of things. And his greatest discovery was completely by accident. And he never made any money off of it. Goodyear Tire, this is Charles Goodyear. Goodyear tires, this is the company. It was called the Goodyear Vulcanite Company. Goodyear called it vulcanization. And the reason is Vulcan is the Roman god for fire. And he accidentally left it on the fire. And so that's called vulcanization because of that. So this used to be the Goodyear Vulcanite Company. It's now Goodyear Tires, right? And what happened? What happened is this low sulfur bonds that came about accidentally, the covalent bonds, held that material together. And it gave it a controllable flexibility and hardness right? That's what they were after. That's what they were after-- flexibility and hardness. And so what is actually happening in this? Well, what's happening is you're adding these covalent bonds between these very, very long strands, right? And so you can imagine that if you pull on that material now, well, the long strands can detangle. So in the rubber tree, these long, long, long, long strands, they're all wrapped up. It's a spaghetti bowl. And I start pulling. And they don't break but they start to kind of slide and detangle, right? And eventually, you just break it all apart. But now, I've got a lock on it because each little piece of spaghetti is tied to another one somewhere. So I can untangle them only to a point. And then these linkages kick in. And they're like, no I'm sorry. You can't stretch anymore, right? That's what the crossing does, it gives it that strength. And so you can imagine, if I had more and more cross-linking, I might have a stronger material, right? The stress strain curve could go like that. Not much cross-linking. A lot of cross-linking, right? Because more and more of it is going to be dictated by these linkages here between the strands. And then there's something really cool that can happen with cross-linking because this is an example of a covalent bond. This is covalent. But cross-linking can be-- let's see, it can be covalent, it can be hydrogen. We already saw that there's sort of natural cross-linking in nylon because of that hydrogen bond. You can introduce hydrogen bonds as well. You can be purposeful about it. By the way, you can have ionic-- you can put any kind of bonding that you can dial in, and there are different ways to do that, can become a cross-link opportunity, right? So you can just imagine how much this opened up polymer design. How much this cross-linking opens up polymer design. Now, there's a downside as well. But first, let's say positive. Let's go to your goody bag. This is the chemistry inside your goody bag, right? Now, this is a very purposeful. Right, these are the Borax cross-linkers you've got that in your bag and you add it in. And what I want you to do is get an actual feeling for the cross-linkers, right? And so you can add more and you'll feel the material change because you're linking more and more of it. But see, in this case, it's a hydrogen bond. You see that? See, those are the Borax cross-linkers. Those groups keep their OHs and then those OHs, there's the polyvinyl alcohol. I drew that on the board somewhere before and those OHs can now bond to the hydrogen or the OH and the PBA and that can form a cross-link. But there's something different about this one and it's so cool. And that's why slime is so cool and Silly Putty, right? Because now there is a timescale involved. There's a timescale involved. So now you can pull it very slowly and the hydrogen bonds that are cross-linking haven't have time to change, to move, to reform. Yeah, they break and then they reform, right? But if you pull it really fast, those hydrogen bonds don't have time, right? And I'll show you a picture in a second. And so what can happen now is the mechanical strength of this material, the brittleness of it. The properties of it can literally just depend on how quickly you pull it. Because you are up against the speed of hydrogen bond forming and reforming. And that is incredibly cool. You are literally feeling the formation, the destruction and formation of hydrogen bonds. But there's something else that's going on in this that's also so cool. This material, slime, that polymer is below its glass transition temperature. Sorry, above. I meant above. What does that mean? It means that without the cross-linkers, that polymer is a viscous liquid. We drew it somewhere. I erased it? I erased it. If you're below the ground the glass transition temperature, you're basically a solid, like a brittle solid. Well, like I drew, polymers are amorphous, maybe with some crystalinity but a lot of it's going to be amorphous. Maybe all of it. And so it's going to have a glass transition temperature and if you're above that, it's this viscous liquid. It's this like super cooled viscous liquid. But what I've done with the cross-linkers is I've said hold on. You can't go away. So I'm literally making a solid out of liquid. We say yeah, just put like, water in the freezer. No, that's not-- the phase of that polymer is liquid. It is a viscous liquid. But it is not allowed to flow or if it does, it maybe can change its shape a little. Bit it can't just flow all over the place because of those cross-links, right? And that's an elastomer. And if you do it just right, you get a viscal material. And what do I mean by that? Well, all these things have to line up, right? The thing has to want to be a liquid so it's got to have the right glass transition temperature. But the cross-linkers are sort of holding it in place. And the cross-linker chemistry has to be the right strength. It can't be too strong, can't be too weak, right? And again, by accident, you might make cross-link silicone. Oh there's the goody bag, which is Silly Putty. And that's why I wanted to give that to you because this hits all the right combinations. In this case, the backbone is made out of silicon, right? Not carbon. But it's the same thing. It's the cross-linking that does this. So there's an example of Silly Putty, which you all have the goody bag. And if you stretch it slowly, those hydrogen bonds-- the strands can kind of pull away. And those hydrogen bonds have time to slowly form and reform. That's why it kind of feels the same. Nice, elastic right? Depending on the material, on the backbone, it might go back as well. In Silly Putty, it kind of just keeps on-- those strands just keep on sliding and the hydrogen bonds keep reforming. But if instead I pull it fast, the hydrogen bonds don't have time to react. That's why, if you pull it really fast-- like OK, don't do this. Somebody shot a bullet through it. It breaks like glass. And you can feel it in your hands, just pull really fast. The hydrogen bonds don't have time to reform, right? And so what you're left with is you're just breaking apart this material. Now, how elastic it feels, right, how viscous it feels. All these things, these are going to be determined by the glass transition temperature of the temperature of the room, right? And so that's going to depend on whether you actually have a very vicious liquid. Or maybe you've got a solid, then it's going to be more brittle. All of these things are now at your disposal to tune the properties of the polymer, right? They're all at your disposal. And it's incredible to think about, for me at least, holding together a liquid in place like that. And that's leads to viscoelasticity. Viscoelasticity, right? That's just viscous liquid in elastic, right? OK. Now, there's a couple more things that we have to talk about. And on Friday. What I want to do Friday is I want to come back a little bit to some of the ways that we're trying to maybe engineer polymers to be more like nature, right? We were talking about Monday, if you want to try to solve the problems, try to do something big. And I gave an example of an ocean cleanup project. But the ocean cleanup project wouldn't it matter if we keep dumping more and more plastic in the oceans that doesn't degrade. All right, so we have to think very carefully about what kinds of plastics we can make. And the problem with cross-linking-- this gets to the negative side, right? And this is a way of categorizing these things that I've been talking about is if I cross-link it a lot, we call that a thermoset. We call that a thermostat. Why? Because when you heat it up, "thermo," it's set. So you mold it and then you've got your little cross-linker. You know, you pour your sulfur in and then it's got a very high number of cross-links. And you can see it there, right? And it gives you very strong material. But you can't break it back apart. So it's very hard to recycle. It's almost impossible. If you try to heat these back up, they just burn usually. And if you melt them, they're not usually that useful because they've got all this other chemistry and it completely messes with the polymer. On the other hand, a thermoplastic has this kind of plastic deformation. Think maybe like a garbage bag. So it can really stretch and stretch and stretch. But it doesn't have the cross-linkers. That's why you can do that. And so this is actually much easier, heat it up, melt it, and you can reuse it. So that's a recyclable polymer. But the issue there is it's very difficult to get all of the necessary mechanical properties from these thermoplastics. And so elastomers, which are maybe lightly cross-linked polymers that are above the glass transition temperature often, those are somewhere in between. But most of the time, even if you cross-link lightly, it's very difficult to recycle. And that's a really good problem because maybe we can come up with cross-link chemistries that are reversible, right? They can give us an opportunity to have the best of both worlds. It's elastic when we want. It's deforms when we want and it's solid when we want. Oh and by the way, you can degrade all those cross-links and reuse it. That's like one of the dreams in making environmentally friendly plastic. See you guys on Friday. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 8_Ionization_Energy_and_Potential_Energy_Surface_PES_Intro_to_SolidState_Chemistry.txt | We left off last Wednesday, last week Wednesday. So it's been awhile. So I wanted just to pick up where we left off. And then cover some new things today. But where we left off was with this diagram, OK? And, you know, I kind of went through it. And I went through it sort of quickly. But I want to make sure you understand it very well. Because this is such an important kind of diagram. And we'll see it again. And you may see it again in other classes. This is a diagram of attraction and repulsion coming together in a balance to form a bond. So, you know, last Wednesday, just to remind you, we made our first bond. All right, we made the ionic bond. And I showed you how that worked. Because, in this case, the attraction, the force of attraction is the Coulomb potential, right? Remember that goes as minus a constant times-- I'll just write it down because why not. You're using it in your goody bag. All right, so the Coulomb energy here goes as minus some constant times the charges, the two charges, divided by the distance between those charges. That's an energy. It's an energy of attraction. But, you know, so here's the thing though. All right, so if you have a plus and a minus charge, those are each 1, right, charge of electron. And then the distance between them, when it's really far this, energy is pretty weak. So that's this green curve here. But as it gets closer and closer, that energy gets lower and lower. And remember, happiness is lower energy. All right, these atoms want to find their happy place. And they see each other and they're like, well, we can be happier if we're closer, until a certain thing happens. Which is then, they've got similar charges like the electron shells, negative charge. Negative charge, so kind of coming too close. That's the forces of repulsion. And so that nets in a total energy curve, which is this white line here, the green and the red, and then you got the white line there, which is the total energy. Because if they came too close, then you're going to be on this thing. And you're just gonna be-- you know, all those electrons are going to be overlapping each other and very unhappy. So this is what we do. And we said, OK, this is the bond of an atom and another atom, an ion and another ion, a cation and an anion. We did all this Wednesday. I'm just getting us back in the mood. And what I told you is that that bond energy is related to the lattice energy, all right? I mean, now, technically, the lattice energy is-- and this is something you're playing around with-- the lattice energy would be the energy that it takes to go from those ions, sodium-plus in the gas phase, right? We put that little guy there. Now, that is something that's going keep coming, too. It's in a notation I've been sneaking in. Gas phase, all right, plus chlorine ion gas phase, and they came together to make NaCl-- ran out of room-- solid. But I put it underneath, solid sodium chloride. That's salt. Now, the lattice energy is the energy that it takes to go from these ions to the solid. And as you can see, well, here I just did two ions. But the lattice energy is going to be related to that bond energy, all right? OK? So the lattice energy of the solid that you make is going to be [INAUDIBLE]. And ionic solids, there's a whole bunch of materials that form solids this way. They're ionic solids. And they have properties that are general. They're not always these properties. But for the most part, these are the properties of our ionic solids, right? They tend to be solid at room temperature. They tend to have high melting and boiling points. They're often transparent in the visible. They're mostly electrical insulators. They're hard and brittle, soluble in water. But in your goody bag, you're seeing that's not always true. Because sometimes, well, most of the time there, but if this lattice energy is too high, then it may not even dissolve. It may still be an-- it just may be some really, really strong ionic solid. But these are the general properties of ionic solids. We'll come back as we learn about other solids Wednesday to the differences in properties between different types of solids. OK, so that's where we ended. Now, to do this, to get to here, we had to make ions. And so we've been talking about ions. And we're going to keep talking about ions because that's a very important part of how atoms see each other, all right? Am I seeing you as a neutral atom with all your electrons in your shell or am I seeing you with a charge missing or not? That makes a big difference in how they approach each other and therefore how they bond, all right? This is a critical part of what we need to learn this semester. And so, you know, we showed this, which is sometimes atoms actually don't mind losing an electron, don't mind. Don't mind is like happiness level which has to do with energy. And so if we want to look at this more formally, we would look at, for example, this chart from [? Avril ?] which is the first ionization energy. Now, that's something that we talked about. So the first ionization energy, which is how much energy it takes to remove an electron from the outer shell, the most outer shell electron, the first one you would take away. That's the first ionization energy. All right, and we saw, again, getting us back in the mood. Oh, I've got colors there. So if you started, for example, let's just take a look at this. OK, so here we are at lithium. I'm going to start here. And that's first ionization energy. And this is just the atomic number. So there's lithium. And remember, we looked at this last week, lithium and beryllium. OK, so if you look at that, we said so for lithium, you had three electrons in the core. So there's lithium, right? And there's the lithium. Oh, this is so fun because now we know what to call this-- 1s2. All right, and then we have another shell. These are just pictorial. The electron is not in an orbit, it's in an orbital. It's not moving around like this. That's our classical minds fighting. And it's got one more electron. Why did I draw a dash? It's got one more electron there in the 2s shell. And that's lithium. But now, we're going to go from lithium to beryllium and we talked about this. So beryllium now is going to add, OK, it's going to add another proton. It's got all of them. But I'm showing the one that was added. And it's also going to add an electron here. But see, for beryllium this electron is really not shielded any more, not shielded more for the most part. You know, it went into the same shell, the same N, all right, and the same L, same shape, same quantum number. So there's nothing else shielding it. And yet, there is a whole other proton. And that's why beryllium is smaller. This is all stuff we talked about Wednesday. But see, look, that's also why it's harder to pull the outer electron off of beryllium-- there it is-- than lithium. So those really go together. All right, all the radius stuff we talked about holds, you know, that same kind of logic in terms of how these electrons are with respect to their nucleus holds for ionization energy. Now, let's go to boron. So if we go to boron, now, we're going over, look at what happened. Look at that. We went from there to there. And the ionization energy got lower. Well, that's also something we can understand. Because now, we're putting it into the 2p. OK, so if I do that, it would be, I'm going to put it under here, 2s2, right? And now, if I go to boron, I'm going to put it here in a 2p shell. So I'm starting a new L. Now, remember, that means the quantum number is still 2, but the shape of the orbital is different. And now, we know also from something we learned last week that because of orbital penetration, the energies of the 2p and 2s orbitals are not the same. All right, the s-energies are a little lower. And it lets those electrons get a little bit closer to that positive charge that they want so badly. So we know that this is going to be a little further out. And so there's a little bit of shielding, a little shielding that you get for this, plus orbital penetration that we talked about. And that puts it further out, all right? And therefore, the attraction is a little less. And therefore, the ionization energy went down. And now, we're going to fill those up. But it's the same thing as before. Now, you're not going further out, you're just adding and you're not really screening while you add. And so you're adding electrons into this 2p orbital. But you're also adding all the positive charge. Salute. And so, the ionization energy is going to go up, except look at that, half filling, right there, half filling, a little extra stability. Remember, we talked about exceptions, right? Half filling. Half filling. And so, you see, now you can see, we can understand these trends that were being measured. All right, and at the time, these ionization potentials, but now we really can understand them. And you can group them. And you can see if you group them, remember, we also grouped the periodic table a little bit by electron filling, quantum numbers. All right, you had the s and p blocks. Those are also often called the main group elements. They're in the first couple of rows of those. They're the most abundant elements. But those trends, now, you fully understand from the quantum mechanical solution to the atom, all right, and the electron filling of those orbitals. We can actually really understand it all. And then the same thing happens with the d. But it's a little more complicated. By the way, these are often called transition metals. The d-block, I told you about the d-block last week. They're also called transition metals. And the word metal is something we'll talk a lot about in a few weeks when we make metals. But they're called transitions because, literally, they transition in the filling. They transition. You're filling d-orbitals, all right, transitioning. And that makes it a little more complex in terms of what happens. But the same trends hold. The same trend holds. As you add more d-electrons, the ionization energy keeps going up. OK, now, this is extremely important, this ionization energy. The first ionization energy is very important because it tells you so much about the outer electron, whether an atom is going to be an anion or a cation. But all of the ionization, you could throw a whole lot of energy at these atoms and they'll all come off, all right. And, you know, so we played with this kind of energy. This is visible light. You had your goody bag, too. You got a spectrometer. You can see lines in here. This is a continuous spectrum. But you see, you've got this whole, all these different wavelengths, right? And now you see this and you don't just think about colors, you think about energies, all right? And so if you think that way, because I want to know how can I measure these things? Ah, stop, better. All right, and so, you know, if I look at energies, energies, I've got like the visible, which is around, oh, I don't know, say 2 to 6 EV. That's pretty high energy. And then you've got, well-- now, can I shine visible light and kick out, you know, a lithium electron? Well, the answer is roughly yes. You know, the lithium energy ionization, well, let's just go back and look. That first ionization energy, 520 kilojoules per mole, I know that's 5.4 EV-ish. Because I don't mind going back and forth, right? Kilojoules per mole of electrons, Jules, EV, moles, you can go back and forth. But that's right around here. So I could shine visible light and potentially knock that out. But see if I wanted the 1s electron of lithium, well, it's much, much lower in energy. Because, remember, we talked about this. It goes in and in each time I add charge. And so it would take 122 electron volts to knock that out, right? So but we have that kind of light, all right. The UV would be something like, oh, 10 to 100. And then, if we really want to blast these, we could go x-rays. Oh, are we going to have fun with X-rays later in the semester. And this would go to something like 100,000. And so I got it. You know, if I wanted to, I could shine light and ionize the whole thing. And that is, in fact, one of the single most important experiments we do to characterize materials. And so I want you to know about it. It's called photoelectron spectroscopy, photoelectron spectroscopy. And, for short, we will not write that out ever again. I will write out PES, photoelectron spectroscopy. Now, this is a characterization tool. I'm basically saying, well, you know, I need energy. I'm going to get it from some photon source. But I'm going to get enough to just blast all the electrons out of this material, out of this atom. Because, for now, our material, they're just atoms. And I want to know about these atoms. So let's take a look. So what happens when I do this? Well, let's see, first of all, I've got some energy that we know is h nu. It's dependent on the frequency. There are the frequencies. There they are. No, there they are. OK, so now, I've got some atom here. And what do I do? Well, I look, did anything happen? No. What about now? I don't know. OK, what about now? Oh, all of a sudden, they come flying out, electrons. All right? And I measure their energy. It goes right back to the photoelectric effect that Einstein was doing. He was shining visible-ish light onto metals and seeing what electron-- but now I'm shining all sorts of light onto an atom and I want them all to come out, not just the outer one. And so we will. You can see if I measure, so now I'm going to measure the kinetic energy of these. And the ionization energy of that electron is going to equal whatever the energy of the incident photon was minus the kinetic energy of the electron. That's what I want. I want that ionization energy. This is, just to be clear, ionization energy. And like I said, I want it all. Not just the first, I want the second and the third. So now, you see. So this is the experiment that you do. What are the results? Well, the results are really just what we've been doing. But you turn it over. So let's look at hydrogen, OK? Now, what have we been doing? What do I mean by that? Well, what we've been doing is we've been going like this. What's that? OK, this is hydrogen 1s1. I could do that or I could just label it the 1s and then you see the one electron right there. Now, if I shine light on this and I measure, the way that you do a PES for hydrogen is you would look at the ionization energy on the x-axis. And this would be the relative electron count on the y-axis. So you're just counting, all right? And what you would see in hydrogen is there would be a peak. It would look like this. That's it. If all the electrons in hydrogen, 1, are in their ground state, 1s, then you would see a peak and this would be, you know that it would be at minus 13.6, right? That would be the energy that it takes to ionize it. And this would be the 1s peak. And that's what you measure. Now, it gets more fun, because now I can also look at other elements. So let's draw helium underneath it. I'm gonna try to squeeze it in. If I look at helium, well, helium is also 1s. And its PES would be-- so this is now ionization energy. This is the PES plot. And the PES for helium would be also the 1s. But the relative counts would be twice as high. So the peak of helium would be twice as high as the peak of hydrogen. Because, you know, for the same number of photons in, I'm getting that many more. I've got both these electrons coming out. So the relative peaks, I've got two more that can come out. Now, if I go to-- OK, now, it gets really interesting, because now, oh, I even have room, which is making me very happy. Because now I can do lithium right here. And if you look at lithium, lithium goes like this, all right? And if I look at a PES plot of lithium, the ionization energy and the relative electron count, OK, I'm going to have those 1s electrons filled. And there's going to be another energy where the 2s electrons are. So this is energy here. These are the orbitals. And somewhere over here is going to be a peak that's exactly half as high, exactly. I mean, that's makes a lot of sense because there's twice as many electrons. It's the relative electron count in the atom. There's twice as many electrons coming out of that 1s orbital and the 2s. So I'm going to get a half as high peak here, all right. And I just told you, the lithium atom, this electron is 5 point something, 4-ish electron volts to remove. This is 122. So literally, for lithium, this would be 122. This would be 5.4. And you know, that's the ionization energy in EV. And what you would do is you would draw the axis like that. So you don't have to put this all the way over here, all right? And all this means is I've broken up the axis and it's continuing to count here in a different scale, all right? These PES plots are absolutely essential in understanding atoms and materials. Because look at what you get. You get, literally, the electron filling plot turned over. It's incredible. You get it turned over. And you can write this as, oh, let's just do this. You know, I could write this as 1s2. We have so many options now, 2s1. All right, I could write this as helium 2s1. That would be like using the noble gas. I could make what people, you know, there are so many-- I can make box plots, oh, box plots. You can do this. Some people really like putting things in boxes, 1s, 2s, box plots. So these are all meaning, these all say the same thing-- Lithium. They all say lithium, all right? Photoelectron spectra. So I could ask you questions like this. There's one. Ionization energy, megajoules per mole, relative number of electrons, this element has a charge of 2-plus and the PES shown below. What is it? So, OK, I know, 1s2, 2s2, and since this is going up three times as high, that sure looks like 2p6, doesn't it? That looks like 2p6. Remember, it's this turned over on its side, all right, where the peak height corresponds to the filling. That's what that experiment gives you, 2p6. Well, then it must-- But then you have your periodic table, ah-ha. Gesundheit. Neon, it must be neon. No, who said no? No. Don't just-- Yes, it can't be neon because of the question. It's got a charge of 2-plus. Oh, tricky, tricky, tricky. Magnesium, it must be magnesium. That is the PES, right? But it's magnesium 2-plus because I said it had a charge. Thank you, shout out, I appreciate that. OK, so that's the power of the photoelectron spectra. And we'll be using it in the class. And this tells us about how electrons leave atoms. It tells us about ionization energies, not just the first, which is the one all the way out here to the right on the PES plot, but all of them. Because I shine enough energy light to get them all out. Now, you can also-- so that's about losing electrons-- you can also gain them. And so I mentioned this a little bit, but just for completeness, I want to come back to this. You have ionization energy, which is how much energy it takes to pull an electron out. You also have electron affinity. Because some atoms also might want electrons. And so when you look at those plots, it actually all makes sense again. Look at this, some atoms like chlorine-- this is electron affinity by atom number-- some atoms like chlorine really want another electron. Why? Because it's got an incomplete shell. It's so close. It just needs one more electron to fill that outer shell. And it wants it. And that's the electron affinity is how much does it want it. Now, want-- happiness, happiness-- lower energy. That's what this is. So if you're not going to lower your energy with another electron, then you're just saying no. You just say no, 0. I will not take an electron because if you give me one, my energy is going to go up. And I will be a less happy atom. Now, that makes sense, too, right? Filled shells. Nitrogen, nitrogen has got a half-filled shell. "No" to that electron, I like my half-filled shell. It gives me a little added stability, a little extra kick. If you give me that electron, you know, I'm gonna not have that added stability. I don't want it, 0. All right, 0. Losing electrons, gaining electrons, losing electrons, gaining electrons, why does it matter? Why does it matter? Well, we made a solid last week. Today, the answer is obvious. It matters because of Danish wind, obviously. 43%, in 2014, 43% of all the electrical energy in Denmark came from wind. It's higher now. But see, now, this is a quarter of the Danish wind. I only gave you a two minute hemodialysis why this matters on Wednesday. So I'm going extended today just a little bit. Danish wind, people are very dependable. So over a three-month period, this is how much we need, energy, electricity. But look at the wind supply. Sometimes it's really there for us. Sometimes it's not at all there. Sometimes it's predictable. Sometimes it's not at all predictable. With that much of your electricity coming from this type of unreliable resource, that is a huge challenge. I mean, even if you go out to Arizona and you talk about solar subsidies, Arizona is sunny all the time. No, not sunny all the time, most of the time, more than Boston. But this is the sun in Arizona. That's the power you're getting from it. And look at this, these are just clouds. Because Arizona does have clouds. And they passed by and they blocked the sun. Do you know what a nightmare this is for a grid operator? If a lot of your customers get their energy in this way, you know, and all of a sudden, half of the your supply of energy just turns off, I mean, that's a huge problem. And this is one of the most limiting factors for increasing, to a large extent, the amount of renewables we have on our grid. And so, of course, I know a lot of you are thinking, well, just store it. And that's what we need to do. But it turns out that really one of the only ways we have to store energy at this large scale is pumped hydro. And you see, we're pumping water up a hill. When I have access, I pump water up a hill. And then when I need, I roll it back down and I turn a turbine with it. I'm literally just trading energy, potential energy. And then I bring it back, kinetic energy. And then I make electricity. The problem is that, you know, well, as you can see, pumped hydro is going to be good where there's hydro, where there's water. So that's limiting. But it also is a very low areal density. And there are also a lot of environmental challenges with making this work in a way that doesn't harm the environment. So there are a lot of issues with scaling up pumped hydro, a lot of issues as a storage. And so I ask, well, what else can we pump up hills? And we know what the answer is because it's what we're talking about-- ions. Where are they? Ions, we can pump ions up hills. There's my picture. Look at that. Ions, and you know what this is, it's a battery. A battery is two different materials, two different metals, a and b, where one of them has an ion, I don't know, like lithium, for example, that can go back and forth. And the electrolyte is this thing in between that only allows that ion through, OK? Now, here's the thing. So if I'm a metal and I lose a positively charged atom, well, then I got to stay neutral. And the only way to stay neutral is to pump an electron out, all right? So if I want to draw electrons out of this, that's fine as long as I draw ions out of it. And then both of them can do work. And then they come back and they roll down an energy hill. That's what they're doing and going back and forth, shuttling back and forth, back and forth. They're rolling down a hill, literally, of energy. You know, when it's in one metal, it's higher in energy. And then, when I, you know, plug my phone in, it rolls down that energy hill. And as it does, it travels across to the other metal, gets lower in energy, and the electron has got to come around and do work for me. Because otherwise, it wouldn't stay neutral. That's what a battery is. It's all about ions, all right? It's all about ions. It's like a ski lift. I like to think of it, you know, it's like these ions are getting into a ski lift and they're just getting pumped up the mountain. And then when you plug it in, they're ready to ski down and they go down. And that's just cruising across this electrolyte. And I don't know what the electron is in this analogy. But that's all a battery is. You plug it in and the ski lift takes it up when you charge it. And now, you power your phone on and it rolls back down. Well, see the thing is that batteries have seen a Moore's law of themselves. If you look at the Moore's law for batteries, it doubles. So this is the energy storage of batteries over the last 150-ish years. It doubles every 60 years. That's not a very good Moore's law. But that's completely changed. That's completely changed recently. And the reason is all about ion shuttling materials and ion storing materials. This is why this has happened. This is why we've had a revolution in electrochemical energy storage, all right. Because 150 years ago, we only made batteries out of about 10 different materials. And today, there's well over 80 that are commercialized. There's many hundreds in research labs. What is making the difference is that those materials now allow the shuttling to happen. It allows the shuttling to happen more easily, maybe faster. And it allows more of them to be stored per volume. It's all about the chemistry that houses those ions. It's all about the chemistry of the ions. So that's why this matters. Now, we roll things down hills to power our world. I told you this already in the first or second lecture. We roll things down hills all the time, all right? So like methane is the core ingredient in natural gas. And what we do-- and we talked about combustion already-- is we light that methane on fire. But see, the thing is that what nature has done is it has put all this stuff up the hill for us. So over tens of millions of years, nature has pushed the chemistry up a hill. That is literally what it's done in energy. Because, you know, it's made it so that if I light this on fire, it will go down the hill. It will roll down the hill and give off energy. Nature has done that for us. But see, we need to be able to do this as well as nature. And so in the last minute of why this matters, I want to explain why this is so challenging. We need to be able to match. Because if you look at the energy per weight versus the energy per volume of a material, that's a very important metric. We plot this all the time if you work on energy materials-- the volumetric versus the gravimetric energy density. This is good, a lot of energy per weight and volume. Well, look at where gasoline is. I want to do some math here. Because I think this is very important. Because I want you to see, now, if I take one liter of gasoline, so if I take one liter of gasoline, now, the cost is around $1. Well, it fluctuates, but, you know, in this country right around now it's about $1. OK, now, if I look at how much energy is in that gasoline, the energy stored in those bonds that nature has pushed up a hill over millions of years, the energy stored is about 33 megajoules. OK, good. Now, I'm going to make a comparison. One MIT professor, one MIT professor operates at around 60 watts. Now, this is something like-- well, this is equal to 60 joules per second. Now, I know some professors who can operate a little higher than that and maybe some a little lower. But that's like the average-- 60 joules per second. Now, the thing is, if I want to get 33 megajoules out of this professor, then I get, at this rate, 33 megajoules takes the prof 153 hours. Now, here's why this matters. Because 153 hours will cost about $1,530 at the MIT professor salary, which is around $10 per hour. And, OK, but look at this. I dug something out of the ground that nature spent 100 million years making, fine, OK. And I burned it. And I got it for a $1. I got the same amount of energy that it would have taken me literally $1,530 to pay for it. That's our challenge. Now, you can plot other things here. But look at this. ethanol, wood, OK, liquid hydrogen, we're still trying to compress it. But look at this, batteries are way down there. This is why this matters. Because we're not even close to being done. We're not even close. See that great uptick in electrochemical storage. We need another order or two of magnitude still in storing energy, efficiencies, costs, et cetera. So that's my "why this matters" for today. Now, back to ions and electrons and atoms. What I want to take the last 15 minutes of class is to introduce a new way of looking at this. And some of you may have seen the Lewis dots, OK? I'm going to introduce them today. And then we're going to make a whole bunch of molecular structures on Wednesday. And the reason this is so important is that it gives us a sense of these outer electrons. It gives us a sense of how those outer electrons look for the atoms, and then, very importantly, how they come together in bonds. OK, how they come together in bonds. So if I just very, very simply ask, you know, what does the Lewis dot structure look like? Well, first of all, the number of dots is equal to the number of valence electrons. And this is equal to the last digit of the group, so of the element group. I'll show you that in a second. OK, so there's Lewis. Now, Lewis, you know, so there's one of his drawings in one of his original papers in the early 1900s. He actually was a professor here, as well. And then he went and did a lot of this work out in Berkeley. And so they named a hall after him. And his contribution was absolutely profound because in thinking about atoms with dots to represent electrons in the outer valence, it gave us a way to so easily think about bonding. Now, we can think about ionic bonding for sure, which is what we've done. But more importantly, what Lewis helps us with is to think about covalent bonding. And that's the subject of Wednesday, which is a whole other type of bonding that we're going to talk about. OK, so now, the last digit of the element group, OK, good. So there it predicts number of bonds formed by most elements in their compounds, good. And there is the dots. Look at that example, fluorine, or using noble gas notation, helium, that really saved us time. I didn't have to write 1s2. Actually, this looks like four strokes of the pen. 1s2 would have been three. I don't know, honestly. But anyway, OK, there is fluorine. And look, one dot? No. Two dots? No, because you add a dot for the valence. So fluorine has seven dots until you reach the valence, until you reach the valence. That is the key of the Lewis picture. And we talked about this before. We talked about this before. Because it showed there were three really important things that come out of the Lewis picture. One is it tells us, as I've just told you, it tells us about bond formation. It tells us about bond formation. And it makes the assumption that the valence electrons are what matter for chemistry. And this is critical. And I've alluded to this before. You now have a real sense of this, though. From these energy diagrams, you have a sense of this. Because look, if I want to rip an electron off of lithium, I got to spend 5.4 electron volts. If I want to rip the next one off, I got to spend over 100 electron volts. And we said, well, we'll do PES with all these energies. It doesn't matter. I'll get 'em all. But what if I just want to say, well, what electrons are going to be available to participate in something like a bond? Which ones? It's going to be these ones. It's not going to be the ones all the way in close to the core that don't want to be bothered and they've got so much energy. Just to get their attention takes 100 EV. And even that's not a-- But here, a little visible light and I got lithium talking to me. Well, chemically, it's the same. That's light, right? These ones are the ones that want to and can participate in bonding. The ones all the way down in energy, way, way, way low, they're inert. They're inert, all right? So most of chemistry happens in the valence. And Lewis really nailed that. So these dots are only valence electron dots. And there's one more rule. Now, OK, so there is the seven dots for fluorine. And then, OK, sometimes you'll see this. Oh, this is just the number of dots. It's this classification of the columns, all right, the groups in a pair. But, no, that is not OK, not standard. Remember, the IUPAC, which throws the best, most lively parties and conferences, they decide on how to name elements. And many, many other aspects of our lives are decided by them. And they said, no, that's an old classification. It's confusing. So we're going to go with the standard, which is that we still count 1 to 18, 1 to 18. 1, 2-- s-block. 13 to 18-- p-block. All right? OK, that's cool. 3 to 12-- d-block. We're going back and forth. But you can just take the last digit there, IUPAC standard. Even the periodic table talks about it. It shows it. It shows the old classification there. And then it says, no, don't use it, it's confusing. Well, then why do they show it? Because some textbooks still show it. So I want you to be aware of that. OK, now, you can use Lewis, you can use this idea of only thinking about the valence to understand when atoms have similar chemistry, all right? So if the valence of carbon is the same as the valence of silicon, you've bumped down one quantum number, but they both have 4 electrons out there in the valence. In this case, it's 2. In this case, it's 3, in terms of the quantum number. But they both have those same kinds of and same number of electrons. So you can maybe expect there to be similar chemistry. It turns out, in this case, there is very different chemistry. But we'll learn about that. Carbon-- hybridisation. Silicon-- no. We'll get to that. But still, this is how we see silicon now. We don't see it this way with Lewis. We see it this way. This is how we see carbon. And so I want you to start seeing elements this way. OK, and so you can look at periodic tables and take images randomly found online with Lewis dots. And you can start thinking of atoms as their Lewis-dot representation. And like I said, on Wednesday we're going to make a whole bunch of structures, molecules, with Lewis dots and show how that teaches us, not only about how to draw the bonds, but, literally, about which bonding-- You know, if I have sodium and chlorine, it's pretty obvious. If there is a bond, it's between sodium and chlorine, all right? But as I go to more and more atoms and I'm trying to form molecules, it's not so obvious what is bonded to what and how. All right, and thinking about this and thinking about the third part of what Lewis did that was so important is what allows us to draw a whole bunch of molecules-- Octet rule, which is that atoms like-- we now know "like" means lower energy-- to reach 8 electrons in their valence. OK, well, we can already see this for the ionic bond that we've talked about. And you can write this. And I want you to write ionic bonds using Lewis. But in this case, it's actually pretty straightforward, right? If I had, for example-- let's get some more board space over here. If I had, for example, an ionic bond between cesium and fluorine, so if I had cesium and fluorine, then the way I would think about this in the Lewis world would be this way and this way. Because I've got 7 valence electrons in fluorine and 1 in cesium. And then, if I write this as an ionic bond, the way I would write it is cesium-plus with fluorine having taken that electron. And so you would write this still with the minus sign there. That emphasizes. It emphasizes what happened. But I see my Lewis here. You know, I see it there. But I'm emphasizing that this is an ionic bond that's happened. Because fluorine wasn't supposed to have 8, but then it did. It's got a super-high electron affinity. Cesium has a very low ionization energy of its first electron. And so fluorine just took it. It literally just took it. I'll take that. Thank you very much. You know, I'm not being judgmental. But ionic bonds are kind of, you know, the relationship in a bond-- and I don't want to judge-- but the relationship is not even. It's not even. And that's OK. But fluorine said, take, take, give. And cesium atom, OK, fine. Now, on Wednesday when we do covalent bonds, it's not so-- then there's a lot more sharing that happens. And that's why we can model it the way that I-- yes, I still left there-- we can model it that way. Because it is like a charged atom and another charged atom, all right? And it's because fluorine took it. Fluorine took it. It's OK. Any kind of relationship, you know, they get along. They make it work. Now, what we can also do just in thinking about this, just in thinking about this, I can go to another example like calcium. And if I think about calcium and fluorine, well, I know again fluorine likes to be f-minus. But see, calcium, when I think about it as a Lewis atom, you know, calcium likes to lose both of those. Calcium, often, because, remember, they want to get to the Octet rule. They want to reach their 8 electrons. But calcium has got 2, what do I do? Either somebody gives me 6, not going to happen, or I can lose 2. And so what you get is that calcium will go, you know, to Ca2-plus. But fluorine is still just f-minus. Right from this I know that the balance stoichiometry, that is to say how many calcium atoms I need to stabilize this whole ionic bond situation, is going to be 2, right? I got to go to calcium F2. And if I were going to write this with emphasizing the ionic bond, I might put brackets around just so it's really easy to see, 2-plus. There's no dots left. Because now I'm writing it in a way where I've emphasized that-- gesundheit-- that F took it. It took it. And it is minus. And there's two of them, you see? So that would be like thinking about an ionic bond, all right, but from the Lewis-dot vantage point where I really see that fluorine took the charge, OK? OK, so I think-- oh, ho, you could go further. You can go further. I think we'll stop there. And on Wednesday, like I said, we're going Lewis all the way and we're doing covalent molecules, covalent bonds. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 4_Atomic_Spectra_Intro_to_SolidState_Chemistry.txt | [SQUEAKING] [RUSTLING] [CLICKING] PROFESSOR: Today, we're moving on, and we're moving into what's really happening with those electrons. And I framed this on Monday by telling you that the Rutherford atom wasn't going to make it more than 10 to the minus 11 seconds-ish because classical E&M would tell you that the electron, if it's going around like this, is radiating energy, and so it spirals down into that positively charged nucleus. So Rutherford got it right in the sense that you got all this positive charge concentrated in this one tiny little volume in the middle, and the electrons are way out there. That's the planetary model. But we needed some new theory to understand why the whole thing was stable, and that's where Bohr comes along, and that's what we're going to talk about today. Now, we need to feel our oneness with waves. Now, I think probably most of about waves, but since we're going to be talking about them a whole lot today and as we go forward, I want to make sure we're all on the same page. So here's a wave. Well, we know about water waves. They move a meter a second. Sound waves a little faster, right? And you've got these characteristics of a wave-- the frequency of the wave, the wavelength. And so I just want to make sure that we're all together on this because we're going to be talking about these variables a lot. So if you have a wavelength, that might be in meters. How many meters do you have per wave? We're getting somewhere here. If you have the frequency-- and I'll use this notation here. You can use f. You might see f sometimes. I like using nu. And if you have the frequency, well, that's going to be the number of waves per second, for example. And so the speed is going to be the two of them multiplied together. And I know a lot of you have seen this, but again, I want to get it up here. So it's going to be meters per second if these are the units that I'm using, seconds and meters. Now, if you're light-- light as in light-- then you have a speed that is a constant in vacuum, and we know that. So the speed of light-- so for light, let's put that right underneath. For light, the speed is c, which equals 3 times 10 to the eighth meters per second. OK, just getting to some good stuff. So if I wanted to know, for example, how to go back and forth between frequency and wavelength for light, well, I can just use this relationship. So let's put an example up there. So for example, if you're red light, so if you're red light, then you might have a frequency of around, let's say, 450. There's a range, but 450 terahertz would be a frequency. And so now I can get the wavelength because I have that relationship. So the wavelength for red light is c over nu, which is going to be equal to 3 times 10 to the eighth meters per second divided by 450 times 10 to the 12th per second, right? That's the measurement of hertz is 1 over seconds, right? That's frequency, 1 over seconds. That gives me a number that seems like it's pretty small, 0.000000667 meters. And so we could use a different unit like nanometers, which is just 10 to the minus ninth. So we could call that the wavelength of red light, 667 nanometers. Now, I'm doing this all slowly and carefully because we're going to be doing this a lot, and so I want to make sure that you're comfortable with this. And we're going to add something to this in just a minute. If I was blue light-- where's blue light? Let's put blue light-- I'm going to leave red light here. And we'll put blue light right above it so that you can see them together. So blue might be something like, oh, a frequency of 650 terahertz, so higher frequency. And that would give you a wavelength of something like 460 nanometers. I'm putting these both here because I'll be comparing other aspects of these colors later. Now comes back-- no. That's our oneness moment with waves and light, and now we come back to our story, our detective story of what's happening inside of atoms, and it really requires us to go back. I said Bohr had some postulates about quantization. There was work going on at the time already related to quantization. So I want to share with you that work because it's what inspired Bohr, and then we'll do the Bohr model. But also just I want to get you thinking about what this all means. So if you think about music, you can think about these two instruments. This might feel like-- the violin might feel like something where you can just play continuous notes, whereas here you can't. It's discrete. I can't play any notes in between. Something is discrete. It's quantized. Well, that's pretty obvious. You look at those. You're like, OK, that makes sense. There's some things that aren't as obvious, like here. Oh, notice what's in the calendar. That's a screen. Now a screen from far enough away might not look quantized, but then when you look at it closely, it is. It's pixels. There's nothing in between. There's nothing in between. The thing is that at the time, people thought about the world in a certain way. There's some things where it's just obvious they're violins like light. And then they're like, wait a second. These things are acting like xylophones. What's going on? So Planck's contribution was really-- he was one of the first kind of deep thinkers to talk about quantization, and it was quantization of light. It was quantization of light. And so what he said-- and this is together with Einstein, and I'll show you what Einstein did in a minute. So it's called the Planck-Einstein relation. And what he said is that the energy of a photon-- so Planck-- why did I just go all caps? I don't know. I don't know. There's a reason, Planck-Einstein. They said that the energy for light-- the energy is related to the frequency through a constant. And so in particular, E equals h times the frequency. Now, this was revolutionary. It might just look like a simple relationship. Oh, and by the way, this is a constant named Planck's constant. I'm going lowercase-- Planck's constant. And, like other constants, it has a value that doesn't change. So that value for Planck's constant is in-- I'll give it to you in two different units so we have it up here, 6.26-- no-- 6.62-- sorry-- 6.626 times 10 to the minus 34 joules-second. Or it's also equal to 4.136 times 10 to the minus 15th electronvolt-seconds. Notice the units are in energy times time, Planck's constant. And you can see that it must be because this is energy and this is 1 over time, the frequency. So it's got to have those units for that to work. But this-- did I make a mistake? I'm worried. The dates are still hitting me hard. I'm still thinking about that, having nightmares. 100 years off, 90 years off. I'm good. So what they saw is that light had a frequency, but it also had somehow this unit of energy, a unit, a discrete unit of energy that you can now calculate using Planck's constant. And that was revolutionary. That was not what people thought. And it really all came home in what Einstein did, which was the experiments that he won the Nobel Prize for-- his deeper thinking was in relativity, but he won the Nobel Prize for this because it was absolutely essential to further this idea of quantization and show it with an experiment. And because it was so important, I want to share that experiment because when Einstein did was really quite amazing. He shined light on a metal, and what he saw-- so this is called the photoelectric effect-- photoelectric effect. And this is where you take a piece of metal- and they knew there were electrons in there, right? That discovery had happened. So they knew there were electrons in-- and they also knew if you hit this metal with energy, the electrons could come off. I mean, after all, that's what the cathode ray tube was. It was electrons coming off of a piece of metal. And in that experiment, you hook up a voltage, but here he shined light. And so if you take, for example, different frequencies of light and you shine them on this same piece of metal-- so let's go red and then let's go green and then let's go blue. Now, if you shine red light on the metal, nothing happened. Nothing happened. But OK, we know if you just crank up the intensity of the light, something's got to come off. You're putting all this energy-- no. Why? Because the particles, the packets, the quantized amounts of energy that that kind of light can have is determined here. So what Einstein says is, well, OK, then these must be particles. They must have some energy associated with a particle that follows this relationship equals h nu. And notice that doesn't depend on intensity. It just depends on frequency. So I might be hitting this metal with more and more of them, but each one only has the same amount of energy limited by that relationship. So if I want to hit those electrons with more energy, I've got to change the frequency. And so what was found in this experiment is that, well, maybe here you get nothing. So here you get nothing. How am I going to write that? OK, nothing. I just won't write it. Here, maybe you get electrons. And here, maybe you get faster electrons because these fundamental particles of energy of light have different energies, right? So now we can calculate it. We can calculate the energy of that red light, and we calculate the energy of the blue light because I have now E equals h nu. And if I plug it in, which I won't do, you get something like 1.8 electronvolts. And if you do the same thing for blue, from the Planck-Einstein relation you get the energy is 2.7 electronvolts. So what Einstein did in this experiment which was so crucial is he measured. He changed the frequency of light that you shine on this metal, and what he saw is that there was nothing, nothing, nothing, nothing. And then all of a sudden you literally had, if you plot the kinetic energy of the-- I'll write it out here-- ejected electron-- so that's what's being plotted there. You plot that versus the-- I said it wasn't going to use f-- versus the frequency of the light, nothing, nothing, nothing. And then all of a sudden there's a linear relationship between them. That explains it because these things were made of these particles that were called photons. So photons. Now, while this was blowing their minds and winning prizes and all that, Bohr came along. Now remember, this is what I mean by dates. Apart from the dates, we know now there was a lot of time between those experiments, especially Dalton and Thompson. And that's where we were on Monday. So Bohr was looking at all this. Remember, he wrote the paper where he said it is with great interest the Rutherford atom, meaning it's wrong. It needs fixing. Meanwhile, Bohr's looking over at Einstein and Planck and he's like, wait a second. Let me get some of that quantization stuff because if it works for light, what if it also works for electrons? What if it also works for electrons? And so he's looking at this stuff going on with Planck and Einstein, and he applied it to the electron in Rutherford's atom. That's what he did. That's what Bohr did. He said, you guys got your light thing. I'm going after the atom. And what I'm going to do is I'm going to make just actually a fairly simple postulate. Why is that up there? Let's go. I'm going to say that for me, the quantization is going to happen with the angular momentum. So Bohr came along, and he said that the angular momentum of the electron-- so L equals mvr, right? Remember, we listed the Bohr postulates, and one of them was that Newtonian mechanics would hold. Angular momentum, mvr. But there's something added to it. He said that that angular momentum could only have discrete values, quantization. So he said that this is quantized, and he was specific. He said it could only be some number, some integer times Planck's constant divided by 2 pi. Why not? When in doubt, divide by 2 pi. As you all know, that has to do with just making math easier. Now, that's the angular momentum, and he said it's quantization. And n-- n is the key here because n is this thing that says that it must be some unit. This is like the unit. It's like that photon energy. It comes in units, but it comes in units of integers. So you could have 1 or you could have 2 or 3 but not 1.5. That's quantization. So n had to be 1, 2, 3. It had to be some integer. And oh, here we go. Watch this. We're going to call it a quantum number. I should have written that in all caps. [LAUGHTER] Thank you for that. It's our first quantum number, and we've only known each other for a week-- a week. Here we are. We're quantization things. Quantum number because it's a number that counts quantumness, quantization. This is how you get the quantization from this. So it's a quantum number. Now here's the thing. I will not derive Bohr's model for you. I will tell you what it is. And if you want, you can look up the derivation. It's actually really fairly straightforward. It basically has to do with this one assumption, quantization of the angular momentum of that electron, combined with F equals ma and the electron is stable. It's not accelerating in. And when you do that, what you get-- let's put this-- no. Let's put that back up. So what you get with Bohr's model is the following. You get three extremely important outcomes. First, you get that the radius of that electron is quantized. Then you get that it's energy. The energy that it's allowed to have is also quantized. And finally, you get that its transitions are-- you got it. Thank you-- quantized. So if you plug stuff in and you do the Bohr derivation for the hydrogen atom, what you get is that the radius of that electron is equal to n squared times a constant divided by the atomic number. And this constant is called the Bohr radius. It's got a value of 0.0529 nanometers, and it's also called the Bohr radius if you want. But that's its value. And then if do the same-- now if you look at energy-- so energy, the equation for that comes out to be minus 13.6 times the atomic number squared divided by n squared in units of eV. Those are the two key results that come from Bohr's postulates and from Bohr doing this-- doing that. And we're going to talk about each one of these. All right, so if I-- OK. So let's go one at a time. No. So let's talk about the radius for a minute. Let's talk about the radius. So if I look at the radius and I'm a nucleus here-- so I've got like some positive charge in the middle and some nucleons and stuff, what I'm saying-- I'm sorry, some nuclei neutrons-- is that the electron can only be at certain values. This distance r can only be at these values, and that's dictated by the quantization number, the quantum number. n equals 1 would be the closest. You can see that from the formula. And n equals 2, and so on. And this is not necessarily drawn to scale. You can see that for n equals 1 and z equals 1, the electron is simply at the distance of the constant, 0.5-ish angstroms, 0.05 nanometers, right? At n equals 2, it's 4 times that. Oh, yeah? STUDENT: What's the z? PROFESSOR: The z is the atomic number. And that comes in-- let's put that here. And that comes in because it tells us how many positive charges there are, how many protons there are in the nucleus. And that comes in when you do F equals ma. That's how it comes into the Bohr equations. And r is a distance, and n is the quantum number, and a0 is a constant. z is the atomic number in both cases. This is mind-blowing because it tells you that that's where the electrons are in the atom orbit, in the atom, but they're only those distances. Remember, quantization means nothing else. Think about that. I'm saying that that electron can't be anywhere else. It must be this distance, or it must be this distance. So this was a big deal. People were like, what? It didn't compute. It didn't make sense. But this is what they were starting to see, that this was the only way to explain what they saw, which we'll come to when we talk about transitions. So this would be the radius. And when you look at the energy, it's just as-- I'm going to use-- yeah, I'll put it here. So you take that equation for the energy of an electron in an atom, and you take z equals 1. If z equals 1, then the energy is quantized, and it's minus 13.6 divided by n squared electronvolts. And if I just draw this as energy going up-- so if I just draw this, then what that means is that I've got one line down here showing kind of the energy level. So this would be minus 13.6 eV. And then I've got another line here, and this would be minus 3.4 eV. And then I've got another one here, minus-- let's see-- 1.5-ish eV. And I'm going to start to get crowded. As you can see, they're going to get closer and closer until you get to some level here, which would be like n equals infinity. And then above that, zero. Above that is free. The electron is free. When its n is infinity, when it's gone through all the quantum numbers up to infinity, it's free. And that is called ionization, taking it out of the atom and making it not part of an atom anymore but just out there free able to do what it wants. You've ionized it. You've taken it out. We'll come back to this concept multiple times. These are the energy levels for an electron, and they are also quantized. They could only be this value or only that value or only that value. They cannot be anything else in between. That was revolutionary at the time. So this would be like n equals 1. Sorry. That's not E. That's eV. n equals 1, n equals 2, and so on. Now the third point that I made is that it also tells us about transitions. This was also absolutely essential, and this was the key piece to understanding what they saw. And so we'll get to that in a few minutes. What do I mean by a transition? I mean that an electron is in an atom at some level, and it moves to another level. But now the point is that if it can only be in these discrete quantized levels, then when it moves from one to another, those changes in energy are also discrete values. There are only certain changes in energy that that electron can experience. And there's even more which is that what can cause those changes or what happens as a result of those changes is the interaction with light. And we're all the way back to here, photons. Think about it this way. If I take a ball and I have it up here, it's got some mgh in it, right? And I drop it, and now it's got some KE. More energy was in gravitation, was stored in mgh, gravitational potential energy, and then I let it go. Well, for an electron, this is its ground state. This is the kind of happy place. So this would be like a ground state, and these would be all excited states. It would be like excited states. So that would be like lifting the electron up and letting go. But when you let it go, it's not kinetic energy that that transfers into. It's photon energy. Photon energy gets emitted from electrons changing their levels in an atom. Light is an absolutely intricate part of this system. And not only that, but everything is quantized. I almost need a moment, but I'm OK. I'm going to keep going. This was our big deal, a really big deal because now I can tell you-- OK, there we go-- because now I could tell you, for example, that if that's the second orbit in hydrogen and that's the first orbit, I can tell you what frequency light is emitted if you go from n equals 2 to n equals 1 because the change in energy from n equals 2 to n equals 1 is nothing more than minus 13.6 eV times-- well, it's the final state minus the initial state, right? The final state is 1 over 1 squared, and the initial state is 1 over 2 squared. All I've done is pull out the 13.6, but these are just-- these are just the Bohr-model energies, and I'm just taking the difference. This is equal to Ef minus Ei. It's the change in energy. So in this case, it's minus 10.2 electronvolts and going from the second orbit, quantized orbit, down to the first. And I can now tell you because of what we did in the beginning that that would emit a photon of frequency 2.5 times 10 to the 15th hertz. That's the energy difference, and that energy goes into producing a photon-- goes into producing a photon, and we can tell you exactly what that frequency is. So these are the kinds of transitions that the Bohr model explains, and it explains energy changing from electron transitions to photons. So what are the implications of this? Oh, there's the question that I just answered. If this is hydrogen, z equals 1. The implications of this are groundbreaking, and it explained what people were seeing because you could be talking about angstroms already. Angstrom looked up, and he's like, well, you guys think it's a rainbow, but I'm not seeing that. I'm seeing discrete lines all over the place. And that's what he saw when he looked at the sun. Don't look at the sun. It's not a good idea, but Angstrom did. And now he said, well, but there are all these lines, and the wavelengths that he tabulated happened to come on this sort of thousands of angstrom kind of scale. And so he liked that, and chemists love it, and that's why the angstrom is named what it is, 10 to the minus 10th meters. Chemists love it because a bond is a few angstroms. Spectroscopists like nanometers. They're like, thousands are too big. We don't like thousands. We want to work with hundreds. So spectroscopists use nanometers. Other people, chemists, whose angstroms. I say can't we all just really get along? And it's fine. Don't make spectroscopists angry, really. I did it once. Now on Friday, you are going to get the most accurate spectroscope that has been made by humans in your goodie bag, and you'll be able to look at the fact that the world is not continuous. These frequencies, wavelengths that are coming out are not continuous. If you take a tube of H2 gas and you apply an enormous voltage to it, you will excite electrons in the H2 molecules. As they go back down, they will emit photons, and what will you see? Only certain values. Well, this is what people were doing. They were looking out into the universe-- which, by the way, is 75% hydrogen. And they're like, really? The hydrogen economy. [LAUGHTER] We-- we-- they were looking now and seeing discrete lines all over the place. How to explain it? The Bohr model is what explained it. You will be able to see these discrete lines yourselves with the goodie bag on Friday. But if you do this, this is what you might see from hydrogen. And there were a lot of people trying to understand this at the time, Balmer and Rydberg. And I'm not going to go through-- they came up with sort of partial formula. They were empirically based. But Bohr-- it was only the Bohr model that could explain it all. It was only the Bohr model. And the reason is that all the transitions were out there, and you need the complete picture of these changes in energy between these levels. You need to know about that to understand all those transitions. And at the time, it was a real thing. If you were like, I want to get something named after myself in science, you'd be like let me go and find a new set of transitions, which are just basically lines, right? It's that you're not seeing a continuous spectrum in some part of space. You're instead seeing discrete lines. And look at this, right? So like Lyman-- you could go down to n equals 1 like we did there, 2 to 1, so from here to here. But you could also go from 3 down to 1. You could also go from 4, and that's a series. Then you could also transition not from to 1 but to 2 or to 3, and each one of those is a series. And then I think finally they're like, OK, no more names. After Pfund, we're done with naming. But all those series are still there. Those are the allowed transitions. So you can also imagine that it's not just emission. So we've been talking about emission. You get these lines when an electron goes down in energy. Remember, like the ball, the mgh goes down. It gives off photons in these quantized energies, but it can be the opposite. You could shine light. So I could get a continuous spectrum here and shine it on an atom, and that atom absorbs light literally by exciting electrons. An electron is in some state, maybe the ground state, and some energy comes along that exactly takes it up to one of these states. But it can't sort of take you up there. It's got to take you there-- quantization. That's why you see these lines here. These are the places where that atom can absorb a photon because it exactly corresponds to a transition from one discrete level to another. So it works both ways, absorbing photon energy, promoting electrons up. And when we do X-rays, we just ionize everything. But right now, we're staying within the potential of the atom. So we're staying within these levels, absorbing or emitting energy. So I could answer questions like this. If you look at a question like this-- I forget. I think this might be from a couple of years ago quiz, so you might get something like this tomorrow. No, you won't. I just looked at a few faces, and no because we're talking about it today. So you wouldn't get it tomorrow because, remember, you only get quizzed on things that we did ending on Monday. It wouldn't be fair otherwise. But next week, OK. What's the lowest-energy photon shown? Which one is it? These are photons now. We think of these as photons. Red. High wavelength, lower energy. OK, good. That was over here. Now what about what transition give us the line at 639 nanometers? Hello, [? Humana. ?] 639 nanometers. So I want to know what electron transition did that. Well, to solve that problem-- to solve that problem, I need to do some erasing. Maybe I'll just draw it in here because what you have is minus 13.6 eV times-- I'm assuming z equals 1 because did I say it was hydrogen? I did, hydrogen. z equals 1. So z goes away, 1 squared. So minus 13.6 eV times 1 over the final state squared minus 1 over the initial quantum state squared quantum number squared. So that's going to equal 639 nanometers, which we now know we can convert, thanks to Planck and Einstein, into an energy. We now know that, a photon energy. It's that photon, that unit, that discrete unit that does it, and this is 1.94 eV because it comes from 639 nanometers. Back and forth, back and forth. We've got to get into that mode and into that mood. Back and forth, energy, frequency, wavelength. And so I won't do the math here, but the answer comes into something like it's the 4 to 3. 4 to 3 is the transition that gives you that. We're not going to go-- some day. We're not going to go kind of all-- this is a fairly simple one to see. We're not going to go like n equals 528 to n equals 7,000, but some simple kind of back and forth. Just get into that mode of looking at these transitions as changes in energy, which means the emission or absorption of a photon. That's what I want you to get out of this. And you can go further than that and answer questions like power questions. So here's a question. If your favorite radio station broadcasts at a frequency of 107.9-- I'm not saying that's mine, but you got to groove out-- and a power output of 50 kilowatts, how many photons are emitted by the transmitter each second? I just want to give you this as another example of thinking about this. Light, energy-- light to energy, photons, particles. And here, you can get that by simply knowing what power is. And if you know what power is, it's watts per second. Sorry, watts is joules per second. And so the energy here is h nu. And I won't go through all the math, but I'll leave the answer here. You get 7.12 times 10 to the minus 26 joules per photon. Now, how did I get that? I got that because I know the frequency. Why am I using joules? Remember, I put them both up there-- joules, eV. Ah, because I know my power, which is 50 kilowatts, which is 50,000 joules per second. And so I've got joules in the power. I just decided to stick with joules. Joules, eV, joules, eV, back and forth. They're just units. If you're not familiar with changing back and forth, you should do a little practice. And this gives you something like 7 times 10 to the 29th photons per second. So you can think about power, photons, light, energy. If I shine that many-- if I shine that many on Einstein's medal, will the electrons come off? That seems like a lot. We don't know because we need to know how much energy it takes to liberate the electron, and then that ties into the frequency. Radio waves are much lower energy, much, much lower energy. Now, this gets me to why this matters, and that's how I want to end today's lecture in the last five minutes because we talked about emission, electrons transitioning from their discrete orbitals as a way of emitting or absorbing. And so you if you have different atoms, there's hydrogen. These are the absorption lines-- absorption lines. Carbon has absorption lines. Oxygen, nitrogen, sulfur, so lots of things have absorption lines. Why does this matter? Well, of course, because of the refrigerator. I love this plot. This is a plot of the refrigerator. This is the size of the refrigerator as a function of year. The date only goes to 2002, but it's roughly stable now. Does anybody know what's limiting the size of the refrigerator? It was going up and up and up a lot, and then it stopped. Does anybody know what limits that? STUDENT: The average height [INAUDIBLE].. PROFESSOR: It's the width of the doorway. It's the width of the doorway. That's why we don't-- that's why people get a second refrigerator. And there's energy use. Energy Star program, big deal. Big deal. But I'm talking about refrigerators for a different reason. I'm talking about refrigerators because until the 1980s, the way we did cooling was with a chemical called chlorofluorocarbon. This is a chemistry class, and we're going to be talking about-- when we talk about why this matters, we connect to the chemistry. Well, the chemistry there was critical. There was a molecule there called chlorofluorocarbon or CFC. And the thing about that molecule is that its absorption, the chlorofluorocarbon-- so let's write it down here, CCl3F. There's a gas phase, so we put the little lower script g. That molecule has its own absorption. It has its own absorption, and it has its own reaction, and so do other things. But it has a really important reaction because when it absorbs UV rays, then it doesn't just like then like electrons get pumped up. No, the whole thing reacts. And so this thing degrades into CCl2F plus chlorine gas, and that's once it's up maybe in the atmosphere, which it was getting released all the time because of the refrigerators. And the thing is that the chlorine gas would then react with ozone, and this would go to ClO plus O2, and here's the problem. I mean, we're already at a problem, but now it's a big problem because-- you can already tell. But I've got ClO, and that's really reactive. So what happens is ClO, which was a gas-- I'm just being careful here with my subjects-- would react with atomic oxygen to give you O2 and the chlorine atom back. And here's the thing. Now I've got my chlorine atom again. These are very reactive atoms in the atmosphere. They love ozone. And so this cycle for one CFC molecule would happen 100,000 times, roughly, before that chlorine kind of goes away and does something else-- 100,000 ozone molecules per CFC. Why does that matter? Well, it matters because-- I love this plot. This is the energy of light from the sun hitting our planet. So this is the energy that you get, and look at this. This is the sun at the top of the atmosphere. Here it is on Earth, the red. Look at these things here. If you could only see in this spectrum, the world would be dark because there's no light on the planet. Why? Because it's all absorbed in the upper atmosphere by water. Absorption-- it's the same principle, Bohr. Bohr's idea of electrons getting promoted when they absorb light is happening, and it's protecting us because right here, see, that's the UV. Low wavelength, high energy. UV, O3. Look at that. Look at what it's taking out. That little sliver of yellow is critical. And ozone was doing the job, and it still is. But if we tear all that ozone out with these CFCs, we have a crisis. And that led to what is one of the greatest policy decisions which was made in 1987-- it's called the Montreal Protocol-- to essentially globally ban-- the US led this. This is a great example of how policy is so critical. The US led this effort to ban CFCs, and we're almost back to normal ozone levels, and we avoid this. And by the way, if this had happened, the prediction at the time was roughly 280 million additional cases of skin cancer over this generation. This is a very big deal, and it all comes back to absorption of electrons by atoms and molecules. See you guys on Friday. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 28_Introduction_to_Aqueous_Solutions_Intro_to_SolidState_Chemistry.txt | Today, we are going to continue thinking about reactions. Just to remind you, what did we do on Friday? So we introduced-- here's what we did last time. We talked about constant-- we talked about reactions. And there, on Friday, we were really interested in rates-- how fast is it happening. So we talked about things that affect the reaction rate-- concentration. Concentration versus time, and how you can measure that. And you can have rate loss based on experimental measurements. And that can tell you the reaction order. We talked about zero, first, second-order reactions, the rate constant, and then we also talked about the temperature dependence. And at the very end, the role of a catalyst in lowering that activation barrier. Today, we're going to talk about how things dissolve. So we're going to talk about solubility, and this has to do with reactions because something dissolving is a reaction. But what's going to happen is we're going to talk about how that reaction finds its happy place. And we know a happy place is a good thing, and it's called equilibrium. And so that's going to involve talking about things like equilibrium constants, dissolution, we're going to look at some examples with ionic compounds-- so salts. We're going to be dissolving salts in water and looking at how to think about how much of that dissolves in terms of the equilibrium constant for the reaction. So that's the goal today. I want to start, though, by talking about why this matters, and that's directly related to your goodie bag. So this is something some of you may have seen. The data here in from the '50s is from the Mauna Loa measurements, the top of a volcano. And this is CO2 concentration, parts per million in the atmosphere. But if you dig into the ice-- we're going to be doing a lot of that here in Boston over the next three months. But if you dig deeper into the ice, in the cores, you go back in time, and it's a beautiful thing. The ice as you go down is ancient atmosphere. That's cool-- because there's little bubbles. Remember, we talked about the half life of carbon-14 that's trapped from a living thing breathing it in. Well, you can trap directly the little bubbles of atmosphere in the ice, and that's what happens. And that's why we know what CO2 levels-- we're going back almost a million years here. It's 800,000 years of data. And this is a graph that I think many of you have seen. It's going up, a lot. And here it is. Now, this is just zooming in on the last 50 or so-- this is just the data, the Mauna Loa data. And The reason is because here we also have another kind of data. So there's a lot of attention paid on CO2, and there should be. There is a lot less attention paid to what that is doing to our oceans. And this is a reaction that we will talk about more when we talk about acidification. So when we talk about acids and bases, which is coming up later-- later being Wednesday and next week-- we'll talk more about this. But I just want to tell you that when you have CO2 in the atmosphere going up, then you also have CO2 in the ocean going up. So the CO2 in the ocean-- the CO2 in the atmosphere is getting absorbed by the ocean. So what? No, it's a lot of what, because what happens is the CO2 in the ocean-- the more CO2 in the ocean-- is making the ocean more acidic. So if you just write this number down-- because it's so astonishing. Past 200 years, the ocean has become 30% more acidic. Now that is the fastest-known change in the ocean chemistry in 50 million years. That's a big and very sudden change. So the ocean is absorbing about 22 million tons per day of CO2. And that number is going up. Why does that matter? Maybe it will do to the oceans what it's doing to the atmosphere and making things a little warmer. No, it's doing more, because when it acidifies, then it has a direct impact on things like this, which is a pteropod. You look at the food chain of the ocean-- there are three things at the bottom. That's one of them. I mean, there's plants. There's photosynthesis. And then there's the bottom of the food chain. Now you know what's going to happen when the bottom of the food chain disappears. This is one of the main kinds of animals that lives at the bottom of ocean because they've got these very small calcium carbonate shells. If you make the ocean just a little more acidic, like we're doing, those shells don't survive, and those animals don't survive. So that's what happens to a pteropod. At the rate we're going-- like decades of time from now-- that's what happens. So what you have in your goodie bag is a way to see this a little bit more accelerated. So here's your goodie bag. So I wanted you to-- I wanted you to see what's happening in the ocean, and you have now the tools to do that. So in your goodie bag, you've got citric powder. Imagine you take a lot of limes, and you squeeze them, and then you dry it all out. And that's what you got-- citric acid. Now you're going to mix that with water to change the ph. And you're going to dissolve this powder. That's the topic of today's lecture-- dissolution and equilibrium. So you're going to mix that into water, and you've got a scale from a previous goodie bag. So my hope is that you've been using that on a daily basis. You take it with you places, and you can use that again here. And you'll need that, because I want you to see how these shells dissolve. And if you lower the acidity more-- we don't want to wait 40-- well, you can wait 45 days, but I want you to be able to do these experiments in minutes. So we're going to go a little lower in acidity, in ph, but you're going to get the same result. You're going to dissolve these shells. Some of you are going on planes, and there's white powder and a scale involved in this goodie bag. Please don't put that in your carry-on. It's just not a good-- it really isn't a good idea. We have data-- students sending me pictures with the airport screening person. So I love the excitement of thinking about this over Thanksgiving, but put it in your suitcase if you're going to take it home. That's my why this matters. So we're dissolving stuff today. That's the topic. How do things dissolve? There's a citric powder, that citric acid that's going to make water more acidic. And again, we're going to move into acids and bases next. Today we're just talking about dissolution. And so here's citric acid. There it is. Now over Thanksgiving, I thought I'd throw this in, because you might have sugar at the table. You know to call it a c12h22o11. Don't ask for the sugar. If you want to add some sugar to something, ask for it by molecular name. But the question is why do these things dissolve in the first place. There is acetic acid, citric acid, there's sugar. These are molecules that look like that. Why do they dissolve in the first place? And the answer is because of the things that we learned about bonding and about things like whether a molecule is polar or not. Or in this case, whether you've got the possibility to hydrogen bond or not. And so if you look at both of these, you can see that they both have a whole bunch of oh groups that look a whole lot like things we've seen that like to hydrogen bond. You see those ohs on the end there, both for sugar, around the outside, and for citric acid. And so what happens is if I put a sugar molecule into water, here's a picture of it. There's a picture of it. So there's sugar molecules there. Now they start in a crystal of sugar-- you pour your sugar in, a teaspoon of sugar. And it's in a solid, but then it goes off, and it breaks away into the water. And those little ones are the water molecules. Why? Because the water was bonding to itself. The water has dipoles-- dipole-dipole. Remember that? The water has hydrogen bonds, but the sugar does, too. The sugar does, too. And so the water sees this crystal of sugar, and it says well, hang on. If you come out, we could bond together in much the same way that we're already bonding with each other. The water molecules are already bonding with hydrogen dipole. The sugar is like, well I've got hydrogen. I've got dipole. And so it can break off, and it doesn't cost that much energy. In fact, it gains energy by doing this. They can gain. And so this can actually be a lower energy state overall. Now that's because of something that many of you have probably heard, and now we can put some bonding to it, but it's because like dissolves like. Oh yeah. That's technical. That's technical because now we know what like means. Like means a type of body. like means a type of-- It's like water. It has hydrogen bonding potential. Therefore, it's able to dissolve in the water. So water has-- well, what does water have? It's got dipoles, so it can do dipole-dipole stuff. It's got h-bonds. Oh, everything has London. Everything has London. So if you've got those ingredients, if you've got that ability, if you've got those likes, then you might be able to dissolve. Well, this is called a polar solvent because the molecules that make it up are polar. They've got dipole. Actually, and so we're very interested in understanding how things mix with other things and why they mix with other things. Let's take a look at another example. So here we have ethanol. I keep on bringing up ethanol. There might be some served, and if you're 21 or over, you can have some. Otherwise, it's illegal. Ethanol up there at the top. Thank goodness, you will say, at your Thanksgiving table, it is not methoxymethane. And because if it were, it would be the same chemical formula, but the oh-- the o would be in the middle of the two cs with the ch3s on the end, and you wouldn't be able to hydrogen bond it. It would all just be vapor. It'd be vapor. No, but it's ethanol because it has hydrogen bonding capabilities. But look at what happens if it's heptanol. So all I've done now is I've taken the same end group, and I've given it more of a carbon chain. How is that going to affect how it dissolves? Given what we just talked about that like dissolve-- they both have hydrogen bond potential. They both have hy-- but look, more and more of it, as you go from ethanol to heptanol, more and more and more of it is something else. It's only London. So what's going to happen now if I put this into water? Well, we can see this going series. Here's methanol, ethanol, propanol, butanol, pentanol, hexanol. Heptanol is down at the bottom, and there's the water solubility. We've got to write that down-- solubility. So solubility is kind of what it sounds like-- the ability for a solute-- that's the thing we're dissolving-- to dissolve in a solvent. That's the thing we're trying to dissolve it into. That's the solubility. And we'll be talking about the maximum solubility in just a few minutes, which is what we usually-- what's the solubility? We often mean the maximum. Well, that's what's listed here-- the maximum. Methanol, ethanol miscible. So you can put as much as you want, and it'll always keep mixing in with the water. But now look, once you get to butanol, pentanol, hexanol, it starts going down. You can't mix as much as you want. In fact, it's less and less and less, and the reason we now know is because of what I just said. As you go down, the hydrogen bond and dipole-dipole becomes less important because you've got more and more of the molecule that's only London. And so it's less like the water environment, the bonding environment that the water liked with itself. And so it has more trouble dissolving and being energetically favorable isolated by those water monitors. By the way, does anybody know the origin of the word proof, speaking of ethanol? If you buy a bottle of ethanol at the supermarket-- so there's gin or vodka or whatever, and it says it's something percent alcohol-- 20% alcohol, 40 proof. Does anybody know where that comes from? It's this. It's just-- but what they did, they didn't know if you were making good alcohol back in the time. And they said, well, I don't have all these detailed abilities to measure and quantify it, so I was going to light it on fire. That is exactly what-- is this a good bottle of gin? I don't know. Let's light it on fire. That's what they did. And it turns out that if ethanol and water are at 50/50 mixture or more, it will light on fire. Don't try that, please. Don't do these experiments. But so that meant is this a good bottle of gin? Well if it's 50/50, I'm going to say it's a good bottle of gin. So let's light it on fire. Oh, it lit on fire. That's 100% proof that it's good. That is true. That is the origin of the word proof. It is proof that it's mixed well, and it's good stuff. Dissolution-- dissolution-- well, what else can we try to dissolve? We're dissolving alcohol. We're going down to heptanol. We're lucky it's not methoxymethane. We talked about sugar. So we've got to have a framework, by the way. Solubility-- the ability for a solute to dissolve. And now, there's a context here, right? Let's suppose I'm dissolving sugar. So sugar-- how are we going to write this dissolving down? Well, sugar is c12h22o11, and if we put it in water, we could we could formally write the whole thing out. We can write it out. We can say I'm going to add water to it, and that's going to go to-- it's going to go to c12h22o11 plus more water. But there's something very important that I'm leaving out, and when you think about dissolution, you've got to keep track. What was it? Was it a solid? Was it a liquid? Or was it something dissolved in a liquid? So this started as a solid, so we put an s there. This started as a liquid. This is the dissolved thing, so we put an aq. And the aq means aqueous. It is in an aqueous solution. It is a dissolved molecule. And so here, this is a liquid. Now the other thing that happens is-- well, the h2o is kind of there. It's all over the place. It's that solvent, and so we often leave it out because it's everywhere on both sides. And so often, what you'll see, and we'll write more of these coming, is that you might write a dissolution of sugar as this. Solid, and it just goes to something that is dissolved. And you know that what happened there is sometimes you'll also see it like this. What did you do on the arrow? I added water. That's what the arrow did. We're just getting comfortable with writing these dissolution reactions. Now what else can I dissolve? Well here's another good example-- vitamin A. There's some vitamins you can have as much as you want, pretty much. Why? Because they're water-soluble. And other vitamins are fat-soluble. That means they don't like water as much as they like the hydrophobic molecules of fat cells. And so they go into fatty tissue, where they don't come out. That's why you not have unlimited amounts of vitamin A. That's not good for you. But vitamin C-- no problem. No problem, because it's going to dissolve in water. And so your body can take in and get rid of as much of it as you want. But if it's dissolved in fat cells, that's a different thing. So vitamin A, vitamin C-- and you can see it right there from the chemistry. This is it. Look at all those oh groups in vitamin c. Look at that. Oh, that's going to love hydrogen bonding, and on vitamin A, you've just got the one all the way down there at the bottom. It's hydrophobic. By the way, in this class, we do not say the words hydrophobic interaction because that doesn't make sense. There is no such thing as a hydrophobic interaction. No, hydrophobic interaction is misused. What it really means is that something that's hydrophobic likes itself more-- being together with itself-- more than it likes being in the water, or something hydrophilic. That's what is meant, but there's no actual interaction there. Hydrophobic, hydrophilic. Well, those are other things you can think about dissolving. You can also dissolve salt, and this is what we're going to use as examples. And this is a picture of a salt crystal. So we show the sugar, and here's salt. Now, so the thing about salt-- where's my sugar? There's the sugar, and let's put salt underneath it. The thing about salt, which is really just a general term for when you have an ionic solid that can be made from a mixture of acids and bases and dissolves oftentimes a little bit, at least in water. So a salt or an ionic solid-- let's say for now-- this gives something different, because this breaks apart into positive and negative ions that they contain. So in the case of sodium and chlorine-- sodium chloride-- well, that's going to dissociate. If you pour salt in water, then here's the reaction for that. You've got sodium chloride solid. We put it in water. That's going to go into ions. So I put that in water, and it's different in the sense that it doesn't break down into the molecules that made up the sugar. It has a particle of sugar that's got millions and millions of molecules in it. Now they dissociate. That's what the aq means. Those are neutral. Here, it's a solid, but you can understand this from looking at it and thinking about it. Look at the way those water molecules are oriented. They're pointing to the cl with their positivity, with positive charge. And they're pointing to the sodium with the negative, and that's how they find happiness in water-- because they're ions, and water can be either way. It can point its dipole in either-- and so it can actually bond. Remember, the hydrogen has got a little positive charge because the oxygen took it. And so that's what the-- it has a dipole, and that's able to turn around and bond and be happy with each of these ions. That's why salt dissolves-- because it can find a happy place. Other reasons exist outside of this class, like entropy, thermodynamics. But this is how we're going to picture it as energetically happy. It can become happy. So we've dissolved some things. We've got to start thinking about the thing that I mentioned before, which is how much can I dissolve? How much can I dissolve? And in that, it gets into this table here. There it is. So that's how much. And you see it's listed there as moles per 100 grams. And so that's one way to write it. You could write it as-- it's often the max solubility is often what we're interested in. And you could write it in moles per 100 grams. You could write it in moles per liter. Well that's something that we've already learned. That's m form last Friday, the molarity. Or it could be in grams. Sometimes you'll see it in grams per liter. So don't get bogged down by the units. It's just how much of this stuff were you able to dissolve? If we look at sugar-- here's sugar, here's salt-- how much sugar can dissolve? I'm not going to cram it in. So for sugar, the max is 1,800, so I'm going to do both those units-- 1,800 grams per liter. And if you convert that, it's 5.26 moles per liter. That's how much sugar you can dissolve in water. By the way, I did a quick calculation, and that's 429 teaspoons. That's 429 teaspoons. I don't know what everybody is saying-- Coke has so much sugar. Coke's got only 26 teaspoons per liter. You could put a lot more in, if they wanted. I'll go the other way. I challenge them to go near the solubility limit. They're not even close. Too much sugar. So salt-- you can also look at salt. For salt, what is it? Salt is 359 grams per liter, and then that's it. That's 6.14. This is nacl. It's interesting-- if you compare sodium chloride with sodium bromide-- so if you look at sodium bromide, it's going to be 8.9. And sodium fluoride there's only going to be 1. Why? Because it comes right down to the picture. I've got to make these happier. I've got to make them happier. If you've got the ions there, and the water likes it, but what if they really didn't have a strong ionic bond to start with? You can imagine you might be able to get to a higher amount of dissociation. Or what if they had a very strong ionic bond? You might imagine that less of it-- you've got a lower maximum solubility. So it all kind of goes together with the things that we've learned. But what we're really talking about is what I need to tell you about next. And that is this idea that we've reached a dynamic equilibrium. We've reached a dynamic equilibrium. So what does that mean? Well, if it's unsaturated-- so if it's unsaturated, I'm way before this limit. This is salt. That's the limit max. Max. What does it mean? What does it mean to be below it? Well, what does it mean to even have reached the maximum itself? What it means is not that things aren't dissolving. What it means is that things are reforming, recrystallizing at the same rate that they're dissolving. That's what the maximum means, and that's an equilibrium. And that's why we're going to talk about next. So if it's unsaturated, there's a nice little molecular picture. There's the thing you're dissolving, and look at them. They're going away into solution. Maybe it's salt breaking up into ions. Maybe it's sugar. They're going into solution and finding a happy place around the water. But once in a while, one of them comes back. You see that little particle there coming back. That happens. But much more, they're dissolving. Then, you reach this point of saturation where it cannot-- where it's going to recrystallize, reform the solid at the same rate that you dissolve. And that's the place where you can't dissolve anymore. So what's really happening is that you have a solute plus a solvent, and it's not going one way as we've been writing it down. It's going both ways. So it's actually the solute plus the solvent goes like this. That is what we've been talking about. That's dissolution, and that's the predominant mechanism on the left before you've saturated. But see, and that goes to some solution. But see, you can also precipitate. Precipitation. And so at the saturated solution, you have-- if it's saturated, then that's another way of saying we've reached the max possible concentration of the solute. Remember, the little brackets means concentration, molarity. It means that these rates-- thinking back to Friday-- are the same. It means these rates are the same. This is a dynamic equilibrium. And so that's what we need now the tools to talk about. And so we're going to talk about three things-- the reaction quotient, the equilibrium constant k, chemical equilibrium, and then the solubility product. Now if you think about-- I just said this is like the forward reaction is going in the same-- it's happening in the same rate as the reverse. And so if you think about this as like a graph of concentrations-- so I'll just put here concentrations and time-- then maybe I have started-- I've put my salt. I pour a teaspoon of salt in water, and the salt is dissolving. So it's going into the products. It starts out at zero in solution, and then it becomes products. At the same time, maybe I've started it up here, and these are the reactants-- Gesundheit-- that look like this-- Gesundheit-- and those would be the reactants. So the concentrations of these things-- at a certain point, they're changing. And then, at a certain point, they're not changing. But they're still reactions happening back and forth like I just showed you. It's a dynamic equilibrium. Now what we do is we write down for some arbitrary reaction-- aa plus, so let's go back to this one. But now, we're going to write it going both ways-- cc plus dd. You can define the reaction quotient q-- that's the first thing-- as the concentrations c of the products-- c to the c, d to the d, over a to the a, b to the b. Please do not confuse this with the rates. Yeah. I've been talking about the rates being equal. That's not from the rates. Remember, the exponents of the rate law come from experimental measurements. The reaction quotient comes from the stoichiometric coefficients. Why? These exponents here have to do literally with probabilities of reaction, of reacting. It has to do with the fact that if I've got two of c, then you multiply the amount of c times another amount of c in the same volume. It has to do with probabilities of reacting. That's where you get those exponents, and they're directly from the stoichiometric coefficients. They do not need to be experimentally measured. Once you have this, you have the reaction quotient. Now, why is this useful, this reaction quotient? Well, it's useful because there is a very special place where now it doesn't change anymore. These concentrations aren't changing. Reactions are still happening, but the concentrations aren't changing-- no change. If there is no change, then-- Gesundheit-- then q isn't changing because it's just a ratio of concentrations. It's a ratio of concentrations equal to this whole thing, not just the top line. The reaction quotient is related to the concentrations. So if the concentration zone changes even if reactions are happening still, it's a constant. So at this point, q equals a constant, and that is an equilibrium constant. So we'll call it keq. It's a dynamic equilibrium. k represents a dynamic equilibrium. And what's really useful about k is that for a given reaction, k is a constant. I just said it. But it doesn't matter-- if we're-- it depends on temperature. But if I keep the temperature fixed, k won't change. It's a constant. It's the constant that tells me where equilibrium is. It's the constant that tells me where equilibrium is. Now there's one more thing that we've got to write down-- that third one there. I'm going to put it on here. So k is an equilibrium constant. That's a general constant. But now, I'm going to turn my attention from any old reaction-- my k is actually very general concept for equilibrium. We are in this class going to use it for solubility. That's how we are using the equilibrium constant. And so the ksp is the solubility product, and it's an equilibrium constant. Equilibrium constant that is related to solubility. It is literally the product of solubilities in equilibrium. That is literally what it means. So we're going to do some examples to see that. It's an equilibrium-- so it's a constant of equilibrium. And it's called-- Gesundheit-- it's called the solubility product, because it's related to the equilibrium of dissolving something, dissolving some solute in a solvent. Now we're going to do examples with salts. So we're going to be dissolving salts and talking about equilibrium constants in that dissolution, otherwise known as the solubility product. It's just an equilibrium constant. There are many. This is the one we're going to care about, and we're going to do it for salts. And so one of the things you've got to-- so nacl-- that was kind of easy. That became a cation and an anion in solution. But salts can be made of many, many different types of cations, and there's lists that go on and on and on. Here are some, and they get named. So here's some of the anions. Oh, chlorine-- we just did that one. Bromine-- you can also have a carbonate. You could have chromates, hydroxides, oxalates, sulfates, sulfides, and the list goes on and on and on. There are many, many of these. They don't have to be a single atom. It can be a part of a salt that breaks off and has a negative charge. That's an anion, and here are some of the cations-- sodium, potassium, calcium. And you know about these things. We've been talking about these things. And you know if it's sodium, then you know it's only going to lose-- it's only going to be na+. It doesn't have a second weakly-bonded electron to lose. So it's really only going to be na+. But then you get-- and calcium is going to be calcium 2+. It's got two, magnesium two. And then transition metals are a little more complicated. They can be a little more complicated. Iron can be 2+ or 3+. It depends, and on and on and on for those. And so we'll do a few examples. We'll do a few examples. We'll get into the mood. And I want to do it-- first, I'm going to do an example that's actually a classic example, because it doesn't dissolve very well, but it's silver chloride. Now sodium chloride is-- sodium chloride goes all the way to the point where it basically dissolves so well. So you can imagine the equilibrium for sodium chloride goes way to the right. But for silver chloride, it's different. For silver chloride, it's different. So we're going to look at this curve. And I'll talk about this curve. Don't worry. We're coming to it. We are coming to it. Let me start in the middle again. Because what the equilibrium constant does, given the definition that I have just hidden, unfortunately, is it allows us to go back and forth. If I know the equilibrium constant, then I can get how much one of these ions is soluble. I can get the maximum solubility of it. So let's take an example. So here's a question. How much agcl dissolves in 1 liter of water? That's such a good question. I don't know. But I do know that the solubility product that this equilibrium constant for sodium chloride is 1.7 times 10 to the minus 10th. Now if I write down the dissolution reaction, sodium chloride solid is going to go to-- sorry, sodium silver-- silver plus in solution. So we put the aqueous plus chlorine minus in solution. And it is going to go back and forth. And so the ksp for this is going to equal-- I'm going to go with the equation that I wrote down-- the concentration of silver to the 1 times the concentration of chlorine minus-- these are ions-- divided by the concentration of silver chloride solid. Now we get to another thing. Remember, I got rid of water, because water is everywhere. It's the solvent. So you don't need to put it into these equilibrium calculations. Well, it's the same thing with the solid. So if I think about this a little bit, this equilibrium is between dissolved. It's between the ions that are dissolving. But there from this reaction, from the way it's written there, I've got a solid form of silver chloride. So this is not-- so this is the concentration of solid silver chloride in the solid. This is just a constant. That's not going to change. That's not going to change. Common misunderstanding. It doesn't change. So we get rid of it, because it gets folded into the constant. That's always the same. The concentration of this is always the same. And so we get rid of it. So now we have the ksp is equal to, in this case, ag+ times cl-. Talk about solubility product. Solubilities concentrate. The solubility is-- when I add this in, it reaches some equilibrium where they're dissolving and reforming. I wanted to know at what point that is, what concentrations is that going to be. This is them. So solubility product-- right there. I used the definition of equilibrium constant. And now I can use this reaction. I can use this reaction because the stoichiometric coefficients tell me-- they tell me both what the exponents there are-- 1-- but they also tell me the relative molar amounts of these things. It's the same. So if x equals the concentration of silver plus, then it must also equal the concentration of cl minus. If you just let x equal the concentration, that comes from the reaction. 1 in front of silver, 1 in front of chlorine. And so that means now that you can solve it because x squared is equal to 1.7 times 10 to the minus 10th, and x equals 1.3 times 10 to the minus fifth, m, and that equals these concentrations. That equals the concentration of silver plus. So that is how much, in equilibrium-- this is an equilibrium constant-- that is how much you will get of these ions dissolved in water. Now there is a nice way to set this up that is often taught in high school. I like it. I think it's a good way to keep track. And so if you were wanting to keep track, and you like tables, there's something called the ICE table. And in this case, it's a little bit boring, but as you'll see, it can actually wind up being pretty useful, especially when we add other ions, which are going to do soon. So ICE is initial change equilibrium. Initial change equilibrium. It's just a way of keeping track of what I just did, because I started with some amount of solid, and I had zero. I put it into water. I had zero, but then I added 1.3 times 10 to the minus fifth, and I added 1.3 times 10 to the minus fifth. And so my equilibrium is a little less, a little less solid. And over here, it's just these numbers. That was indeed boring, but have no fear. It becomes helpful. It becomes helpful because what we're going to do next-- not right now-- but on Wednesday, what we're going to do is we're going to mess with this equilibrium. Oh, equilibrium, I'm going to throw you off. And equilibrium is saying, no you're not, because I'm equilibrium, and I will always be found. And that's called Le Chatelier's Principle, and what that does is it allows us to really go back, and we get to actually add a bunch of chlorine ions to this, and we're going to see what happens to this balance. It will find equilibrium. And we have the tools now to know how to find it. That's that green curve there. This is what we're going to work with. There's the concentration of chlorine. There's a concentration of silver in ions. You see the dissolved chlorine on the x-axis, dissolved silver on the y-axis. And look at that. I've got that point b there that is where these things have found equilibrium when it's not messed with. Now the thing is, if you were a reaction happening, and you hadn't gotten to equilibrium yet-- so I hadn't added much silver chloride-- if that was the case, then you might be at point d. Notice that the reaction quotient, q, is less than the equilibrium constant. q is less. That means it's going to be driven to dissolve more than it precipitates until it gets to point B on this curve. You could also be way, way out with a much higher q point, e, and you know it has to find equilibrium. It's going to find equilibrium, which is what we just calculated. Those are the concentrations of equilibrium. It has to get back to point B. It's going to do it by precipitating out solid. It's gone too far. But you can see that equilibrium can also be had and will be had. It shall be had. Because k never changes if you fix the temperature. k is the same. So I can mess with this in other ways. I can go and add a whole bunch of chlorine, and it will come back down onto that green curve. It will come back down onto that green curve, which is the curve that guarantees k is the same constant. That's what that green curve is. And there's one last point I'm going to make today, and that is units. Units of k depend-- it was like the rate law. It depends on the reaction. Look, a concentration is a concentration. That was deep. Units are capital M moles per liter. But that means that k-- k is two of those multiplied together. So in this case, k has units of m squared. It must, or it wouldn't work. k can have units of m cubed. We will pick up here on Wednesday, and we'll cover more of this, more examples before we move on to acids and bases. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 3_Atomic_Models_Intro_to_SolidState_Chemistry.txt | Welcome to the third lecture of 3091. Let's get started. Today is a very exciting day, because we are going to discover the electron together. That's our goal today. It's right there, discovery of the electron. And we're going to start talking about models of the atom. Before we do that, I want to count. I want to count, because that's what we did on Friday. And we'll use this problem actually twice today. Either way, like if I do a problem in class, and it's in a goody bag and stuff, I don't know, it might be the kind of thing you might be asked on a quiz. I'm just saying. So that's good to know. If I ask you this question, because we counted on Friday, we learned how to count. And we learned about the mole and Avogadro's number and how that gets us into and out of the atomic world, right? So if I say, how many gallium atoms are this strip of gallium? Well, we can do that now, right? Because well, how do you start? This is where you start. And you are all taking it with you everywhere now. Remember on Friday, we talked about how you got to bring it with you everywhere and tweet about it and send me pictures and all that. I need the density, right? I need the density. Density in the periodic table. Let's start with the density. So if I have the density of gallium and I looked that up as 5.9 grams per centimeter cubed. OK, that's good. Well, you know, from the strip I can measure it with the super most accurate ruler in the world that you have now in your hands from the goody bag. And you see those are the measurements. And so if you measure it, you have a volume. OK, it's those three numbers multiplied. I won't write it out. It's 0.5 centimeters cubed. And so now with the density and the volume, I get that that strip has 2.95 grams. OK? Now that sounds pretty good. It's promising, but I want the number of atoms, right? I want the number of atoms. And that's where, again, you go to here, because in there is the grams per mole and also the atomic mass units, which, as we talked about Friday, are the same thing. Right? And so if I look that up, I have that for gallium, we have 69.723 grams per mole, right? And so that means that in 2.95 grams, I have 2.95 divided by that number, 69.723-- let's raise this up a little-- which equals 0.0423 moles. I still have not answered the question because the question says, how many atoms? Right? But that's where n sub a, or the mole, or this dozen, right? But it's not a dozen. It's a much bigger number. But this number, Avogadro's number, comes in. And so in that strip of gallium-- ah! No. In that strip of gallium, you've got, in the GA strip, you've got 0.0423 times na atoms. That's something like, you know, 2.5 times 10 to the 22nd atoms. That's a lot of atoms in that strip, but I was able to do. But the secret is the-- right, remember? One atomic mass unit gets you back and forth between one gram per mole. And by the way, the atomic mass unit is nothing. It's just a mass unit. I mean, one AMU happens to be, by the way, oh, roughly 1.66 times 10 to the minus 27th kilograms. Huh! That works because now you can multiply this by a whole mole, and it's a gram. Back and forth, OK? Good counting. Starting with counting. We're going to end with counting. But in between, we're going to discover something beautiful, because last week-- OK, the thesis of this whole class, remember, is electronic structure holds the key to everything. Everything in your life can be understood if you understood the electronic structure of the elements. Week one, last week, we talked about this. We built it on Friday. We talked about elements. We did a few reactions. We talked about balancing. We made sure we have enough nitrogen to grow plants for 100 million more years, eliminate reagents. This week, we're going to answer the question, why are these things different? We're switching gears, right? We know that there are these indivisible atoms now. And we built a whole table of them, right? Mendeleev. But now we want to know why they're different from one another. What causes that difference? And that starts with the discovery of the electron. And really this is such-- so let's write this here. We'll have this board. This is such a great story. This is a detective story that starts with an atom looking-- so these are atom models, OK? So back in, oh, let's say, 18-- ah, OK, 460. If it's Democritus, it's 460 B.C. And then remember, we sort of went all the way to Dalton, who, in 1803, he said, yup, Democritus, we're going with that. We're going with that. And oh, there are these indivisible things called atoms. So that was 1803. But we need to understand them in order to explain this. We need to understand them. And so this is a detective story. It's one of the greatest ones ever, ever, ever played out in science. It led to nothing less than the understanding of all of nature. OK, now, how did it start? Well, it started with J.J. Thomson. Now, so J.J. Thomson-- by the way, he won the Nobel Prize for this discovery. He's credited with discovering the electron. Also seven of his students won the Nobel Prize. That's cool! And I remind my graduate students of that all the time, without trying to put any pressure on them. How did he do it? Well, you see, he had these things. There is his lab. There's Thomson, there's his lab. And these things were around. They're called cathode ray tubes. Cathode ray tubes. What do they do? All right? What do they do? Well, your grandparents would know something about them. I'll talk about that in a few minutes. But Thomson, Thomson was interested in seeing what he saw on this thing here. And that's the photographic plate. So what we're going to do is we're just going to draw the cathode ray tube, draw it as best I can here. So there is a cathode ray tube, and over here we've got a cathode, and over here we've got an anode. So this would be a cathode, and this here is an anode. OK, good. Now, that means that if I hook the other two pieces of metal, two pieces of metal, and if I hook them up to a voltage supply, then I can get charge on them. We know that already. But what they didn't understand is what happened next. You see, because if you crank the voltage up really high and you put a phosphor screen here-- so this was a phosphor screen-- and if you crank the voltage up high enough, and this is what Thomson did, you pump the gas out of it. Right? So you get it so there's not too much stuff in there. Then what happens is something shot across and lit up the screen. Something. What? They really didn't know, and when you don't know, you experiment and you apply the scientific method. And one of the things that Thomson did is he said, well, what if I put some charge plates here, like this and this? So I have a [? set, ?] right? So it is said, this is going to have charge that flies off. We know that, because we kind of know what happens. But he didn't know what was happening. He's just cranking the voltage up, and all of a sudden the screen lights up. So he's like, well, let me put some charge plates here and see what happens. And what happened there is it went like this, right? And it struck the phosphor screen there, and it glowed there. So that's with charge, with charge plates. Right? No charge plates, just to be perfectly clear. And he could play with this, right? And people do play with this, to my great surprise, if you go onto the internet and you Google this. You find people make these and play with them, and that's a magnet. This is what it looks like, right? But we didn't know what this was, right? So that's a magnet, and this person is just turning over whether which part of the magnet, where the magnetic field is oriented. And you can see, you can really mess with this. What could it be? It's responding to an external field. It's responding to an external field, so we know right away, this tells us that atoms are not the ultimate form of matter. Atoms were not the ultimate form of matter. That was a big deal, because the atoms here were neutral. They didn't respond, so this must be something else that's flying off and responding, that I can see visually on this photographic plate. It must be something else. Well, he was also-- Thomson was-- oh, ha! Going down here, I'm going to use this so it's right next to it. He was not able to measure the mass of these things, but he was able, using a combination of magnetic and electric fields and the little bit of Maxwell, some ENM, he was able to deduce the charge to mass ratio. So this falls over here. I'll put it here, from Thomson. So he was able to get the charge to mass ratio. And he was able to measure that at minus 1.76 times 10 to the 11th coulombs per kilogram. Now, he knew it was negatively charged because of this, right? He put the plates on, and it wanted to avoid the negative plate. So they knew their ENM at the time. They knew that the negative charges would repel. So he knew that it was a negative charge. Now, the last thing, oh, is this one important for chemistry! He would swap in different materials, different metals for the cathode and anode. For the cathode especially. And the results were the same. The results were the same. Oh! Independent, independent of metal. Independent of the metal. That meant this thing that he was observing for the first time was fundamental. There's something fundamental about it. Fundamental! This was a really big deal. This was a really big deal. And it really opened up the idea that the atom had something else in it. Now, see, Thomson knew that the atom was neutral, like I said. So you've got these charged things coming off of these atoms, which are neutral. That means that it must consist of these charged particles. But if it's neutral, it's got to have the other charge of particle in it as well. And so the detective story goes on. So this would be Thomson around 18-- oh, things really picked up. 1807. And he said, well, OK, these electrons must be inside of the atom. That's how I'm liberating them. Crank up the voltage, and they must be in there. But so must positive charges. So must positive charges, OK? And that was Thomson's model. So at that time, you could really have this picture of stuff inside the atom. Now, like I said, he couldn't measure-- they didn't have a scale good enough to measure the mass of the electron. He could get it out, all right? But it was Millikan and his very famous experiments called the oil drop experiments that gave us the actual charge. And then from this ratio, we can get the mass of this mystery thing that was coming out of atoms, OK? And instead of trying to draw the Millikan experiment, I found a wonderful, short, like one-ish minute video, which I'll play, because it shows how the Millikan experiment worked. It's really cool. Oh! [VIDEO PLAYBACK] - Robert Millikan, working at the University of Chicago, succeeded in measuring the charge on the electron. That's not happening. - He allowed the fine spray of oil to settle through a hole, into a chamber where he could observe their fall. The top and bottom of the chamber consisted of electrically charged plates. He introduced a source of x-rays which can cause creation of charges when they strike matter. Charges reduced by the x-rays attached to an oil droplet, and producing one or more charges on the droplet. When there is no voltage of light, the fall of the droplets is determined by the mass and the viscosity of air through which they fall. When a voltage is applied, the droplets that have a negative charge will fall more slowly, stop falling, or even rise, depending on the number of charges on them. By adjusting the applied voltage and observing the droplets, both with voltage off and voltage on, Millikan was able to determine that the charges on the droplets were all multiples on a smallest value, 1.6 times 10 to the minus 19th Coulombs. He took this to be the charge on a single electron. [END PLAYBACK] All right, so now you see that that would have been-- I hope that was-- could you hear that in the back? Sort of, OK. All right. Oh, I got this. You know, that would have been hard for me to draw the animation, but what a brilliant experiment, right? So spray some oil in a container, OK? Try to get the drop small. You don't know how small they're going to be. And then as they fall-- OK, have an electric field inside there. And then as they fall, zip them, zap them to charge them. You also don't know how much charge you're putting on them, right? But you're charging them. And if they're charged and they're in a field, and there are these tiny, little microscopic droplets of oil, they're going to maybe slow down or suspend or maybe even go the other way. And what he observed by doing this over and over again, that there was some multiple that you never got below, right? And you couldn't say necessarily for 100% certainty that it was the fundamental charge of electrons. It might have been a multiple of that. But it was pretty clear. You could never get below that. So that was the charge discovery in this detective story, and it allowed us to understand that the electron had a mass and it had a charge. And that we knew both of them at the time. So at the time then, we already had a much deeper understanding, if you think about it, just four years later than the time of Dalton because of these two experiments. Yeah? So we said that Thomson was 56-- he'd be [INAUDIBLE] 1940s, so wouldn't that be 1907? 1856 to huh? Thomson was alive. Alive. [INAUDIBLE] I think I read in my-- he was ahead of his time. [LAUGHTER] Thank you very much. I think I read this wrong, and that's because I don't have my glasses on. So now that I look at that, it wasn't that small of a distance. [LAUGHTER] Thank you very much. You make mistakes. You make mistakes, and that is how you learn! [LAUGHTER] You don't make progress. Progress has almost nothing to do with success. I mean this. I'm using my own mistake here as an example. Progress has almost nothing to do with success. Progress has only to do with what you choose to do with failure. Did somebody just say, whoa? Thank you. Thank you. That hit me here. All right. OK, let's do a why this matters. Let's do a why this matters. Why does this matter? Well, because, see, these guys were trying to figure out what was inside of an atom. But other people, like John Baird, said, well, wait a second. You just gave me a paintbrush. You gave me a paintbrush. Look at this. That's a paintbrush, painting with a magnet. It really is. And the screen would light up over here. And so he said, well, I can paint. I can paint pictures. And this really is the first television screen. This was the first TV screen, and all they needed to do, there-- OK, right? There it is. Look! That's a cathode ray tube. Maybe that's one of Thomson's students who won a Nobel Prize. And but now, you put this down, and you put these sort of things around it. What are those things? Magnetic fields, that's all it is. It's just magnetic fields. It's Thomson's experiments, right? But now they're using it to zip the beam around faster than you're [INAUDIBLE] keep up with so that it looks like a picture. Now, electron painting had never been done before, because we didn't know that we had these electrons. But as soon as we knew, boy, did that launch a completely new era of screens. Right? The era of screens. We don't use cathode ray tubes. I was going to say, ask your grandparents. They'll tell you about the cathode ray tube TVs, which they all had. But you know, we don't use cathode ray tubes to paint with electrons today in that way, but we still paint with electrons today. Right? Youre OLED screen is still simply an electron-based painting tool, right? OK, we're just pumping the electrons into the phosphor in a different way, and we'll be talking about that as we go through the rest of this week and we understand how electrons interact with light coming in and out of an atom. OK, so that's my why this matters. And by the way, a side note here is that when TVs first came along, green was pretty easy. There were a lot of chemistries that were used for this screen. You put a different chemistry here, and it lights up differently when electrons hit it. Why? Wait until Wednesday. Green was easy. Yellow was easy. Red was hard. Red was hard. They couldn't get a good red. And of course, that's essentially the reason there were no color TVs until the '60s, right? And the answer, of course, was here. The answer was that there was a phosphor that worked, but it was yttrium orthovanadate with a little bit of europium added to it. Just a little bit of europium. Why did that work? Again, we need to understand how electrons interact with matter, which is where we're going. And speaking of yttrium, speaking of yttrium, this is also a side story, but it's kind of worth noting. You know, elements are named often-- well, elements can be named after many things, right? In this case, it was named after Ytterby, Sweden. Ytterby, Sweden, is a pretty cool place. They had this one cave. You got to go to it. It's really cool, because four elements of the periodic table were all discovered from Ytterby, Sweden, in this one cave. Four! And I keep thinking, where is that cave in Cambridge? [LAUGHTER] Hm? They cannot discover any more elements, because there's no other way to mess with the name Ytterby. [LAUGHTER] That's pretty cool. That's pretty cool. That's a cool cave. That's worth visiting. All right, so the structure of the atom in the 1900s-- thank you for the date correction. 1900, the structure of the atom was as follows, all right? The atom is electrically neutral, but there are these negative charges. These negative charges called electrons. The electron has a very small mass. We know that from the oil drop experiments, right? Which means that the bulk of the atom is positive, because the bulk of the mass of the atom-- you know, the atom weighs one thing. The electron is just a small part of that in terms of the mass, so most of the mass is positive. And there was a question. OK? So we're getting somewhere, but how do these charges really look? How do these charges arrange inside of this atom? Is Thomson really right? And that was the next part of our detective story, was to really understand what these charges are doing inside the atom. I'm carrying this around like my security blanket, which is making me feel very secure right now. And you should all carry it too. This part of the story relies on another type of energy that was being discovered at the time. And you all may recognize this as radiation. Now, radioactivity, radioactivity is nothing more than ray activity. This is what they named it after, ray activity. That's radioactivity. And it was these three people who really pioneered the understanding-- gesundheit-- of radioactivity and in particular, the fact that some elements seemed to just be radioactive. What did that mean? Well, it means they gave off these rays of energy. That means they gave off these rays of energy. You had Henri Becquerel. You had Marie Curie and Pierre Curie. Notice I didn't even try. Where's Jerome? I didn't even try with the French. But anyway, they took these materials, like uranium ore, minerals made out of uranium, and they said, this stuff looks like it's got something coming off of it. Let's see if we can find what elements are causing that. All right? And Marie Curie, she found polonium and radium this way, those two elements. Polonium named after her home country, her native country of Poland. And this stuff gave off all this energy, and it was so energetic they could literally put it in a cup and have some writing there and put it in a super-dark room, and it would illuminate the picture. Right? It would illuminate the picture. But this is not a recommended way to light your pictures. So Pierre carried a vial of radium, because it was so cool and it glowed! He was like, look at this stuff! Right? And he actually did have bouts of radiation poisoning, although he died, tragically, when he was struck by a horse-drawn carriage in 1906. Marie lived much longer and won two-- but also died because of radiation poisoning. But she won two Nobel prizes. First woman to win a Nobel Prize, only woman to win two, and for only person ever to win one in two different disciplines, physics and chemistry. She was quite a brilliant scientist. And they gave us this radiation. Now, OK, but see, OK, they were really into radiation, but it was Rutherford who said, well, wait a second. Maybe I can use this stuff to keep going along this story. Right? Rutherford was like, well, maybe there's something we could do with this radiation. And so what happened? Oh, there it is! And so I'm going to put it right above it so you can see. So he said, well, OK, if I take something that glows, maybe some radium, and I carve out a little thing here, and maybe I put some plates to kind of protect scatter, I can make a beam of this stuff, OK? So this would be like maybe radium, let's say. And then what he did is he took this photographic plate and he curved it, just so he could really collect as much as you could, right? So you can really collect at really high angles. And he did the same thing. He put charged plates on either side, and he found that you had three different types of beams. He had one that went like this, one that went like this, and one that went like this. And we now know that this is beta, this is gamma, and that's alpha. Let's put this right above. This is so cool to watch. So what Rutherford did is kind of like this. He had stuff and he made a beam of it, and then he put stuff around it. But if you got a beam of something that might be charged, you're not going to smash it or burn it. You're going to put some plates around it, or a magnetic field or electric field. And that's what he did. And notice, alpha bends down, towards the positive charge, towards the negative charge. So alpha must be positively charged. I said, OK, that sounds cool. Maybe I can use this. Maybe I can use this on this, on this question of how these things are spatially distributed. So then he said, OK, let's just screen out those alpha particles. And let's just take these alpha particles and let's shoot them. So now comes the famous experiment, so you had stuff in here, radium making a beam. And you collect it and you make it just alpha particles. OK? So now he's just got all alpha particles, and what he did is he also had a photographic plate, or this could also be a phosphorus screen, and you can watch where-- You know, if it's a photographic plate, you record events, and you can see it like a picture, a film. And if it's a screen, then you watch it. Either one was used. But what he did is he put a very thin strip of gold in there. 0.7 microns of gold, so that's a 0.7 thickness is 0.7 microns. Really, really thin, like a tissue paper of gold. And he shot these alpha particles at it. Now, why is this important? Because-- gesundheit. Because if this model of the atom was correct, then, you know, he'd isolated these positive particles that he's shooting at these atoms. If that model of the atom is correct, then this is kind of like distributed all over. Right? And so it's kind of neutral almost anywhere you look. It's kind of neutral. And so you would expect that a positive charge-- you'd maybe feel a little bit, but it kind of feels mostly neutral because it's evenly distributed. So it just kind of come out, but that's not what he saw. That's not what he saw. What he saw is that sometimes that would happen. You'd get a little bit of deflection. And other times, you'd get a signal out here, or there, or even right back at the radium. How is that possible? This did not work. That model of the atom did not work. And Rutherford himself said, it was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you, because they just didn't expect that a positive particle could bounce backward off of an even distribution of charge. And so this is a picture from your textbook, Avril, that I know you are actively reading on a daily basis. So there's the alpha particles. That's what I drew, that they would expect-- or that what I mentioned. They would expect it to just kind of pass through. This is what was found, where they kind of come back at very sharp angles. That's what he actually observed, right? And it led to the Rutherford atom. And the Rutherford atom was that, look, the only way that this can happen, that you get these events, where a positive charge comes back like that, is if there's a very, very strong concentration of the positive charge in one small place. And so he said-- let me see if I can read my date correctly-- 1912. And so Rutherford, he said that the positive charge has to be in the middle, and then the electrons must be far away. And that is also sometimes, it was called the planetary model. And in fact, there was another scientist from Japan, Nangaioka, who, five years earlier, had predicted-- although he sort of predicted this, because he loved Saturn. [LAUGHTER] He really did. So he said, you know, it seems that maybe we have a Saturn situation. So he called it the Saturnian model. But I want to mention it because it was part of the way there. Right? But in this planetary model, which came from Rutherford's experiments, you had a very different picture now of the atom. You had a very different picture. You had all the positive charge really concentrated in the middle. Now, the thing is also that the atom, what they were starting to understand is that the atom isn't just-- like, there isn't just all this space here. Right? They knew the electrons weren't distributed evenly in here. They knew they were sort of far away. There is a lot of space. There's a lot of space, that the radius of the atom is 100,000 times bigger than the radius of the nucleus. Right? So just to put that in perspective, so here's a picture. There would be the nucleus, the head of a pin. And this would be the atom, a stadium. OK? If you took all this stuff, all the stuff in a human body-- that means the protons, all the stuff in the nuclei, the neutrons-- ooh, I'm getting there! And the electrons, and you take 7 billion people, and you put that all into one volume, it is the volume of a sugar cube. So you know, so things are pretty empty out there. Things are pretty empty out there. How empty are they? But this is where you find true meaning. This is deep! How empty is stuff? Look, if I weigh the universe, you know, the universe itself, we're going from the atom and the electron, all the way up to the universe. The universe weighs something like, oh, 10 to 60-ish kilograms. That's over roughly $30 billion lightyears. OK, now, if we just do the math, so you weighed the universe. You know how big it is, roughly. Plus of give or minus-- plus or minus. Then that means that 10 to the minus 20% of the whole universe is stuff. And the rest is empty. That's deep, because we find meaning in all of it, don't we? That's deep. Think about that. That's not just the universe, but the atom. It's the same. You know, so much of everything is nothing. OK, planetaria, but there was a problem with the planetary model. There was a big problem with the planetary model, because classical ENM told us that if, you know, if a charge is accelerating, if a charge is accelerating, which it is, because it's ughhh! We think it's going around. We think they're going around here. They are not! Friday, we'll know more about that. But we thought they were. But if it's accelerating, you know, to keep the circle, then it's got to be radiating energy. That's what classical ENM tells us. An accelerating charge loses energy. So if you go with that, then the electron-- let's see, the stability analysis that we get out of something like that-- hang on-- is that the atom would be stable, four-ish 10 to the minus 11 seconds, five times, roughly. That's not giving us a lot of time. All right? And so we were getting so far. We had this great model, but it didn't go with classical ENM. So atom's stable for that with classical ENM, oh! Because you know what's coming. We're going quantum. Not now, I'm just preparing you for later-- Did I hear an oh? Thank you. Friday. OK, so Bohr comes along, and on Wednesday, we're really going to go into Bohr. And we're going to talk about how Bohr's model, how Bohr's model of the atom allowed us to understand how light and matter interacts. That's a big deal, right? But we'll go into that on Wednesday. For now, I just want to tell you what he did, which is he thought about this problem a lot. He's like, this can't be. We need some way out of this. And he wrote this paper, where he said the following. He said, let's go over here. He said, OK, "In order to explain the results of experiments on scattering of alpha rays"-- we just did that, scattering of alpha rays-- "by matter, Professor Rutherford has given a theory of the structure of atoms." And then Bohr goes on. And he says, "Great interest is to be attributed to this atom model." Now, that is a serious diss. Great interest? If somebody calls your model of great interest, you know you are in trouble. He might as well have said back then, you know, this model is of great interest because it's totally wrong! [LAUGHTER] And if you want to know what's up, come over to me and I'll tell you. And that's what Bohr helped with in his work. And what he did is he wrote down some postulates. He said, look, the Rutherford atom is correct, but the problem here is classical ENM theory. Right? You cannot apply classical ENM theory to the orbiting electron. You can't, because that's what happened. So something's wrong with literally nothing less than classical physics. Right? He said, Newtonian mechanics still works, right? Newtonian mechanics works, but not ENM. So he said, we can go classical on how things are moving around. F equals MA stuff, but not ENM. He also postulated that the energy of these electrons, the energy that these electrons sit at or have-- we'll get into this a lot-- he hypothesized that that's quantized. Quantized. Quantized. We're saying that word. It's a beautiful moment. Quantized. Quantized means that it could only have certain values, certain values, which means that if this electron were to be in one value and then change into another, that the transitions are quantized. The transitions of an electron from one energy level of the atom to another can only have certain values. And that is what we get from the Bohr model. And boy, did it explain things! That's going to be the subject of Wednesday, as I mentioned. Right? But this gets us to the Bohr model. And here's Bohr. We're now-- oh, I got to draw this. We got all the positive charge in the middle, and then these electrons can only have certain radii, certain energies. They cannot continuously roam around. They can't. They're quantized. That's the Bohr model. And this is closed in 1912. This is closing in on the last and final model, which we'll get to this week, which is where quantum mechanics comes in. OK, now, we got one more thing to talk about. So this is where we're going. We're building up the full understanding of the atom. But there's one more thing that I've got to come back to, and that is that, look, we just said there are these positive charges in there and these negative charges out there. But it turns out that atoms can have different masses and stay neutral. All right? So we're now in a position to talk about this thing that we've been calling the nucleus and the fact that it's not just protons. We're in a position to talk about that. And so, you know, Rutherford gave us this idea of the proton, the particle, the proton. And it wasn't until 12 years later that Chadwick found the neutron. What the neutron does is it adds mass to the atom. See, the mass didn't work. So again, you know, if you know what the mass of the proton is, then you can add it up, and you get the mass of the electron, although it's like 1,000th as much. But it doesn't work. There's something else in there that's adding mass. But it's not adding charge, because these things are electrically neutral. So those other particles are called neutrons, and what it does is it gives us ranges of masses for the atoms. Right? It gives us ranges. So because, you know, hydrogen now. If you think about it, 99.99% of all hydrogen is just that thing on the left, a proton and an electron. But sometimes, it has a neutron. So it weighs more. It's still got neutral charge. It's called deuterium. And then you've got tritium, which is unstable. You can make it. And these are called isotopes, as many of you, I'm sure, already know. These are called isotopes, and so the isotopes of an atom, the isotopes of an atom with the same atomic number, different number of neutrons, different number of neutrons. So atoms have these three things in them. They've got the neutrons, the protons, and the electrons. We've been talking today about the protons and the electrons, but they got to have these other things in them, or the mass doesn't work. And so if you look at something like carbon, you see the same thing. And now I want to get into how these things are written, right? Because here's carbon, and there are those electrons. They're now put out there. OK, we didn't draw circles, so maybe this is around Rutherford's time when Avril made this picture. And there is the pluses, and there's other things in there. Right? And those are the neutrons. And so what we do is we have a nomenclature for this, which is the standard. And that is that if we write an element like this, this would be the element. And we put something here and something here, this is the mass number. OK? So that's how many protons plus neutrons. Neu-- ah! Neutrons. And this is called the atomic number. Now, the atomic number, I've been referencing this thing, this atomic number. I've said, Mendeleev and many others, they put things in order of atomic number, maybe mass sometimes. Mendeleev did the thing where he did those things and properties, right? And the atomic number was just a number, until later. And once we discover another type of ray, x-rays, then this has real physical meaning, right? But right now, it's still just a number. It's a number of the element in the periodic table. Right? But that mass number tells me how many neutrons I have and what the isotope is. So now I go back. I can go back, and I can look at carbon. Oh, by the way, carbon-- I wanted to write this down before I say that. Carbon, when you look it up in the table, you have 12.011 AMU, or grams per mole. That's what's in the table. But now you know why, because it's just an average. And in fact, the IUPAC, which is a rocking body of the International Union of Pure and Applied Chemistry, I'm sure you've seen them on the feeds. They hold meetings and in Vienna on a nonstop night of intense meeting and partying. And they said, hey, maybe in 2011, maybe we should actually do this with the periodic table, just to show that actually there really is a range of-- if I take carbon out of the ground here or carbon out of the ground there, maybe they have different AMUs, right? These are averages, and there are ranges. So they're actually talking about different ways of even putting this into periodic tables, right? But this is it. And this gets us back. Oh, there's boron and nitrogen ranges, right? But what we do now, at least, is we just look at the one number, because that's sort of averaged over, you know, basically anywhere you can dig this stuff up. And the stable isotopes. Stable isotopes. And it gets me to my last slide, which just comes back to the beginning, as I promised. Right? Because now I can say not only how many gallium atoms are in this strip, but I can tell you, gallium has two stable isotopes. By the way, it's got a bunch of unstable ones, but it's got two stable ones, gallium 69 and gallium 71. And I can now ask you, how many gallium 69 atoms are in this strip? How? Because I can again look up in the periodic table and find that in a gallium strip, I know how many atoms I have. But if I say, well, 69 times x plus 71 times 1 minus x, that's going to equal 69.723 AMU. That's what's in the periodic table. But I now know that that's simply an average over all the-- right? So this would be the fraction of 69 GA, right? Because that's how simple these isotope problems are, right? It's simply averaging over the potentially stable isotopes. And I leave you with one more great mystery. As you are pondering this, think about this. There are two elements, ah! There are two elements for which there are no stable isotopes. Technetium and promethium. Nothing! Technetium, we use all the time. 80% of all MRIs use technetium, but it's not stable. We got to make it, right? That's pretty deep too. How did that happen? Why did that happen? For the answer, you'll have to stay tuned. Have a good night, and see you guys on Wednesday. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 1_Introduction_Intro_to_SolidState_Chemistry.txt | OK. Let's get started. How's everyone doing? Wonderful. Come on, let's do better than that. How is everyone doing? Wonderful! Thank you. Welcome to 3.091. This is the first day. And I'm really excited. I hope you are too. In this very first lecture, we're going to do a little bit of administration, tell you a little bit about the class and how it's set up. And then we'll give a little mini-- we'll have time for a mini lecture in the second part of the lecture today. So I thought I'd start by inducing me. My name is Jeff Grossman. I am in Course Three, that's the Department of Material Science and Engineering. My background, I went to Hopkins, undergrad, moved my way towards the west coast, Illinois PhD. And I did a post-doc at Berkeley. And then I came here to MIT about nine years ago when I joined the faculty. My own passion, my own interest in research is in materials for energy and water and chemical processes and separations. So I thought I'd just give you a little example of what I mean by that. So I get really excited about the ability to take the material and make it do something that it can't do now, but that it needs to do to solve a problem. Maybe it needs to be cheaper. Maybe it needs to be more efficient. So one example is in this material. This is a barrel of oil. And what we do today with this barrel of oil is we mostly burn it. A lot of what we do is burn it. Now if you do that, and you think about it in terms of energy, that's 159 liters of this material. And if you think about what you're carrying about as energy, you get 1.73 megawatt hours of energy out of that. But see, I like to think about materials, again, and ask them what they can do differently for me, for these problems. And so if you think about that and you say, well, OK, let's just take 1% of the carbon in that barrel of oil and do something with it. Let's make thin film solar cells. Well, if you do that and they're not very good, they're 5% efficient, they completely die in a year, you get 10,000 times as much energy over that year than burning it. So it's an example of what you can do when you think about using elements differently. That's what I get really excited about in my research. And that's just one example. You can take that same carbon and you can make a thermoelectric, that's on top. Or you can make the world's thinnest filter. That's a piece of graphene with a hole in it. And that's still working with just one element, carbon. All right? So there are many, many things you can do once you understand what these elements are and how they combine together. And that's what we're going to talk about in this class over the fall. How many elements are in your phone? Listen, we just talked about one element. Anybody know how many elements are in your phone? How many? Take a guess. Seven. 27. OK, it depends. It depends. Do you have the Samsung? If you've got the Samsung whatever, it might light on fire. If you've got the newer one, it won't. Do you have an iPhone? Well you know what? In the iPhone, there's 63. 63 elements. So we have these lectures in Course Three, they're called the wolf lecture. And I gave one a couple of years ago. If you're interested in hearing a little bit more about what I do or what some other people do who think about materials in this way, you can use Google wolf lecture and you'll get to some of the videos. And I'll mention it when they come around during the year. OK, so that's a little intro to me. What about the class? So as I said, the class-- at the heart of the class is kind of what I just described. But in order to do that, we need to know some very basic things. We need to know how atoms are arranged, the atomic arrangement right there. But we also need to know what atoms are there in the first place. So if we know that, if we know those things, then we can build, then we can build. We can get properties. We can get structure. We can get processing. We can get performance. And we can talk about how all of these things relate to each other. They're all correlated. If I change the processing of something, I change the properties. How do you know how it changes when you've got to know these things? And at the core of it all is really the thesis of this class, which is that the electronic structure-- Paul will be talking about this-- structure of the elements holds the key to understanding. That really is what this class is about. Holds the key. The electronic structure of the elements holds the key. That's what we're here to talk about. And the first part of this class is really the basic foundations of chemistry. Some of you may have seen some of this before. We're going to build these elements. We're going to talk about them. We're going to learn about the electrons and the electronic structure. But then we're going to make solids out of them and we're going to talk about what those do and how the chemistry relates to the solids and to the properties of those solids. Now, I want you to know, again, this is the sort of administrative introduction. You are not alone. On this journey, you have many, many resources. And we are here to help you, to help you learn this material and do the best you can. So first you have me. You have your TA. We'll talk about them in a second. You have Laura. Oh, we're going to talk about Laura in a second. You have the textbook and the internet. I hear there's stuff on the internet. And you have each other. And I really want you to work together. Thank you. Who said --that was... I was like, that hit made right here, right in the first 10 minutes of class. So the textbook is Averill. It's a really good textbook. It's a really good textbook. Please use it. Please. It's available here. Notes, like I said, will be posted on the same day of each lecture. I'll post whatever you see on this screen. I will post. All right, good. The goody bag. So the goody bag is another part of your homework. And these will be given out nine weeks. So sort of like the quizzes. And in each goody bag, which I'm going to talk about in a second, there are things to do. This is a hands on compliment to 309.1 lecturers and all the other materials. Now, these goody bags are really important. And one of the two quiz questions, every week that there's a quiz, there are two questions. And one of them will be directly related to something that you're supposed to do in the goody bag. And in fact, we will most times ask you to bring something into the quiz that is in the bag. So don't throw it away and please do it. Because one of the two quiz questions will be very related to the goody bag. And the other will be related to problems, you know, lecture. Everything's related to lecture. But at least one will be related to what's in the bag. So this, we really take this seriously. This is a very important part of your homework. And oh, there it is. Some of which you need to bring to the quiz. And we'll tell you what that is before the quiz. I want to tell you a thing, though, because here's the thing, goody bags are-- it's a little bit of what's going on here? This is a lecture class. Why are we getting goody bags? And it's because I believe in the soul of MIT. I believe in the soul of MIT. And that is that we learn best by thinking and doing, by thinking and doing. And even though this is a lecture class, I still want you to do. I want you to have stuff in your hands so that you can play with the chemistry that we're learning about. Now, this goes back. I said the soul of MIT. What do I mean? This goes back, all the way back to before MIT was born, 1850s. You've got a group of really smart people. They're getting together and they're meeting. And they're saying, OK, we're going to start up a new university. What should we do? And they wrote out a plan. They wrote out a plan. And it's called the institute plan. That makes sense, 1860. It's a great read. But I want to pull out one part that's really important, that what they wanted to do with this great new institution was to do something that would serve the interests of the commerce and the arts, as well as of general education, call for the most earnest cooperation of intelligent culture with industrial pursuits, intelligent culture, industrial pursuits. That is Mens et Manus. That is Mens et Manus. Mind and Hand. It's so important to MIT that we put it on our logo. We put it on our logo. That's how important it is. We don't put some animal on our logo. We put what matters, mens et manus. We don't put the word truth. Veritas, veritas, I mean, I'm not going to name names, Harvard. But I mean, isn't that setting the bar kind of low? You know, were they lying before? I don't know. I'm not-- look, honestly I don't know. I don't know. But what I know is that what we know is that, of course, the goal is truth. The difference is, we know how to get it. Mens et manus is how to get it. Let me ask you a question. Why are you here? Not here in this classroom. I know you're in the classroom because you signed up for it. Why are you at MIT? Why are you here at MIT? I can tell you, you are here because you are some of the brightest, most gifted, most talented students on the planet. Right? Thank you, whoever said that. Agreement. That was like a like online. But you are here because you want to use those talents to make the world a better place. You are here because you know how to answer any question. But you are also here because you are going to experience a transition. You are going to experience a transition here. You are going to make the transition from knowing how to answer any question to knowing which question to ask. And that is the transition from student to scholar. That is MIT. That's mens et manus. Now, it's not easy. You don't come here-- nobody comes to MIT to phone it in. If universities were restaurants, this wouldn't be that fancy one where you go in and you order, and then somebody cooks and brings you your food. This would be the one where we all go back into the kitchen and together we make the best meal we've ever had. That's the MIT way. That's the MIT way. And it's not about-- you don't roam the halls here and bask in this reputation and think of it as some privilege to be here, because you are MIT's reputation. Freshmen, raise your hand. Yeah. Starting-- There's a few of you. Starting today, you are MIT's reputation. Starting today. It's on you. So we don't walk around these great halls and feel privilege. We feel responsibility. That's what it means to be here. OK? All right. Good. Well, now that we've got that all straightened out, let's move on. That's the end of my administrative stuff. And with the last 20, 25 minutes, I want to do a little bit of an introduction to chemistry. And I'm not going to ever test you on history. But I want to just kind of get ourselves in the mood by going back. So I'll give you a little bit of history here in a few slides. So we are interested in solid state chemistry because chemistry is that essential ingredient to understanding the natural world. And the solid state is the link between that and materials, and engineering. That is the link. And so we'll talk a lot about this all the time. Oh oh. Come on out, Jerome. But where did chemistry begin? OK, now hold on. I've got a pen. Oh no, I got a-- OK, that's better. I thought it was a pen. Where did it begin? Where did it begin? Well, people were mixing stuff a long time ago. But really, and there's some debate about this, you know, the word chem itself, you know, it may have come from chem, the land of chem, which means sort of the rich soil in Egypt. Or maybe it came from chemea in Greek, which meant sort of mixing and pouring together. But the point is that what they were doing was they were taking stuff and making other stuff out of it. That dagger there is from ancient Egypt. It's from around 3,000 BC. And the way they made that was by taking stuff from a meteor. It was iron and some nickel and some other things. But from a meteor. And they were able to make a really, really strong weapon out of it. They called those daggers from heaven. But the point is that what chemistry is about is it's about how you mix these things and what are you mixing in the first place. What was it that you took out of the meteor? And why did it make that instead of that? How did I get that dagger? That's really what this is about. And so if we go back to sort of the first people who really started discussing this, we start with Plato and Aristotle. And Plato had this idea, probably some of you may have heard it, so what is stuff made of? What is like the essence of things? They really thought about this and they debated it. And you know, Plato-- well, Plato said there's four things. And some of you may have heard this. There's earth. There's fire. There's air. And there's water. And everything is made out of that. Now, you can see that's a little limiting. Aristotle came along and said, wait a second. Hold on. If I look out into the stars, they don't seem to change much. So there must be something else that's not quote, "earthly and corruptible." And he called that ether. But anyway, the point is, it's hard to kind of explain everything with this. How are you going to build a world out of-- We can't even explain the Boston weather with just these four words. So it was limiting. But then along came these guys, Democritus and Leucippus, who was his teacher. And Democritus said, OK look, there's something fundamental. So Democritus, he said, look I believe there's something more than these four things. Democritus. And he said there are these things called atoms. So he was an atomist. And atom is indivisible. Indivisible, that's the meaning. Atomist in Greek, it meant indivisible, atom. And they fought about this. They fought about this a lot. And Plato, it's said, was so upset about Democritus that he wanted all of his books burned. That's a-- back in the day, that was a serious diss. It would be like if I blocked somebody on Instagram. Can I do that? Is that a thing? No, Snapchat. What? I don't know. I was right. See! That's how-- I'm blocking you. Burn your books. Seriously dissing each other. Democritus. Now, it really happened quickly from there. A mere 2,000 years later we get to modern chemistry. Why did it take 2,000 years? It took 2,000 years because we were missing something. So we had a lot of alchemy. The thing about alchemy, there are actually some really interesting discoveries in alchemy but they always tied it to something very non-rigorous, like oh, this works because of the phases of the moon or the tides. And so it really needed some rigorous way to study what things were made of, what things were made of. And that came, oh, inevitable that that might happen. Here we go. That came with the scientific method in the 1600s and Sir Francis Bacon. And I think a lot of you have seen the scientific method. But it was pivotal for chemistry. It was pivotal. Because it allowed people to think about this question of what stuff is made of but using a rigorous approach. Making observation, form a hypothesis, do an experiment, record. And that's what people started doing. That is what, for example, Robert Boyle did, one of the earlier ones to think about things and talk about the element. They were all going back to thinking about these same issues, what are things made of? An element can't be broken down into two or more simpler substances by chemical means. So he was at that. He was going for that core. What is at the core? Oh, and then you had Priestly. Priestly discovered oxygen and he did it by burning things. He did it by burning things. Just mentioning that word makes me want to put those on. And so he really studied combustion. He studied these reactions that were happening with oxygen and carbon containing matter. It was combustion. So he burned stuff. He burned this. He burned that. He burned that. And he said, what happens when I do that? That was another way. Priestly-- sorry, Boyle played with pressure and volume. Pressure and volume relationships for gases, that was his way to try to get at what things are made of. Priestley wanted to blow stuff up. Oh, and he also worked with beer. He actually worked with beer. And he discovered that the same gas that comes out of fermenting beer is the gas that comes out of combusting. So you got a guy who's working on lighting stuff on fire and beer. And you can imagine maybe that wasn't going to be a good day at some point. And what happened is, actually, his experiments did-- this is true, they slowed down when he fell into a vat of beer during one of his experiments. Now, the thing is, though, that he studied combustion. And that makes me think about combustion. And I feel like oh oh, I feel like this is a good time for my goody bag, which is to illustrate a point. Oh, thank you, Jerome. That's going to some feed somewhere. OK, now the thing is that when you go to a restaurant and they have real candles, I get really happy about it. We'll tell candle stories later in the term as well. And this is what you're doing. You are lighting a candle on fire. Now, when you do that, you see, here's the thing. You're actually-- I don't want you to think about it as lighting a candle. I want you to now from now on think about it as lighting C25H52. And in fact, if you go to a restaurant and you want to ask them if they have real candles, I don't care if you're out with friends or maybe you're on a date, raise your hand. Ask the waiter and say, do you have any C25H52? And see if they know, see if they've taken some chemistry. What you're doing is you're doing that. You're combusting that fuel. See, the whole world runs by lighting things on fire. You could light propane on fire. Or you could light hydrogen on fire. That's where we said, yo, hands on is a good way to learn. So let's see what that's like. There's the candle, which is tilting. I'm watching it. Really? Oh, OK. Or you could light hydrogen on fire. And if you do that, this is what happens. So, OK, let's do that again. These bubbles aren't-- oh, I should have kept this on. There we go. That's really, really fun. But I'm going to stop. Oh, I'm not going to stop because you just turned it on again. Now you're coaxing me to do more. Well, these are bigger bubbles. These are bigger boulders. Let's see if this gets a nice flame. Oh, thank you. There we go. OK, now that's my goody bag for today. We run our world by doing this, I mean, not by lighting hydrogen bubbles on fire, but when you put your phone in to charge it, you are lighting a fire. Think about that. You might not feel it. You are. No, I'm-- I love that reaction. I love that reaction. You are lighting a fire. It may not be you, but somebody else's down the street at a power plant. This is how we run our world, we burn things. And so this study of combustion was extremely important. And it's going to be our first reaction. So what's happening? When you light C25H52 on fire, what's happening is that you're reacting. You're doing-- remember, Priestley discovered H52 plus oxygen. He discovered oxygen. And he also, remember, this comes off of beer too. He also discovered these other gases coming out. Right? And that's the chemical reaction. Is it the chemical reaction? It's not done yet. It's not balanced. It's not balanced. Mass is being lost left and right. Not OK. Not OK. So we must, when we write down things that happen in chemistry like a reaction, we must find balance. It's also important in life. But it's very important in chemistry. And so if you balance this, you're going to put a 2. Because here's the deal and I'll talk about this more in a sec, C25H52 plus-- anybody know how many O2s? 38, whoever said that, 38 is going to go to 25CO2 plus 26H2O. These are the kinds of things that are perfect, perfect to keep doing exercises on though your problems, through your goody bags, in your recitation. These are exactly the-- how did you do that? If you don't know, you will soon. You'll get help, practice. I know, I know. I wrote this down. I'm just testing you. I made a mistake. Because the 2 was supposed to go to the propane one because I wanted to write that one and I got excited. And I went to that instead. And that would be 13O2. And that would be 8CO2, and so on, 10H2O. These are balanced. These are balanced. And balancing reactions is important. Why? Because it balances the mass and it tells us something else. Once we count atoms, you'll see. We're going to count atoms on Friday. But it's telling us that you can't just lose stuff. You can't just lose stuff. You've got to have the-- Oh, and that, by the way, is what-- Lavoisier, Lavoisier. Where is Jerome? Oh, yeah. No. Shoot. Isn't that butane, not propane? Butane not propane. It very well might have been. C4 H10, yes. I believe that is butane. Thank you. Oh, Jerome, come back. Jerome helps the class and he helps my French. Lavoisier? No. OK, say it. Lavoisier. OK, I tried. But now he said, look, you've got to conserve mass. You can't lose-- you can't create or destroy matter when you do chemistry. When you do this, when you do this you cannot create or destroy matter. Conservation of mass, Lavoisier. It means that if I-- you know, you get a little more than that, see this is balancing. But you also can think about whether you have anything-- I probably should blow that out. Whether you have anything-- did you have too much, too little? Did I mix it just right? So like if you take-- let's take another reaction just as an example. If I mix iron and oxygen to make ferric oxide, oh, not balanced. Two, three, four. And I tell you, for example, that I've got 10 grams of iron reacts with, let's see-- OK, I'm going to give it to you a different way. I'm going to say that it gives you, reacts with O2 to give 18.2 grams of FE203. This is an example from the textbook. Then I know because of conservation of mass from Lavoisier that 8.2 grams-- if this reacts fully, right, if iron reacts fully, it all goes away. There's none left. Then I must have, if I got 18.2 grams of ferric acid, I must have reacted 8.2 grams of oxygen. That's conservation of mass. But there's another thing you can do with this. So I must have reacted 8.2 grams of oxygen. But there's another thing because if I started, you know, if I started now with 10 grams of O2 and 10 grams of iron, aha, I'm going to have excess. I know that. I know that now. I'm going to have excess. And the thing is that there's something that's limiting here. What's in excess? The oxygen because I started with the same amount of iron. There is more-- but that means that we've got another term, which is that iron is the limiting reagent. You see that? Because now I'm limited, meaning this reaction goes and goes and goes and something runs out first. That is the limiting reagent. And this was just by thinking about Lavoisier's conservation of mass. I can't create or destroy atoms in a reaction, not in this class. You can take nuclear somewhere else. But not here. Here we don't destroy or create matter. Limiting reagent, balancing reactions, very, very fundamental first chemical concepts. Now, these guys were playing around with stuff and really trying to figure out, again, they're going back to Democritus. What are these indivisible elements, atoms, what are they? And all these guys were starting to mess with that. Once they had the scientific method, they were willing to go very far. Here's Lavoisier's list, 33 elements. And he tried to organize them. And in some cases, he succeeded fairly well. Look at this. This is a, OK, where is Jerome? This is a Tableau des Substances... No, not even close. But look, "Simples..." Simple, fundamental atoms. What are those things that we're mixing together that we've been mixing together for thousands of years? What is that thing I'm pulling out of this oar and making stuff out of? And he was trying to classify them using these experiments. And some of these are actually really good discoveries. I want you to experience this. I want you to go back to this time. And that is what the first goody bag is about. So what I have given you in this goody bag is the most accurate measuring device ever created and branded 309.1. It's a ruler. I've given you five metal strips. And I want you to pretend that you don't know what these are. How do you find out? You do nothing with fire, nothing. But I gave you vinegar. And you do lots with vinegar. Pour it on them. Measure it, weigh it. Look at it. Shiny, not shiny. Different color, densities. Think about what the differences are. And I want to put you back in that time. And I want to make a point here, goody bags are not just about the questions I ask. I hope that you think even beyond the Goody bag. So if you use that vinegar and one of those things reacts, let's just suppose, with the vinegar, think about what that is and maybe vinegar is a test. Maybe it's a test and you can react-- you can pour vinegar and make reactions with other things. Maybe you should be thinking about where that thing is in the Infinite Corridor and the whole Infinite Corridor should smell like vinegar. But we won't tell President Reif about that. But that's what I mean, maybe not the Infinite-- but explore. These are meant to be-- I want you to really use these as an adventure. You know, think outside the box. So you've got this most accurate measuring device. You've got these strips of pipette. And you've got some gloves. And that is your first goody bag. OK, now though the last thing that I'm going to tell you about is why this matters. And just like the goody bags, when I started teaching class three years ago, I wanted to also protect a certain fundamental part of each lecture. And I call it my why this matters moment. Sometimes it goes on for more than a moment. But I really, really want every lecture to connect what we just learned to a big picture. Most of the time, it's some application or some global challenge. Right I want you to see those connections, that what you're learning is directly relevant to some big thing. So my why this matters moment is really related to these discoveries themselves. Please give me till 11:55. I will always let you go on time, 11:55. But please don't start putting stuff away because it's distracting to everybody. So two and a half minutes. We name the age we live in often by the element, by the atom, by the material, by the material that was most useful at the time. The Stone Age, the Bronze Age, the Iron Age. I would say we've moved through the industrial age, the age of plastics, the age of silicon. As a material scientist and engineer, I love this. I love that you name the age you live in by the material that mattered. But I also love that we will never do that again, ever. And the reason is that we live in a truly unique age now, a different age, one in which we can put atoms, we can realize Feynman's dream and put atoms anywhere we want. The question is not, can we make it as much any more as it is, what should we make? We live in the age of atomic design. And that is really important. And I mentioned the phone and the 63 elements. You know, look, this is called a revolution. This is called a revolution. You went in 50 years from $1 per transistor, eight orders of magnitude cheaper. In 2012, it became cheaper to print a transistor on a chip than a character in a newspaper. That's a revolution but that revolution, it started as a processing revolution with one really important element, silicon. And now it's a materials revolution, with 63. It's a chemistry revolution. And the reason this matters so much, what are these things, is because so many of the problems that we face in this world today, so many of the global challenges, will rely on new chemistry and on new materials. Those are the bottlenecks. Those are the bottlenecks in costs, in efficiency, in processing, in properties. And those are the kinds of things that we're going to be talking about all throughout the fall. And that is our construction set. And we will build this on Friday. So see you guys all on Friday. [INTERPOSING VOICES] |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 30_Acids_and_Bases_II_Intro_to_SolidState_Chemistry.txt | Thank you for that. Happy Monday after Thanksgiving. I hope you guys had a good break. We are back in action here on acids and bases, second lecture on acid and bases. Before we get into that, there's a celebration happening on Friday. You know that's why we're all feeling like a little extra goodness right now. Because we know there's a third midterm coming up Friday in this class, right? And that's nothing but excitement. Here's a concept map that I made for exam 3. This helps you sort of see the kinds of problems, right, that you could get, the topics that we've covered, and how it's all flowed and connected together, all right? So here are the quizzes. There's the optional one. Oh, you could still do it. So please, if you want to do the 8.5 and get that graded-- and then if it's higher than your lowest quiz, we'll swap it in for that. And you still get to drop one-- please do that and hand it in by tomorrow's recitation. There's that one. There's quiz-- there's the other two quizzes. And here are the psets, and the goody bags. All of this is the kinds of problems-- this has the practice problems and the topics that we want you to know about for Friday. Oh, and also, on Wednesday, instead of introducing a new topic that you won't be tested on, I'd prefer just to talk about these topics. So that's what I'm going to do on Wednesday. And we'll do some additional problems and kind of review. OK, but we still got to get through the rest of acids and bases. Now this is where we left off. This way of thinking about acids and bases is where we left off on Wednesday, all right? And this is the Arrhenius-- oh, Svante. Oh, Svante-- he was the first-- remember, [INAUDIBLE] over thousands of years would say, this is bitter. Put it in that category. This is sour. Mmm, right? Acid-- no, base. Acid. And there are ways that people categorized acids and bases for a long time. But nobody really started to try to understand them at the atomic scale until Svante came along and said, look, it's all about the H+ and the OH-, right? So acids-- and this is what we talked about on Wednesday. Acids are donating H+ ions, protons, into solution, into water. And bases are donating OH-, adding more of those types of ions into the water. And that's what makes them have the properties they have, OK? Now, so if you have-- so let's see. So H-- I'm going to just write this one on the board. Because we're going to talk about these two reactions-- plus H2O-- and remember, the water is just a kind of spectator there. It's a liquid. It's everywhere. It's the thing you're dissolving in. It's the solvent. That's the water. So you don't include it in these-- if it's just the liquid that you're dissolving something, it's not what you include in things like equilibrium constant expressions. OK, so that's going to be-- oh, but these guys, these are dissolved in the liquid, all right? These are the ions-- plus CL-. OK, now in this particular case, you can write the equilibrium constant, right? And that's the acid. It's also [INAUDIBLE] the acid dissociation constant. Remember, we had the solubility product constant. And we also can write the Ka, right? So Ka for this would be-- following our normal K equilibrium constant procedure, it would look something like this, right? Now here's the thing-- for this particular reaction of-- for this particular dissociation of HCl, of this acid-- and remember, it's an acid because it's giving us H+. Oh, H+ doesn't stay H+. Remember, that's also what we talked about Wednesday. H+ is actually H3O+. But the H+ is not stable in pure water. But sometimes-- actually, very often-- you'll see it-- in textbooks as well-- you'll see it just written as H+. But we know that it's the same thing. H3O-- if it's in water, it's H3O+, right, OK, so don't let-- but this is something like 10 to the sixth. That's huge. That equilibrium is so far over. That equilibrium is so far over given this enormous equilibrium constant, that we often will say that this reaction has gone to completion. Now we know that it's not infinite. So it hasn't-- technically there's still a little arrow going that way, all right? But-- and so, but since it's so much almost to completion, you'll often also see it written with just one arrow, all right? So I wanted to make that clear. But we know that it's an equilibrium. This has found an equilibrium. So there's still a little bit happening the other way, but just so little, all right? So you'll often see that going to completion. OK, good-- getting us back into the mood. Now for the base, we can have NaOH-- let's do that one-- plus H2O goes to Na+ in solution plus OH-. And this is the Arrhenius acid-base definition-- gesundheit. And you could write a similar expression for the base, Na+, OH-. And that's going to go over NaOH. Now this is also very large. Now these are both strong. And we're going to come back to this at the end of the lecture. I want to talk about what makes an acid or a base strong or weak. These are both strong. And just as a little preview to where we're getting, that's because they dissociate so much. So this also has a very high Kb-- Kb, because we're talking about a base-- Kb, Ka, Kb, right? Oh, but some people don't like these large exponents, these large or small 10 to the minus 5th, 10 to the positive fifth. And so we take the log. You can take the log of these. And then you'll get, like, a pKb, right, or pKb, which is the power of that. So log of Kb would be low, right, would be low. If this is going to be high, then minus the log of the power of that is going to be low. So you can talk about pKb, Kb, Ka, pKa. It's all just thinking about these equilibrium constants. And we'll come back to this in a second. If I had a question for you-- if I said, OK, how would you use this-- if I said that I've got point-- I'm going to give you a problem now-- 0.05 moles per liter of NaOH, and I asked you, what's the pH, well, you can do it now, all right? Why? Because I've just told you it's a almost fully dissociated base. So the equilibrium constant for that dissociation of that base is very, very high. It means it lies very far over to the right. So basically, what does that mean? If I've got 0.05 moles per liter starting out, it's mostly dissociated. [SNEEZE] Gesundheit. So I've got 0.05 moles per liter of Na+ and OH-, all right? OK, but the OH- is what matters. So full dissociation-- oh, we're going to do this because I want to be precise here-- put that in quotes. It's almost full-- means that I have 0.05 moles per liter of OH--- by the way, and Na+. But Na+ is a spectator in terms of the pH. It's not going to change the pH, right, the Na+? Oh, but the OH will, all right? And so now we go back to these Ks. We know-- right, we know-- that the equilibrium constant of pure water is equal to the concentration of H+ in solution, which is H3O+, times OH-, and that that's equal to 10 to the minus 14th, all right? That's another thing that we talked about on Wednesday. So if that's true, if the-- which it is at 25 C-- the equilibrium constant for water, which is-- remember, water can be either one. It's this times this. And we know what this is. Now you know what that is, all right? And if you know what H+-- if the concentration of H+ is known, then the pH-- power of-- the p is the minus lower-- the pH of that is something you can calculate. And so the pH winds up, here, being 12.7, all rigth? And that's the kind of thing that you can have fun with. You dissolve some base, and you can measure a pH, or a pOH, if you wanted-- all stuff we talked about last week. OK, but now we move on. And now we're going to introduce the next definition of acids and bases by first talking about what happens when you mix them together, all right? These are not mixing together yet. They're just-- a base dissolves in water. An acid dissolves in water. But what if I had an acid and a base, and I mix them together. OK, well, so if I did that, then I would have HCl plus NaOH. Now I'm just going to-- I'm not going to include the water. It's there on the left, and it's going to be there on the right. I'm just going to write all these ions down that happen, right? So you're going to get-- oh, you're going to get H+-- oh, we know it's H3O+-- plus Cl- plus Na+. OK, these are all dissolved in water. I'm emphasizing that with the little aqueous subscripts. And you can see what's going to happen now, right? So if I have these there, well, look at this. The Na+ and the Cl-, boy, do-- they see each other, and they're like, well, maybe we could form some salt. Maybe we could form some salt. And then the OH and the H, and those form water. And so you can get H2O plus NaCl. And so what happens is what's written there, is that when you mix an acid with a base, you will get water and some salt. Now remember, salt is a general term, all right, for these cations and anions, right, coming together in a solid. So OK, now here's the thing though, all right? So this was the acid and I'm writing this explicitly. Because you'll see-- and this was the base. Because what is neutralization? Well, you think about it in terms of pH. I had something with maybe a low pH, or maybe a high pH, and I want it to become neutral. Neutral would be-- if I'm in pure water, Kw is 10 to the minus 14th. Neutral would be-- it's the same acid and base ratio. So these are each 10 to the minus seventh, pH 7, neutral, all right? There's no excess of protons or OH-, all right? So if there's no excess, it's neutral. So neutralization is basically the act of making it neutral. So I had something that gave a lot of H+. And now I got something that gives me a lot of OH-. If I have the same amount of them, all right-- so they're equivalent-- then you can make it neutral. That's neutralization, OK? But see, the thing is I can take HCl, and I can mix Windex, or ammonia, with it and neutralize it. And back then, they were like, but that doesn't make any sense. But first of all, because we don't have Windex yet. But second of all, because Windex doesn't have any OH-. Gesundheit. And so that got them really thinking, all right? So if I take-- so let's take NH3. So if I take NH3-- dah. If I take NH3 and I go like this, NH3 plus HCl-- now this is not-- I'm not writing the liquid dissociation of the thing. I'm mixing it, acid and something else that I know neutralizes it. And in fact, what ends up happening is you get-- gesundheit. You get this. And so what we know is that this is neutralizing. It's taking-- what's happening in this reaction? What's actually happening? What did the-- what did that base do effectively to neutralize HCl? What it did is it took H+ from the solution. That's what it did, all right? So NH3-- in this reaction, NH3 takes. It takes H+ from the solution. That's exactly-- HCl was like, I want to make it acidic-- H+, H+, H+. And NH3 is like, no, give me the H+. You cannot make this. I'm neutralizing you. I'm taking your H+ back. If it takes that, then that's neutralizing. That's going to neutralize the HCl as well. But see, we have a problem. Because this is not an OH-donating molecule. But it still neutralizes. And so we need a broader definition. And that came from-- that came from Bronsted and Lowry. And so this is another way of thinking, a more general way of thinking of acids and bases-- more general and more correct, because the Arrhenius definition does not cover classes of molecules that could be a base. And so in this way, in the Bronsted-Lowry definition-- in the Bronsted-Lowry definition-- it's all about the proton. So let me write that down here. So in Bronstein and Lowry, their definition is that acid-- and he did it by looking at acid-base neutralization reactions like I just described. Acid-base neutralization-- I'm not going to write it, because there's no room-- reactions is all about the proton transfer. These are proton transfer reactions. This is how they saw acids and bases, as exchanging a proton. Taking it or giving it, that's what it's about, all right? And so in our sort of general-- you know, in the general HA lingo that we introduced, you would have HA plus B. And that's going to go to BH+ plus A-. That's an acid-base reaction. And so you can see that these are-- this is a proton donor. This is a proton donor. And this is an acceptor. It's a donor, H+ donor, H+ acceptor. That defines them, according to Bronsted and Lowry, as acids and bases. OK. OK, well we can also-- OK, so now let's see. Let's now go back. That's a broader definition. Let's now go back to the ammonia, OK? And what I want to do now is, instead of mixing it with an acid to think about whether it trades a proton, I want to just dissolve it in water. So now we know it's a base, OK? So what happens? So if I dissolve NH3 in H2O-- I want to point something out here, that if I do this, and I get NH4+ plus OH-, that's the-- now I'm taking a base. So there we had-- there's the acid. Here's the base. That's the Arrhenius base. But now I'm taking a Bronsted-Lowry base, right? It took an H+. Well, that means that this is a base. But look, this-- OK, this gave a proton. So this must be an acid according to Bronsted-- according to Bronsted and Lowry. And similarly, over here, we know that this is going to give one. And I'm going to write that in a second. And this would love to accept one. So that's a base, right? And remember, we have a name for this. In the same reaction, we have a name for this. This would be the conjugate, all right? Those are conjugates. That's why I'm writing those, all right? So there's a conjugate base-acid, conjugate acid-base. Now we have a Bronsted-Lowry understanding of what conjugate acid-base means. It's just a proton transfer, all right? It's just a proton transfer. Now notice I could take this-- so I've taken this base, and I've put it into a solution of water. I could take this acid and do the same thing. So if I do that, I've got NH4+. And I'm going to put that in water now, OK? So I've taken this out. And now I'm putting that acid into water. And if I do that, look at what happens. I get NH3 plus H3O+. Notice what happened. Well, I've switched acid. That's my acid. But this took the proton, so it must be a base, right? And over here I've got the base. And this must be an acid. Proton transfer, that's what's happening here-- conjugates. I took the base, and I got an acid. So this tells us that water-- and it tells us what we said on Wednesday. Water can be both. Notice water is an acid in the one hand and a base on the other. That makes it amphoteric. So H2O can be both. It can take or give protons. And so it is amphoteric. Just for fun, if you take the conjugate-- you know, if you take any acid-base pair, and they're conjugates, you can also write out the Ka's and Kb's, all right? And so if I take a conjugate acid and base, well, you know that you could write Ka, all right? OK, so let's write the acid dissociation again. So for that one, here it is. Ka is NH-- is H3O+. OK, I'm going to take the bottom one, H3O+, times NH3, right, concentrations, divided by NH4+. But if I took the base dissociation constant from the conjugate-- I'm taking the conjugates now, right-- then I would get that it's-- OK, it's OH- and NH4+ divided by NH3. Like I'm saying, just for fun, you can now look at this. You go, Ka times Kb-- Ka times Kb. That's reassuring that Ka times Kb is equal to H3O+ times OH-. And I mean concentrations, which is Kw. That is fun, for the Bronsted-Lowry acid-base conjugate. NH4+, NH3, right? OK. OK, that's Bronsted-Lowry Lowry acids and bases. Here's a little recap. Here's a little recap. I'm not going to read through this in detail, but I just wanted to give you-- OK, what have we done so far? We've talked about amphoteric. We've talked about conjugate acids and bases. These are now written in the Bronsted-Lowry form, right? They accept or give a proton. We've talked about the conjugate acid-base pair, what I just drew an example of right there, neutralization reactions, and salt. OK, good. These are some concepts that we have covered that I would like you to know about. Now but there's a few more concepts that we have to come back to-- a few more concepts that we have to come back to. And it has to do with what I started with. In the beginning, I said, you know, these Ks are huge. So the equilibrium-- if an equilibrium constant lies-- is very large, then you know you're mostly making product, right? And so it's essentially to completion. And I also mentioned that those are strong, strong acids, strong bases-- strong acids, strong bases. Why is an acid or base strong or weak? That's what I want to talk about next, all right? And actually, there are-- there really aren't that many strong acids or strong bases. In fact, they're mostly here. These are really, pretty much most of them. Minor [INAUDIBLE] I don't want to go into. But these are really-- if you think about the general category of strong acids, strong bases, this is it. Why? Why is that? Well, it has to do with what we've been talking about, all right? So those are strong acids. Those are strong bases. It has to do with dissociation. It has to do with dissociation. So if I take HCl and I put it in water, it almost completely dissociates. That's why it's a strong acid. Now there's confusion around this. And I want to make that-- I want to make it very clear. There is a big distinction I want to make. Because if something is a weak acid, meaning it's not one of these, what that means is it doesn't dissociate very much. That's what it means. It doesn't have anything to do with concentration, all right? So let's take a weak acid. Let's take vinegar, all right? So if I take a weak acid-- we see vinegar as CH3COOH. You know what's nice about writing them out this way? You say, why didn't you put the H in there, and make it CH4, and CO2, or put the two Cs together? Oftentimes you'll see molecules written out this way, because this tells us a little bit of a hint about connectivity. And for acids and bases, it's actually useful. Because you see the H there? That's the one that's going to trade. That's the Bronsted-Lowry transfer H, all right? So often, for acids and bases, you'll see that written out in this way. Makes it easier to think about. And if I mix this into water, then what do I get? Well, I'm going to get H3O-- I'm going to write this down here-- H3O+-- that's an acid, right-- plus CH3COO-. There's the H that came off-- proton transfer. So why is it weak? Well, it's weak because here, the acid equilibrium constant is something like 10 to the minus fifth. So even if I have a lot, you do even if I had like 1 mole per liter of this stuff in solution, it wouldn't-- most of it wouldn't dissolve. In fact, much less than a percent of vinegar actually gives me H3O+, all right? But you think, oh, just add more vinegar. No. No, because we're talking about concentrations, right? It's the concentration in that water. I'm locked into that by this. This is an equilibrium constant. And it's talking about the concentration that I get in the water. I'm stuck. I can't be-- that's how much you dissociate, right? That's how much you associate as a concentration. So it's more acidic than water for sure, all right? It's 100 times more acidic than water, all right, 10 to the minus fifth. But it's never going to be a strong acid, because I don't give very much of the protons. So this is the key that I want to make clear. This is the key, OK? It's very important. It gets its own board. The strength of an acid, OK? So acid concentration is not equal to acid strength. More, and more, and more vinegar-- no. It doesn't matter, because it's about association. See, this is about-- this is a function of the solubility. So how much can I get in there? We talked about all that a lot already. That's a different thing, solubility, the Ksp. How much of this can I dissolve? Great. I can dissolve a lot of vinegar in water. And it doesn't crash out, right? That's wonderful. Doesn't make it a strong acid. This is about dissociation, all right? And so if you think about what makes an acid a strong acid-- or for that-- or a weak acid, or what the relative strengths of acids are-- and the same holds for bases, just the other way around-- it has to do with how easy it is to take that H off or add it on, right? That's what it has to do with. And so if you look at-- for example, if you look at just a little series, all right-- if you look at a series there, even the strong acids have differences. They're all strong. They go mostly to completion. But if you think about it in terms of what we just talked about, the acid strength is going to go roughly often opposite trend as bond strength. That's of the proton transfer of the H, all right? There is an H-- HCl, HBr, HI. Which will be the strongest acid of those three, right? Well, we now know. This must be the strongest, right? Opposite-- because this is going to be easier to get the hydrogens off in solution. And you can think about the weak acids in the same exact way. These are the strong acids. This holds for all-- you know, it's a proton transfer. It's all about the strength of the proton. Neutralization is a proton transfer. Whether you make an acid strong or how strong it is depends on how many of those protons go into solution. And so you can look up charts like this. And I love looking at this stuff, because it makes you think about the chemistry, all right? So here you go. You've got, OK, acid, different acids, molecular formula, all right, structural formula, conjugate-- notice they're putting that H there in blue. Oh, that's so helpful. And you just look at the blue H, and you think about it as dissociating, just like Bronsten and Lowry did. You think about that as coming off. And then you think about the conjugate base, all right, that's formed on the other side. And then you go over here, and you think, oh, this tells me how much it's coming up, how often. This tells me the equilibrium of putting this into solution. You say, this is-- which one of these is going to be weaker or stronger, all right? Well, and what does that mean about the bond strength itself? What does that mean about how easy it is to dissociate? We now know. And we can think about that, and understand that. These are not-- these may look similar, these Hs attached to carbon or oxygen. What's the difference? Big difference when you put them in water. A lot more or a lot less might come off. Changing the acid strength-- the more that comes off, the more the dissociation is to the right, all right? Oh, and the lower, right, the higher this number, because the equilibrium moves over to the right, all right? And the less that comes off, the lower this equilibrium constant is, Ka, right-- the lower it is. The higher the pK-- we don't like these small, tiny numbers. You take a minus log, and you talk about pKs. That's fine, all right? OK, good. This is the kind of fundamental understanding of acids and bases that I want you guys to have. And so if we go back to neutralization, now you say, well, OK, this makes sense, right? We had a strong acid and a strong base. If you had them equivalent, you'd get a pH of 7. Now you know, if you had a strong acid, weak base, weak acid, strong base, it's going to-- this is fairly self-explanatory, less than 7 or greater than 7. And now you know also, if it's a weak acid and a weak base, and you're neutralizing these together, it would depend on the Ka and Kb. Because it depends on how many of those ions you put in solution or can take out. And that's given by the Ka and Kb, OK? Good. Good. So we could answer questions-- we're now armed to answer questions. And like I said, I want to answer-- we're going to do some problems on Wednesday. We'll talk about the topics that are going to be on exam 3. But we can answer questions like this. Here we are talking about acids and bases. What if I mix a bunch of them together, and they've got different Ka's? So think about this, what if I mix all of these together? Can I get my pH, right? So here's a question. You have 1 mole per liter of HCN with that acid dissociation constant and 5 moles per liter of HNO2 with that one. What's-- OK, you write that dissociation reactions and calculate the pH of the solution. That's a good question, right? So let's see. Now how would I answer it? Well, oh, Bronsted-Lowry, don't want to erase you, but I have to. I have no choice. So let's answer that question. The way that we think about this, it's telling us what to do. Write the dissociation reactions for each species. OK, so I'm going to write it down. So I have HCN. HCN, right, is going to go to H+ plus CN-. Now Ka, for this, is equal to H+-- oh boy, did I drop my subscripts-- times CN- divided by HCN concentrations. But we're back to solubility product days, right? This is an equilibrium. And so it's going both ways. But you know from the coefficients here-- all right, you know from those coefficients that if I dissociate x amount of this, then that's how much of this and this that I make, all right? So now I know, right, that-- well, OK-- Ka times-- if I just bring this over there-- HCN is equal to x squared, where x-- let's let x equal concentration of H+, which equals the concentration of CN-. Been there, done that, right, from the solubility product problems that we've done. And so you can get, from this, that the concentration of H+, which is also H3O+, is equal to 2.5 times 10 to the minus 5 moles per liter. Good. OK. But you can keep going. And I'm not going to do all the math. But you could go through the same exercise with HNO2. So for HNO2, you get that the concentration of these ions that you get from that dissociation is around 2 times 10 to the minus 3 moles per liter. And by the way, maybe I got less than 10 to the minus seventh out of these things-- maybe. Maybe if I did, then, if I did, then I'd also need to take into account pure water, right? So from pure water, we know, if you just had water by itself, then you'd have, the concentration here is 10 to the minus seventh moles per liter. But now you just look at this, and you say, OK, I got my answer. Because everything's a lot lower than this one thing. So this is going to dominate. I mix it all together. But you see, it doesn't matter. The other things aren't able to disassociate nearly as many H+ ions-- they're just not able to-- as HNO2. So this dominates. And you can just calculate the pH from this. So you're going to get, like, two, three digits down some effect. But it's OK, right? From this, you can get the pH. And the pH is equal to 1.35. The total contribution of H+ ions in that solution is roughly given by this contribution here, OK? Might not have been, but it was in that case. OK, how about this one? So here's another-- OK, so we could take another example. Let's stay over here. OK, so in this example, I'm barely giving you anything here, it seems like. But I give you enough. I've got a 0.2 moles per liter solution of a weak acid. And it's 9.4% dissociated. That's the key. That's the key. Because remember, this is-- now we've learned the Bronsted-Lowry definition. It's all about H+ in or coming out. That's what-- there it is. Now I erase it though. It's all about the H+. So if I know 9.4% dissociated, I know something about what kind of acid that is, right? So if I have, now, HA plus H2O, and that goes to H3O+-- I've switched back-- plus A-, then how much dissociated? Well, 0.2 moles per liter times 0.094, right? So 0.094, that's the percent-- is that right? Yes, 9.4%-- times 0.2 moles per liter. And that equals 0.019-ish, right? That's how much HA-- HA-- dissociated. But if I know that, and I know my reaction, and I got the stoichiometric coefficients from the reaction, then I also know how much H3O+ and A- formed. If that much dissociated, then that much formed, right? Because they're-- right? That's the reaction. Those are the coefficients. And so I can get it all. And I can calculate Ka, which equals 2 times 10 to the minus third. I didn't give the pH. All I gave you was how much of it dissociated. I didn't even tell you what acid it was. But that's OK, because we now know that this is what counts. This dissociation, how many of these Hs per liter, how many of those go into solution is what matters. OK, now there's another thing that I want to talk about. And we'll do some more-- you know, a little bit more, like I said, problems on Wednesday. There's just one more thing I want you to know about related to acids and bases, and that is that you can have more than one. All these examples, if you go back to this list, those all had one hydrogen that was ever going to leave, let's say. I mean, OK, ever-- maybe if you went up to the middle of the sun or something, you'd get those. But this is-- you know, in normal conditions, these are the ones that are going to leave, and no more. But you can have molecules like phosphoric acid where-- look at all those protons. So here's acetic acid. That's the one that's willing to leave. Those are not. They're on the carbon. Way too hard to get them off. This one-- OK, in solution, sure, to a certain extent. In phosphoric acid, I've got three hydrogens. And so that's called a multiprotic acid. Why? Because protic-- proton-- how many protons, right? It could be monoprotic, like the ones we've talked about, or it could be multiprotic, like phosphoric acid. And so if you think about this the way we now think about acids and bases, we think about an acid as a dissociation, right, with some dissociation equilibrium constant. And so if you think about phosphoric acid, H3PO4-- oh, let's put the water in, H2O, liquid. And that goes to H2PO4- plus H3O+. Well the Ka for this one is-- so if I put the Ka's in here, this one is 7.5 times 10 to the minus 3. But see, what happens is, now, this one has two more hydrogens that could come off. So you could write those dissociation reactions as well. So H2PO4-, if you put that in solution, which it is, then that could give you HPO4 2-, all right? H2, H-- it's a proton transfer. Minus, 2-, all right-- plus H3O+. And there's one more, plus one more. This dissociation constant is much, much smaller, 6.2 times 10 to the minus 8. And the third one is 4.2 times 10 to the minus 13th. So they're smaller, but they happen. And that's important. And it can be very important. It depends on the molecule. But in this case, it's not just the parent, that acid there that's losing a proton. But then, this one can also lose a proton. It can also be an acid. It just has an equilibrium that's not as far over. So it's not going to give as many H+ into solution, OK? It's important to know about. Then finally, the last thing I want you to know, you will not be tested on this, but it's important. Because here's where we are. Here's where we are. Arrhenius said, right before Thanksgiving, it's all about the H+ and the OH-. That got us really far, got us through Thanksgiving. But then today, we're like, no, it's got to be more general. It's just proton transfer. No, don't tell me about OH-. It's just proton transfer. And so then comes Lewis. Oh, we know Lewis. We've already talked a lot about Lewis. Lewis said, why are you guys all obsessed about protons? And he said it in just that way. An acid is much more general even than that. An acid is any species, anything that except a pair of electrons. Remember, Lewis was all about those electrons and the octets. And a base is any species that can donate a pair of electrons. So Lewis correctly took a much more general view and said, no, acids and bases are not even about proton transfer. They're about electron transfer. And again, I'm not going to test you on Lewis, acids, and bases, but I wanted you to have that full picture. That means, by the way, aluminum 3+-- take an aluminum atom, make it 3+. It wants electrons. It's an acid. It's an acid, all right? Doesn't have a prot-- it wants electrons though. That makes it an acid according to Lewis. So that concludes acids and bases. And I'll see you guys on Wednesday. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | Additional_Lecture_2_The_Chemistry_of_Batteries_Intro_to_SolidState_Chemistry_2019.txt | [SQUEAKING] [RUSTLING] [CLICKING] JEFFREY C. GROSSMAN: I want to start with this sort of big picture, all right. So we're going to kind of introduce energy storage, and then we'll talk about sort of why electrical storage is kind of a thing and important, and then we'll kind of go into batteries and then talk about the chemistry of batteries and throw in a couple of, why this matters. So if you think about this on a planetary scale, actually, the planet is kind of a storage device. It's an energy storage device for that thing over there, right, and so like if you think about all the energy flows that go on on planet Earth, well, then you've got-- there's a picture I like. It kind of shows you the different energy flows, right. You've got the sun, so you've got like solar radiation. You've got tides, there's the moon, you've got wind. You've got all these kinds of energy happening on the planet, all the way down to life happening, rock formation, right. And then sometimes those things, you know, rocks can form alloys over millions and millions of years. And the thing is, if you think about this, where you draw the line is somewhat arbitrary in a sense, right. It depends on our use. I mean, if you think about it, fossil fuels are actually a renewable source of energy on geological scales. The problem is that it's like 10, 20, 30 million year old sunlight, right, and it'll happen again, but the question is how do we store energy on timescales that matter to us. And so you can draw this line and then think about what that means, and typically, most sort of energy storage needs are going to be around a year or less, right. OK, so now why electricity? Why electricity, right. So the thing is, electricity is more and more important in our lives, and if you look at just sort of like the major sectors, so you have the power generation, you have transportation, you have industry, you have buildings, and this is a plot of these sectors and the CO2 emissions. You can plot all sorts of things about these sectors. This is CO2 versus time. These are kind of the major contributors, but the thing is electricity, because of things that are happening as well, the opportunity-- either the use of electricity now like power to your home, or the opportunity to use it a lot more later, like transportation, that means that electricity has sort of about a 70% role, about a 70% opportunity of all of this. That's how much electricity matters here, but the thing is that electricity, as you know, in an electric car it's not going to be useful if you have to have it plugged in all the time, right. And so most of this 70%, most of this 70%, I want to emphasize, is not currently-- it's the potential for the electrification of our world that is happening. That's how much electricity will play a role, but most of it is useless unless you can store it, so storage of electricity is clearly really, really important. And so we get to sort of how do you think about storage technologies, and again, I just want to kind of gently introduce us to energy storage and we'll go into batteries. Now so the kind of plot that I'm showing you here, talking about Ashby plots, right, but this is actually a different kind. It's a special kind because it has to do with energy storage, and there's a guy who in the 1960s, David Ragone, he first-- And it's actually fascinating. He wrote a paper in 1968, a review of batteries for electric vehicles in 1968, right. And he's saying, how do I put-- there's all these possibilities. We want to electrify our transportation fleet, but which batteries make the most sense? And so he decided to plot them on these two axes, energy stored versus the power and that's a very common way. These are called Ragone plots now, so it's the energy versus power. And this is how you'll see a lot of discussions. If you look online and you're interested in energy storage as a field, you'll see a lot of plots like this where you got, OK, so energy is like watt hours, right. For example, this would be like watts times hours and this would be like the watts. So one of the nice things about plots like this, mostly, it's always on a log scale so you can stretch out and see all the different technologies on one plot, right. And one of the nice things about a plot is that this is energy, this is energy, and this is energy per time, right. And so that's energy per time times time, so we can have lots of fun with that. But so what that means if you think about it a little bit, if you plot energy storage technologies on plots like this, then these diagonal lines are one number of time. It's like if you had a given density and you're using it at a given rate, a giving power, then that's how long it would last. So these plots are really convenient. They're really convenient ways to compare energy technologies. OK, so those would be like, I've got energy storage technology and it's got this much density, this much power-- well, then it's going to last if I use it at that power for 41 days and so forth. Now the interesting, though, there's lots of stuff on here. Why don't I just want high and high? That sounds like a good thing, right, so shouldn't we be pumping water up hills all over the place? Well, obviously, if you think about it, you don't do that in your phone. You don't even do that in your backyard. You don't even do that in a town. Why? Because pumped hydro is extraordinarily limited. You have to do it at very large scales. It's very low energy density, right. It's geographically highly constrained. It is one of the oldest ways we have of storing energy, right. I'm just thinking of the waterwheel, but it's very constrained, so we can't use this in most applications, pumped hydro. But OK, oh, so this is compressed air. Compressed Air, CAES. That sounds good. That's got a whole lot of limitations as well. We've got, just to give you the abbreviations, Superconducting Magnetic Coils. Superconducters, right, just keep it going. And then you draw the [MUMBLES] and it just keeps going forever. It's a great battery, right, except for all the problems that it has and how much it costs, but it's a really cool concept and it's here. All of these things exist. All of these things exist. They all exist and they're all very interesting and important energy storage devices. They're interesting, important, electrical energy storage devices, right. So that's the main point. So like here, you know supercaps, you build up a lot of charge in a solid. Here you've got flywheels. Well, I'm just rotating something heavy really, really fast. I get worried about that. I don't want to be in that building, but they do build flywheel energy storage device where you slow it down and then you generate electricity from that. So these kinds of plots are really, really, really helpful for comparing one energy technology with another. Now, the thing is as I said, there's a whole list of these and it goes on and on. Oh, that's really hard to see. They've all got a lot of limitations, all right, and so capacitors have a low density. You can convert energy into some fuel and store it and then combust it. That's got low round-trip efficiency. This is what we're going to talk about today, electrochemical. There are challenges there too. Kinetic, pumped hydro, there's so many different ways of storing energy. The thing is there is an advantage that I want to mention about using electrochemistry. There is an advantage about using electrochemistry, and what is electro-- I just said the word electrochemistry. I should tell you what that is, right, and it's the relationship between electricity and chemical reactions. I love saving time, rxns. That's equal to electrochemistry. This isn't it. Now electrochemistry, OK. And so what you'll see is like things like electrochemical energy storage, and this isn't just one type of thing. It's not just one type of battery. It's not even just batteries. It's this broader definition, but we're going to talk about batteries today. Now, so there's something else that's really useful. So I mentioned we're getting to the point where we're electrifying everything. We need electricity almost everywhere, even in your combustion car. You need electricity a lot more than you used to, than 20 years ago. And so as we electrify our world, and then I said I've got to store that energy, you start to think about what Carnot did. Now I'm paraphrasing here, but this is basically what he said if you think about how it applies to this conversion, which is that the second law of thermodynamics makes you pay a penalty of energy if you convert it into heat and then back into electricity. Well, even if you just have heat in the first place and now you want to generate electricity from that, you're limited. We talked about this before, right, like that the energy density of gasoline, it's so high and it's such an amazing energy storage material because you can transport it safely, right. Yeah, but the thing is, I just gave you the example the car. You put it in your car. Does anybody know how efficient, how much of that energy your car actually uses to move percentage wise? Take a guess? Yeah, it's 10% to 20%, maybe, 10% to 20%. It depends on the car-- my car probably 5%. It's an old car. So most of that gasoline's energy is going into waste. Why? Carnot. Now this is not a thermodynamics class, but this comes into this so I wanted to mention Carnot and the fact that there is another way to do it. So you could do chemical reaction to heat. That's how we power most of our world today. A power plant gets up to a higher temperature so the penalty is less, but still most power plants, you lose about 50%. Most power plants at best are around 50% efficient in terms of converting the thermal energy into electrical energy. Yeah, but I can use a different way, and that's electrochemistry. And it's one of the reasons why electrochemical energy storage is so appealing, because you don't really pay a penalty. You don't really pay a penalty. I can go back and forth without the second law hurting me. So I want to mention something about Carnot, about the second law, and again, this is not a class on thermo, but the second law involves entropy so I had to say something, especially because it's in Averill, right, and that's your textbook that you're diligently reading every day, and there it is. And by the way, Averill has a beautiful chapter on batteries, chapter 19. I'm sure you know. It's really well written. This is what he says about entropy, though, and for now we can state that entropy is a thermodynamic property of all substances that is proportional to their degree of disorder. Averill, such a good book, but let me ask you guys a question. So people think about entropy as disorder and they think about entropy as sort of more disorder. Let me ask you, which one of these has a higher entropy, the one on the left or the one on the right? So how many of you think the one on the left has a higher entropy? How many of you think it's the one on the right? So the thing is it's the one on the left. Yeah, right. I love that you guys are blown away. Why? Because entropy is not about smoothness and it's not about disorder. That is wrong. Entropy is about accessibility to states. It's about how many states you have to be in. Now, that can appear to be like disorder, right, but a messier room does not have higher entropy, and in this case, it's just simply the rules of the algorithm. I have rules here about the dots being able to touch each other. They are randomly selected but with the constraint that they cannot touch each other. That means that those dots had fewer possibilities. They had fewer states that they could be in. This system has much higher entropy. It's about the number of states. I just want to make that-- again, this is not the topic of this class, but I've got to make my case about entropy when I can. And I brought up Carnot, so there you go. Entropy is not just disorder. Now OK, so we're back to batteries. Now so electrochemistry, there's a lot of reasons why electrochemical storage is-- here's the chart again, right. It's this one here. You see that, electrochemical batteries, all right, that we're going to talk about. There's a lot of reasons why this is so appealing. No penalty on the second law is one of them. You can have transport easily, right. We drive electric cars. We see that. No pollution at the point of use-- pollution can be focused at the point of production. Also, the batteries can be charged by renewable technologies like solar wind, and they're very highly efficient. Electric cars, for example, have a very high efficiency depending on the car, but upwards of 90% in terms of converting that electrical energy back and forth. The metrics that matter, I wrote a couple of them here, all right, watts, hours. These are the things that matter in batteries, but so many other things matter, and this is why this is a complicated problem. And it's also why batteries are not in any way a one-size-fits-all. I mean, you know that from like AAs and AAAs and 9-volt, but it's so much more than that, because it's about the application change not going from your computer to your phone but going from your car to your phone to a house to a grid. And so the needs are all going to be different, and then which ones of these things you care about is going to depend on what chemistry you think about. It all comes down to the chemistry as we're going to see. It's all about the chemistry. And just to put a few numbers down, the cell phone has a few watts of power. Light bulb is 10 to 100. A AAA battery has 1.5 watt hours of storage, all right, so that's how much energy is inside of a AAA battery. Needed to drive a car 200 miles, 100 kilowatt hours. It's just fun to think about these things, right. Powering your house, 40 kilowatts max, depending on how many AC units you're running. The world is this many watts as of a few years ago. So if you had, I think it's something like 20,000 billion AAA batteries, you could run the world for an hour-ish, something like that. These numbers are just fun to play with, so I wanted to put a few examples down. So where'd it all begin? It all began with an argument. It all began with this beautiful argument that I have to tell you because it's really a cool piece of history. And it took place in Italy. And it was Galvani who was a physiologist, and he really loved the topic of why animals move, right. So what is motion? How is it that animals are moving? You know, how am I moving my muscles? How is anything moving? Well, there had been theories about this going all the way back. You remember Socrates and Plato, the elements, right, and aether? There were similar theories for motion of the human body. In fact, the word pneuma was used to describe the essence that goes through all living things that allows them to move. Well, Galvani didn't like that and he wanted something more concrete, and so he was really trying to make a connection, especially between things that move and electricity. Now, it was difficult. There wasn't like a constant source. So you had electric generators in your lab at the time, but Galvani wanted to go big, so he hooked up his lightning rod to a frog. The frog was not alive. Well, the frog was not alive, seriously, because then you'll see why-- at least I don't think so. I really hope not. Now I'm thinking about it, I'm getting a little bummed honestly, but I don't think they were alive. That doesn't look alive. So he hooked them up to lightning rods and he watched and he waited as the storm's coming, and he's like, the lightning rod hits and he hooked him up to some metals, and the frog went crazy. It was like, ah-ha, motion, electricity. He'd been jolting them in his lab with his own electric generator, but he wanted to go big, and also, they didn't know. They didn't know at the time what electricity was, right, so they didn't know if the electricity that he generated in his own thing was the same as the electricity that came from the sky. He saw that you get the same result. The frog still moved. So Galvani said the motion of a frog, and all motion of all living creatures, generates electricity. That's what he said. He deduced that the motion itself is something that generates electricity. This is a really cool little article about this debate, because in Galvani's case, if you have two metal wires hooked up to a frog, like we say, if you close the loop, then what he said is that the charge that was already in there moves around the loop and back and makes the leg move. Volta got really interested in this and he said, you know, let me study this. He was a physicist at the time and he said, this looks fascinating, let me study it-- but he said, you know, why do you keep using two different metals? So I think this is about the metals, not the frog, and they had a big huge argument, right. So this is Volta's picture, is that it's when you connect two different metals that you get a charge flowing through the frog. Volta, I think, as many of you know, went on to win that argument, and we have the volt named after him, and he also created the first stable battery, the voltaic pile. Galvani on the other hand really opened the doors to the idea that we have electricity in our bodies, which is pretty cool. He also is the reason why the story of Frankenstein was written. It's actually true because they went around and took all sorts of things and electrocuted them and showed that they moved, and often they weren't alive, and so Mary Shelley, I think, saw that. You went around like the countryside shocking people, and I actually meant that as like not literally, but just shocking them by what he's doing. And I don't think they ever used humans, but still, people's imaginations and so forth, and it was quite a thing. OK, very interesting story. Now, why does this matter? I've got two why this matters. Today my first one is on a personal scale, smaller scale, and then my next one is on a grid scale. So this is called a little sun. There's over a billion people who can't read at night because there's no access to electricity, and of course, a lot of the studying that you could do would be at night, right. So this is a really cool program. Why does this work? This is a solar cell but this wouldn't work unless you could store the electricity, right, so I wanted to make you guys aware of Little Sun. You just press this and you can read at night, right. This is already making a big difference in millions of people's lives. That doesn't happen unless you've got electrochemical energy storage, right, and there's some stats up here. $25 billion liters of kerosene are used to meet the basic lighting needs in a lot of these places. That releases tremendous amounts of toxic fumes, and you can make a huge difference, But you know, it's sort of the same as cooking, right. So wood fire cooking is a huge problem. So the challenge is, but you don't cook. If you have a solar cooker, that's great except that most people don't cook during the day. They cook at night or in the morning, right. And so you've got to store energy to make these things actually useful, right, and I really like this program so I wanted to make that one of my, why this matters. And we'll go big on the next one, but let's get to the chemistry. So what happens in a battery? The best way to understand this is with a classic example of two different metals. It's Volta's two different metals. Volta did lots of metals. That's what he did. And he showed that the frog moved with most of them, so it's the metals. So let's take these two classic metals and show what happens. You've got copper and you've got zinc. Now, in this case, I'm just going to have a zinc. This is the zinc piece of metal, and this is a piece of zinc, and I put it in a solution of copper ions. And so in this case, what do we have? Let's write this down. In this case, I've got zinc solid. Well, before I write this down, let's write down what's in the solution. The solution is copper sulfate and this goes to everything that we've been talking about. So I've got a solution, a beaker, that has copper ions in it. How much? I don't know. Look up the K sp, right? Now we know how to do that. So 2 minus in solution. So you put some of this copper sulfate powder in that beaker. It turns blue and it gives you these ions that are dissolved in solution, and then you put a piece of zinc in there. So what happens is there is a transfer of electrons, right. Now we talked a lot about electrons. We're talking about transfers of electrons. I mean, just think about it, right. We went from H, right, like this. Look at that, right, the bonding and then we did this, OK. Here we go. It's a little bit less sharing but still. All right, and then we did this, where it's like, OK, there's almost nothing there. It all went there. These are where my electrons are, all right. Well, then we put this in water and we got this, OK. And we got them sort of separated and this one just broke off with its electron, leaving a positive charge behind. So we've done all this. We know this, all right. So what's happening here is special because what's happening here is a trading of electrons. You see, in all the bonding that we've done, we're talking about how electrons share in bonds and then these things break apart, but now I'm going to do reactions that trade them. That's electrochemistry. That's electrochemistry, and what happens when I put that zinc strip into that solution is this. The zinc that goes into solution is 2 plus and 2 electrons. And the copper in the solution plates onto the zinc, plus those 2 electrons goes to copper like that. And if you look at the full reaction, the full reaction would be then zinc solid plus copper 2 plus in solution goes to zinc 2 plus in solution plus copper solid. That's the trading. Now I like writing it like this because you really see explicitly the gaining and losing of electrons. Oh, those have words associated with them, gaining and losing electrons. But this is the reaction that we're kind of used to, right, so this is just the same thing. It's what happens, right. It's what happens. It's a trade of electrons. Now it turns out that when something loses electrons like that we call it oxidation, and when something gains electrons we call this reduction. And so when two different materials-- in this case, these two metals-- trade electrons and one of them is oxidized and one of them is reduced, that's also got a name. It's called a redox couple. Those are a redox couple, redox, redox, right. So these are a redox couple, is what they're called when you trade electrons like this. OK, so what's happening? What's happening is at the surface of the zinc, I see these copper ions in solution. And it's like, well, if I gave you two electrons, then you'd be able to become a copper solid, right. So if I give you two electrons, you're a copper solid, but now if I lose two electrons as a zinc atom, I'm a zinc 2 plus. That's this. That's just this, right. Why does that happen? Well, we'll get to that, right, but that's what's happening. So a zinc atom at the surface, all right, a zinc atom at the service sees a copper 2 plus and it says here, have 2 electrons. Oh, and then it's like, wait a second. I gave two electrons away. I've got to go into solution. And that's exactly what you see over here, right. The copper is plating the zinc and the zinc is coming off of the strip. They're doing this. All right, why is that happening? Because if you put that strip in there and the thing just heats up, this is exothermic. That means that this reaction will go. This is spontaneous. This is spontaneous. Where can I write that? Nowhere. Spontaneous. It didn't have to be, as we'll see, but this one is. But all that's happening is you're going down in energy. You're giving energy away and you're heating up the environment. Now the question is, how can I not let that happen? How can I stop the heating from happen and instead take advantage of this electron trading to do work? That's what a battery is. That's what a battery is. It's taking advantage of this. And really what it comes down to is just a different construction of the device. It's this. The Fundamentals of a battery is this kind of thing, but now instead of allowing this plating to happen, I'm going to separate that and I'm going to construct it in such a way that those electrons get traded through some wire that I can do work with, OK. So for example-- where is it? OK, maybe not for example. Maybe for example, maybe I need to turn this on and off. Here we go. This is the picture that Averill gives for the battery. So what you saw wasn't a battery because I wasn't stopping the electron flow and doing anything, but if you construct it like this, is called a galvanic cell. All right, if you construct it like this, now I can do work with that electron trade. I can do work with the electron trade. That's the difference. So let's go through this. So on the left, so what did I do? Well, I got my zinc strip but now I've got a copper strip. Just trying to see. OK, I've got a copper strip over here. I've got a zinc strip over here. But I'm putting the zinc not into the copper directly where I just short it and the whole thing just heats up. Instead, I'm putting the zinc into a solution of zinc ions. Look at that. Zinc has zinc ions in solution, right. Now, copper, copper is in a solution of copper ions. So these are now the two beakers. On the left I've got zinc nitrate. In this case it's a nitrate, not a sulfate. It doesn't matter. The whole point of the solution is that I get these ions in a beaker. Got to get these ions in a beaker, so you've got the copper ions in a beaker on the right with the copper solid and you've got the zinc ions. So let's look at what's happening and blow up those different sides, right. So at the anode, it's called the anode because it's where the charge comes from. It's called the anode. What's happening? Well, what's happening is what I just showed you. The zinc is leaving the solid. The zinc is leaving the solid and it's becoming 2 plus. It's becoming 2 plus. But the thing is that now, you've got to be careful here. So the zinc is becoming 2 plus. What about those two electrons? Where do they go? A zinc atom leaves its metal strip and goes into that beaker of zinc ions, right. But now you've got two charges, two electrons on the strip because it left a zinc 2 plus. Nothing more will happen. Nothing more will happen unless those 2 electrons go away. That's the whole point. What's so cool about batteries is that it's all about neutrality. If you think about batteries as current flowing, batteries is all about neutrality. It's all about keeping these things neutral because if I just build this charge up here, I can't have another zinc atom go in so the charge has to go somewhere, but that's what that wire is for. That's what this wire is for, right. This wire allows the charge to go over to the other place, the cathode, where the copper is sitting there saying, you know what, if I had two electrons-- if I had two electrons on the copper strip then I could use those as a way to lure in a copper 2 plus ion, right. I could use those as a way to lure it. And if I did that, then I'm doing what this did right here. It's the same thing that we just did with the plating except now those electrons are going through this wire up here. The zinc is trading electrons with the copper. That's keeping this neutral, so now the electrons go away and now another zinc can go into solution. Then the copper grabs them, plates a piece of copper, and then more electrons can come, right. It's the same thing. It's the same thing. This is what it looks like. If you take those electrodes out of the solutions after running this for five, 10 minutes, look at what happened. The copper plated. You got copper ions from solution plating that and zinc ions from solution leaving that. And so those are the reactions that are happening. One's happening on the zinc side and one's happening on the other side, but there's something else. There's something else because there's just one other piece of it that you've got to understand to understand how a battery works, and it's this thing here. So up here, this is my conduit for electrons to go. Each time it goes, I can plate. I can lose the more it goes, right. But notice if I lose 2 plus-- so I've got these 2 plus. It's not charged. Look at this. Copper sulfate in that case is copper nitrate. Copper sulfate goes in a solution, it's not a charged solution. These things are balanced, right, 2 plus, 2 minus. So now I'm taking only a 2 plus out of solution. No way. It's the same thing. You've got make it neutral again, all right. It won't work otherwise. You can't start building up charge in the solution. So that's what this thing is about. That's what this is. This is called a salt bridge and that completes the circuit. See, this lets the electrons flow. This gives ions. It doesn't matter what it is, right. That's the electrolyte. It's just a source of ions. Here is an ACL. It's an ACL, so now for every copper 2 plus that plates onto the copper electrode, two NA plus atoms go into this beaker from the bridge, because here I've got dissolved salt. Dissolved salt, sodium chloride is in water, so I got all these sodium ions. I got all these chlorine ions. And when a charge imbalance happens because something plates here, I can draw sodium atoms out of the salt bridge and I can draw chlorine atoms out of that side, right. That completes your circuit. That's an iron neutralizer, the salt bridge, and this is allowing the electrons to keep everything neutral, and that's how you get the current, right. It's pretty cool. It's pretty cool stuff. Now the thing is, you can do this yourself. That's why I bought these for, is that you can-- somebody sees the the potato battery. I like the human battery. So these are just volt meters and these are set to the right thing, yep. And what you'll find, you can see what happens, right. Like when you touch the two electrodes, yeah, and sometimes you've got to squeeze. Oh, look at that current. You have electric-- you are providing electricity. You are the salt bridge. You are the bridge, so what that means is you are actually plating and taking metals off of these two electrodes right now. Your hands are doing that and you're trading electrons. You're a conduit for that to happen, right. You're the salt bridge. It's pretty cool, and you can play with these. You know, you have volt meters. You have lots of items on the table tomorrow maybe at a dinner, and I'm just suggesting that you play around with this idea, because look. This is what Volta did. This is literally what the name of that device, the guy that that's named after, this is what he did. He hooked up two different metals. I just put some paper clips in there, it works. It can be almost-- not almost, but lots of different metals will work. Why? We'll get to that in a sec. Here I've got another paperclip and a copper wire, right, two different metals. The power isn't coming from you as Galvani thought. No, the power is coming from the difference in potentials of those metals wanting or not wanting electrons. Everybody wants electrons, let's be honest, but who wants its more? It all comes back to the same stuff we talked about. It's just that now we're talking about the metals and whether they're more interested than the other metal in having those two electrons. That's what a potential is. That's what a potential is. Oh, now so OK. Here we go. There's the volt. So what determines it, right? What determines the potential? What determines the potential? Well, it's what I just said. You know, intuitively, it's the difference in these things wanting and saying, well, OK, I got copper and zinc. How do I know? Well, if you look at zinc-- let me see. Where do I have it? Here we go. Zinc, I want to make sure I get this right. So zinc is argon and 3d10 and 4s2. All right, OK, copper, oh, we got numbers 30 and 29. Copper is the same thing in terms of its argon core, 3d10 4s1. Which one of those wants to give up 2 electrons or take 2 electrons more than the other? That's what determines the potential. And you know, when you look at this, you might say-- this is from Averill so he actually is missing the f's because this is just taken right from the textbook, but anyway, so that's the potential. Look at this. This is like an energy diagram but these aren't electronics. These are just potentials, but the potential is related to which one wants the 2 electrons more. It's 2 plus. The oxidation state is 2 plus, so you look at this, you say, well, I don't know which one might give up electrons more. It turns out it's the zinc. And you might say, well, but wait a second. If I just take an electron out of here, won't it be harder than one out of here, because this only has 4s1? The answer is yes, but I'm taking 2. I'm not taking 1. I'm talking about the oxidation state of 2 plus. That's going to be harder here. It's harder to take a 4s1 in a 3d from that beautifully filled 3d shell than it is to take these 2s electrons. That's why that's higher. That's why that's hire, right. So what you can see is that the potential difference is all about chemistry, right. The delta v is all about the chemistry. It's all about all of the things that we've learned. It's more than that because of these solutions, so it's more complicated than that, but at the core of all of this is all the topics that we've learned, how strongly are electrons bound to an atom, to a metal, right. And so well, if I took cobalt, what would happen? If I took cobalt instead of copper, the voltage here would be 1/2. And you can sort of look at the table and think about why that is. Cobalt has a 4s2 but it's got less d electrons, and so is it going to be easier or harder to pull them out? Well, it's going to be easier than copper but harder than zinc so it goes in the middle, right. And so the potential difference is less for cobalt and zinc but you still get a potential. And so that's what Volta did, and he did all sorts of different metals. And you can understand why zinc and copper does this but copper and zinc doesn't. That's copper in the zinc solution. That's zinc in the copper solution. Nothing happens. The reason is the potential. It's just this. It's just literally the potential difference that we just drew. This is going to be easier to take. So I'm not going to go the other way, right. I'm not going to go the other way, so nothing happens when you do that. And then finally, the last kind of piece of this is that we need a reference because you can only get the changes in potentials. It's like enthalpy. You can only get the changes, not the absolute, so you've got to have a standardization. And the standardization in the world of batteries is hydrogen. So what you do when you think about battery potentials is you write down all of these reduction-- reduction is gaining electrons, right-- you write down all. It doesn't matter. You could write reduction, you could write oxidation. They're just a flip of the sign, right. It's just a flipped sign of the potential, but what you write these down with respect to a common electrode, and it's the standard. It's called the standard hydrogen electrode. So sometimes you'll see it as the SHE in the battery world. And what it is it's just a very nice platinum electrode that doesn't change. So it doesn't plate or lose atoms. And it's a reaction that on that electrode happens with hydrogen, hydrogen gas. And so what you do is you hook this up on one side. You hook that up on one side, and then you put all the metals against that. And you measure their potential. What's so powerful about the SHE is that now I've got all of these with respect to the same reference, right? So now when I look at my zinc-copper battery, well, it's just the difference between those that's the potential of the battery. So hydrogen electrodes gives you that ability to standardize it and create huge tables. And if you look up potentials of reduction, they're all related to hydrogen. You get thousands of them. How much does somebody want to lose, how many electrons? That's going to be in these tables. And it's by a setup that is very similar to the one I showed you that is related to hydrogen. OK. Or you could go the other way. If I had zinc-- I just showed you zinc. Copper in zinc doesn't happen because it's the wrong sign. But then what if I put copper-- instead of in zinc, I put copper in silver. So there's copper in zinc, copper in silver. It works. Because now silver is down here. How did I know that? Because I hooked it up to hydrogen and I compared it to copper. And so that's how you do battery chemistry is you look things up. You look up these potentials in tables, and you think about which voltage you need. And that gets us to this Nobel Prize. You say, well, why are lithium-- why did the Nobel Prize go to the-- not discovery but the development of lithium ions as a storage technology? And the reason is this. You saw-- it was the highest one on the list, too. Look at that. So now, I don't know what the potential of a lithium something battery is going to be, but I'm starting out with a pretty negative voltage with respect to hydrogen. That makes you think you might get high voltages, and you do. Yeah, but you have another thing that's really cool about lithium is how light it is. And those things combined make it a beautiful battery technology. And that's why the Nobel Prize was given. Because if you look at it on a plot like this, it's all the way up here, if you look at energy density versus volume. And a lithium battery works much the same way in the sense that you have the separator here. That's the electrolyte. But now it's rechargeable. And I don't have time to go into rechargeable batteries versus what are called primary batteries, which is the one I showed you. But all the chemistry is the same. You're shuttling ions back and forth. You've got that external circuit. And what I wanted to-- and there's a whole bunch of chemistry. So what has exploded in the last-- well, that's a bad word for batteries because that is a danger with lithium batteries. [LAUGHTER] But what has really made the field take off is you know the word. We're in the class. It's chemistry. Because the flexibility of where you can put that other electrode and what properties it has gives you enormous room. That's why lithium-ion batteries can go in everything now, all the way up to the grid. And that was my last Why This Matters in the last 2 and 1/2 minutes. This is the grid now. So we went from small to very big. The growth in solar and wind has been incredible over the last 10, 20 years. The problem, as I think I've shown you, is the variability. And so this is like-- if you zoom in here, that's the load, the black line. And this is what these two resources give you. And you can see that it doesn't match. If you zoom in on a day, then you really can see it, right? There is the load. And there's what you would get from solar and wind. And so here's the question. I said, well, OK, how do you fix this? You got to store it. There's no way to use renewables at large scales unless you store it. In fact, we're really at essentially the tipping point. Because if I look at the countries in the world and what their SPV penetration is-- and this is a couple years old, but it's still pretty much the same-- Germany, Greece, Italy at the top. Sad frowny face here for the US of A. But Germany is at 7.1% as of a few years ago. That's a huge penetration of PV. Well, here's the problem. If you look at-- this is Germany, now. Huge penetration-- these are these six years, seven years of adding PV. This is how much they had. They're now at 7%. 2016 is 7%. This is how much the price of electricity was in Germany when they added. So notice that here is the sunlight electricity during the day. They can sell it for a lot of money. But as they add more and more of this renewable to the grid, its price goes down when it's available. In fact, it's now not economical for them to add more PV unless they can add it with storage. Because actually the price has gone down so much and it's gone up here when they're not generating any PV electricity. So you can come up with-- and they did in this paper-- this sort of value factor which, when it goes below 1, means there's no economic outcome that's positive here. You can't make money any more. And look at where that happens. It's right around where they are. It's at 5%. You can't solve renewables at the scale of the grid without storage. And no option exists today. It might be batteries. It might be batteries, but there's nothing that actually does it today at the scales that we need. So that's a real challenge for chemistry. And I hope you guys have a great Thanksgiving and maybe hook up some different metals. And I'll see you all next week. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 36_Diffusion_II_Intro_to_SolidState_Chemistry.txt | This is unbelievable. So I just learned that YFN means your friendly neighborhood hackers. And look, we brought it with us. Unbelievable. This is making my day. This is making my week. This is incredible. And really, I don't know what to say. This is amazing. So whoever you are, thank you. This is awesome. So where's Laura? So Laura, next year I think we should give every student one of these. This will be in our-- so we've got some candy. We've got T-shirts. We'll get to those. We've got welcome to the last lecture. Oh, there's some T-shirts. We're going to let TAs throw them out. We're going to let TAs throw them out. But you guys got a-- so we got a little bit more to cover on diffusion. I got a why this matters, and then I got a couple of things I want to say, and then we'll conclude. And like I said, the plan today is to conclude a little early to give you time to fill out the course evaluations. We really take this seriously. We really value any and all input that you have. And so please do. If you haven't done it already, please use the last 15 minutes that we'll give you in the class to fill those out. It means a lot to us. This is still-- it's hitting me right here. It's slowing me down. I can do it. I'm going to get through it. So we were talking about Fick's laws, and you're only going to be tested on the first one, but I do want to tell you about the second one. You are not going to be tested on the second one. That is correct, unlike last year. Because we didn't have time to get the lecture in on the second law before the recitation, which was yesterday. And so look, there will probably will be a diffusion problem, but not on the second law. But still, I do want to finish the example, and I won't go through it in detail, but I want to set it up and just show it to so you have a sense of what Fick's second law lets you do. And we ended with the first law, and I wrote the second line down. Now, the first law, that is something I want you to know about. And so remember, with the first law, so let me go over here so you have a whole board to use. So in Fick's first law, what that meant was it's a law that describes the steady state. So Fick's first law was steady state. And this is now bringing back memories from two days ago, so steady state, which means no time dependents, no time dependents. And so we had some examples of that. In the law itself-- I'll write it down here-- was that the flux is equal to minus this diffusion constant times the change in concentration over some distance. And so we had this example of a membrane right, and so you could imagine we had the glove. It was the remember, the butyl glove. And so we had some distance. Maybe this is like a delta x, and there was some concentration up here, C1, and some concentration down here, C2. Now the key in Fick's law is you're going to hold these concentrations constant at the source, at each of these points. And so you can imagine that it's going to change as you go down the hill, and we frame this in terms of Brownian motion. We're able to understand why concentration, why diffusion happens down a concentration gradient. Now that flux, that flux. Remember. So J is an amount divided by an area times time. That's J. That's the flux through the membrane, and that is also going to be constant. Now, the d, what we assumed in the d is that it's a constant. So in the Fick's first law problems, we often assume d doesn't depend on concentration. So that means that if I'm over here at a high concentration, I'm over here at a low concentration, well then-- by the way, this would be like-- oh. Would you like some candy? I would like some candy. Thank you very much. Do you mind if we walk up and down the aisles? No, not at all. No. Please do. Yeah. That's awesome. This is just getting better. I'm feeling less and less like talking about Fick's law, but here we are. I'll go fast. Now the thing is, look, this is x. By the way, I really want to know where they got the costumes from, but this is x, and this would be like the concentration. Now d could depend on concentration. By the way, it's actually quite interesting. It does have a small dependence on concentration. We often don't include it. So this would be like a straight line. But if it did, if it did depend on concentration, then you might have a deviation. You could have something like this, or you could have something like that, and you don't need to have a straight line to use Fick's law. But in a lot of the problems that we do, we assume d is just independent of concentration. It's a constant. So I'm adding more and more carbon into iron. Does the diffusion of the carbon now depend on how much I added? Well, yeah, it might a little bit. By the way, that doesn't mean that it depends on time. That's the distinction I want to make. So again, the problems we do with Fick's law, we'll assume d doesn't depend on concentration. We'll just assume it's a constant, but the main determining factor to see if it's a Fick's second law or Fick's first law problem isn't whether d changes with concentration. No. It's whether there's a time dependence. It's whether there's a time dependence, and that was the problem that we left off on on Monday. We say, well, you've got to do this thing called case-hardening to a lot of materials. You make the outside harder. The inside assets a ductile, otherwise, either it's not strong enough, or it's too ductile. It just breaks down either way, or it's too brittle. So you want a case-harden, and that was our Fick's second law problem. And again, I'm just going to kind of set it up. I'm not going to go through it all. But in the second law, what you have is an ability now to look at the time dependent. So you can imagine if I have position x and concentration C and I start somewhere here, so I have some source. That's why we call it Cs. So what did we do? We took this piece of steel, and we asked the question. I'm case-hardening the steel, so this is iron with carbon. And I want to put carbon into it but only up to a certain point in a certain amount. How do I know how long to put it in the oven? That's a Fick's second law problem because it's a time-dependent process, and I want to know about concentrations into the piece of iron, into the piece of steel as a function of time. So you can answer questions like this with Fick's second law. So I had it at 0.25% weight percent carbon, and I carborize the surface. I carborize only a certain amount of it. I case-harden it, how long should you keep it in the oven. And by the way, also what should your carbon, your concentration of source be on the outside. And all these things kind of go in. And so you could imagine if the source is there and I'm holding-- maybe I've got some constant, C0, that's the 0.2. That's the piece of material I started with. Well. You can imagine at t0, you've got this. So this would be time equals t0. And then as soon as I expose this to this higher concentration of carbon, you might have something like this. So now you've got t1, and now I wait a little bit longer and it's going to be able to penetrate more in. So the concentration inside the material changes as a function of time and so forth. That's what Fick's second law tells us, and the second law is basically a heat equation, which somebody noticed after a lecture yesterday. And you [? can ?] let's see, dx squared. So this is a partial differential equation. But solving it is just talking to our mathematician friends, and they come back and say that's an error function. And an error function is just a function, And It gives us a solution, C, the concentration at any position and at any time minus the initial concentration divided by this difference in concentrations between the source and the initial is going to equal-- oh, boy. You'd think I'd have learned. 36 lectures, and I still worked my way into a corner. I'm going to do the equals this way 1 minus error function of x over 2 root dt. That is what you get when you solve Fick's second law. That is what you get when you solve Fick's second law. And now the error function is just a function. You don't need to worry about the details of this. You can think about it as like any other function. But one thing that I want to share with you about the error function is-- and this is something that, yeah, you're not going to need it on the exam, but you might need this in life. You might get an error function in life, in IRF. And if you get into IRF, I don't want you to panic because IRF is just a function, and it just looks like this. It just looks like it's a linear plot roughly up to about 0.6. This trick-- and then it kind of goes up to 1-- it's just a function. Is that 1? Yeah. And so if z is less than 0.6 then the error a function of z is roughly z. Talk about a valuable thing to learn, and that saves you a lot of time, by the way. But it's just a function and it's a solution to fix second long. So again, this is how we solve time dependent diffusion problems. And this is how you could solve this one right here. And what I'm going to do is I'm not going to actually solve it. I'm going to give you the answer to that question. So the time is seven hours. That's the answer, and I'm also going to leave you with-- oh, there's error function tables. Look at that, error function tables, and there's this setup of the problem. So I'm going to leave you actually with a sketch of the setup. And again, this isn't something we'll have on the exam, but I want you to have this kind of problem at your service. So these are all embedded in the question. There is your CS. There's your c naught. At this distance, this is what the question asked. This is the concentration we want. We know when we look up the diffusion of carbon, we assume it's not dependent on concentration, that D. And so we just use a constant, and then we plug it in, and we get a time. You better believe that this is really important. This is really, really important. This is how we make stuff. This is how we make stuff that we make our world out of. So that's the picture I showed you. And this gets me now to our last why this matters, and I'm doing it on the backdrop of this beautiful periodic table. So that's a lot of energy, by the way. First you had to make these iron pipes, and now you've got to case-harden them or maybe you've got to put even more iron in. Look at the other 900 C is really hot. How much energy do we take to make stuff as simple as concrete? It's just liquid stone. Didn't you just grind some stones down or something and then pour it out? Actually, if you think about concrete and other materials like steel, the use of them is skyrocketing. This is how much we're making. So look at this, the millions of tons. Let's see. There's steel, and there's cement. Look at that. There's the world population right there, and this runs out in 2009, 2010, '11, but it's continuing. It's continuing because it's not just that the population is growing, it's that the places that the population is growing the most are also industrializing. And that means building. You go to some of these places and there's a new building. Literally every block, there's a new building being built. That takes this stuff. That takes a lot of ovens at a lot of really high temperatures. Just the cement alone is accounting for 7% of all CO2 emissions, all CO2 emissions on the planet, 7% of that goes into simply making this magic liquid stone, this powder. And at the core of it, what is at the core of it? It's chemistry. It's clinker. It's called the clinker. What is a clinker? It's this thing. It's simply a calcium silicate group. Making that is what takes all this energy. It takes half of it at least, making that right. And as always, the chemistry gives us the chance to get out of our constrained optimization. And so you've got to make this mixture of different types of synthetic rock. You've got to make mixtures of alite and belite. They have different names. And you might be able to make more belite, which takes less energy. You can cool your oven down by 300 C to make belite compared to alite. But then even though it's structurally OK, it takes 90 days to dry. So you can't do that. So you've got to spend the 300 extra Celsius to make more alite. Why It's in the chemistry. It's in the chemistry. And understanding, what seems like not a very interesting-- cement, is it really that interesting? You bet it is. 7% of all CO2 emissions is just making this one material, and we're making a lot of it. We're making a lot of it. There's a 164 tower in Dubai that opened in 2010 that took a billion kilograms of this stuff. Now, this makes a lot of CO2, and I've talked about CO2 throughout the semester. And so one thing I wanted to leave you with was also some things that we're doing about it. What do we do about CO2? So how is carbon capture and sequestration going? Can we take it all out? We put it in, can we take it out? And so I'll share a couple of things. So one of the problems with CO2 capture is that if you don't do it right where you make it, then you have to transport it, and it turns out that's actually really hard to transport CO2. And there's a lot of problems that have happened over the years. In the '80s, there was a leak that killed thousands of people. There have been massive leaks in Canada in transporting this material. So transporting is actually really hard to do. There's metal pipe corrosion. This is a great chemistry problem unsolved. And then the next part of CO2 sequestration is one is transporting it that's challenging, and the other is storing it. What do you do with it? Let's suppose we could capture it. Well, then what? And there's ideas around this. So you could pump it maybe underground, geosequestration. That's not a sure thing. We don't know, we don't understand at those scales exactly how well we can in case this material and how long it lasts, so a lot of questions with that. There's a whole line of thinking on putting it in the ocean. Right? Really? So you want to fix the CO2 problem by acidifying the oceans even faster? OK. Interesting. So transport and sequestration is a big problem, but what about technologies where you simply sequester it right there where you make it, like clean coal. You may have heard of clean coal, so clean coal. So I'm going to look at a case study of a clean coal project in the US. In 2005, I got really excited because eight companies joined, and it was called Future Gen, and the US government was going to put a lot of money into it. They were going to capture 90% of the CO2. You get a little bit less efficiency. Depending on how you think about it, it could be 30% less efficiency, I mean, less power out of the plant, but you sequester the CO2. And in 2009, construction was planned in Illinois. In 2010, it was canceled. In 2010, it was restarted. In 2012, there was all sorts of discussions with the DOE that agreed to pay $1.1 billion. It was all located, and then the companies dropped out and the DOE canceled projects. So that was 10 years. 10 years and $200 million of taxpayer dollars and absolutely nothing has come out of it. So one of the things I want to point out about this direction, I'm not saying that it's not a good direction to keep pursuing. But the timescales are long. So we need to be able to make decisions that can cover longer timescales. That's a huge part of the challenge with this. It's a policy challenge as much as it is a technical one. And tying this back to the cement, there is some really interesting research that's going on in actually instead of-- OK, so you've got the CO2. It comes out when you make it because the ovens and you're burning stuff, but what if you could take that CO2 and just put it into the thing you're making. And by the way, if what you're making is a huge scale like cement, then maybe you can put a lot of it in, and maybe that could have a real global impact. So there's a lot of interest in substantial global carbon uptake by cement carbonation. So maybe you could get some of it in the cement. I like this paper. I'm pointing it out because they say that they used Fick's diffusion law. They modeled the carbon uptake by applying Fick's diffusion law. Yeah. It actually matters. Right? You can do this too now. And just a few weeks ago on CNN, this concrete, yes, concrete is going high tech. Why? Because this company has managed to figure out how to put a bunch of CO2 into the concrete while they make it, and it doesn't change the mechanical integrity of the concrete. And it's in there. It's actually in there. This is all chemistry. This is all chemistry. So this is, I think, another very interesting direction where technology can play a vital role in this particular global challenge. And now I'm putting the chalk down, and I come to the conclusion I can't look there. I'm going to get sad. You're making me-- the first thing about this that I got to do is I got to thank people. First I got to thank the concepts, got to thank the material. This is what we have talked about all semester. This is what we have learned. This is underneath the hood of so much, and I hope that I've conveyed that to you. This is a starting point for so much. What have we learned? Well, we've learned how to go back and forth. This is what I showed you on day one. And now what have we done? We have filled this out. We filled it out with balancing reactions. We filled it out with the periodic table, that one. We filled it out with electrons, Bohr model, Schrodinger, atomic orbitals, molecular orbitals, ionic solids, Lewis dots, VSEPR, Van der Waals, London, H-bonds, all sorts of inner intermolecular forces, metallic bonds, semiconductors, bands, crystals, symmetry, lattices, planes, x-rays, x-ray diffraction, defects, amorphous glassy materials, reaction rates, rate laws, diffusion, solubility, Arrhenius, acids and bases, polymers, and their properties. That's you guys. That's you guys that have learned that material. It blows me away. That's a lot of stuff, and we've learned it all, and we've integrated it into our lives. But you couldn't have learned any of this, we couldn't have done any of this without some people. So now we thank the subjects. We thank the people. Laura, would you stand up please? Yeah. Many of you have interacted with Laura, and you know and now sometimes you might get a short answer. You might get a not a short answer. But why is that? It's because Laura cares deeply, deeply about you all and about your learning and your experience in this class and at MIT. She cares deeply. And that's the kind of thing that makes this class what it is. We wouldn't have any shot at it without Laura. So Laura, thank you very much. It's sort of like when you drop your kids off, if they go to sleep-away camp, there's like the camp parent who makes sure that your kids-- it's like they make sure that your kids will brush their teeth, get to bed, have a blanket. And they check in on your kids all the time. And I feel like we have that for this class. It's Laura. So thank you, Laura. But now the thing is now the other, now, I introduce these people to you. Isaac, I still don't have your picture. But can the TAs please stand up? Can the TAs stand up? Come on. Thank you. Thank you. Thank you. I am so incredibly lucky to have this quality people, this quality of staff, these kind, caring, passionate, amazing people as our TAs for this class. And I am so lucky I get to meet with them every week to talk about you guys-- it's always good; don't worry. it's always good-- and interact even much more than that about education, about teaching, about how to make this a fantastic experience for all of you. So thanks to the TAs. And that's why I want the TAs to throw the T-shirts out, and we'll do that in a minute, well, a couple minutes. No. One? But see, it's the TAs that have been here for you, but it's also you all. There is you. It's you all that have been here for each other. And so I see it because I talked to some of you. I don't talk to all of you, but I see the community that forms, and I am very appreciative of that, and I'd like you to be appreciative of that. So let's thank each other here for just a minute. It's really pretty-- this is actually getting me revved up a little bit. Now, I skipped over. The goodie bags are amazing, and so I'm thankful to the goodie bags and to the department, if anyone is here from the department, for helping us support these goodie bags, thank you. Thank you. Yeah, thank you. That's good. So I'm very grateful to that. I'm grateful to all of you. And last but not least, I guess just last, I am thankful to Harvard. I was going to say something else. But anyway, I'm thankful to all of these resources that we have used. And so I'm going to give you some closing comments. I started this semester off in the first lecture, and I asked you all a question. And the question was, why are you here? And I said really, really, really, why are you here. Why did you come to MIT? And I asked you, did you come here so you can just phone it in, get a stamp in four years, and go out? Did you come here so you can walk around some place in privilege and look down on other people? That is not why you're here. That is not why you came to MIT. You came to MIT because you know that this is where we will make the transition from being able to answer any question to knowing which question to ask. That's the transition from student to scholar. And you know that you came here because you have a passion, and you want to come to a place where you can take that passion and you can try to solve really hard problems together to make the world a better place. Those are reasons why you came here. You came here because you know that progress does not happen because of success. It doesn't. It happens entirely because of what you choose to do with failure. That's MIT, that's the MIT way, and that's why you're here, and I asked you to think about that on the first lecture. And then I gave you a whole lot of examples throughout the class. I tried to connect our learning and solid state chemistry to global challenges, to global challenges that we face. And I tried to do that. So I had 36 opportunities. There's many more, so I had to pick and choose carefully to try to give you a sense of some of these things. And the thing is about these challenges, one I just talked about in many others, it's not that these are things that we need to do soon. There is an urgency of now about these things. There is an urgency of now. There's no more time to talk about it. And so these challenges, it's not like something happens over here, happens over there. These are planetary challenges. These cover the whole thing. So it's all on the table. It's all on the line. And by the way, if it's a planetary challenge, it means the challenge doesn't care if you're in a blue State or a red State or if you live in Europe or Africa or here. It doesn't care. So we better figure out how to get our act together and work together to solve these very hard problems. And that's why you're here. That's why you're here. And so on the first lecture, I asked you to think about why you're here, and on this very last lecture, I'm asking you to think about what you want to do about that. What do you want to do about that? And there's a lot of things you can do. I don't mean that you have to work on one of these things we've talked about in this class. That's not what I mean. I mean, think about what it is that you really want to do and how you want to do it, and then go for it. You might have figured that out before you came here. You may know-- gesundheit-- already way before you got here. You may be figuring it out right now. You may have no clue what you want to do when you graduate. All of those are fine. All of those are fine. But what I want to leave you with is a framework and some thinking that what can I leave you with as a suggestion, as a framework. And it really comes down to three ingredients, three ingredients that I'm asking you to think about when you go off and do, when you leave here, when you go off and do. What are you going to do? How are you going to frame this? And so the first ingredient is kindness. Now OK. Bear with me. Because you're all thinking, what is he talking about? This is a chemistry class, and you're talking about kindness. But you see, the thing is, I'm trying to think about what is a framework that if you follow it you will do good. You will help change this world for the better, which is what we do at MIT. It's how we think. And the thing is I don't care if you define kindness as love or as helping someone or as community or as on one of those dates when the candle was in the room and you did the calculation and you saved the day because you knew oxygen was going to run out. That's all kindness. All of that is kindness. But the thing is that if you have kindness in what you do, it's really hard to do bad. Is that under the hood? Is there an element of kindness? You can't have too much of it. We need that vector. We need that vector in this world. So that's one ingredient, and the second ingredient is knowledge. Now, what I mean by this is acquisition of knowledge. And we have done a lot of that here in this class, and you do a lot of that at MIT, and we celebrate these acquisitions. What I'm asking you to do is think about this as you go forward. How are you acquiring knowledge? How are you learning? That sounds cliche. Always learn. You guys have to keep doing that. You have to keep doing that because, for you, it's nourishment, and it's another one of these vectors. You can't do any harm by learning. You can only do good for yourself and for everyone that you care about. So keep doing it. The rate might change of learning acquisition as a reaction, the rate constant might be lower. We know places where that's already the case. And that's OK. Maybe the order is lower too down there, but I think they probably are learning something. Keep that going. And the last thing is passion. And this involves your talents. You all have talents already. You all have them because that's part of who you are and why you're here. Those are things that you bring a passion to. What are those talents? Maybe some of them are hidden, but work on them and pursue them. Do things that you literally are so excited about you can't sleep sometimes because of the excitement. That's what I'm talking about. Follow that. Feed it. And while you do, lose the fear. Avoid greed, and embrace challenge. That's what MT is. That's why we're here. That's why we're here. So I feel like these three ingredients, I've thought about this, and I thought about these three ingredients, and I've thought about-- we have these already. You all have these ingredients because that is also MIT, and you are here, and you are MIT. And so what I'm asking you to do is to hold onto them and keep thinking about them. There's a lot of reasons why I love to teach, but one of the big ones is when I think about the challenges that I share with you guys and I have 36 chances to look up in the room and I think about those challenges and I think about these ingredients and I think about all of you, I actually feel like we have a shot. I actually feel that. And so with that, good luck on the final, and have a fantastic break. Thank you so much. Thank you. Thank you, guys. Thank you, guys. Thank you. Thank you. Thank you so much. Thank you. You guys are amazing. Thank you. Thank you. Thank you all so much. I'm going to have to leave here, or I'm going to get really emotional. I've clearly got a lot of other dance moves the TAs have helped me, so I can practice those. Thank you so much. That's just incredible. You guys are all incredible, and this has been incredible. So we do have 10 minutes. I'm going to leave. So you guys can fill evaluations out without me being in the room. And so you can say whatever you want. Again, don't let the free T-shirts being thrown at you influence your opinion or the free candy. But thanks, everyone, for a fantastic semester. Thank you. Thanks. That was amazing, you guys. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 35_Diffusion_I_Intro_to_SolidState_Chemistry.txt | As I mentioned last Friday, there is one more topic that I want to cover. And that is the topic of diffusion. And so today, I'm going to talk about diffusion, which will be in the context of what diffusion depends on and how do we describe it. And in particular, there's two laws, Fick's first law and Fick's second law, that I want to talk about. And I want to talk about how this looks in the context of some of the chemistry that we've learned, some of the things that we've learned. So diffusion, this is kind of a standard picture people show for diffusion, although really, this is probably mostly convection, which is the movement. It's not diffusion. But diffusion kind of has taken on this broader meaning. And so what do we mean by diffusion? Well, diffusion comes from the Latin word diffundere. And that means to spread out. But in this class, we're going to start by thinking about it as a particular movement down a concentration gradient. And that is what's happening here. You see there's a very concentrated droplet of dye that you put in there. And then it's less concentrated out here. You probably don't have any. So the concentration of the dye out there is zero. And in here it's very high. And so it wants to go down a concentration gradient. So that's movement down a concentration gradient. OK. How do we describe this movement? And it really goes back to experiments that Robert Brown did. And this is the original paper that he wrote way back in the early 1800s. I love the title of this paper, "A Brief Account of Microscopical Observations Made Specifically in the Months of June, July, and August 1827 on the Particles Contained in the Pollen and Plants and on the General Existence of Active Molecules in Organic and Inorganic Bodies." What a nice, detailed title. That was Brown. So he started with pollen. And he looked at it. And it looked like this. [MUSIC PLAYING] Oh, that's very loud. There wasn't supposed to be music. So that's what he saw, except this isn't what he saw. This is what Koshu Endo randomly posted online and I'm showing you. But if you look at pollen in a microscope, and you're Robert Brown back in the early 1800s, you'd see this. So he thought, aha, OK, there's life. This is must have to do with biology. And things are alive. And so they move. And it must be something like that. And so he put other things in. And they did the same thing, they jiggled around. And then he said, well, OK, what about if I put something that was once alive but is not alive anymore. And it did the same thing. He said, OK, wait a second. Maybe there's something to do with having ever been alive, the spirit of the particles. And he said, OK, the ultimate spirit, I'm going to break some glass-- which is what he did, I love that. I like breaking glass too. And so he broke the glass. And he took little pieces of glass and put it under his microscope. They did the same thing. They all did the same thing. So they were really perplexed. What was going on? This is the microscopic basis. This is now called Brownian motion. It's random motion due to thermal energy and collisions. And it's the connection of this microscopic random motion due to thermal energy and macroscopic observables that led to a big advance in our understanding of diffusion. And so the question of how far took 80 years. So one of the first questions, you say, how far do these things go. How far are they going? They're randomly moving around. I got that. I see it in the microscope. How far do they go? So how far? Well, that was a question that you can answer if you can get the mean squared displacement. I don't know. I say tomato. And so displacement, it's OK, mean squared displacement. And it really came from Einstein in 1905. Actually, this is what Einstein did for his dissertation, his Ph.D. work was on this topic. It was on understanding how to go from these kind of microscopic random fluctuations to some macroscopic observables. And so, the mean squared displacement, well, that's just the average of some measurement of some distance the particles go. So if you go back to the dye, there it is. But now it's in atom form or molecule form. And it's diffusing around, diffusing around. And OK, you say, well, how far did it go. That would be the circle. And that's something that Einstein tackled in his Ph.D. and found that the mean squared displacement is-- well, let's just say it's equal to 6 times some constant times time. This is in three dimensions. If you're in two dimensions, it's 4. If you're in one dimension, it's 2. But the main point isn't that. The main point is that the mean squared displacement is a function of some constant times time. And so if you just wait long enough and you know this constant, then you can figure out how far things diffuse. That was pretty cool. That was pretty cool. But you see, now, what we're going to be more interested in in this class is not how far but how fast. We're interested in how fast. And again, these are questions that Brown couldn't answer. Brown observed these things and wrote about them. And that's why it's called Brownian motion. But it was later that people said, well, let's try to think about how to write this down and come up with theories that explain it. And so for how fast, now we have the flux. So we have the flux or the diffusion flux J. And what we want is a way of describing how much. So it's an amount of something, the amount of the substance per area. You're going to normalize it. So you say, I'm going to take this sheet. And I'm going to say, how much of it is crossing this sheet area per time. So amount of a substance per area per time, that's the flux. So it could be grams. The amount could be grams, mass, kilograms, micrograms, nanograms. It could be number of atoms, moles. That's also an amount, moles. So we just say amount, it's general. Grams, moles, et cetera, n number of atoms. That's what moles tells us. Grams, et cetera-- The time could also be-- well, the area could be centimeter squared, meters squared. That's an area. And then the time is usually seconds. Usually, it doesn't have to be. And so this is what Fick's law gives us. So this was first established. How do you get a dependence of this flux? How do you get a dependence of this flux? And again, we've got this constant in there. And Fick's first law is all about that. Fick's first law says that this flux is proportional to-- and I'm just going to write it like this for now-- concentration gradient. So that's C divided by the change in X. C is concentration here. We'll draw an arrow. Concentration is C. Now, we could use brackets. Bracket A, concentration of A, mass per liter, something like that. I'll use the letter C, delta C over delta X. So that's the concentration gradient. Oh, by the way this wasn't as far. It didn't take as long as Einstein. But this was 1855. OK, good. So Fick came along maybe 30 years later. And he said, well, I want to know how fast things are going. I want to understand how to describe the flux of these randomly moving things. And I notice that it depends on the concentration difference. And in particular, I like it when diffusion is positive. So J is equal to minus D times dc/dx. And that is Fick's first law. Gesundheit. And D is a constant. D is this constant again. So D is the diffusion constant. We'll talk about this, diffusion constant. There's the concentration gradient. And we got a minus sign. Notice, movement down a concentration gradient. But if I didn't have a minus sign, then I wouldn't have a positive flux. Then the droplet wouldn't spread out. That wouldn't be good. So we're going from high concentration to low concentration. When I've got a bigger change in the concentration, I've got faster diffusion. That's what Fick's first law tells us. OK, good. Very importantly, Fick's first law means it's steady state. Notice, there is no t here. There's no time. For that, we have to wait for the second law. So steady state diffusion, and that means not a function of time. The kinds of problems where Fick's first law matters is-- and I know some of you have probably seen Fick's first law. I get that. But in a little bit, we're going to connect it to crystals. But the kind of diffusion where the first floor matters is the kind where you're holding concentrations equal. Let's say you're holding a concentration gradient, and it's not changing. And so you're wondering, well, if I hold a concentration at one thing here and at another thing here, how fast do things diffuse. That's what Fick's law tells us. And so let's do an example. Here's my example. I found these gloves, because I was looking for copolymers. Here's a copolymer that doesn't have the nitrile groups. Remember, the nitrile groups last Friday that were so important when we were doing our polymer design. This doesn't have it. This is butyl rubber. OK, fine. So there's the copolymer. It's copolymerized with these two groups. But now, it's got those groups, because it gave me the right flexibility and some of the other factors that I want in the glove. But is it strong enough chemically? And that's a very important question. We talked about this last Friday that these chemical groups change the mechanical properties. They also use the chemical resilience. And so here I have it. I have a toxin. This is why we wear gloves, because we don't want to get methyl chloride into our skin. Methyl chloride is toxic at 50 parts per million. Keep it out. Be very careful. So I don't know. I look up online and say, well, OK, butyl rubber. These feel comfortable. But how are they? OK, with methyl chloride, which is a toxin, it's a molecule, it has a diffusion constant of 1 times 10 to the minus 6 centimeters squared per second. The gloves are 0.04 centimeters thick. And I'm working with paint remover. I'm working with paint room. The concentration of paint remover is a gram per centimeter cubed. Am I safe? How much am I going to be exposed to this? At my skin, I'm starting out with zero. What's the flux through the glove? Can I stay safe? Well, first of all, one thing that we can do is we can look at-- just to make sure we understand. So there's the diffusion constant. Why does it have those units? You can see this. The units of J are an amount-- this is what we already said-- divided by an area times a time. So if the units of C are an amount-- OK, I stayed with grams-- over a volume, then you can see right away that diffusion must be area over time. It has to be. If it's not, then the units don't work. I've got concentration over a length. This is a 1 D. I'm going across some membrane maybe or some area. This is a membrane, the glove. And so you can think about this for yourself. The units have to be area over time. Is it area over time? Centimeters squared per second. They were, OK. So that worked. Now, another thing is, before we do the math, why does this happen anyway? Why is there diffusion? And it's actually quite straightforward to understand, even when you go all the way back to Brown's experiments with pollen. You say, well, things are moving randomly. Why is there now a direction? Why is there now a direction? How is that possible? How can there be a direction if it's all random? How can there be a concentration gradient? And then you can get that, because if you have some high concentration-- Let's suppose that this is a high concentration, and this is a low concentration. And you've got the Brownian stuff going on. It's all in something. There's Brownian motion. Whatever containers these are, whatever is in whatever, it's got Brownian motion. And so that means that, because it's random, it's equally likely to go that way as it is to go that way. But see, at the high concentration, I've got more particles. You can see that, because I drew bigger, longer arrows, so it's obvious, more particles, because it's more concentration. There's just a lot more of them. So there's more going across. If I tried to draw a boundary here or boundaries inside here, and I look at those as my areas. There's a high concentration. There's a lot of stuff going randomly back and forth across them. Down here, you can do the same thing. But there's a lower concentration, so there's less stuff going back and forth across those barriers. And now I bring them together. I bring them together. I bring the methyl chloride to the glove, to my hand. Or I bring something with a high concentration next to something with a low concentration. And I keep the concentrations of the endpoints fixed. I hold these fixed. Fick's law tells me now how this happens. And you know that if I bring them together now, well, look, this was going just a little bit back and forth. And this was going a lot bit back and forth. And so there's going to be a net diffusion that way. Right there at the edge, you can see it. More goes to the right than goes to the left. And so this starts to create a diffusion gradient. That'll create a diffusion gradient. So that's why it works. That's why concentration gradients cause diffusion. And Fick's first law tells us how. OK, well, if we go to the question-- so let's see-- so the J-- so now I've got my glove. And so the J is equal minus 1.1 times 10 to the minus 6 centimeters squared per second times-- and this is going to be-- OK, I started with what? I started with 1 gram right per centimeter cubed. And on the skin I've got 0. So it's going to be time 0 minus 1 gram per centimeter cubed. And that's going to be divided by the distance 0.04 centimeters. And this goes to something like 2.5 times 10 to the minus 5 grams per centimeter squared second. That's the flux. That's what Fick's law tells us, because my skin is good enough. My body is good enough that any time a toxin, a methyl chloride molecule touches it, the skin just takes it away and poisons me. But that's important. It's not good, but it's important, because it keeps this concentration fixed. Otherwise, this would build up. And it would be changing. So you can't use Fick's law. No. Fick's law applies when these two concentrations are fixed, and there's not a time dependence. Instead, there's a rate. There's a flux that you're after, J. And so now you say, well, can I handle this. Can I handle this? Work it out, 50 parts per million. I should have used the nitrile. Should have used the nitrile copolymer. Go back and make new polymers. Maybe you need much, much better chemical resilience, which might lower this diffusion constant. This is serious stuff. This is serious stuff. And it's so serious that I want to talk about eggs. The rubber glove stops this toxin. Well, it slowed it down. Did it slow down enough? Use Fick's first law. But you can make membranes that are actually very selective. It's the coolest thing. You make a membrane that only allows water to go through but not salt. That's a selective membrane. And that's called osmosis when you can allow one fluid to go through. It's semi-permeable, permeable to one type of fluid but not another. And the egg is such a great example, I had to show you this video. I made this years ago in another class, because I wanted to show it. This is what you can do with eggs. They're so cool. Here's a video. So you soak them in vinegar. OK, it's even got directions and everything. We don't need any volume. And there they are. OK, you soak them in vinegar. Now, the vinegar dissolves the calcium carbonate. You know this. You've done these reactions. So there you go. You soak them in vinegar. Leave it overnight, I'd say, is probably best. Then you peel off the vinegar. And look, there's a membrane. The egg has a really cool membrane. This isn't cooked. This is just a raw egg. But now I add a bunch of syrup. Now, this is almost pure sugar. So the concentration gradient is enormous, because the liquid inside the egg doesn't have any sugar in it. But outside, the liquid has huge amounts of sugar in it. So the water inside the egg wants to diffuse out. Eight hours, well, it's kind of enough. 18 hours, you pick that membrane up. It's a really cool membrane, the egg membrane. It's a semi-permeable membrane. And it's hard to see here. But you see, all the water is kind of gone. It's totally deflated but completely stable, because the membrane's that strong. The water inside has left. And then if you get some green dye and you put it back into some green water, then you can make green eggs. And you can make green ham that way. And that's osmosis. But it's semi permeable. You've got to give the credit right. You got to give the credit right in these things. This was years ago when I thought I had a career in that direction. So the egg is just such a cool membrane to play with. And I though, most of us can access an egg. If you want to play with a really cool semi-permeable membrane, that is cool. And you really will see Fick's first law at play. You will see it, and you will feel it. Now, there's one thing I want to comment on about all this stuff. And we are not going to learn the details of entropy in this class. That is really something for a thermodynamics class. But I got to just correct something, because it's in your textbook, because a lot of this diffusion stuff, and in fact dissolution-- remember we talked about salt dissolving? We did whole lectures on this, the dissolution. Remember, these all have to do with the total energy of the system. But we really focus in this class on one part of that, on one part, which is called the enthalpy, the bonding, the potential of the bonding. But there's a whole other part that's related to the entropy. And we're not going to go into this in detail, like I said. You won't be tested on this. But I got to bring it up. So entropy can be really important. Why does something dissolve? It may be dominated by entropy. It may be dominated by entropy. That goes into the total energy still. So we still have our happy place, lower energy better. But as you learn more about what that energy entails, you've got to start including entropy in the future. So Averill says, "...for now, we can state that entropy is a thermodynamic property of all substances that is proportional to their degree disorder." No! That's not true. That is not true. So I have a question. So you see this all the time. Entropy, higher entropy, more disorder. My room is messy, it's got a high entropy. So which of these has a higher entropy? Which one of these has a higher entropy, and which is lower? This is a very simple grid. They've got the same exact number. How many think that the one on the right is-- it's the smoother one, right? How many think that this is the lower entropy system? I've set this up. I've set this up, because you know that's not. But why? The reason is the real reason for what entropy. It is not simply, because in this-- these are computerized experiments. So we told a computer to pick randomly a place on a grid a certain number of times and then fill it in. But in this case, you restricted it. You said, no. I can't have any two squares next to each other. And in this case, you simply did it randomly. So this looks smoother. This looks less disordered. But this has a higher entropy. Why? Because you have fewer options. And that is what entropy is about. Entropy is about the number of possibilities that you have. Now, that might look more disordered, it might not. But under the hood of entropy is possibilities. It's number of states, of accessible states. A salt atom in the crystal has very few possibilities. If it were a perfect crystal, one, it's got to be in the lattice. But once it goes into the liquid, it's got many more possibilities. It's those numbers of states that matter. That's why its entropy goes way, way up. It's the number of possibilities, the number of possible states. OK, good. That was my aside on entropy. Now, we talked about Fick's first law. We talked about diffusion. We looked at drops. And we made sure the gloves we choose are going to be safe. What about solids? What about solids? Well, things diffuse around in solids too. Here's a red atom that is going to diffuse. Watch this super high-tech video. How does it do it? A vacancy comes in. Look at that vacancy, moving around, moving around, bam! That atom moved, it moved, self-diffusion. It moved. It's an atom of the same type that was able to move in the lattice because a vacancy diffused through right. And so there's the beginning. And there's the end. You see it moved over by one position. But now we're all thinking. We're all thinking, and we're going back to our days of, well, OK, hold on. What really happened? So here's the vacancy on one side. Here, it went over to this side. There's the atom going through. So the vacancy moved that way. The atom moved that way. But as you know, that atom had to experience some barrier. So it had to get over some activation. It had to get over some activation. And that really helps us answer a question that has been burning in our minds this whole lecture, which is, what else does flux depend on. We saw the concentration gradient thing. We looked at it in very simple terms. We understood it. But what about D itself? Well, when you see something that's an activated process, there's only one person that ever comes to mind, right? First name S, last name A. And it's Svante, because you know that if it's an activated process, then there's some pre-exponential factor times the activation energy divided by k B T if it's for a single atom. Or if it's a mole, you use R. There's your activation energy right there. But so now, we can start connecting this. So one thing you know about that diffusion constant is it will have a temperature dependence. And you also know now that if you plot the log of it versus 1 over T, you will get that. Oh, we're back. We're in our happy place with this. It's Svante. So this is an activated process for an atom diffusing. These kind of atoms diffusing through their own lattice, remember, these have high barriers. This might be kind of hard to do, but it's still going to happen. It's still going to happen. You might also get something like this. And this goes back to our discussions of defects in crystals. You might get an interstitial diffusion. So this is diffusion. This is the topic that we're covering. But now we can relate the equation that governs diffusion, the D, to the crystal and to the chemistry, because what is the barrier that that feels when it goes through. So remember, an interstitial, like carbon and iron to make steel. That would be an interstitial defect or an interstitial atom that you've added to the system, to the iron lattice. And how does it move? Well, that's absolutely critical, because it tells you how to make it. And we'll get to an example of that. So if I look now at these plots-- and these are plots I think I showed you back when we talked about interstitial defects. You can really connect it to what we've learned. So just to start, these are the log diffusion coefficients. So that's the diffusion coefficient that we're talking about right there. Versus 1,000 over T-- so it's 1 over T scaled by 1,000. And these are those beautiful Svante Arrhenius dependencies. And these are experimental measurements. And look, here's hydrogen in BCC iron. Here's hydrogen in FCC iron. Now, notice, at the same temperature, there's a huge difference. And then, I have FCC up here, and then I've got BCC over there. Why? Well, those are the stable phases at those temperatures. But notice that when they're at the same temperature, there's a huge difference. Why? Because you know from the crystal structure that there's more room. So the activation barrier will be lower in BCC. Those voids are bigger. There's more space. It's not as well packed. We've covered that. And look at this. So there's hydrogen, it's really small. There's iron, iron in BCC Fe compared to hydrogen in BCC Fe. Look at those orders of magnitude. So there's an iron atom diffusing in its own solid. And here's hydrogen atoms. You can see there's carbon and so forth. So this is the picture that comes back. If we want to think about diffusion in solids, we can connect it to the packing and crystal structures, because we know that these atoms, these interstitial atoms are going to diffuse through the interstitial space. What is the interstitial space? Well here, as just and example, I showed just BCC and FCC. Here they are. Here's is an octahedral site. This is BCC down here. So notice, the shaded in atoms are the lattice sites, BCC, at the corners and in the middle. And now, you see, the red atom is in the middle. Gesundheit. Let's say the red atom was here. And it's moving to there. And now it's in the middle. So it might take that path as it diffuses. It's going to find the voids. Diffusing atoms find the voids. And so if you have something that's trying to diffuse, it's going to want to find these voids. This is an octahedral site, because this is an octahedron. And you can have one of those in an FCC crystals as well. You can have an octahedral side. So in FCC, you've got lattice sites in the face but not in the center. And so its octahedral site is here. You've got tetrahedral sites. Those are other voids in the crystals. We touched on this a little bit when we talked about defects in crystals. But now I want to relate it to diffusion, because how a crystal has voids is critical to D, how something diffuses in it, whether that something is something in the crystal itself or something you've added to it. And how it has voids depends on the bonding, the chemistry, and the crystal structure. So you could answer questions like this. These are some just fairly straightforward questions that you could answer. What if I had carbon-- this is what I just showed you-- in alpha or gamma iron? Well, so I'll just write the answer. It's alpha ion, because it's an open-- I don't need to calculate stuff. Which one is BCC, which one is FCC? And it's alpha, because it's an open BCC structure. And so the diffusion barriers are going to be lower. Iron in alpha iron at 500 or 900? Well, now I've got a temperature question. I know D is going to be higher, so 900 degrees C because of influence of temperature. Iron or hydrogen in alpha iron. It's like what we just talked about. So this would be particle size. H is smaller. These are conceptual questions. What's the fourth one? Silicon vacancy in silicon or helium in silicon. Well, for a vacancy to move, you need those kind of same atoms move. And if a same moving through those sites, that's harder. That's harder than if a small interstitial like hydrogen is moving through those voids. That's why, when you look at this, iron is orders and orders of magnitude slower. The D is orders and orders of magnitude slower than carbon or hydrogen. And so that's going to be He, because the vacancy diffusion is so much slower. And then five, Mg, this is a good one, because hydrogen in platinum or hydrogen in magnesium. Well, magnesium has a larger atomic radius. And you look it up. And you say, they've got the same close-packed crystal structures. But one of them has a larger radius. What does that mean? Well, it says something about the voids as well. The voids are going to be larger. Remember, this is happening through the voids. This is happening through the voids. This is how batteries are engineered. This is why we have a revolution in batteries. So this is a couple of years old. There so many great review articles on battery technologies. This is the cathode. By the way, what is a battery? Well, a lithium battery is basically two sponges. And one of them is usually carbon. And when you're running your battery, lithium ions are going in. And the carbon is so strong and able to hold those lithium in and not break down. So we often use carbon on the anode side. And then when we charge it up, we push them back over to where? Where they came from. That's the cathode, which is a lithium compound. But it has to be a material that can hold lithium in it, hopefully at very high concentration, but also be OK to lose it and not completely collapse. And so you've got lithium ion phosphate, those are these ones. You've got manganates. You've got cobalts, cobalt oxide materials. There are so many battery materials. They all work the same way. If it's a lithium battery, the lithium is going to leave some crystal that has lithium in it. But the crystal has to stay. So it's like a sponge that loses something from it but stays intact. And that's something goes over to the anode. It gets soaked up there. And you can recharge, discharge, recharge, as you're just pushing these lithiums back and forth. Oh, but you're pushing them back and forth. You are diffusing them. That's what you're doing. You are diffusing them through whatever structure you're putting them into. So if it's the cathode, and you're trying to get the lithium to come back in or to go back out, then you can imagine that the barriers and the dimensionality itself are crucial. What is an activation energy for a lithium atom in a given material? And does it take a 1 D path? It turns out these can be extremely efficient. Look at those. They're practically holes, they're lines in the lithium ion phosphate. It's a really nice cathode material, because you've got these channels. These high conduction channels, these highways. But then you say, well, but maybe I need 3D voids. Maybe it shouldn't have channels. It should have voids, so it can choose more paths. Maybe we have planes. Maybe it's a layered structure where they have these 2D planes they can travel through. What's the best? Well, it's not trivial, because it's one of these, as usual, constrained optimization problems, where, when you make the pathway faster, you might need to have more voids where you'll lose material. Or maybe you make the material unstable. So you want it to have a really high density. The lithium diffusion and a battery material is related to the charge time. So here's another paper from a few years ago. If you plot materials-- now, here's the specific power. And so that's how much power, watts per weight. Here's the watt hours. So that's the energy per weight. And this is the charge rate. This is the charge rate. And notice, so you want it to be very, very fast-charging. And these are good materials. There's the lead-acid battery that's still in your car. These are nickel metal hydrides and so forth, other technologies. Here are the lithium ion batteries. This is what's in your cell phone. Notice something, as I get up to faster charge rates, more open voids in material, I lose energy density. I go this way. I don't want to go that way. I want to go this way. Constrained optimization, these are the things. What goes into solving these problems is calculating diffusion barriers and energy densities and all the other things that matter. But if you want to target fast charging, you've got to know the diffusion barriers. One of the number one challenges in battery design is to keep that high while continuing to push up that way. Polymers, you could imagine that a polymer is slower. It makes sense, right? Because it's harder-- as we talked about when we talked about polymers, it's harder to make crystalline polymers, to keep a polymer from having some amorphous region. After all, you've got these 100,000-unit long strands of spaghetti. So it kind of makes sense that the diffusion is going to be lower in a polymer than in one of these crystals, like an olivine, lithium ion phosphate. So that makes sense. But polymers would be great, because they're lighter. And they can be flexible. And so we like polymer batteries a lot. We want them for many applications. So these are the kinds of tradeoffs that you think about. And D is right there in the center, because none of us want a battery that takes two days to charge. And now we get to the next part, which I'll just introduce, which is, now we're going to think about time. And I want to set this problem up. And then we'll solve it on Wednesday and then and then finish. But I want to set it up, because the next thing that's related to diffusion is time. And I want to give you an example of how that's so important with something called case hardening. So if you buy a gear-- and you do buy gears all the time. Whether you like it or not or know it or not, lots of stuff has gears in it. And you look at the gear closely, it would look different on the outside. If you do a cross-section, most gears are going to look different on the outside. And the reason has to do with exactly what we just talked about. If this is a piece of iron, pure iron, it's going to be too soft. So you're now turning this. And it's got to have mechanical integrity. And so you've got to make it harder. But the thing is, if you put it in and it's pure iron, it's too soft. But now if you make it this really ultra-high strength, carbon-infused steel, it gets too brittle. It gets too brittle. And so you need a balance. And one of the ways to balance the mechanical strength of something like a gear or many, many other materials is to harden just the outside. That's cold case hardening. The case, it's the case. How do you do it? Well, you get carbon. Remember, carbon going into the iron makes it a lot harder, but it also makes it more brittle. So if you made the whole gear the same amount of carbon infused in iron, is going to be too brittle. If you don't put any on, it's not mechanically strong enough. So what you do is you very, very delicately expose this piece of iron to carbon. You harden it, but just on the outside. Now, how does the carbon get in? Diffusion. It gets into the material through diffusion. So then the question at your gear-making plant that you have to know is, what temperature and how long should I expose this gear to how much carbon. You've got concentration. You've got temperature. And you've got time. And if you know that, you can actually process this material in just the way you want. And you have to know those parameters. And that is Fick's second law. So these are steel pipes. And they are going to be case hardened. So what's happening? You've got an iron with a certain amount of carbon content. But now, I need to add more carbon but just the right amount to the exterior of those steel pipes. How long should I do it? And that is Fick's second law, which I'll write down. So how do we figure that out? Well, we use the time dependent version of Fick's law. And Fick's second law says that the change in concentration with time is equal to the diffusion constant times the second derivative of the concentration with position. That's Fick's second law. And so this is non-steady state, which means-- and we're not going to derive this. But I do want you to know that this is the Fick's second equation and how you might apply it, which is why we'll do a problem with it. But it's non-steady state. And that means that the concentration that you're going to get is going to be a function of position and time. It's going to be a function of both. So now, I can answer the question. Remember before, I had to keep the concentrations fixed to apply Fick's first law. Now I can answer the question. Well, OK, if I have certain parameters, what is the concentration going to be. What is the concentration going to be at some time and some place? And that's exactly the question I have to answer. How much carbon am I going to have at what position in the case hardening? In what position of the pipe exterior, how much carbon am I going to have at what time? How long should I leave it in the oven? That's the question I'm trying to answer. And by the way, what temperature? And so the thing that Fick's second law helps us understand is, if I start with some C S that's the starting concentration, and I have a starting concentration C 0, and I expose at the surface some higher concentration-- remember, before, I kept these fixed. I kept these fixed. And I got a flux. Now, I'm going to be able to get this. So this line here is time equals 0. But now I can actually get this profile. Time 2, time 3, that's the position through some surface. So this is what Fick's second law tells us. It tells us these profiles, the concentration in the material. And it just simply is a time-dependent version of diffusion. OK, what we're going to do is we're going to pick up with this. We'll do this example of case hardening. We'll do a few more comments on the class. I'll tell you a little bit about the final. And please bring your laptops on Wednesday for the last 15 minutes. We'll reserve for you to do evaluations. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 11_Shapes_of_Molecules_and_VSEPR_Intro_to_SolidState_Chemistry.txt | Let's get started. We'll cut right to the chase. How did it go on the exam? I crushed that. I heard I crushed that. Yeah. Look at that. So you guys rocked it on exam one, and I'm very proud of you, and I think this is awesome. The average was actually really high, which is great. It means, to me, that you guys have learned the materials that we want you to learn in the first third of this class. You know, this is also a chance that if you didn't get up into this range, you're here or down here, or even if you're up here, and there are a couple of problems you didn't quite get, this is the time to go back and say, what did I miss? Right. How did I misinterpret? Or what did I miss, and go back over it now. Solidify this knowledge. Why? Because of what's coming next. And I really want to contextualize this with something really important to me. And that's bread. Now, if you want to get the best ride in town you go to Bricco. All right, and I really actually strongly encourage you all to go to Bricco, because it's true. It's really the best bread in town. There's Frank. He founded Bricco. It's in the South end. It's amazing. But you notice his ingredients, there's only seven ingredients in the bread at Bricco. There's salt, right, yeast, water. There. Anybody know what farina is? Flour. OK, crusca, it's like kind of wheat stuff. Oil and passione. Passione. Passion. It's the thing he's holding. That's ingredient number seven at Bricco, which is the best friend in town. Now, here's my point. I need you guys to keep bringing the passion to this class. You should be bringing the passion to everything you do, because that's what we do at MIT, right? That's how we approach everything we do. But I need you not to say, well, I did really well in exam one, and now I can kind of phone it in. That's not the time to do it. Trust me. Because now we're going into the next phase, the next third, a little bit less than a third of the class where we're going to come maybe out of some of our comfort zones. All right. So some of you have seen a lot of the material that we've already covered. Some of you haven't. And some of you may have seen some of the material we're about to cover in the next week or two, shapes of molecules and molecular orbital theory, hybridization. But then we're going to go into crystals, and we're going to take molecular orbitals, and we're going to make solid orbitals out of them, which are called bands, which is going to give us semiconductors. And then we're going to dope those with chemistry. All sorts of stuff. So keep up the passion, please. That's why I was showing you Frank and Bricco. Keep up the passion. OK, now today, we're going to cover a really important follow up to Lewis, which is, how to predict what shape a molecule will be. And so, you know, if I take, so for example, H2Be and H20, now those look-- right, OK, so, H Be H and H O H, right, from Lewis, we don't really know why they both looked that way. Are they both linear? Are the ones below both linear? How do we tell what shape they are? And as I'll tell you about today, so I'm going to give you a way to do that and then a recipe, another recipe, just like Lewis, and then, we're going to talk about the goody bag, which allows you to touch and feel the shapes. You can't-- we're going beyond 2D. We're coming out of the board. We're going 3D. You got to hold it in your hand. Now, unfortunately, we couldn't buy like 500 kits that are this big. But I'll pass these around, because you can just feel-- in these molecules, you can just feel that if I do this, it's a different molecule. If I rotate that, it's a different molecule. It's actually going to have different properties. And so that's what we learned today. Lewis, this is just all flattened into the board. Doesn't tell me enough. It doesn't tell-- I need to know what those-- is it like that? I don't know. Do the yellow things want to be that close, or should it maybe be like that or like that? You see how when you can do this? I could do this for hours honestly. So I'll pass these around. And I'm certain at some point we're going to hear loud clinking sounds of these balls falling apart. They're not held together too well. Please try to not-- just play with it a little bit. Right, and then your kit is a smaller version of that. OK, so how do we do this? How do I tell you what shape that should be? Well, we have a way to do this, and it's got a name even. So just to contrast, Lewis is what we did last. And Lewis gives me the numbers and type of bonds, OK. The number and type of bonds. But now, I've got a way that's called Valence-Shell Electron-Pair Repulsion model, VSEPR, VSEPR. But see, chemists are-- if I haven't conveyed this before, chemists are really good at naming things. So it could be valence electron shell pair repulsion model. It's not. But it could be. And if it weren't we'd call it VESPR. It would sound a lot better. Right, and so that's what we call it even though we write VSEPR. So we write it like this, but we say vesper, because this is chemistry. And this gives us the shape. Now, it's based on actually a very straightforward premise, which is that electrons repel each other, something we already knew. OK, electrons repel each other, and the stable arrangement-- I hear that molecule changing shape. It makes me happy. OK, the stable arrangement minimizes-- now, this makes sense right? Minimizes the repulsions. OK, that's really the whole premise of VSEPR. VSEPR theory is a way-- it's a very simple recipe that we will learn and apply that is based on this premise. We already talked about electrons repelling each other in atoms, right, and in a bond. And now we use the same idea that electrons repel each other for a molecule, right, within a molecule. OK, good. But we need to rank order this. We need to rank order this. And so we have a rank order. And you'll see kind of why, as we go. I'll tell it to you right now. And I need abbreviations, because I don't want to keep writing bonding pair and lone pair. So I'm going to say that a bonding pair, a bonding pair, oh what's a bonding pair? Well, it's the two electrons in a bond right there. That's a bonding pair. OK, good. So a bonding pair, to save time, we're going to write as a BP. And a lone pair, what's a long pair? Well, we already know that. A lone pair is right there. A non-bonding pair, a lone pair, an LP. OK, good. Now, this allows me, now that I have this very important key, I can tell you what the repulsion order is without writing out bonding and lone all the time. Repulsion order and this is what we follow in VSEPR. OK, so we're going to go the lowest repulsion is between two bonding pairs, bonding pair to bonding pair. And the medium medium, OK, so the next in our list is bonding pair to lone pair, and the highest is between two lone pairs. All I'm telling you is in this ordering is that OK, electrons repel each other, yeah, got it, but there's an ordering to it. If I'm a lone pair, and I see another lone pair, that I'm more repelled. Those two things are more repelled, then a lone pair and a bonding pair, which are more repelled than two bonding pairs, and we got one more thing we got to think about, because you see the bonding pairs can be single, double, or triple. And so those have a rank order, which also makes a lot of sense, right. So like one bonding pair would have a single bond. With a single bond would have a lower repulsion than a double bond, which would have less repulsion than a triple bond, right? But that makes a lot of sense, right? You're putting more electrons in the bond. And so there's more stuff to repel. Right, so if I'm talking about bonding pairs, there is also a sub order. OK, that makes sense, right? So this is like a single bond line, two lines, no room, three lines, right? All right. Now this frames the VSEPR model. Now let's apply it, and the rules are actually really pretty straightforward. So what we're going to do is we're going to build our understanding by applying this to some examples, because it's exactly the same thing we did with Lewis. OK, so I'm going to start my recipe with the first three ingredients. Passion is always there, too. But these are the first three ingredients of my VSEPR recipe. First, I write the structure. We know how to do that. We learn how to do that. And then I'm going to classify the pairs of electrons as bonding or non-bonding. So I'm going to just label them like that. And then I'm going to maximize the separation between domains and pay attention to these rules, and I'm going to start a little bit of a table over here, number-- because we're going to fill this table out-- number of electron regions. OK, let's do this. You know when I do this, there's nothing but fun coming. Nothing but fun. I went all the way, all the way over. Why? Because I need some space. Now, here, we're going to say electron pair geometry. We'll be talking about this today, geometry. OK, good. And then, here, OK, now this is sort of the molecular-- molecular geometry. So I'm going from-- I'm going from counting electron regions and labeling what kind they are to a geometry of the electron cloud around things to an actual molecular shape, which is what VSEPR gives us, and to know why is this so wide, well, because there's options in here. Right, because I could have no lone pairs. All right. I could have one lone pair. And this will all make a lot of sense. Two lone pairs and so on. We'll keep going as we get there. OK, so I'm going to start with a very simple case. I'm going to start with this. Right. And then we're going to put some stuff into that table as we go. All right. So look if I have H, Be H, then that's already 0.1. I'm going to write, just like I did with Lewis, I'm going to write the number of the recipe next to what I'm doing on the board. OK, so one, write Lewis structure. I did it. OK, good. Now two, I've got two electron pairs. OK, and they're both bonding. Both are BP. OK, and so I've got two BP domains. And as you can see, I've got no LP domains. And so three, this is going to be max separation, max sep will have to be linear. There's just no other way about it. I've only got two electron domains. I've got no lone pairs. So two bonding domains, two electron domains. It's got to be linear. That's the only way I can go. So if the number of electron regions is two, which it is for BeH2, OK, and and then the number of-- the electron pair geometry is linear, right, because that maximizes the separations. And there is no lone pairs, and so it's linear. You say, well, that really seems redundant. It is in this case. It's not going to stay that way, not going to stay that way. OK. Good. One lone pair, no. No lone pairs. I mean, if I had two electron regions total, OK, and one of them was a lone pair, it's kind of boring. I mean, you know, if this were a lone pair here, instead of a bond, well, clearly it's linear. There's only two atoms. That's kind of the simplest case. We won't really talk about that. Good. OK, now it's the same kind of thing. Let me just make sure that there's no confusion here. And let's not leave this up. But if I had-- let's say I go back to my first slide there. OK, so what if I had this one. It looks more complicated. Right, it looks more complicated. So now, OK, so OK, step one O and here you go, here you go. And there's a point I want to make here, because if I count up the bonding pair domains and the lone pair domains on this one, all right, say well, OK, I've got the bonding pairs here. I've got some BPs. I've got two BP domains right? Here and here, just like I did in H Be H. But oh, I've got all these lone pairs now. Don't I have four lone pairs? No. And the reason is we have to pick an atom. In VSEPR, we have to pick an atom. We're doing this around an atom. VSEPR applies around an atom. You pick a central, and I picked carbon as my central atom. And then you apply VSEPR. So there's no lone pairs around the carbon. That doesn't matter for the shape around carbon. OK, that's really important. So I pick my central atom. So now I know there's no lone pairs, no LP, because central atom. You know what I mean, central atom. And then finally, three, we're going to get linear because, again, there's only two electron domains. There's going to electron domains, and the only geometry really that you can take in this case is linear. No lone pairs, two electron domains. Good. And now you know it's about to get fun, because now we're going to go to something more. Let's go back to our recipe. There it is. OK, so I'm going to make room here and do my next one. I'm going to go a little bit more complicated. So now I've got BF3. So B, and we're going to go F, F, F. And we know we got all these lone pairs out on the F's. Remember this is one that's electron deficient. We talked about this last week, electron deficient. So the boron is happy even though it's only got those three bonds. The question is, what's the shape? We've been drawing these structures like this in 2D. What's the real shape? Now we can apply VSEPR because that's step one. Step two is that I've got three bonding pairs, 3 BP, right, and no LP. OK. And so now if I have three electron domains, three electron domains, which I have, then the max separation for this case, max sep is going to be for them to spread out in a trigonal plane. OK, so that's what they're going to do. And that is called trigonal planar. It has a name trigonal planar. So you know, it's kind of-- planar. So it's kind of right, but it's not quite right because I drew with 90 degree angles, and that's not what they are going to do to maximize their spacing here. They're going to go at angles like that. All right, they're going to find 120. So if I have three electron regions, here we go, then-- now I had three-- OK, so the electron pair geometry. Why is this different? You will see in a minute. The electron pair geometry covers BPs and LPs, but in this case, there's only BPs. Fine. There's still only three. Right, and so it's trigonal planar. Trigonal planar. And guess what, if there is no LPs-- OK, so the structure that the molecule takes or those electron domains takes is trigonal planar. The structure that the molecule takes if there is no LPs is also trigonal planar. But now, the moment we've been waiting for is what happens when that's not true, and you've got a lone pair. So again, we're going to do this by example. OK, and so my next example is formaldehyde, which is something we love to talk about when we did Lewis. So CH2O. OK, so the Lewis structure for this looks like this. Right, we drew this before, O, it's got the lone pairs out here, and then there's H and another H. That's formaldehyde. OK, now hold on. But there's no-- are there any lone pairs here? There's no lone pairs. There's no lone pairs. So hold on. If I go to two, I've got three bonding pairs and zero lone pairs. It looks a whole lot like BF3, but it's different. It's different, because now this matters. Now this matters. Right, and so now-- so now, I've got to add-- so it's like-- so 3 would give you trigonal planar. Right, 3 gives you trigonal planar, because I've only got those three domains, but now I need four. And there's four. You knew there was something else, right? Give more space to non-bonding domains and to bonding domains with higher bond order. That's the fourth part of the recipe. This just goes with what I wrote here. Right, so I've got differences in the molecule between the BPs. So you know now that I got to give more space to this double bond single bond repulsion than to these two single bonds. All right. So step four for this case is that the bond order is important. And this molecule will bend. So the shape of the molecule is going to be bent. It will bend, because this-- I didn't draw it this way. If I wanted to be right about it, I'm going to go like this, still not quite right. Right, and this repulsion is stronger than that repulsion. It bends the shape. So it's not trigonal planar. The electron domains give you a trigonal planar framework, but the molecule itself is bent. OK. And there's another example of that that we could do. There's another example of that, and that is-- let's do it here. And that is if we had the lone pair, which is what I mentioned before. So I'm going to take another example here, which is SO2. And in this case, uh-oh. Am I making a mistake? I can hear-- I can hear stuff. Did I make a mistake? No. I see this. I did. I do, and that's wrong. Bond order is important, but it is not under one lone pair. It is not under one lone pair. It would be under here. And it would be-- if there is a bond order, it would be best if bond order. Sorry about that. Bond O. Trigonal planar, trigonal planar, slightly bent. Look at this. I should have just kept this. Bent, bent, because now, I've got-- look at this. In this Lewis structure, I've got two BP and one LP. And finally, we have the case of the lone pair, and we can fill this column in, and it's bent. And it's bent because, again, it's the same. I go back to this as my key. OK, so I did bone pair, bond pair, but there was a slight shape difference because of the ordering of the bond pairs. Here, it's going to be even higher because I've got a lone pair. Actually it's a lone pair double bond, but it's still stronger than a bond pair bond pair repulsion. Lone pair bond pair, right in the middle is stronger than bond pair bond pair. That's the rank order. So this will be bent. This will be bent. So I've got three domains, and hold on, four LP more repulsive. Oh boy, repulsive, and so that gives me bent. Trigonal planar, trigonal planar, bent. OK. Did I get through that without making more mistakes? So we're going to go further than this, which is why this is only the beginning. But before I do, why does shape matter? I told you shape mattered in the very beginning, but why does it matter? Isn't it just important to know what the chemistry is and not the actual shape? Why does the shape matter? And so I thought I would give you an example of that with smell. And yeah, right? Exactly. And actually, I think we should do this more often. You know, I feel like this is inspiring. We talk about stopping and smelling the flowers, but do we actually do it? And look at that. They've got one arm around the other. This is a moment. You can't share this moment on Instagram. You have to put your phone down and be there to have this moment. And I highly encourage you to be inspired by this. Now, smell and taste are actually related. And smell is, you know, it's actually a fascinating thing. We can smell about 10,000 different smells. It's remarkable. A dog can smell between 10 and 100,000 times more. Right, so if you do the analogy that people do with vision, we can see a third of a mile. A dog can see about 3,000 miles. Right, that's pretty cool. But the question is why do we smell? How do we smell? How do we taste? And it turns out that the way that we distinguish from one smell and one taste and another has to do with the shape of the molecule. And so just like a key right, with a key, it's all of the same material. It's the same chemistry in the key. But the shape is different that can unlock the door or not. That is literally how our receptor cells work for taste and smell. This is a cartoon I found that I kind of like. See, OK, shapes, circles, squares, right. OK, and some shapes come in, and they make happiness, happiness, flowers and stuff. I don't know. Is cheese happy? It looks neutral. I would get happy with cheese. But anyway, fish for some reason, but you know, it depends. But it's shape dependent. Now the thing is, taste works that way, too. So this is what's happening in your tongue. You've got these taste buds. We've all heard the word taste bud, right? But what is really going on? What is really going on is a combination of chemistry and shape recognition. So the taste bud, if you look at the taste bud, that's inside of the little pores inside your tongue. There's a blow up of it. And so this is what the surface of your tongue looks like. And there's a little pore. And there's little filters in there that helps certain molecules kind of get in there. And what happens when they get in there. Well, what happens is you've got these taste receptor cells that are like lock key pairs. They actually only look at like the circle or the diamond, and they can tell you which one is which. And that's a major part of how we distinguish from one shape to another, and it's actually so-- it goes back a long way. Why? Because shape and smell are literally survival. They are literally like you can-- if you taste something, and it tastes poisonous, don't eat it. You live. Right? So it is a very emotional thing to smell and taste, because it is actually coupled to your very survival. And Democritus himself, Democritus, our friend Democritus, said that shape must be involved. He thought that because things that taste bitter are sharp that the bitter molecules were sharp. They must have sharpness to them, like shards of glass. That's how he imagined. And the sweet molecules were sort of the soft, fluffy spheres. That's not that far off from being sort of what happens a little bit. But when we look at it like, you know, glucose and quinine, we say, OK, those are very different chemistries. So it might not be as obvious why one of them taste so different than the other. But check out this example. This is the same molecule, the carvone molecule that is called an enantiomer, which means that it has handedness. It's the exact same chemical formula and the exact same structure, except for one is like this, and one is like that. That's handedness. Right. And that difference makes one of them taste and smell like spearmint and the other like caraway seeds, right? It's incredible. It means that in our tongues in our noses we must have chiral, the ability to determine the chirality. It's pretty cool. Right. So shape is critical, and this is one example of why. This is one example of why. OK. Back to my VSEPR recipe. Now, we've got to go a little further, because this is three electron domains, three electron domains. They can be messed around. They can be messed around. Right, especially in this table, we're talking about lone pairs or not. OK, I gave you the example, which I kind of fudged in there with bond order. But what happens if I go to four? So if I go to four, let's do an example with four-- if I go to four, I need some room here. Why don't I do it in the center here? Let's do an example of all possibilities with four, with four domains. And there's three really good examples, right? And you know them. Let's do CH4. So we'll do these kind of more quickly, CH4-- gesundheit-- which will be CH-- never leave enough room on top. And H. There it is. OK. Anyway. And NH3. OK, so N H H H. And H20 to all O H H. This is how I drew it on the first slide. Or this is how I drew it up there before. Now the thing is that in each of these cases, what I want you to see in these three examples is their similarity first. And their similarity is that if I pick my central atom, carbon, nitrogen, oxygen, if I pick my central atom, and I look around it, each one has four electron domains. Right. So in this sense, the kind of overarching electron geometry is identical for all three of those cases. It's identical. And it's tetrahedral, tetrahedral. OK, that's the electron pair geometry that's exactly the same. But I need to now count-- oh, there's the recipe. I need to now classify the electron pair as bonding or non-bonding and then maximize the separation between domains while giving more space to non-bonding domains and bonding domains with higher bond order as we have now talked about. So if I do that, here, what I have is four bonding pairs, zero, zero lone pairs. Here, I've got three bonding pairs, one lone pair. And now I really get to fill in stuff in my table, because now, I've got the examples of zero, one, and two lone pairs for the same electron domain cloud. OK, so there is zero, one, and two. You can see in methane, OK, the only way for this to maximize its repulsion, they're all the same bonding pair, they're all single bonds, so there's not going to be any kind of funny business of a double bond pushing harder, and there's no lone pairs. And so this is just going to have the tetrahedral shape. That will be the shape of the molecule. That is the electron domain shape as well. So in this case, it will be tetrahedral, tetrahedral. OK, but see, now, in this case, I've got three bonding pairs and one lone pair, and we know because of the VSEPR recipe, which has-- oh, it's up there. I keep forgetting it's up there. OK we know that I want to maximize the separation, but I got to give more space to the non-bonding, because the lone pair bond repulsion is stronger. Good. So what that means is it's going to be sort of like tetrahedral, but in one place, there is no bond. There's just this lone pair pushing down. And so the shape that you're going to get is called trigonal, trigonal pyramidal. OK, because now, that's just the way that shape looks. So this is going to-- I'll show you actually a picture of this in a second, OK, of what this looks like. And then with water, we now know that this will not be the most stable configuration, because these lone pairs are pushing on these bonds too much, so water is actually going to look like this, where these guys, the lone pairs can come out like that, right, that angles like that, be away from each other, right, and then the bonds are down there, and that's going to maximize the shape of the water-- I mean, it's going to minimize the repulsions that the electrons feel in the water. And that is back to the same name, which is bent. I've got a different number of electron domains, but I wind up with the same molecular shape. See that? Bent. Now, I know this is kind of small. I've got tables that have all this at the end that you'll have in the slides. OK, so I know this is a little bit crammed in here, but the process of writing this is very, very fun and fulfilling. Now, we can keep going. But I want to tell you about a labeling system that's actually really helpful. So if we keep going, we kind of want a system to be able to think about these things. And so there is a system. Again, this is chemistry. We know how to name things. Oh, but I promised you I'd show you that first. So here it is. There it is. I didn't know why. I click to that already. That's this. And the reason I wanted to show you this, this is from [? Avril, ?] your textbook that you're all reading so carefully. And look at that. This really brings it home. Look at that lone pair. Now you're seeing it as the electrons feel it, which is a probability distribution cloud. That's the lone pair, and now you can see the lone pair wants space. It wants room, and it's pushing down on these bonds, which are pushing away from each other. And that's how you can figure out what shape it is, trigonal pyramidal. That's trigonal pyramidal, OK, the molecule. Right, as I said, we need a way to label things in chemistry. So we can keep going-- linear, trigonal planar, tetrahedral, trigonal bipyramidal, oh, five, octahedral, six. Don't get confused octa, eight, but octahedral is for six electron pair domains, right. But we need a system to name this. And so the chemists have brilliantly come up with AXE. Now A is the central atom. X is the bonding pair regions, and E is the lone pair regions. X and E. OK, and so if I go back to like BF3, well, this would be-- oh there you go, AX3. I say, what is the shape of AX3? I know I've got to central atom, A, and I've got three bonding pairs around it and no lone pairs. That's what that means. I could go back as SO2. Where is SO2? Did I erase it? Maybe I erased SO2. I did erase SO2. Oh, I didn't erase SO2. Happiness. Look at this. SO2 would be AX2E. Or if you want, you can put a 1 there. It's a highly sophisticated, advanced labeling system. That simply allows us to keep going, to keep going without writing so many words. But we know now, A is the central atom. X is the number of bonding pair regions. E is the number of lone pair regions. And we know about our rules, and we know about bond order. So now if I look at those same-- really? What happened to-- oh, there they are up there. If I look at those same ones, well, OK, look those are tetrahedral. OK, the electron pair distribution is tetrahedral, but-- gesundheit. But as you can see in these pictures, you can get three different molecular shapes. You can get tetrahedral, same as the electron pair. You can get trigonal pyramidal, or you can get bent. Those are the names of the molecule. Notice that those aren't the names that include the lone pairs. Lone pairs are how you find which one of these categories you're in. But those are the names of the molecules. Those are really important, those names of the molecular shape. Now, if I move over one-- OK, oh there they are, names, tetrahedral, trigonal, pyramidal, bent. Now, let's move over one. So now I'm going to stop. I'm not going to keep on writing everything down. Let's see one last example. So here's trigonal bipyramidal. This is what happens when you have five electron pair domains. Again, and I keep saying this, but I know that this can lead to confusion later. So I keep on reiterating, the electron payer domain count, right, the five here, the five here, which gives us this overall kind of electron payer domain shape, that would be five there, right, trigonal bipyramidal, that includes a single, double, or triple bond is a domain. It's a BP. Right, it's a BP domain. A lone pair is a domain. How many of those do you have floating around your molecule, around your atom? Sorry, your central atom. And then you go down and you sort of decompose it in terms of how many of them are bonding pairs, X's, versus how many of them are lone pairs. So let's do one last example. And it'll be for five. It'll be for five, right, so five. And I want you to really-- I'll go with the center again. I want you to really kind of get a feeling for why this works. And this is, I think, a nice example. If you had the molecule, if you had the molecule SF4, I'll give you two examples today that violate the octet rule, right, because chemistry lives in the fast lane. If you have SF4, S is willing to have an expanded octet, all right. As you go down in the periodic table, atoms are more willing to live in the fast lane and break the rules. S-- we've drawn it like this, F, F, F, all right, we've drawn it like this, violating the octet rule and all that, but still being happy because it's lowest in energy. And these are all my electron non-binding domains. OK. If I count-- remember, those aren't going to matter. They're on the outside. I'm picking one atom. A is S. Right, A is S in this case. And in this case, I've got-- you know, I've got, OK, how many body pairs? I've got four bonding pairs and one lone pair, and one lone pair. And so you can see-- I can just look this up here AX4E1. It must be a seesaw structure. But I want you to see it, and this is where playing with these models really helps you see it. Because if you think about this, it seems, at first, that if I had four bonding pairs and one lone pair, why can't I just write it like this? This seems like a good idea at first, right? S, and then you've got a fluorine down here. And then you're going to go out. I'm using a kind of cool new notation here. Right-- and then you're going to go. Did somebody just say whoa? Or is that just in my mind? Like that, but I'm saying whoa. That's a terrible rendition of this notation. But this means-- now I've got to come on the board. And for me, that's challenging because I'm not very good at drawing 3D things. And so we use these little shaded in kind of sticks like that, and it comes out of the board, and then that's like the dashed one, and it goes into the board. So there's one-- that looks terrible. I'm going to try one more time. So maybe if I just do this, F, F. And this is in the same plane here. These three are in a plane, and that's going down. And you've got your lone pair up here. That looks pretty good. Why is that better than something like this? And now, one coming out of the board and the other going into the board, all right, and now this would be-- gesundheit. This would be my other possibility. I mean, if I think about, I think why can't it just be this? These look-- I don't know. These look nice and spread out. Why not? It's the lone pair again. It's the lone pair. It's got to be, because, you know, I put up there, somewhere, there's not there, that the lone pair repels the bonds more, and I've got more than say a bond and a bond. But the point is I've got three of those here. This has three, one, two, and three. This has two. This is a happier structure. Right. So I wanted you to see-- so this is the way to think about it. You're using VSEPR. You're applying these principles. I mean, it's just a four-point recipe, but I'm hoping that in thinking about it this way and having the little models in your hand, which you have in this goody bag, that you guys will also kind of have a feeling for it, so that yes, you have the tables that are written out much better than how I wrote them out. And here's one in the lecture notes that I'll leave you, right, total domains. It's very similar, notation, structures, OK, shapes. But also that you have a feeling for why this is the case. All right, shapes of molecules. See you guys on Wednesday-- no, Friday. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 15_Semiconductors_Intro_to_SolidState_Chemistry.txt | We are going to be leaving the territory of-- some of you have had high school chemistry. We're going to be starting to leave that territory and get into some of the things I've talked about before, that you may have never seen. So for example, today we'll be talking about semiconductors and what happens when you take all these atomic orbitals and you put them into a solid. And then, by the end of the week, we'll talk about metals. And then next week, we're going to talk about crystals. So that's what's coming. OK, now there's one type of bonding-- Before we get into semiconductors and solids, there's one type of bonding we covered but I didn't put the stats up on the column. So I just wanted to kind of make sure that we finish that, and that was hydrogen bonding. And there's a little video-- you know, water, we're not going to go into all the amazing properties of water. But if you're interested, this is a really cool website that talks about all the anomalies of water. And many of water's properties are anomalous. And much of the reason for that is the last type of bonding we talked about, intermolecular bonds we talked about last Friday, which is the hydrogen bond. So remember our table, I just wanted to fill out the table because I didn't fill out the table, right. That's the type of bond. And the model was some kind of electronegative-- it was some kind of highly electronegative atom. Like a was oxygen or nitrogen or fluorine. So that you drew a lot of charge off of that hydrogen and you left it with a really strong delta plus so that it could then bond to some other atom. It might be the same, by the way. It was in the cases that we talked about. Actually, I'll just use the same atom because we use HF and H2O, so then it would be the O or the F. But it's to a lone pair on that atom. That was the hydrogen bond. And the range of attraction is in kilojoules per mole of bonds is something like, let's see, 10 to 40. So it's pretty strong, you know, compared to some of the weaker things like London, the hydrogen bond can be pretty strong. And one of the examples that is the classic example would be Hydrogen, hydrogen, there's the hydrogen bond right there, you see? Hydrogen bond between another oxygen with its hydrogens as well. Now, we talked about how water is so special because in the case of water, not only do I have exactly two hydrogens in each molecule that have caused this strong delta plus, but I've also got two lone pairs. So the numbers work. In HF, that's also a hydrogen bonded-- it can hydrogen bond to itself, but I've got three lone pairs on each F and only one hydrogen that I've taken that charge away from. So you still get the hydrogen bonding, but this is the beautiful balance. And there's a video showing water bonded to each other. Oh, there it is. Because there's nothing like just imagining what a hydrogen bond looks like. And so of course, a hydrogen bond will-- OK, charges. Oh, it's French. [SPEAKING FRENCH] OK. Oh, that's what it looks like. Really, if you could get down there, it would be a ray of a laser beam light. It's a liaison hydrogene. And there it is. And this is what's going on in the liquid. They're kind of seeing each other. And they're talking. But they feel this strong, strong hydrogen bond. And that's happening all over the place. And in a liquid, they are making and breaking all the time. Question? STUDENT: Why do they separate? Yeah. Yeah, because-- well, it depends. It depends. It's a great question. Why do they separate? Well, they don't in ice. So in ice, I mean, you know, they can. But in ice, it's a solid, so they're sort of locked in. And so in there, you're going to have pretty much each water molecule has four-- look at this. It's you know from VESPR, right? Four possible bonds. You can make tetrahedral. But there's actually lots of ways you can orient them in a solid. There's many forms of ice, 12 different ways to make a solid of ice. But see, here-- gesundheit. So here in the liquid at room temperature, with the laser beams, they're coming in and they're bonding, but they have enough energy to break the bond, too. So they stick and they unstick because there's enough thermal energy in the system to continuously break them and remake them. That's when it's a liquid. When is it a liquid versus a gas? Right, the boiling point, remember, that is a proxy we talked about last Friday. That is a proxy for the bonding strength between these molecules. If it's a stronger bonding, then the boiling point will be higher and higher. Oh, water should have boiled at a much lower temperature if it didn't have this very special hydrogen bond, which is not a dipole-dipole. People try to write the physics of the hydrogen bond as 1 over r cubed. That is not true. That is not just as simple as, it's another dipole-dipole type interaction. It's complicated. Complicated. Remember, we had covalent bonds were complicated too. So we're not going to put a simple one over r something dependence for the hydrogen bond. OK, good question. OK now, all right, so that's an intermolecular-- now we're going to move on to solids. So that was our last intermolecular interaction. Oh, before I do, there's actually an example question I put there that you could now answer, which this is actually-- I just want to show you the type of-- so we covered all these intermolecular bonds. Here's a question from a quiz last year. List the intermolecular forces in each substance, methane, and then you've got that CH3OH. And then you've got the CH3CL. And so you can do this now because for methane all you've got is London. It's not a polar molecule. All you've got is London. There's no hydrogen bond. But for CH3OH, do you have hydrogen bonding? Yeah, you do. Because if you draw-- because you've got an H that's had the charge taken from it by oxygen, a highly electronegative atom, and then and then you've got lone pairs on that same oxygen. So you can hydrogen bond and you've also got London. All right, now is that it? Is there anything else? CH3OH. What else? Is there any other type of bond? Well, the last question you need to ask is, is there a dipole moment in the molecule? Because if that molecule had a permanent dipole, then there is another type of intermolecular interaction, like we talked about, dipole-dipole. So you'd have three. And for CH3CO, well there, you also-- you know that for CH3CO, you're going to have a dipole moment. Remember, we compared dipole moments of molecules. And if you have a diaper moment in a molecule, then that dipole moment can talk to another dipole moment and be interacting with it. So that's a kind of bond. And is that it? London, London is always there. It's always there. We've also got London, don't forget. So you'd have dipole-dipole plus London here. Dipole-dipole London and hydrogen, just London. So this is the kind of way I want-- and you might be able to think about, well, would a trend-- if you think about, like, OK well, if a molecule just has London, then what does London depend on? All right, well, it depends on the polarizability of the charge. So that can be something [INAUDIBLE].. It depends on the size because that tells you how much-- if I say, well, what about SIH4 compared to CH4? Which is going to be more strongly bonded? You can answer that because you know the only force there is London. And you know for London, it's how much contact area you have is going to be one of the factors. The more contact area, the more chances these things can get close enough and polarize. And then polarizability would be the other. All right, now we're going on to solids. Now, the big picture is pretty awesome. If you look at the big picture, we have done the electronic structure of atoms and molecules. So we've gone through and we've worked our way through the Bohr atom, the atomic model, quantum numbers, Aufbau, what happens when you go from one electron to many, and octet stability, which is Lewis and Vesper. We have also now talked about how molecules come together in different ways, how the atoms come together in the molecule, and how the molecules come together in different ways. Dipole-dipole, London dispersion, hydrogen bonding. And then there's these forms of aggregation, gas, liquid, and solid. And of course, we're excited about this. So I put my little emoji there, because that's what we're going to do now. We're actually going to take-- we talked a little bit about solids already, but now we're going to take the electronic structure that we've built up, these orbitals. And we're going to make solids out of them. So this is the week where these are the next two weeks, right, where we're going to talk a lot about solids. And this week, this week we're going to do the electronic structure. And next week, we're going to do atoms. So that's our plan. And next week Friday, we're going to break a bunch of stuff with parents. So this week we're [INAUDIBLE]. What do electrons do in solids? Next week, how do atoms arrange in solids? And together, that gives us so much understanding of how chemistry effects properties in solids. So that's the big picture. Now, here's the thing, why does that matter? Well, because so many properties can vary by so much depending on what I just said. How does chemistry affect properties in solids? Well, it's complicated, but look at this variation. 28 orders of magnitude of connectivity, electrical conductivity. 28 orders of magnitude, that's a lot of zeros. [INAUDIBLE] These are the electrical conductivities of different atoms, different materials. And you can see that it's going to depend on what you have, but it's also going to depend on-- so that's the chemistry. But it's also going to depend on the bonding. It's going to depend on how the electronic structure comes together. And it's going to depend on how the atoms come together. That is what will explain 28 orders of magnitude. That is a lot of variation. So we start by thinking about how electrons see these solids. That's how we're going to start. Now, the thing is that these good old days of molecular orbital theory are going to end. They're going to have to end today. No, they're not ending for like quizzes and exams. Don't forget about them. But we've got to move on because this is just two atoms, two atoms. But what happens when we have like a solid? How many atoms do we have? We've got a lot. We've got a lot. So let's go-- we'll go here. OK, now, if you think about that picture, that was our-- I'm going to take 1s orbitals. And let's just think about this for a second. If we take 1s orbitals and you have two atoms that come together, well, they're going to do just what you see there where you've got your 1s and your 1s. I'm not even going to fill these out, and they're sigma and sigma star. And so that would be like two 1s orbitals that I have either-- OK, so up here, I took one. It was a positive and I subtracted one. And here, I took one and I added one. And remember, this gave us this and this. And this gave us the bonding orbital, right? Bonding, anti-bonding. It all came from LCAO. LCAO is the linear combination of atomic orbitals. That's the theory we use to build up our [INAUDIBLE] theory. But now, we've got to go a lot further. So I'm going to-- because I don't want to keep adding things here and here, it gets complicated. I'm trying to find the MOs for more, and more, and more atoms because I've got to put a whole lot into my solid. So I'm going to write them just on one side. If I write the two here like that, and that would be sigma and sigma star, then you know what I'm talking about. I'm taking the two and I'm combining them in two different ways. But what if I had three? Well see, if I had three-- so I've got three s orbitals now. So if I've got three s orbitals, then if you think about it, I could also add them like that. That would be a very nice bonding orbital. If you add them, you see that they're going to have the same kind of really nice bonding. But I can also subtract one in the middle. And if I did that-- so I had a plus, minus, plus there. So those are the signs, plus s, minus s, plus s. Then you can see that in-between those, as I add these together, you have nodes. So this is not going to look like a very happy bonding situation. I'm going to have something in there and then something in there. That's what that's going to look like as I bring them closer and try to add them. I made two nodes here. Well, there's one other way you could do this. And we know that the number of molecular orbitals has to equal the number of atomic orbitals. We know that. So there's one more that we mean, because I started with three s's. And you can kind of see that. You can see that because if I had a plus and a plus, if I had done it that way or maybe the other way, maybe this was over there, but you see I got a little bonding going on in here between them. And I got an anti-bonding there. It's almost like it kind of cancels out. So you can imagine that that combination of s orbitals, if I add and subtract them this way, it's going to be somewhere right in the middle. Maybe not so bonding, maybe not so anti-bonding. OK, good. All right, well let's do four. So if I had four, if I started with four-- this is like four hydrogen atoms, let's say, although I'm not filling it yet. But so four 1s orbitals. Well now you see, I could do this. I can add them all up and get this super awesome bonding state. I could also alternate them so that everything is anti-bonding. Look at that. These are going to cancel as you bring them together and you're going to get anti-bonding, anti-bonding. But there's also two other options that are in between. Right here, I can add 2 and then subtract 2. And you can see, these are both negative. And as they come together, they're going to overlap constructively, so will these. But in the middle here, they're going to be kind of destructive interference. And so you're going to have this one node in between. But in general, you can imagine, this will be more bonding than anti-bonding, so we draw it down below the middle. Whereas up here, you could also imagine something like this, where you start with a plus and you put two minuses in there and a plus over there. I'm just adding and subtracting. I'm just taking LCAO theory and moving it to more than two orbitals. OK, so now you can actually understand this from a number of nodes, right? This is like three nodes. This is two nodes over there. This is one node. And zero nodes. This is exactly the kind of thing we talked about already, but for only two orbitals. Now we're just continuing it along. OK, you can see why the good old days are over, because see, now we've got to jump this up a level. And we've got to think, how many s orbitals do I have in a typical piece of matter? How many do you think? Take a guess. Roughly? Take a chunk of hydrogen this big. How many atoms do I have? A mole, exactly. That's why we invented the mole, so that we could count big numbers by just saying 1, 1 mole. But if I have a mole, then I've got like 10 to the 24th of these. I just went to four. I just went to four. But it just keeps on going. So [INAUDIBLE]. So here is like, OK, 1, orbital of atom. Boy, I labeled that well. 2, bonding, anti-bonding orbitals of molecules. And then you see, OK, there's 3. There's 4, 5. Then you keep on going. OK, they got a little tired and then they wrote 30, 30 orbitals. 30 is still way, way far away. But if you had 30s orbitals, you might still be able to kind of see these lines discretely. Remember, this is energy space. There might actually-- remember, like, these energies are really important, the lines between the deltas. But if you have a mole, if you have a mole, it looks like one continuous set of lines. These don't really look discrete anymore. These don't look discrete. And so if you have a mole of atoms, so if you have a solid, we call that collection of orbitals that are all stuck in here a band. We call that a band. OK, so if I take-- Now, I'm not going to draw 10 to the 24th. 10 to the 24th s orbitals. But pretend, pretend that that's what I have here. Then what happens? What I do is I form a continuum of states, a continuum of levels for those electrons. And this, they're so packed together, we don't even draw them as lines. We can't. So this is called the 1s band. Oh, band. It's a band. It's a band of states. It's a band of states. It's the 1s band. Now, there are some important rules that are really just the same but we follow them here too. And so I want to emphasize them. We've already learned them for molecules. They holds for bands and solids. All right, so now, for example, let's think about this. Each of these orbitals could occupy how many electrons? Two. So I have two electrons for each s orbital, s orb. Well, you know, if you think about that, then for n atoms, and here what did I say, 10 to the 24. Because there is one s orbital that comes from each atom. So for n atoms, then I would have 2n electrons can go into the one s band. So if I have-- that makes sense, right? Because I had two for each orbital, and I had that many. So if I know how many atoms I have that make up this solid, then that's how many electrons that band can have in it. All we did is go from the atomic orbital picture to the solid to the mole. OK, why am I saying that? Because filling is incredibly important And it happens in the same way-- just like it was so important for molecular orbitals, it's also critical for bands. So filling is like MOs. It's just like in MOs, start from lowest energy. Then fill up. So you know, if you think about this, if I had-- now I'm going to actually say, well, OK, I made the one s band by combining these orbitals. Now I'm going to see what I have, which tells me how to fill it. Now I have hydrogen. So if this is the one s band and I have hydrogen, hydrogen has, for each hydrogen, I've got one electron. But I just said that in the one band, for each atom I could have put two electrons. So you know if this is made of hydrogen, it's half filled, which is exactly what that picture shows. That's the filling. You see it's filled half way. The band is filled halfway. It's like the MO picture, but it's not because I've made a continuous band out of it. So instead of putting lines here, lines there, and then occupying them with arrows, I fill them up with 10 to the 24-ish electrons. So we fill bands up like that. And we know that it's half filled because it's hydrogen. Another thing that we know is that the energy-- remember this? This is also important. That energy is related to interaction and overlap. This is something we talked about, interaction and overlap of the orbitals. How much does that band go up and down in energy? This is something we've talked about. How much that band goes up and in energy, it's just like in the H2 molecule. One s, one s, sigma, sigma, star. How far apart are sigma star and energy? Well, it depends on the interaction strength and the overlap between those orbitals. The same is true in solids. Same is true in solids. So what is the width of that band? How spread out over energy is it? OK, so those are some ground rules. Now, what else can we say? OK, so there's the absolute least bonding state, anti-bonding, all the way up there, the most bonding. And then you've got almost infinite number of possibilities in between. But the thing is that filling determines the properties. You can't just know the bands or the band structure, as it's called. But you have to know the filling. And when you have a situation like this-- so let's write this down. When you have partial filling, partially filled band is a metal. And I will talk about metals on Friday. So we will come back to that concept and the electronic structure of metals on Friday. But today, I want to talk about something else that happens, which is when you fill not in the middle of a band, not in the middle, but maybe all the way up to the top. And there's a gap. That's a different kind of material. So if filling stops somewhere partially in the band, in the middle of a band, if you're filling, you're counting, and you didn't go all the way up, then you have a metal. But the thing is, here we've only talked about one kind of band. This is the s band. But every single one of these-- look at this, I can take a mole of orbitals and make an s band, like that. And then, oh, that was 1s. Here's 2s. And look at this, I can take the p electrons and make a p band. 2p band. I can do this with every single-- 2s, 1s, right? So depending on a bunch of factors, these bands could look different. What does it depend on? So this is-- the way these bands are organized, right, we know now that these orbitals will come together like I've drawn there and there and form bands. What is the difference between these? Did they overlap, maybe? Are they very far in energy? These things depend on atom-- depends on the atom and the bonding and the structure. How do those bands look? Well, it depends. And we'll be talking about some examples. But for now, I want to focus on that filling thing again. What happens if I-- if I fill this 1s up to half way, it's a metal. If I keep filling, keep adding electrons, and I fill this up halfway, it's a metal. But if I stop filling, you know, if I fill this and then I fill this, and then there's some space in energy and I didn't fill any more, that's not a metal. That's not a metal. So if there's a gap-- so there can be gaps between bands, which I just showed you. That's between an s and a p band. And then the filling determines the properties. And they're so important. They're so important. So let's talk about that. I want to talk about these. What we're going to do is we're talking about semiconductors today and Wednesday, and metals on Friday. And we're going to see it through the vantage point of these electronic levels. And then next week, we'll see it through the vantage point of atoms. So, if you have a gap between-- well, you always have gaps, the question, again, comes to filling. If the filling stops-- I want to define this very important-- so if the filling stops, then you have this highest occupied level. So let's suppose I have filling-- this is some band. And this is some band. And this is unoccupied. So unoccupied by electrons. This is fully occupied by electrons. So what's going to happen? Well, first of all, let's just get some terms. The last band that I filled, the last band that I filled, is called the valence band. Valence band. And we use vb for short. And the first band that I don't fill is called the conduction band. And as we talk more about these materials, you will see why they're called that. Conduction band. But for now, just trust me in the labeling. So the conduction band and the valence band depend on the filling. Because again, it's, you know, it's where I filled to. Now, the top level, the very, very top level out of these 10 to the 24 level, that very top one has a special name. It's the vb maximum. That makes sense, right? It's the top valence band, or the vbn. And the one down here also has a special name, it's the cb minimum, or cbm. So the valence band maximum and the conduction band minimum are, you know, it tells you, if you think about it, it tells you if you subtracted those energies between those parts of the band, you'd get the distance and energy between the two bands. So the conduction band minimum is this energy level here and the valence band maximum energy is-- Now remember, valence electro-- it's called the valence band. It makes sense because valence electrons, remember, are those ones, as we talked about with Louis, where you've got kind of the bonding and the action and the chemistry happening. And so this is that top most state of the top most valence band. This is the valence band top most state. And there's one more definition we got to know about. This is why we're setting this up, which is that that energy difference here, oh, that is really important. So the energy of the conduction band minimum, minus the energy of the valence band maximum, is equal to something called the bandgap, energy gap. It's the band gap. This is an extremely important property of materials, the electronic band gap. What is the energy difference between the highest occupied electrons and the lowest unoccupied electrons? That's what we're talking about. So if that filling, if that distance-- OK, so here's an element, carbon, in the diamond structure. And just-- purple is filled and not purple is not filled. So you know that that's the connection band. And that's the valence band. And the band gap, which we've just defined here, is the energy difference between those. And for diamond, it's really large. For diamond, it's really large. So you can see that, in fact, the gap is over-- if we converted this to electron-volts, is more than five electron-volts. Now, I mentioned thermal energy in answer to the question about water. And room temperature-- so room temperature, the energy of room temperature, if you think about temperature as an energy of like vibrational motion, and you can do that. You use something called the Boltzmann constant and you can get what the room temperature energy is. And it's something like 0.025 electron-volts. Seems pretty small. So the point is, if that's the energy, the thermal energy I have at room temperature, then if my bandgap is so much larger than that, is so much larger than that, it means those electrons are always going to be stuck. They're never going to, even in some accident, have enough thermal energy to come up and be excited. We'll talk about this later. So they're really stuck. And if an electron is in the valence band, it can't move very well. It can't move-- it's kind of stuck in those bonds. It's kind of stuck in those bonds. And so here, if there is electrons all stuck here, and they never have a chance to go up here where they can travel freely, this is really going to be an insulator. This is not going to carry electrons through the material. So that's an insulator. When you have a very large bandgap like that, we call that an insulator. OK, but see if we compare-- and this is interesting, you're just going down in the periodic table if you compare diamond to silicon to germanium. Well, see, what happens is the bandgap gets smaller. I'm still filling them all the way up to the top of a band. So I'm not a metal. If I'm a metal, it's because my filling stopped somewhere in the middle of the band. I'm not a metal. I'm a semi conductor and not an insulator. And the reason I'm a semi conductor is that my bandgap is kind of small enough so that electrons can kind of jump from here up to there, sometimes. Sometimes. That's why it's called a semi conductor. They kind of conduct electrons. So the electron can get up to the conduction band in a semiconductor more easily than an insulator. And that's why it can conduct electricity much better than an insulator, many, many, many orders of magnitude. But the way that we have talked about getting electrons excited, getting electrons excited, is with light. And we can do that for solids, too. We can do the same thing for solids. And so if I plot the absorption, if I take a solid and I shine a light on it, there's something called the absorption coefficient, which is really just a measure of how far down into the solid did that light have to go before some electron got excited by it? How well does it-- well, OK, different solids can absorb electrons differently. But the main point is, if I plot that versus energy for something like silicon, the bandgap of silicon is 1.1 ev, so that's what it would look like for silicon because you know-- this should be wavelength. Because you know-- because as you get to lower-- so you know that above-- it's exactly like for atoms. If I shine light, like remember the Bohr model, if I shine light on an atom that has enough energy to promote it up, then it can go. But the difference here is that for silicon, for silicon, that's right there, for silicon there's this big empty space with no bands. It's just like the atom. When there's no states, I cannot have an electron there. I'm stuck. So the electron can be here or it can be here. And so the light, the energy of the light, has to be at least the bandgap to get it up. At least the bandgap to get it up. So what that means is that if I had-- so, why does this go to zero with increasing wavelength? What it means is that if my valence band looks like this and my conduction band looks like this, and now my light is coming and I want to take an electron and promote it up. Well, it cannot be-- the electron cannot exist inside the band gap. There is no states there. It's just like we talked about with atoms. But now we've got almost infinite states here, no states there. And then again we start with infinite places where it can be. What that means is that if I shine light on it, the only way silicon, the only way it can absorb this light is if the light has energy greater than the band gap, or wavelength less than the bandgap. That's why I drew it that way. So it goes to zero-- so it's going to absorb in wavelength up to a certain point. And then if I go higher in wavelength, those are energies that are too small. It's going to try to absorb here but there's nowhere to absorb to, so it can't. But if it has enough energy, oh, look at this. This is also different. I could absorb here or here or here. I can take all those higher energies and absorb light kind of continuously, unlike in the atom where it could only be the level to hop you from n equals 2 to n equals 4. But here I have an almost continuous set of states, except in the band gap, where I have no states. So that's what the absorption would look like for silicon. And if you look at the bandgap, this very, very important quantity, there is carbon. There's diamond. There's silicon. There's germanium. And you can see that as you go to heavier and heavier elements, you change the gap. And you can understand this from what you know about the electron levels in atoms. Because in an atom, remember, if I get heavier and heavier, those electrons all the way on the outer edges of the atom, they're more weakly bonded to the atom. They're weaker and weaker and weaker. So the energy is higher and higher. Remember, so instead of being these deeper energy levels that have more bonding, more energy in them, lower in energy, they're less and less energy. So you can imagine, now, that those are what you're starting with. Those are what you're starting with, these. Right these are the atom levels that you're starting with. So if you're going to make your atom have electrons that are more weakly bound with energies that are higher, that aren't as deep and don't have as much distance between them in the energy landscape, then the gaps will get smaller. Now, that is a very simple picture of why the gaps get smaller. It happens to work for this column. So if you look at this-- and some people would even call this a metal because that's such a small gap, but I like to call it a semiconductor. But if your gap is zero, if you're band gap is zero, it means there is no gap. So by definition, you're in the middle of a band. You're somewhere in the middle of a band. And if you're in the middle of a band, you're a metal. You didn't fill that band all the way up. But if you filled it up and then there's a little gap between you and the next level, then you're a semiconductor. Or if it's much greater, like greater than three electron volts, 3 and 1/2 electron volts, then you're an insulator. Now, so you can imagine now why-- so as the interactions get stronger, you can imagine those distances can get larger. And so you could have larger and larger gaps, which can explain this trend. But as I said before, where did I put it, it depends, not just on the atom and the original energy levels, but on how those bond together. And we'll talk about that on Wednesday. And it also depends on the structure, which we'll talk about next week. So it's not just as simple as that but you can understand basic trends this way for some of the columns. OK, we'll get to that on Friday. Now, the goodie bag gives you a way to probe this. So what's in the goodie bag? The goodie bag has a way of measuring conductivity. Now, I just told you that you cannot get connectivity unless you have electrons that are able to make this jump, to make this jump. They've got to be able to go up from the valence band to the conduction band. If they don't do that, they're not free. They're stuck in these bonds. You've got to get them free so they can roam around and not be stuck in this bond or that bond. This measures that. This measures that. This measures whether electrons are in the conduction band. You've also got some LEDs and some LEDs. And this is a critical piece here. Why do you have two kinds of LEDs? Because on the one hand, what you're doing, I'm giving you a way to excite electrons. So I'm giving you a way to actually promote electrons. So you've got 10 to the 24 electrons in this valence band. This would be the vb and this would be the cb. And what you're doing is you're shining light with different frequencies, which means different energies. That's what the right hand side gives you. It gives you different energies of the photons. So you have a way, now, of shining energy on a semiconductor. And that's the second piece. The semiconductor is what an LED is. It's a semiconductor. If LEDs were insulators, they wouldn't work. And if they were metals, they wouldn't work. Because the way an LED works is actually just the opposite. Here, I'm shining light so that an electron gets promoted and I can measure that with my voltmeter. But I could also put electrons in with a current. That gets them up here and then they drop down and emit a photon. That's what an LED is. That's what an LED is, it's just the reverse. And so what I've given you on the left are different semiconductors. Because you see, just like a different-- you know, if I'm running current through like I am on the right, you see, those are different colors. That means that I must have different gaps. You know that. You know that already from the work we did with Bohr and the color or frequency of light that you can get out of different transitions. But it's the same here. Did an electron-- you know, did I put an electron up here and then it fell down and gave off some lambda? That lambda has to equal the difference between the-- it just equals the band gap. It equals the band gap because the electron was put up to here in the cbm and then it drops down to the vbm and it emits a photon. That's what's happening on the right hand side. But see, you've also got, on the left hand side, you've got sensors because it's just LEDs on the left. But if I hook those up, if I hook those up to the volt meter, well, I'm hooking up a semiconductor to a voltmeter. So I'm hooking this up and I'm reading, is there any current. And the answer in a dark room is no, unless you make it really, really hot. Don't, because you have to go to very high temperatures. Well, what if I shine light on this now? Well, if I shine light that has the right energy to put electrons up in the cbm from the vbm, then I'll read something on the voltmeter. So by giving you different LEDs on the left, I'm giving you different band gaps. You have in your hands different semiconductor band gaps. And by reading whether there are electrons in here or not, you can tell what frequency of light I shined on them. So they're both emitters of photons and detectors of photons. And it all comes together because you have a semiconductor, which happens because you took 10 to the 24 states, you put them together, you had gaps between them, and you filled all the way up to one of those gaps, and that gap was around 1 or 2 electron volts. We will pick this up on Friday and go further-- I mean on Wednesday, on Wednesday. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | Goodie_Bag_6_Crystallography_Intro_to_SolidState_Chemistry.txt | [SQUEAKING] [RUSTLING] [CLICKING] PROFESSOR: In this goodie bag, we'll be exploring atomic packing and solids through the cubic Bravais lattices. We'll make three ball-and-stick models of different cubic unit cells. For this, you will need 31 balls and 44 sticks-- 24 for FCC, 12 for SC, and 8 for BCC. If you don't have a fancy crystal modeling kit like this, any ball-and-stick equivalents will work. A delicious alternative would be using toothpicks and marshmallows. The objective of the goodie bag is to explore the different structures of crystalline materials. As you build the models, consider how many nearest neighbors an atom has in an SC, a BCC, and an FCC crystal. So now we're going to construct our Bravais lattices. We'll start with simple cubing. Begin by taking 8 balls and connecting them with sticks such that each ball sits at the corner of a cube. Then for body-centered cubic, we're going to start with a single ball in the center of a cube and connect to 8 other balls at the corners with sticks. For face-centered cubic, it's going to be a little more involved. We'll start with the corner of a cube and form tetrahedra to other balls, which themselves will form tetrahedra to another corner of a cube. If we iterate through this process, we eventually end up with a full face-centered cubic lattice. We've now constructed three ball-and-stick models of cubic Bravais lattices. The simple cubic lattice has atoms at each of the 8 corners of the unit cell. The body-centered cubic lattice has an additional atom in the body center of the unit cell. And the face-centered cubic lattice has an additional atom in each of the 6 face centers of the unit cell. Each of these lattices has a different coordination number. That is, an atom in each lattice has a different number of nearest neighbors. If we imagine these atoms as hard spheres rather than this ball-and-stick model, we can see that each lattice will also have a different atomic packing factor. That is, each lattice will have a different volume of space occupied in the unit cell. In general, the coordination number of a lattice correlates with atomic packing factor, which itself correlates with stability. That's why we don't often find simple cubic metallic solids in nature. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 19_Crystallographic_Notation_Intro_to_SolidState_Chemistry.txt | But I want to pick up where we left off on Monday. We were talking about the three Bravais lattices. There a couple more things that I want to say about them. What we captured, the key essence, apart from they're being these cubic crystals and the different elements that form them, is this idea that you can go from the unit cell, once you know the unit cell, you can get the atomic radius. So that's kind of cool. Because you know information about things like, for example, packing. So then we had the packing fraction, the atomic packing fraction. Calculated that. The maximum, the max packing fraction. Packing fraction. We calculated things like the closed pack direction. Well, you can also go to things like the density. Once you know how to go back and forth. And so for example, if I want the density of a material, that's grams per centimeter cubed, right? That's the density. But you could write the density in terms of the things we talked about on Monday. So again, you know, I want you to be able to go back and forth. Once you know the unit cell, and you can know something about the radius of the atom, then you can get the density. For example, here, the density could be defined as the number of atoms in the cell, times the grams per mole of that element, divided by the volume of the cell, of the unit cell, times Avogadro. Now, you can go through the units and you can work out that this is grams per centimeter cubed. So just as an example, if I had, let's see, the example I have is copper. So example, copper. So you know, everything comes from the periodic table. Everything. Life, class, everything comes from the periodic table. So I go-- I say, I want to know-- I go to the periodic table. And instead of looking up the density, I look up other things. So I look up, for example, that it's FCC. And then I look up that it's 63.55 grams per mole. And I also look up the atomic radius. So I look up these things, the atomic radius. And that's equal to 1.3 angstroms. So I look all this stuff up. And the point is, I can now go back and forth. I can use the information, because I know if it's a crystal and I know the crystal type, FCC, then I can use this information to get other aspects, right? So other properties. So for example, I know that the radius of the atom for an FCC, if it's close packed, if it's-- you know, FCC one element, remember it grows out until they touch. And that got us how we get the radius of the atom. And it's a root 2 over 4. But I also know that there's four atoms per unit cell because it's FCC. I also know that the volume of the cell is simply the lattice constant cubed because it's a cubic cell. All the ones here are cubic. So the volume of these cells are cubic. From this information, I can compute the density. So again, it's going back and forth. The radius of the atoms packed in, the lattice constants, the number of atoms in the cell, these are the things we talked about on Monday. And as you go back and forth, you can do things like this. All right, good. Now we talked about the-- we talked about the Bravais lattice there, right? And those are three Bravais lattices that I want you to know. Bravais lattices. But the thing is that there's something else you need to know in order to know what crystal you have. And so this is the FCC Bravais lattice. There it is. It's over there on the right. It's the face-centered cubic. OK, gesundheit. And I'm going to pass this around. All right, oh yeah, I'm starting over here because I keep on starting over there and it's not fair. That's not right. Yeah, all right. Got a couple-- OK, so that's FCC, right? So we're going to write down FCC. But the thing is is that see, this is our stamp. Remember, it's a stamp of points that are equivalent. That's what the lattice vectors are. They are not the things that you put at that point. That is something else. Now, on Monday, we put the things there were atoms, spherical particles. And they're all the same. That was their introduction to crystallography. But the basis can be more than just a single atom. It can be more. So how do we think about the basis? So here's a lattice. This is a lattice of points. The way you want to think about a lattice is a stamp without putting things there yet. The basis is what you put there. Let's write that down because it's so important. So the lattice is all we're going-- oh, I'm going all caps, how to repeat. And the basis is what to repeat. So if this is my lattice and I decide that I want to get a million downloads on YouTube, I'm going to put a cap there. And this is my crystal. It's a crystal of cats. That's OK. I have done nothing wrong. Because I have simply, according to the rules of crystallography, I've had a lattice and a basis. And I've just defined my basis as a cat. And that's the crystal you get. But see, the basis could also have been, for example, two atoms colored differently to show that they are different atoms. And then this would be my crystal. So the actual crystal that you get may be different than the lattice structure. All right, so let's take a look. So for example, over here, you know, if my basis is a single atom, nickel, copper, gold, platinum, well, I showed you these from the periodic table on Monday, right? We highlighted them when we talked about FCC. Then the crystal that you get is FCC. These are the ones we did on Monday. Those are FCC crystals. But at the same time, if my basis-- check this out. Oh, this is salt. When you guys pour salt on your food, do not see it any more except as this. This is salt. This is salt. It is-- let's see-- simple cubic. Is it, though? Because a lattice, a simple cubic-- remember the definition. A simple cubic lattice would mean a stamp of a simple cube where every corner is exactly the same, exactly the same. That's what a lattice is. So this cannot be a simple cube. This is not simple cubic because they're different. They're different. At the corners, I've got two blues and then two red and then two other blues. So they're not the same. It cannot be a simple cubic structure. This is an FCC lattice. This is an FCC. Yeah, I'm starting over here again. I'm trying to even it out. That's an FCC lattice, but with a basis of sodium and chlorine. So it's an FCC. You can call it an FCC lattice. But the basis is a sodium chloride pair. And this has a name. It's called the rock salt structure. It's called the rock salt structure. It's an FCC lattice with this two atom basis where, look, you can see the FCC lattice here. Look at those blue points, right? Blue outlines an FCC lattice. But now the basis is one atom here and one there. So now I take that, this pair, and I put that at all the lattice points. You see that? You put that in all the lattice points of FCC and you've got salt. And that is how you must see table salt from now on. Well, we talked about diamond. This is diamond. Diamond is the same Bravais lattice. It doesn't look like it at first. It doesn't look like it at first. But see here, I have a different pair. Here I have a carbon-carbon pair. Or it could be, or silicon-silicon. There are actually a number of pairs of that would form this diamond. In fact, we specify because there are different types. This is called cubic diamond crystal. This is the cubic diamond crystal. What is the lattice? It's FCC. It's FCC. Now, let's take a look at that. Look, there is my FCC. Here, here, here, here. But inside of the FCC, I've got this pair of carbon atoms. Now they're the same type of atom. And instead of one being kind of in the middle of this edge, as it was with sodium and chlorine, now the other atom is kind of off in a diagonal. But it's still just a pair of atoms. It's a pair of carbon atoms. But if you look carefully at diamond, and this doesn't go on in this picture, but it does in real life, this is an FCC lattice of these pairs, repeating everywhere. That's what diamond is. It's an FCC lattice. So the symmetry is FCC. But the basis is a carbon-carbon dimer, right? And that gives you cubic diamond. OK, so there's the Bravais lattice and there's the resulting crystal structure. Now, speaking of diamond. And this is a graph that I showed you a while ago when we were talking about the differences between diamond and graphite. One of the things that you can see that's so important, this is the same element, two different crystal structures. One of the things that you can see that is so important is how direction is going to be an important property. You need to know how to identify direction because you know, in this case, it looks like maybe if I point in different directions, the thing would look kind of the same, or maybe have similar properties along-- but look at this. This looks seriously directionally dependent. It looks like if I did something like tried to carry charge or thermal energy this way, then it would be very, very different than if I carried it this way. Well, we have a word for that. And it's called anisotropy. And so this is another important word, anisotropy. If it's anisotropic, then the property, well, it could be like electrical or thermal conductivity, for example. Thermal conductivity could be like how it breaks, the fracture. And so on. But whatever-- but the property is-- depends. I'll just say it's directionally dependent. Directionally dependent. So the reason I'm telling you this is that I now need a way-- we've come up with sort of ways to talk about crystals. But I now need a way to specify where I am in a crystal. And maybe how the crystal cuts. I need a way to talk about directionality in these crystals. And that's what I want you to-- that's the topic for today. So the first question-- well, to jump to the punch line is there are these things called Miller indices. And that's what we're going to learn today. Miller indices and they describe direction and cuts, planes. OK, and the first question you can ask is, where are you? Where are you? Now it turns out that there are these people called crystallographers. And they came up with crystallography over the years. And they have very strong ideas about notation. So we'll be talking about crystallographer notation. And we'll be very careful not to do anything that would upset a crystallographer because that is not something you want to see. But the first question that you could ask is, where are you? So I'm going to draw the planes and we're going to-- the axes-- and we're always going to use the same notation, the same axis directions, just for simplicity. In this class, we're going to say that as we're talking about crystal structures, that's y, that's z, and that's x. x is coming out of the board. y is it going that way and z is going that way. Now, if I want to know where I am, then what I can do is draw the cube. Oh boy, here we go. OK, there it is. And I can just start looking at points. But instead of having all sorts of different numbers, crystallographers like to work with simple numbers. So what we do is we define the position as the fraction of the cell edge. OK, so what that means is if, you know, OK first of all, the origin is 0, 0, 0. So that's 0, 0, 0. But it also means that if I had this point here, that's going to be 1, 0, 0. Doesn't matter what a is. Remember, a is the lattice constant. It's the edge of the cube, which has a real value in different crystals. There's copper. There's a in copper, which we could compute. But no, when we specify positions in crystal, we do it as a fraction of that length. OK, so that-- so this point, for example, would be 1, 1, 1 out there. All right. And let's do one more. This point here. All right, well that's going to be-- gesundheit-- that's going to be over a half now. So it's a half and then 1, 0. It went over a half, over 1, up 0. So it's a 1/2. OK, good. This is just where we are. We're just getting warmed up. But really what we need to know is where we're going. And that goes from a point to a vector. OK, now again, there are some fairly straightforward rules that we follow not that we don't know what vectors are. We know what vectors are. The point is, do we know how to specify a vector in a way that a crystallographer would be happy with? That's what we have to learn. Because again, we don't want to make them upset. So there is a set of very simple rules that you can follow. And I've got this for the direction and I've got this for the planes. So here are the rules for the direction. Here's a vector in a crystal. Origin o and here are different vectors. And what we do is we, OK, we position it to start at the origin. And then we read off the projections in terms of the unit cell dimensions. a is x. b is y and c is z. Now, we know that those of the same length in a cubic crystal. Adjust to the smallest integer values, that's really important. And enclose in square brackets. No commas. So let's do a few. So oa, here are the xyz read out. Well you know, this is 1. This is 0. This is 0. So this would give us this as the notation for the oa vector. You see that there, right? Notice I've already done step two. So the second bullet there, read off projections in terms of unit cell dimensions, a, b, and c, because it went all the way to the edge. So it's a 1. It didn't go halfway. It's not 1/2. OK, good. So ob, OK, well that's going to be 1 along x, 1 along y, and 0. So that direction would be written as the 110. And oc would be 111. This is feeling kind of boring. Oh, gesundheit. But it gets exciting in just a sec. Because all we're doing here is we're going OK, 100, 110, 111. OK, good. But now we go to od. Now, od is one of these cases where we have to be careful. So od, you see that? It's going from here to there. So along the x direction, it's still going a full one of the edge. Along the y direction, it's 0. But along the z direction, it's 1/2. And so because crystallographers don't like fractions when they talk about directions, we have to scale that up. And so we just get rid of it. We multiply everything by 2. We put it in brackets. And everyone is happy. 201. That is the 201 direction. In this crystal. That is the 201 direction. They don't like fractions. Now, they don't like negative signs either. They don't like negative signs. But so if I were doing-- if I were doing oe, OK, that's just going in the other direction. So oe would be-- let's see-- minus-- OK, well if I just jumped to what we think it might be given what we just saw, it would look like that. But crystallographers don't like having negative, you know, these minuses inside of their brackets. So we write a bar. So we write it like this. 0, 1 bar, 0. And that is the notation for going in the negative direction. OK, 01bar0. That would be oe. Did I get that right? oe, Yeah. And you can see that of, if you have the same kind of fun, you would see that of is the 112 direction, that that's how you would write the of vector right because you got 1/2, 1/2 and 1, but they don't like the 1/2. So you multiply through by 2, you get the 112. These are vectors in crystals. Now, there are equivalences here because this is a cubic lattice. So if I look at this and say, well, OK, if I went this way and I went that way, where I wind up are equivalent points. Right, where I wind up is, you know, you're winding up at the same place in a way in the crystal because the definition of that stamp in a cubic lattice is that those are all equivalent. Doesn't matter what's on your basis. Because it's defined by the Bravais lattice. That's to stamp. So we have a way of writing that, too. So the 101. That's the 101 direction and the 110 direction, cool. If I make it a little simpler, you know, I can say that the 100 and the 010 and the 001, oh, let's do them all. And the 1bar00 and the 01bar0, and the 001bar all get me to equivalent places. And we can call this a family, just because we don't want to keep writing them. If they're all equivalent, we can say this is a family of directions. And you've got to be careful here because you can't use a bracket anymore because it would just be one of the directions. Instead, we use-- sorry, you can't use the square brackets. Instead, we use this kind of bracket and we say it's the 100 family. So if you see a direction written with brackets like that, then it means all of these directions, this family of directions. But if you see it written with a bracket like this, it means that one vector. Crystallographers, you've got to keep them happy. You've got to keep them happy. All right, now, that is directions. That is direction. But what about cuts? What about planes? So we've got our vectors, now we need our cuts. We've got to be able to specify these things. And you know, as we'll see, as we'll see, the properties of materials, of crystals, especially the more anisotropic they are, depend very heavily on what's in what direction. So for example, if I have a plane in a crystal that doesn't have a high density of atoms, and then I got another plane that's got a lot of density of atoms in that plane, well you might expect those to have different responses to, for example, mechanical stress, strain. You push it. You pull it. You break it. You hammer it. Maybe one of those planes, if it doesn't have as many bonds in it, might break first. This is just one example. So we've got to know how to talk about these things. We've got to know how to talk about these things. So now with planes, I have another recipe to make crystallographers happy. Now first read off-- oh, well let's put an example up. So I've got an example. I'll do a few. And then as always, you guys need to do more to practice. So I'll do a few. So I'm going to draw a plane and I'm going to make it a nice simple one. These are my axes, x, y, z. And I'm going to draw my cubic crystal-- my cubic unit cell, sorry. OK, here we go. Almost a cube. OK, now I want to take a plane here. I'm going to take this top one. So I'm going to take this and I'm going to go through those-- I want to know, how do I describe that plane that cuts that way? OK, one, these correspond to those, crystal plane algorithm. I hear a few of you like algorithms and computing in course six. We got one whistle. So here's the thing, if I have a plane like this and I ask you, where does it intercept? Where does it intercept the axis? Right, well, it's never going to intercept the x and y axes, never. It's never going to intercept them, no matter what I do. But you say, but what if I put it in the xy plane, then it's always intercept-- no, that's not what we mean by intercept. What I mean by intercept is, it has to reach them eventually no matter where it is, no matter where I position it. But this never does. This never does. And so the intercepts, you know, for the ABC intercepts, so we can write those here. OK, I'll write the same way, x, y, z intercepts, it's infinite. And it's infinite. All right, and then for the z, OK here, I'm intercepting at this part of z. So I'm intercepting at 1. But the thing is, oh man, crystallographers don't like infinity either. They don't. And so what they do is they just take 1 over. That's a nice way to fix infinity. And so step two, this would become 00 and 1 over 1 is 1. Right, 1 over infinity is 0. Now we reduce to integer values and there's no work to do there. And then we are very careful about this. We use parentheses. No more brackets, parentheses. Do not get this wrong because you might be talking about a vector by accident. You don't want to talk about a vector by accident. So this is the 001 plane, no commas. Remember the dislike of commas. It's the 001 plane in this crystal. That is a cut. OK, good. And that's called a Miller plane. Well, if I do another example, do one more, so that is this one. I have it. I've got a bunch of examples here. And they can draw it, you know, better with the shading and stuff there. But you see, that's what we just did. We did the 001 plane, that one there. All right, there it is, nice and big. The 001 plane. What about that one? That's the 110. The 110, if you do the 110, 110, I'm not going to try to draw it again, what you get is step one, you get 11infinity. Step two, you get 110. Step three, you get 110. and step four, you get 110. Those are the steps to identifying and labeling a crystal plane. Let's make it a little more complicated. We still haven't really done something where we take advantage of the rule of step three. So let's make it harder. Let's bump it up. So there's a slice. Remember, I can slice a crystal in any way I want. And I got to be able to write it down. I got to be able to represent it on paper. That's what this is about. That's what the Miller indices and the Miller planes let us do. So this one I'm going to try to draw. So here, I've got my axes. And that's y. And that's x. And that's z. Here we go. It's almost a cube. Now, I've gone up here to 3/4 high. So I've gone up about 3/4 there. I've gone here to 1/2. That's 1/2 of the cell length. This is 3/4 of the cell length. And then over, I've gone all the way to 1. So now if I go through my steps on this one, I've got a 1/2, and I've got 1, and I've got 3/4. OK, I'm going to take the reciprocals. OK, that becomes 21 4/3. Oh, not happy yet. Not happy yet. We don't like fractions. We got to get rid of fractions. So in step three, we simply multiply through by 3. And we get 634. And then we can write down that point. That is 634 plane. And so on and so on. Now, let's go back to-- let me erase this. There's a couple of things about the planes that once we know what the plane is, we know some other things about it. So let's talk about that now. All right, if I-- first of all, a Miller plane is not just one plane. A Miller plane is not just one plane. A Miller plane is an infinite set of planes. OK so when I say a Miller plane, well, very often what you do is what we've been doing, which is you define the plane that's kind of closest, somehow, to the origin. In the unit cell, the one that cuts it. You define it that way. But actually, what Miller planes are are infinite sets of planes. And so I'm going to put-- Now, I'm not a crystallographer so I can write infinity. I'm OK with that. It's an infinite set of planes. OK. They are equally spaced and here's the key, one should always be going through the origin. If you want to look at it as a set of planes, you've got to put one at the origin. That will help you think through what I mean. OK, so if one goes through the origin, then what you see right away is how they repeat. Because you know, it's not just that I go-- in that case, let's take that one, the 001. Now you say, where's the next 1 in this infinite set of planes? OK, well, put that one at the origin and make another one. Then put that one at the origin again and make another one. That's how to think about this infinite set of planes. OK? So if I had one that went half way, a 200 plane or an 002 plane, as it would be if it's properly noted with Miller indices, well then you say, well, OK, I've got one half way. You know, isn't that it? No. The set of infinite planes also would have one at the origin, one half way, one at the top. Because each time you put it back and you do your next one. That's how to think about this infinite set of Miller planes. So you get this infinite set of planes that have-- I'm going to go to this picture. There it is, a 100, that just go on forever. Those are the Miller planes. And what they have is a spacing between them. And this is really important. So this is the next thing I want to tell you. All right, so I've got-- so a Miller plane isn't just one, it's infinite. It's an infinite number of them. And what that means is that when I give you-- when I give you this, right, or that, or any Miller plane, I know what the spacing is between them. That is enough information. Because the distance between planes is-- actually, here it's very easy to see. The distance between these planes is a. But the difference between the 200 planes is a over 2. And in fact, for a cubic cell, there's a formula that you can use for any two planes. The distance between, well, let's see. I'll write the notation first. The distance and the plane there is equal to a. And the distance of a 200 plane between planes is equal to a over 2. But there's a general formula of any HKL. HKL, the Miller indices. Those are just variables. But if those are the variables of the Miller plane, then the distance between two of them is the length of the edge of the unit cell, a, divided by the square root of the squares added together, a squared plus k squared plus l squared. So you see, OK, if it's 100, that a. That works. If it's 200, that works. And it's general. So now, just by knowing, but because I figured out how to write a plane in HKL notation, I also know the distance between these infinite planes. That becomes very, very important once we shine x-rays on crystals next week. That becomes very important. OK? OK, so what else can we know from the Miller planes? OK, so that's the distance between them. Now the other thing that actually becomes fairly self-evident when you play around with this a little bit, and that was on the slide before, is that-- let's see. The plane and direction with the same Miller indices are orthogonal. And you can see this from this very simple picture. There's the 001 plane. But that's also the 001 vector, just as we defined it today. So the 001 plane and the 001 vector are orthogonal. So this is an example, the 001 plane is orthogonal to, oh, now gotta get this right, the 001 vector. Oh, no commas. Brackets, parentheses, no negatives, no fractions, no infinities. I followed the rules. And because of that, these indices have meaning as I'm showing you, because of that. Now, you can play around with this but this can also be fairly straightforwardly derived, things like this. But you don't need to know that derivation so I'm just telling it to you. OK, so these are the kinds of things that you can do with Miller indices. As I said in the beginning. One of the things that you would want to know what planes you're talking about is because you want to know what the packing in that plane is. Just like we talked about packing, right-- well, we talked about it before on Monday, the atomic packing fraction, there it is, the max packing fraction in a volume. We also want to know, especially if a crystal is anisotropy, we want to know about packings in planes. And so, you know, if you take a very simple example like a simple cubic, there's a simple cubic lattice. And now we're all good with our notation. There's a simple cubic crystal. The basis is one atom. I don't have anything else in here. Those are the atoms. It must be polonium. And there's the 100 plane, the 110 plane, and the 111 plane, 111 plane, 110 plane, 100 plane. Now as you can see, the density of atoms in these planes is different. The density of atoms in these planes is different. And that has a real serious impact on the properties. So if we look at this-- so I'll do a couple of quick examples. If we look at this example of simple cubic, that's a very easy case. And so for a simple cubic with a lattice-- this a is called the lattice constant. I've mentioned it a few times. Lattice constant a. Then you know, if we look at the 100 plane, then the area is a squared. The area of that plane is a squared. But the question is, how many atoms are in that a squared. Well, you do the same thing that we did before. So I'm looking at a plane here. And I'm trying to figure out how many atoms are in this-- well, you know this is being shared in this whole big plane that goes on and on. It's being shared by four other squares. And so there's a fourth of each atom in there. So there's one atom, effectively, in that plane. Again, this is in one of the simplest cases. But the number atoms equal to 1. And so the density of atoms per area is equal to 1 over a squared. That's a pretty easy case. If I did the 110 direction, it's a little bit different because the area is now different. So the area in this case is root 2 a squared. You see that? The area in that plane is different. So the density of atoms in the plane is different. So the density in the plane affects properties. And I want to give you just a quick example of that and why this matters. Because in fact, this is one of the things, all the way back to Hook and his cannonballs, and even way before that, people noticed this stuff. I broke something, but it always seems to slice in a certain way. Why is that? Why would something always break in the same way every time? With the same kind of-- what does way mean here? It means like the angles that form. So why are those angles always kind of the same? Well, it has to do with the strength, the relative strength, of the planes inside of these crystals. And in fact, here's a beautiful piece of work on stretching a wire. All right, so here you're taking a wire that is crystalline and you're pulling it. And you can see how it breaks. You can actually see how it breaks. It's breaking along the weakest plane. It's breaking along the weakest plane. So if you want to think, then, about well, OK, how do I know how to make this stronger, for example? So it stops breaking-- you know, the weakest link is the weakest link. That was deep. So then that's the plane of the crystal that I need to think about. How are bonding together or packing in in that plane? This is real stuff. This comes about in many of the properties, not just the fracture, but in many of the properties of these materials. The property, as I said before, depends heavily on which plane you're in. And we'll see that more and more. As one last example of this, you could go a little-- So again, you go back to your periodic table. And you take-- there was copper. Let's do nickel. All right, you take nickel. And let's see, I know that the atomic radius of nickel is 1.52 angstroms. I've looked that up. But I also know that it's BCC. And this gives it away. This gives me what I need. The crystal symmetry, the lattice, gives me what I need. Because once I know that it's BCC, then I also know that the lattice edge goes as 4r over root 3. Remember, for an element like nickel, the packing direction is along the body diagonal. That's what gave me this relationship, back and forth. Unit cell edge, radius of the atom. So that means that in nickel, it's 3.5 angstroms. And once I know what a is, well then I can know what like the-- you know, in one of the planes I say, which plane is that? I don't know, let's look at the densities in the plane. So for the 100 plane, it's-- I'm just going to give you the answer-- 0.59 is the packing fraction. Packing density. This is the-- sorry, that's the fraction. All I did is I used the periodic table to do what I just did for the simple cubic case but now for BCC. And it's because of all the things that we've learned in the last two days, in the last two lectures that we're able to do this kind of calculation. And I can compare this plane with another plane with another plane. And think about how the atoms are packed differently within the anisotropy of the crystal. OK, Friday, bring parents. Bring parents. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 21_Xray_Diffraction_Techniques_I_Intro_to_SolidState_Chemistry.txt | Today, clearly, I am a noble gas. Thanks to-- it was actually the TAs' idea. And I was like, of course, that sounds awesome. So I hope you guys have a fantastic Halloween. Obviously, what you want to do is any kind of trick or treating or partying, you know the drill, you bring your periodic table, because you never know. And that's what happened-- so on Sunday right before the exam, I got this. And so this is Charles and Raymond. And they're saying we wanted to celebrate the Sox victory in Boston, but because the midterm was the next day, we brought along our periodic table to study. There were so many people moving around it felt like we were in a sea of electrons. And of course, the only thing that could have improved on that was actually like a little dance video. But that will probably come. I'm sure there on that next. But you see, they brought it not just to study for the exam, because they didn't know if it was going to be important or not at the rally. You never know. So make sure you have your periodic table with you. Now, OK, oh, yeah, they mentioned the exam. So speaking of the exam. Here are the results. And you can see that there is a pretty wide distribution. You can also see the averages is 77. And just to remind you-- that's about here-- just to remind you, these marks, 85, that's the A range. This is the B range. So the average was in the solid B range territory. And so it goes. The standard deviation was 12. Now I could tell-- you know, the exam 1, the topics of exam 1 many of you had already seen. And I think in this exam, some of you had not seen some of these topics, especially like the crystallography, but also even the molecular orbitals, the band structure. And I could tell that, you know, some people had to work harder, maybe a little bit stressed, and I could sense, some of that. But when I get stressed what I do is I need to kind of de-stress somehow. And sometimes some people listen to music, some of the other things. I always like comedy as a way of relaxing. And so what I do is I'll Google like for articles about Harvard. And so I found this one. This is actually an article in The Crimson from a few years ago. And the most common grade at Harvard is actually an A, a solid A, the most common grade. Suspicion that the college employs a softer grading standard than many of its peer institutions. You think? I mean, if everyone's getting an A, you think? So what I wanted to point out is there's a difference. And it's not just about your privileged, therefore, you deserve an A, whereas we know that that actually takes work. That's not the point I want to make. The point I want to make is that we know also what to do when we miss things. We know what to do when we don't get things right. That's when you do the work. So wherever you are on this curve, go back and figure out what didn't work. What did you miss? Figure that out, because that's where you learn. Thomas Edison who said, I never failed, I just did 10,000 experiments that didn't work. You've got to get through that. You've got to know that it takes hard work. And that's the thing that we know here. That's the difference. So please make sure you do that with exam 2. We are all here to help you continue learning. I'm a noble gas. And speaking of continuing to learn, where were we? X-rays. On Friday, we had sort of some other things going on. But I also was trying to teach you guys about X-rays. And what we did is we learned how they're generated. Remember that? The Roentgen experiments. And so I want to pick up here with the kinds of X-rays that we have. There are two kinds of X-rays. And we learned about those on Friday. And I want to just remind you about those. And I also want to show you a video to kind of recapture what they are. The first kind of X-ray, remember, we plotted this as intensity. And I'm not going to draw the cathode ray tube again and the experiment that Roentgen did and all that. But I'm going to just jump to the two kinds of x-rays that you get. And so if you plot the wavelength of the X-ray versus the intensity of the X-ray, then one kind is the Bremsstrahlung, which comes from that that electron getting slowed down. And if it slows down, it emits radiation in a continuous spectrum. And remember, we sort of drew these. And this would be like the incident electron energy. Let's say incident energy of the electron. Maybe like that's, OK, I don't know, like 10 keV. And then up here the incident energy of the electron was you know something like maybe 30 keV. And so you see that as you hit that anode, remember what Roentgen did, he took a cathode ray tube and he upped the voltage. So he really cranked up the voltage. So those electrons coming off the cathode are going really fast. And then what he did is he put a piece of metal in their way. And what happened is those really high energy electrons, they see those metal atoms and sometimes they get inside the electron cloud and they turn. And that's when they give off this continuous radiation. But we also know that there is a limit here. Remember that? We talked about that? And that this limit is set by this maximum. It's just the maximum amount of energy that you could get out of a photon being emitted this way would be equal to the incoming electron. The incoming electron transferred all of its energy to the photon. So that's why there is a maximum. And it's also why it increases-- remember, shorter wavelength, higher energy-- so it increases as you increase the energy of the instant an electron. That is Bremsstrahlung. Let's watch a video, because this is actually a very nicely done video that captures it with animation. So OK, here's your CRT. Now here they come. Those are those electrons. And that voltage is high. So they're coming out. Look at them coming out really fast. Lots of kinetic energy. And then they go. And then this is what Roentgen did. He put a little piece of metal in there. So now those electrons are hitting the metal. So those are the two things that he did differently. Remember, the room lit up even when all the lights were off. There they are. Nice. Good electrons. Oh, and here they go. And there's a metal atom. And oh look at that. Now what are these? Those are x-rays coming off. Those are the electrons hitting the metal. And here's the metal atom. And as the electron comes in, remember it sees the charge of the nucleus. And it gets deflected. And that deflection loses energy. And that loss of energy goes into a photon. Now, because these electrons have such high energy to start with, the energies of the photons are very high. They're x-rays. And so here's the wavelength. And you can see that-- well, it's sort of a little hard to see this blue range. But see higher energy, shorter wavelength. Lower energy, longer wavelength. Very nice. So that's the animation of Bremsstrahlung. OK. Good. Now the thing is what we also learned is that there is another type of X-ray. There's another type of X-ray. And in fact, if you crank this up high enough, you can get that other type. So now, we go higher. Oh, we didn't see it here. It just looks like that and that and that. And then all of a sudden we go up to 40 keV and we see this. Why? What happened here? And what happened here is a totally different mechanism for generating X-rays. And that was the second type that we talked about on Friday. Those are called characteristic. And the reason is that-- remember, we have these levels, which now that we are talking about X-rays, we give them letters, K, L, M, N. But it's just the quantum numbers. n equals 1. n equals 2. n equals 3. n equals 4. And what we said is that the weight characteristic X-rays are labeled is that if an electron is excited from this lower level, if an electron is knocked out of here, then there's a place for an electron from here to go down. Maybe the electron is excited. Maybe the electron is kicked out. And then something here can come down. And when that happens, just like in the Bohr model, you get radiation. But now, unlike hydrogen 13.6 electrons, these are keV of energy. These are very high energies. Why? Because it's the one S electron. And we know that once you get down to those metals, those one s electrons, you go further down the periodic table, 1s electrons are seriously tightly bound. And those levels have a difference down there that is pretty high in energy. It's X-ray high in energy. That's the point. So now, when that cascade happens, we call that a k alpha. And if it were to have come from here, it would be called a k beta. And those are transitions, that unlike this continuous energy, those transitions only happen at very specific energies, delta energies. The change in energy from L to K-- or for K beta it would be going from M to K. It's k alpha, k beta. We use the k because that's the final place the electron goes when it decays down. So if I just showed you these, well, that would be like k alpha, and that would be like k beta because you know the k beta is going to be a higher energy photon because it came up from a higher level. Well, you would also have on here some other peaks. You would have the L peaks. So you'd have like L alpha, L beta. So as you crank the energy up, then you can knock out those core electrons and these cascades happen and you get these discrete peaks. Notice, they will only come when you have enough energy in that incident electron to knock this electron out from the core. So that's why they don't appear until you get to a certain incident electron energy. They don't appear until you get to that certain energy. And so we have a video on that, which also I will narrate. Is this it? There we go. So there it is. It's a metal atom. I don't know which one. OK, there's the incident electron. You fired it. And look at that. It knocked out a core 1s electron, because it had enough energy, very high energy. That's what Roentgen did, cranked up the voltage, higher keV. And there it is. A cascade down and an X-ray comes out. You see that? Is that all? Oh, yeah, and then it's going to draw-- because of those, you get these characteristic peaks. Now, we call them characteristic, because now you see why. So unlike the continuous radiation, these peaks depend on the atom, because they depend on the energy levels of the atom. And so that's why like if you looked this up, you say, well, OK, let's look at the k alpha radiation. Let's look at the k alpha peaks that come out of different atoms. They're going to be different. So you have very sharp lines of x-rays at very specific energies. For copper, it's 8 keV. For molybdenum it's 17.5. Silver tungsten, it changes. And you can see that it goes up, as that 1s electron is lower and lower energy because I'm adding all these protons. So it all makes sense from the concepts we've learned. Oh but see, now that's really useful. That is really useful, because now I've got a way to have a source of X-rays that is super well defined. It's super clear. It's always this-- that's so cool. As long as I have the same metal, it's always the same. I can increase or decrease-- well, I can't go below the threshold. But I can't go above it. And that peak is characteristic of the metal. So it doesn't change. That's really useful because I've got now a flashlight. I've got an X-ray flashlight where the energy that I'm sending out is always exactly what I know it to be. I can predict what it is. And it always will be that depending on which medal I put in there. So that's a useful thing. Why is that useful? Well, that gets to the topic that is the topic of today and of Friday, which is what are we doing with these X-rays? Well, first, we're generating them. So that's what we've talked about so far. But now, we're going to actually use them to determine the crystal that we have. We're going to actually use them. We're going to use that flashlight. And so you can see why this would be useful, because this is the range-- we've showed this before-- of X-rays. So they have these energies of keV. And they have wavelengths right around a few Angstrom. That is a little less than an Angstrom, maybe 2 Angstroms. See where those wavelengths are? Well, those are atomic spacings. Those are like distances between layers. And so if we could shine these on a crystal and somehow figure out what it is with that light-- oh, there's a way we can do that. It's called diffraction. Because what happens is-- and we know this from many fields. You can think about this just as a water wave, any way. If the wavelength is similar in size to the features, then you get constructive and destructive interference as a result of the interaction between the wave and the features. So that's called diffraction. And you can see it here with this very simple picture of say a water-- this could be like a water wave, a sound wave. And there it is. And it's interfering both constructively and destructively along these lines. You can do this test yourself. You can take-- I highly recommend this-- take a laser pointer. Now if I just had a piece of metal-- I don't have a piece of metal, but if I did, then I would shine it on it. And what you'd see is that the dot would just reflect off the piece of metal. So I just would get the dot back. But now I've got this thing here. Many of you may not know what this. This is called a CD. But it turns out that a CD has features in it. It's got trenches that are like a little less than a micron apart, 100 nanometers. And this is 500 nanometer light. So you would expect there to be diffraction. You would expect there to be constructive and destructive interference. And when I bounce it off of this, look at that. There it is. This never gets old. I'm not getting just one reflection here. I'm getting a whole scatter of them that have constructively interfered, because of the feature sizes being the same as the wavelength. But now, I want to do that with X-rays. And I want to do it onto crystals. So how do we do that? So let's think about that. And we're going to think about it in terms of what the Bragg father-son pair, who won the Nobel Prize and are on a stamp. That's what you get when you win a Nobel Prize I guess. And what they did is they figured out how to do this. So let's go through that just so we understand it conceptually. So I'm going to say that I have a set of atoms. Now, I'm not going to worry about what they are. But I'm just going to say that there's some plane of atoms here. And there's another plane of atoms beneath that. So there is another one and so on and so on. And now, these would be Miller planes. These would be Miller planes in the crystal. And let's just assume they're very, very simple, this plane and that plane. And now what I'm going to do is I'm going to have some x-rays. I'm going to have some light shining on this. And it's going to be incident. And it's going to be reflected. But see, I'm going to have another wave here. OK, let's see if I can get through this drawing, sort of, almost, kind of. Now, here's the deal, these are waves. So if I want to draw this as a wave, I might draw it like that. And if these waves are constructively interfering, let's just complete that, then this wave would look like that. If they're in phase, then that's what they would look like. Now, this one's getting reflected off the surface. So I'm going to do that reflection here. And if I wanted to come back off of the surface, and this one got through the first layer, this didn't get reflected. It's going down. And if I want it to come back up, then this one must also look like that to be in phase. They must be in phases they come out, or else they're not going to interfere constructively. You see that? So those are my X-rays. They're waves. They're just waves. Oh, but this is the whole secret, because if this angle here is theta, then what that means is that this angle is theta. And if that's true, then this distance is d sine theta, where this is d. That's just some simple geometry. So what you know then is, OK, now we're getting somewhere, because you know if I had a wave, this is one wavelength. So if I had a wave come in like this and one of them is going to get reflected off of this lower surface and the other one got reflected on this, but I don't want them to interfere in any way but constructively, that's what Bragg said, Bragg and Bragg. That's what they said. Then the only way for that to happen is if this distance plus that distance-- so d sine theta plus d sine theta is equal to some multiple of the wavelength. It has to be. And so what you get is-- well, that's what they have there, n lambda, some multiple of the wavelength, equals 2d sine theta, where theta is the incident angle of that X-ray. So this is incident-- just to be clear-- incident X-ray. And these are reflected X-rays. Now to keep it simple in this class, we're just going to say n equals 1 for 3.091, just to keep things simple, because I want to grasp the basic concepts here. We're not using X-rays to get the structure of DNA. But we're going to use X-rays to figure out cubic crystal structures. I'm going to show how that works. So this must be true for-- so this is in parentheses-- for constructive interference. Those dots that you saw interface-- interface, no interference, interference. This is true for constructive interference. Of course, you could write any equation you want. But if you want them to be constructively interfering when they come out, that has to be true. And that's what the Braggs said. But we're not there yet right, because now we've got to do experiments. So we've got to do experiments. So what does this mean? Well, OK, I'm going to take X-rays of some lambda. And I'm going to shine them on a sample. And I'm going to measure. So what I'm going to do-- well, I think I have a picture-- I'm going to measure-- there it is. This is what an X-ray diffraction experiment looks like. So I've got some sample. And I shine x-rays. I've got a source of x-rays. We now know how to make that source. And we'll just filter out one of these lines. So I've got that source, and I can change the angle. And then I've got a detector. And I can measure did I get interference or not? It's just like the dots. So now, we know that if I do that and I say I scan-- so I've got intensity. And now I'm plotting it with angle. So now I'm moving the angle around and I'm changing it. And, bam, I get interference. And I see a spot in my detector. So literally it would look like this. You would get some angle where there's interference. And the detector would say ding, ding, ding, ding, ding, I see a lot of x-rays coming off. And now you change the angle a few degrees and I don't see anything, because it's all destructive from these crystal planes. So it seems like then if I just vary theta, am I there yet? The problem is that I might not know d. A, ha, but we do know d. We do know d, because we learned about d. For cubic crystals, we know d. Because for cubic systems, we know that d of hkl of any plane, the distance between those planes is equal to a over the square root of h squared plus k squared plus l squared. And that's something that we learned. This is distance between Miller planes in cubic crystal. I'm saving time. I wrote xtal. I saved a lot of time, which I just wasted by being so proud of it. I know d. Well, OK, so what does that mean? Well, let's take a look, because now I'm going to go back my equation. And I'm going to say, OK, lambda equals 2d-- and I'm saying n equals 1. And I'm not going to be very specific. This is now a d that comes from the spacing between planes that are specified by the Miller indices times the sine of theta. And I'm putting hkl on the theta as well, and you'll see why. Because here's the thing, now I've got constants. Now I've got constants, because look, this is fixed by the source. So this is a constant. If I have copper, then it's 1.54 Angstroms. This is a constant. This is fixed by the crystal. That's also a constant. Fixed by the crystal, because that's also a constants, because it's the lattice constant. We're not changing that. So for a given set of planes that these waves are bouncing off of and maybe constructively interfering with, that depending on the theta, then these are constants. So if I regroup then-- so I'm going to regroup them. And so I'm going to say that-- let's see, d-- so I'm going to substitute in that expression up there down here. And I'm going to use a copper source. So I'm going to say-- well, let me go through this one step at a time. So d equals 1.54 Angstroms over-- let's see-- 2 sine theta hkl. Now what have I done here? This is for a copper source. So I'm fixing in the constant. So that's the Bragg condition. But I also know that d is equal to-- I don't want to write it again. It's also equal to that. OK, good. OK, so let's put that together. We'll do a little division. And what we get is 1.54 Angstroms over 2a squared, the whole thing squared. I'm squaring it-- I don't want the square root-- equals sine squared theta hkl over h squared plus k squared plus l squared for-- let's be very specific-- for constructive interference and a copper source. So now, I'm getting specific because this is how experiments with X-ray diffraction are done. But now the last time I checked, if you got something equal to a constant, then that something also is a constant. And this is a constant. And so what I need to do now is figure out I'm going to measure these data is where I get a signal. Remember, I'm going to now change theta so that I see where I get signals. Now, those thetas divided by the hkl that they are bouncing off of must be a constant. They must not change. That is at the heart of X-ray diffraction. That is at the heart of it. And so we're going to do that with a specific example today and Friday. But before we do that, there's another thing that we can observe. There's another thing that we can observe. Oh, this is what it would look like. So here it is. So now I I've changed-- now, why do we do 2 theta? It's kind of historical. You plot X-ray diffraction spectra. So this is an XRD, X-ray diffraction spectrum. This is the intensity of the peaks. And these are the peaks. This is a beautiful thing. I'm seeing a crystal here. I'm seeing a crystal. And by Friday, you will be seeing a crystal. Those aren't just peaks. Those are planes in a crystal. Those are planes in a crystal, which tells me not only that I have these planes, but it tells me what crystal I have. But that's not how we start. The way we start is we do these measurements and we just read off angles. So we got to get from there to there, to crystal structure. So what I want to determine is the crystal structure and the lattice constant. That's my goal. What I have is a spectrum that looks like this where all I've done here is put these specific angles here. And you have to be given-- so this is the aluminum XRD spectrum. So if you shine X-rays on aluminum, this is what you get if you know also that those X-rays are from copper, which means that lambda is fixed. So this should be like the information you get to start. You'd be given this spectrum, given these peaks. And you've be giving this information here. It's a copper target. And from that, we can determine the crystal structure in the lattice constant. Now, there's something-- oh, why do we do 2 theta? Well, it's historical. It could have been theta to make all those dividing by 2s go away. But instead, you can see that as I rotate this, this changes by theta. The detector changes by 2 theta. So that's why extra spectra are given in 2 theta. There's no other real good reason for it, even though in the Bragg condition, it's not 2 theta, it's theta. This comes from geometry of the planes. And this just comes from historical setups and how you move the detector. So what's measured and plotted is the 2 theta. But before we go, before we do this transformation, where we take an X-ray spectrum like this and we get the information we want, there's one more thing. And that is not all reflections are allowed. Not all reflections are allowed. And so let's talk about that, and then we'll come back to the spectrum. Now, you can kind of understand this by looking at just a simple kind of comparison here. So these are the hkl's. Remember, that's the hkl for a Miller plane. This is h squared plus k squared plus l squared. Why do we put that? Because we know we're going to need it. There it is right there. So we know we're going to need it. But if you look at the simple cubic, that's simple cubic. Any combination of hkl is OK. There is no combination that would give you interference along those plains stacking. You may say, well, OK, yeah, what are you talking about? Why are you even bringing this up? Well, when you see the other two crystal structures you'll see what I mean. So now we have the case of BCC-- when you see this, you see BCC and FCC. And what I'm showing you here isn't the 100 plane. It's the 200 plane. So this is the family-- remember, the family-- of 200 planes. There they are. But now you see that what happens-- and I have a picture here to show you, but I'll tell you first-- what happens is the light comes in. So there's those squiggly X-rays. It comes in. And there is, OK, d, which depends on the lattice constant, is related to the lattice constant. But look. Now there's another plane in between. There's another plane in between. And in fact, with the 200 planes, that plane in between exactly cancels out the constructive interference. Here it is. So there's what I would have had. If you want to think about this as the 100, there's the 200s. But notice when I go from 100 to 200s, I add this plane in here. And because that plane has atoms in it, because that plane has atoms in it, it acts like a mirror, and it can also reflect. And so what happens is I would have had this nice-- there it is. There's a picture I drew. There's that first X-ray bouncing off. There's the second one bouncing off. And those are nicely in phase. So I would see that. If that's the angle that gave that to me, I'd see that in the detector. But now, for BCC or FCC, I've got something in the middle. And that something in the middle is exactly canceling out. You see that? So now, it cancels that out. In fact, that's called forbidden. You won't see a signal. And so these are called selection rules. And for a simple cubic, you can see there's nothing inside. So there's nothing in this unit cell that could do this. So everything's allowed. Whether there's a plane in there or not, it doesn't matter. The selection rule is whether it's ever allowed. And for a simple cubic, everything's fine, because nothing would cancel out. But in here, you see in this 200 case, you can see very clearly from that picture how it cancels out. But there's many other kinds of angles or planes that might also do that. And so I'm going to just give you what the selection rules are. We won't go through and derive them all. But let's see, they are actually quite simple. And so I'm going to write them down. So if we look at allowed reflections, and then we look at forbidden reflections-- so this is what the selection rules tell us-- so if it's simple cubic, then it's any h, k, and l. And there's no forbidden reflections. But if we go through BCC and FCC, then what we find is that for BCC the selection rule is that h plus k plus l equals even. If that is even-- and we won't derive these, but it comes from the same very simple picture I just showed you. If something is in there that can cancel out the constructive interference, it's going to be forbidden. Otherwise, it can constructively interfere. And that's what this tells us. For BCC, it turns out to be h plus k plus l. And so here, what's forbidden for BCC is h plus k plus l is odd. And for FCC it's h, k, l, all odd or all even. And the forbidden FCC is h, k, l, mixed odd/even. These are the selection rules. So if I were to give you the planes that you see, then right away from an X-ray spectrum, you could just use these selection rules right away to know something, to know something about it. And if you work this out and you look at, OK, so we have simple cubic. So h squared plus k squared plus 1, 100, BCC, FCC, you're not going to see it. That doesn't mean that there is no 100 plane in those crystals. It just means that if you shine X-rays on it, you will not see it. OK, so the 110, though, now here, OK, so we see our-- by the way, mixed, even, odd, and it adds to odd. That's why neither one of these works. Here we go, adds to even, BCC OK, but it's still mixed even odd, won't be FCC. 3, it's not mixed, so it can be a allowable reflection for FCC, but not for BBC. Because if you add them up, it's on odd, and so forth and so forth. And look it, 7 doesn't exist, because you can't do it. No matter how hard you try, you can't get 7. That's OK. And there's 8 and 9 and so forth. So 9 also-- so here it's allowed, but it's not allowed in either of these, because you can't get either of these to be satisfied. This is what selection rules give us. And it comes, again, from simple-- oh, OK, well, that that's another thing. I said a simple, I have to tell you something, because the Bragg condition, it relies on an assumption that's mostly true. But the Bragg condition requires that the reflection is independent-- some of you may be thinking I did I draw it onto the atom or did I draw it in between atoms? Where did that thing-- does it have to reflect off an atom? We're not going there. With the Bragg condition, it's independent of the atom positions in a plane. And the second thing which is what I've been sort of alluding to is that the atomic planes are mirror like. I mean, this is sort of an obvious assumption since I've been assuming they've been mirrors. But if you start thinking about atom position, you might go back to that selection rule picture and say, well, wait a second, does it always have to hit the atom? What if this one was over or something like that? No, no, we assume it's just one continuous plane if there are atoms in it. If there's no atom in it, then it's not a reflective plane. But it's continuous in the assumption of the Bragg condition. Now we go back to our picture. So what we want to do is, again, our goal, should we choose, our goal is to go from this spectrum knowing this information-- it's a copper target and being able to read off the peak-- our goal is to determine the crystal structure and the lattice constant. That's our goal. And let me let me just write this again, because it's extremely important. So maybe I'll keep that one. I'm going to erase this and put it right in the middle, because this is what drives XRD. This is what drives XRD, which is that our goal is figure out what makes this constant always. Now, you say what is this constant always? This is the expression, which I'm going to write again. So what I have if it's a copper source is 1.54 Angstroms divided by 2 a squared. That's a constant. That equals sine squared of the theta for some plane divided by h squared plus k squared plus l squared. So I'm just kind of repeating what I've said and what I've written elsewhere, but that is really it. That is what we do in X-ray diffraction. What makes this? Well, by this what I mean is this term on the right. How do I make sure that this never changes its value? Because the thing on the left never changes its value. And just making sure that you've got that concept, that is this, oh, yeah. And it turns out I got a recipe for you to follow to do this. And it's on the previous slide. And so we'll start thinking about it now. And we've got five more minutes. And then on Friday, we will finish this and then talk about what to do with those continuous X-rays. So what do I do? Well, the way you do this is systematically. And the first thing you do is you read off the 2 theta values that generate a set of sine squared value. So that the X-ray spectrum measures 2 theta. But you know from here that I need sine squared. So I'm going to write down the sine squared values. So that's a first step. So for the first one, I've got the first peak. So I would start to log my data. The first peak and the 2 theta is 38.43. And the sine squared theta is-- and I'm just going to do that math-- 0.1083. OK, good. And the next one, the next peak is 44.67. And the sine square of that is 0.1444 so on. So you read off all the peaks and you make those columns. So I've gone through. Now the second one, normalize the sine squared theta values by the smallest value. You say, so why am I doing that? Trust me, this will achieve our goal. This will get us there. This is a nice simple recipe to achieve our goal. So the next column would simply be sine squared theta divided by sine squared theta min. Well, I ran out of room there, but that's going to be obviously 1. And this would be 1.333 and so on. So that's the next one. So I'm just setting that the top row to 1. I'm setting the top row to 1. Good. Now, I'm going to clear fractions. So I've normalized, and now I'm going to clear fractions. And if I do that, I clear fractions. Well, you say, what does clearing fractions mean? It means just what it implies. I don't want fractions anymore. It means I just need to multiply the whole set of numbers by something that gets rid of the fractions. And it turns out that in this case it is 3. So 3 times sine squared theta over sine squared theta min, and that is going to be 3. And that is going to be 4 and so on. We're almost there. We're almost there. And we're not going to finish today, but we're getting real close. That's good. It leaves us with a sense of anticipation and excitement. So, OK, I got clear fractions. What values of h, k, l, if I have this h squared plus k squared plus l squared, would give me the sequence of clear fractions? What values of h squared? Now, remember, you say, why am I doing this? Again, go back to this. This is why. I have a simple recipe for you to accomplish this goal. That's where we're going with this. So I'm now going to see with those clear fractions, now it's very easy to see what h, k, l's would give me-- h squared plus k squared plus l squared equals that clear fraction value. So for example, here, well, if this is h squared plus k squared plus l squared, maybe this is 111. And This might be 200, for example. So we're almost there. We're so close. What we're going to do is I'm going to start on Friday I'll put this up on the board and we'll finish filling it out and going from this matrix to the crystal structure and the lattice constant. OK, have a great Halloween. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | Goodie_Bag_4_VSEPR_Intro_to_SolidState_Chemistry.txt | [SQUEAKING][RUSTLING][CLICKING] CLAIRE HALLORAN: Today we're going to do Goodie Bag 4, VSEPR. Our objective is to visualize the three dimensional structure of some simple chemical compounds. The only thing you'll need is a molecular modeling kit. A conceptual question you should think about today is, what factors determine how bonds rotate in a molecule? So the first thing we're going to do is draw the Lewis structure for this molecule. So we know that Si, S, and CN, since they have between three and five valence electrons, make up the backbone of this molecule. And then we're going to add all of the atoms around each of these backbone atoms. Then counting valence electrons, we're going to make sure that we have the correct number of non-bonding pairs in bonds. And this will allow us to identify the geometry at each atom in the backbone. So this silicon bond here has four bonding domains, and all of them are bonded to another atom. So we know that this is going to be a test tetragonal geometry. This bond here, centered around the sulfur, has five body domains, and one of them is a non-bonding pair. We know that this will be a seesaw geometry. And then, finally, this carbon has two bonding domains with atoms in each of those domains, so this will be a linear geometry. And now we're ready to build our model. So first, we're going to draw the Lewis structure for this molecule. As with before, we're going to identify the backbone atoms in this molecule. So in this case, the backward atoms are C, B, and O. And now we're going to attach the other atoms to them. And now we're going to check and make sure that there are the proper number of valence electrons around each atom. Next, we're going to identify the geometry at each of the backbone atoms. So the first geometry centered around this carbon atom here has four bonding domains, all of which contain bonds to other atoms. So that's going to be tetragonal. The second geometry has three boding domains, all of which are bonded to other atoms. So this is going to be a trigonal planar geometry. And, finally, the geometry around this oxygen atom has four bonding domains, two of which are bonded to atoms, and two of which are occupied by lone pairs. So that's going to be a bent geometry. And now we're ready to construct our model. The Goodie Bag 4 worksheet asks us, whether some different bonds can be in the same place? So first, it asks, whether this H, Si, Si, S, and CN bond can be in the same plane? Remember that the CN bond cannot rotate, because it's a triple bond. But we can rotate around this S, Si bond, and get all three of the bonds to be in the same plane. The plane shown with my hand here. So here is our first bond. Here's our second bond. And here's our third bond. All on the same plane. Next, it asks if we can have this H, Si, S, F, and CN bond in the same plane? And so we can try to rotate to achieve this. But we notice that this isn't possible, because these two bonds are always in the same plane, and we can't get any of these bonds into that plane, because of the seesaw geometry. So for our second molecule here, the Goodie Bag worksheet asks a different question. We notice that this central CB bond can rotate, and the Goodie Bag asks, which configuration is the lowest energy? So remember that the entire VSEPR model is centered around the idea that we want to minimize repulsion between electron clouds by maximizing the distance between them. So we're going to rotate this so that the repulsion between these groups in the molecules are minimized. So as we rotate this, we notice that there are two different possibilities. Remember that these green atoms are CL, and this yellow atom is oxygen. These are the largest atoms in our model, and also the ones that have the most repulsive electron clouds. So we're going to want to maximize the distance between them. So one way that this can be achieved is by rotating the model so that the oxygen is aligned with the hydrogen, which has a very small electron cloud, and thus is capable of only very slight propulsion. So this configuration is low energy, because the very repulsive chlorine atoms are far from the repulsive oxygen atom. However, there's also a different possibility. If we rotate, we can put the oxygen atom in a plane in between the two chlorine atoms, so that the oxygen is not directly facing an atom. However, this configuration puts the oxygen atom spatially closer to a chlorine atom, as shown here, and this hydrogen atom is very close to this chlorine atom. So the first configuration is lower energy. Today, we used our molecular modeling kids to visualize the 3D structure of some simple molecules using the VSEPR model. By playing with these bonds and rotating them, we were able to understand how repulsion between atoms gives rise to the lowest energy configurations of these molecules. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 27_Reaction_Rates_Intro_to_SolidState_Chemistry.txt | How's everyone doing? [STUDENTS CHEERING] It's a goody bag. Thank you for that. It's a goody bag day. It's a good day. Related to the goody bags, once again-- hello. Once again, I want to get these out for you, because you got to have them on a Friday night. There's no real better way to think about spending your time on a Friday night than opening up a brand new goody bag. The topic of the goody bag is related to reactions, which is the topic we're starting today. But it's really also related to acids and bases, which is something we'll talk about on Monday. So I'm giving you the goody bag today, but I'll talk about the goody bag and contextualize it in the lecture on Monday. But as I said, today what we're going to do is we're going to start a new topic that we'll be talking about for the next couple of weeks. And that is the topic of reactions. And what we're going to talk about is reactions, OK? Now this falls under the domain of something called chemical kinetics, which is this study of reaction rates. Kinetic, move, right? So what's moving? Concentration, right? This stuff that you started with is doing something and winding up different. That's a reaction, OK? Now we've already written lots of reactions in this class, but today, we're going to talk about the rates. And the rate in particular is the changing concentration of the reactions and products with time. So there's a couple of terms that we got to know because we're going to be working with them. So I'm going to start by writing those down. So the first thing is the thing that we're going to talk about changing, which is the concentration. So the concentration is equal to the moles per liter. I mean, there are lots of ways to write concentration. This is the way we're going to write it, moles of a substance per liter. Well, that has a name for it. That's called them the molarity. What's molarity? It's moles per liter. And then we can also write it in shorthand, which I'll be doing as simply some substance in brackets. If I say that I have some substance A, and now I say, oh, look, I'm writing brackets around it, then what I mean is the concentration of A, OK? That's what that means. Which is the moles per liter of A. Good, so that's concentration. What else? Well, there's the rate. There is the rate. And the rate is, as you might guess, the change in the concentration with time. That's the rate that we're going to be talking about a lot today, the reaction rate. There's a change in the concentration with time. What else do we got? Well, we have then a law. Because see, here's the thing, the rate is just the rate. It's the change with something with time, right? It's like the change in distance with time would be like velocity. OK, but what if I wanted to say, well, could you tell me what the rate is as some function? Right, as some function? That's called the rate law. That's called the rate law. And it will give me a rate versus concentration. That's what I want out of a rate law. And then, well, I could integrate that. I could integrate that because the rate itself is a derivative with time. It's a change in time. So if I integrate it, well then I've got something called the integrated rate law. Integrated rate law. And what's that going to give me? It's going to give me the concentration versus time. So the rate law is a rate versus concentration. And the integrated rate law is the concentration versus time. And then we've got Arrhenius. Let's put it all on the same board. So I'll do one more here, which is the Arrhenius dependence for reactions. And as you might guess, this is going to give me rate versus temperature. These are the things we're going to be talking about. These are all the things-- all of these things we're going to talk about today in the context of reactions. I wanted to get them all down on the board so you can see them and think about them and feel them as we go forward. So there are a whole bunch of things that impact the reaction, the rate of a reaction. What is it that matters? Well, concentration. How much do you have? That's the brackets thing, right? Temperature, catalysts, the surface and the structure that you're doing the reaction on, the solvent that it's in. All of these things impact how the reaction happens. As you can see, I grayed out, made a little darker those bottom two because I'm not going to talk about those two. But I will talk about each of those other items today. That's our goal. How do those items play into these dependencies? That's our goal, to figure that out. OK, so we're going to start with concentration. And we're going to talk about how concentration plays a role in reaction rates. And we're going to start with something very simple. I have something called A. Oh, it's purple. And it turns into something called B. It's green. There you go. There's a reaction. OK, so let's start-- OK, so aA goes to bB. Now the very first thing that I can think about in this is that nothing is lost. Nothing is lost. So I'm going to assume that nothing disappeared from my container. You see that? The container is there. So I haven't lost any mass. I haven't lost any mass. Well, mass conservation is a big deal, right? So mass conservation-- by taking this very simple case, mass conser-- oh, really? We could have a mass conversation too. That's OK. That's kind of what we're doing. But let's write it as conservation. This means that if I take away A, I must add B with the same amount and not lose anything. And if you look at that in terms of these coefficients, remember, these are stoichiometric coefficients as part of the reaction. Then what that means is that 1 over b times the change in the concentration of B with time must equal minus 1 over a times the change in concentration of A with time. That is mass conservation, just written as changes in concentration. You can see it. Look, imagine just as-- imagine that this is 1, just as an example, OK? 1H2 goes to 2H, right? I'm not going to lose anything. Those are my stoichio-- well, OK, this would tell you that the rate of this if I don't lose anything would be 1/2 the change in H, the concentration of H with time equals minus the change in concentration of H2. That's just exactly what I wrote, but with an example. I didn't lose anything, right? So the rate of change of this must be twice the rate of change at that, right? Stoichiometric coefficients come in and they make sure-- when I write it this way, they make sure I don't lose anything. That's why you've got to have these there. Now there's a couple other things about this, all right? There's a couple other things about this. One is that you'll notice that the rate is always positive. Maybe we should write that up on the top board. Because it's a convention. But we like positive numbers when we talk about rate. So the rate is always positive, OK? Always positive. So when we have-- when we're thinking about a reaction where something's getting consumed, we're going to write it as the negative of the change in that concentration. Because that's a negative, and then we're going to write it as a positive of the change in concentration of things being formed, OK? So the rate is written as positive. Well, there is another thing, which is we can go more general. Let's just think about this. If we go more general, more general, then let's have more than just A to B. Let's have aA plus bB goes to cC plus dD. So now I've got four things. Two products, two reactants, right? OK A and B are the reactants, C and D are the products. And they've all got their coefficients. And so you can look at the change in any one of these and know the rate. You could look at the change in any one of these and know the rate. So if I look at this, then the rate would be equal to minus 1 over a, great convention. OK, it's a positive value. That's got to be equal to minus 1 over b times the change in B with time. And it must be equal to plus 1 over C, times the change in C with time. And that also must be equal to plus 1 over d, all right, times the change in the concentration d change dt. Right? So I could look at the way any one of these things is disappearing or forming and because those stoichiometric coefficients are there, I know what the rate of that reaction is. Again, this is basically a statement of mass conservation. Ah, but there's more because this is just a definition of rate. Rate is changing concentration and it's always positive, done. But I want a law. I want to know not just OK, yeah, I see how you change with time so that's a rate. But I want to know a law, that if I gave you any concentration, if I gave you any set of concentrations, what would the rate be? What would the rate be? Could I come up with a function? And so we write down the rate as depending on the concentrations. Now, you can do this with the products or reactants. We're going to do this with the reactants. And so you would have that it depends on the concentrations of a and b raised to some power with some coefficient. So k times a concentration of A to the m times the concentration of B to the n. OK, so we're raising these-- Now, we don't know what these are, but what we're saying is that there's got to be a way if I just know, and maybe if I do some experiments or something, maybe I could come up with a general function that depends on just the concentrations, wherever they are of a and b. And they're going to be raised to some exponent that I don't know yet and they're going to have some constant. That's a rate law. That's a rate. OK? Now, OK, the thing is that the rate law-- let's go over to here. OK. So the k is, as you can imagine going to be very important here, that's called a rate constant. K is equal to a rate constant. And this is going to depend on conditions like temperature, pressure, solvent. So it's going to depend on things like temperature, pressure, solvent, et cetera. All right? And we'll see how you get k. But m and n cannot come-- and this is very important-- m and n and k for that matter must come-- Gesundheit. --from experiments. They're determined experimentally, determined experiments. OK? This is a mistake that is often made. You see, m and n must somehow be related to these coefficients. No. M and n is something else. I've created a function that the rate depends on, the rate of this reaction depends on. I've put these exponents in there and I'm looking for the dependents on m and n. And the only way to do that, to know that, is to do experiments. You can't just get that from the way the reaction is written. You can get mass conservation and the relationship between rate, rate, rate. Right? Those are all the same. But you can't get the rate law unless you do some measurements. Now, then when you do, then you can get the order of the reaction. So this is the reaction order and that's an important property. There's one more thing I'll say, and then what we're going to do, I'm setting the stage here and then we're going to do examples. OK? So don't worry, we're going to go through different examples that I think will help crystallize these concepts. But the last thing I want to say in terms of setting the stage is that the units of rate, if you look at how I've defined the units of rate, it's a change in concentration with time. So the units of rate, the units of rate are going to be molarity over time. That doesn't change. Right? That doesn't change, those are the units of rate. That's the definition of the rate. But you can see, and we will see, that if that has to always be true, then the units of k may very. And we're going to see that as we go through examples. So the units of k could be different and we'll be depending on the reaction order, which is basically saying it depends on the rate law. OK, setting the stage. Now, let's look at some examples. And we're going to go through this sequential, we're going to do a zeroth order reaction, than a first order reaction, and then a second order reaction. OK? So zeroth order. Well, zeroth order reaction, right? So this is nitrous oxide. So this is a reaction of nitrous oxide turning into N2 and O2. Nitrous oxide is used in many, many applications, not just laughing gas. If we go with what we just wrote down. OK, so let's take this example, I'm going to write it down here. 2 N20 goes to 2 N2 plus O2. OK. Well, so the rate we know from what I wrote down, the rate is equal to minus 1/2 times the change in the concentration of N20 with time. But see, so now I'm doing some measurements and I'm plotting some data here. So I'm plotting here the concentration of all three of these, the reaction, and the two products with time. And what you notice is if I plot the concentration with time, it's a linear relationship. What does that mean? That means that they don't depend on concentration because the rate is the change right, the rate is the change in those with time, right? The change in the concentration of time is always the same because it's a straight line. So if that's true, then the rate from, let's say, straight line, straight line, and we'll talk about plots as we go, straight line plot of concentration versus time. All right, straight line plot, then we know that concentration is independent of time. I'm sorry. Rate is independent. OK, hold on. Rate is independent of concentration. OK. The rate in the beginning is the change in concentration. The rate in the middle is the change in concentration with time and it's a straight line so it's always the same. Well, that means something. It means something in terms of our rate law. Because now I know from my rate law, that the rate is equal to k, the rate constant, times N20 to the m. That's from here, right? You take all the reactants and you put them down and you put exponents on them. And so I've just got one, N20 and it's raised to the m. Yeah, but I know there's no concentration dependents of this rate. So I know that m equals zero and it's a zeroth order reaction, rxn. Oh, I've saved so much time, rxn reaction, which means that the rate is simply equal to k. The rate must equal k, OK? So in this case the rate, right, so we're continuing here equals k. And that means that k must have units of molarity over time. Why, because that's the units of rate, the units of rate don't change, the units of k will. But in a zeroth order reaction, the units of k have to be equal to the units of rate that there's nothing else in there. OK. Again, I want to emphasize, there is no simple correlation between the stoichiometry, the coefficients of these things and the rate law, you've got to do experiments. This data comes from-- Well look at it, there's the mass conservation in action, 02, N2, all right? You can see it. You can feel it. Slopes are different, they must be. Now, here's a reaction some of you may care about. So why does a zeroth order reaction matter. Why does it matter, because that's the reaction of beer. So ethanol, OK, but we'll call it beer. So what does this mean? If I consume beer, if I did, then the concentration of ethanol in it, there it is plotted versus time is linear. That means, this is very important, it's a zeroth order reaction. It means if I plot the rate of the reaction, you see it? There it is. It's a constant, it doesn't depend on the concentration. But that really has a lot of ramifications, because if I drink a lot of beer, the average 70 kilogram person, it takes 2 and 1/2 hours for the enzymes in their liver to decompose 15 milliliters of ethanol, one beer. But look at that, so that means that if I had more than that the body, it's not dependent on how much I have. The reaction of the liver, the enzymes that are breaking this down is not dependent on the concentration. So if I load up a lot more in there it doesn't matter. There is a pipeline and there's a rate and it's not changing, well that leads to consequences to be thought about when you drink beer. But there's another thing we can do with beer, which is we can integrate it. Right? Because if we know that beer-- if we know that minus-- I'm going to say there's some reaction that beer goes to something. So minus d beer, dt OK? Assuming it has a coefficient of 1, is going to equal, that's going to equal k because it's a zeroth order reaction. Right, it's a constant. And so OK, now I integrate both sides. If I integrate, then I've got the integrated right law for beer, which is that the concentration of beer is equal to some initial concentration of beer minus kt. This is the integrated rate law. That's the integrated law. Notice what I've done, I've gone from talking about the rate of the beer being the rate there, which is a constant to a dependence. There it is, integrated rate law. The promise was that would get me a concentration versus time and that's what it did, concentration versus time. How did I get it? Well, I had a rate law and I integrated it. And I'm not going through the math, but you put the dt over here and you integrate, OK? OK. So that's an integrated rate law. Well, let's go to the next order. If I had first order and I looked at the data for that, then you would see if I plot the concentration versus time it would not be a straight line. Why? Well, because for a first order reaction, so now we're in first order. Right, then what that means is that the exponent here is a 1 or if I have two reactants, the addition of them, it's first order total or it might be dependent on two reactants and its first order in each of them, but then second order overall because they're multiplied. Let's stay we the simpler case of just one reactant A. So if there's one reactant, and let's suppose A goes to some product. I'm not even going to write it out. You know that the rate is equal to minus the change in the concentration of A with time and that because it's first order it depends on the concentration raised to the first power. That's what we said first order meant, right. I'm telling you this is first order now. OK. Well, you can see that if it's first order it depends on the concentration. And so that means that you can see from that and you can see from the plot. If the concentration of A doubles the rate doubles. The rate doubles. Why can't our livers do that. No, they're zeroth order, but there are lots of reactions that are first order. OK, here's the thing right, but the rate still has to have units of molarity change with time, right molarity over time. They can't have that unless k units are k units, in this case, are going to be-- So let's see, let me write down-- OK rate has units. The overall units are molarity over time. So now I've got k times molarity, that has to have units of molarity over time and so k must have units of 1 over time. 1 over time, but that tells us what the plot. It helps tell us things about like-- But the plots always come better when we integrate. If we think about the integrated rate law, then it leads to linearity, which is something we always want in plots. Linear lines. Straight lines. Linear lines, really, I just said that? I did just say that. I'm OK with it because it's true. But let's integrate this. Integrated rate law. So now, there's my right law. Now I'm going to take the dt over and the concentration over and I'm going to integrate and I get this, A equals some initial concentration times e to the minus kt. Notice the units, right 1 over time. That's good. OK, or if you want, you could say that that's a Ln A equals Ln a 0 minus kt and this right now, this gave it to me. Because now you know that if you plot Ln versus time, if I have a plot of Ln versus time, plot of Ln versus time, then it's linear. And we like linear and that's what's plotted on the right. So that's a plot of Ln versus time. And so you can kind of go backwards and forth. Like if I had a linear plot with concentration versus time you know it's zeroth order. If I had a linear plot with Ln concentration versus time you know it's first order. And you know that from the integrated rate law. Now, if you have data, how do I know what the rate law-- I'm sorry, what the reaction order is. Often you'll just have data, you study a reaction and you get data. Here's a very important reaction. This molecule is called cisplatin. Now, cisplatin is maybe the most used and one of the certainly, very most important chemotherapy drugs. But it's not active in this state with the two chlorine's there. You've got to get one of them out and put a water molecule there, so that with the water molecule it can go and damage the DNA of the tumor. That's the idea and it can't do that unless this reaction happens in the body. And so you can imagine that this kind of reaction is extremely important. What? And the rate is extremely important here, all right. This has to happen in a time frame that we know very well. So this rate has been studied very carefully for this reaction. And you can see, OK, you can write down numbers, how much did I start with and how much what was the rate, and I'm watching the rate change. Right away you know the rate changes as the cisplatin concentration changes, it's not zeroth order. And the second thing you know is, if you take any two of those and you take ratios, then the ratios are the same so it must be first order. That's one way to know what order you have. All right. Yeah. So if I have data. With data like that, with data you could do this. You could take ratios at two different times. And what does that get, it gets you the order. From that you can get the order and we'll see another example. And then, you can go further than that because then I could plug in data from one of the times, right. Once I know the order I could use data from one time, from one time to get k. Because once I know the order, I know how to write down the rate law, and then I can use any of these lines to get k. So we want to be comfortable with reactions, with rates going back and forth between order, rate law, integrated rate law, plots. So if I had data that looked like this I could do the same. Here's another reaction, I could say what order is it? I'm not going to do it, I'll just say it. So you take any two times and concentrations and what you have is that the ratio squared gives you the ratio of the rates, that means it must be a second order reaction. It means it must be, right. I had something that went into products and the dependence of the rate from this data, the dependence of the rate is that something's concentration squared and you can see it from the data. So well, let's see. I'm going back to the middle here. OK, so now, we just went to second order. So for second order what do you have? Well, you have that minus d of something, some reactant, and I just got one reactant here, if you think about it as A. So the same thing A went to something, OK, so minus the change in concentration of that with time is dependent on k times the concentration of that squared. And if we integrate this one, integrate then what do you get, you get 1 over the concentration of the something equals 1 over its initial value plus kt. You can see that. Take the t over here and take this over there and then that's going to be the integrated rate law. So now, look at that. What do I plot to get linear? I got get to linear, I don't like curves in reaction rate plots, I like linear lines because the linear line will tell me something about the reaction order. And by the way, the slope could tell me something about the rate constant. Right? And so if it's second order, the plot that gives you a linear line is 1 over concentration versus time. So let's look. So let's say I had this data and I try to plot it. Well, the first thing I do is I say, well OK hold on, if I plot the concentration versus time that's what it looks like, not linear. That means it's not zeroth order. OK, I'm going to take the same data and I'm going to plot the Ln of it versus time, not linear. Not first order, but if I plot 1 over that concentration versus time it gives me a perfect linear fit, so it must be second order. Plots, order, linear. OK? We're not going to go beyond zeroth, first, and second order because what I want is for you to feel your oneness with reaction rates and kinetics through these three orders, zeroth, 1, and 2, and through these different concepts that are up there. So one more thing you can do, before I turn to temperature, one more thing you can do once you know the rate law is you can calculate how long it takes for the concentration to be cut in half. If I know the rate law, then I can calculate the half life, that's called. All right, so for example, if it's first order let's suppose it's first order then the integrated rate law-- I'm just going to put the two concentrations on the same side here-- A over A0. OK. So the integrated rate law gives me this. It's just what I had before somewhere. OK, but you can see, well what if the concentration of A is exactly 1/2 of the beginning concentration. So when A equals 1/2 the initial concentration, but then this becomes a 1/2. Right? So if I want to know how much time it takes to get to 1/2, that's a good thing to know just in general. How long do I have until half of this is left? Well, then you know that this is going to give you ln2. When A equals 1/2 A0 then ln2 equals kt. And we're going to call that an important. It's not just any time, it's called the half life, it's the time that it took to get to A being 1/2 of the initial. So I've got it. I've got it, I've got the half life. The half life is a very powerful tool. A number of you may have heard of carbon dating. This is just a picture of wild animals, but what happens, those wild animals breath in oxygen and other stuff from the air including stuff that contains a spread of carbon atoms and we know carbon has isotopes. We know about carbons isotopes, we've already talked about them, c12, c13, c14. C14 has this wonderful thing about it that it radioactivity decays. And it turns out that there's a certain concentration of that in the atmosphere at any given time and we can go back in history and know how much concentration there was in the atmosphere. And we know if these animals are breathing it in. We know that they should have that concentration at A0, except this isn't a concentration this is a number of atoms. But It's OK, it's still a first order process. The radioactive decay of c14 is a first order process. And so, because of that we can date, we can date things using carbon dating very accurately. This is a really awesome tool that relies simply on the half life, on knowing the half life. All right, here's a summary that I thought would be useful. This is it. This is everything we just talked about before we switch to temperature. Rate law, zeroth order, first order. And you're all like, why did I write this all down, it's all on this page. No, it's good to-- Did somebody say, yeah. It's good to write down, it helps you think about things. So here it all is. There is the zeroth order, first order, second order, OK? Integrated rate law that comes from that. The units of the rate constant. The linear plot to determine the rate constant. What gives you what the slope of that line is and then what the half life would be if you just follow this for the other two orders. OK so this is all that we just talked about and the best way to feel your oneness with this is to do some practice problems more than what we've just done. The next-- really? The next topic, I'm not going to spend the same amount of time on temperature and catalysts obviously, but I want to mention how these impacts reaction rates. We're talking about how fast the concentrations change. And to do that, what we're going to do is we're going to use the collision theory of reactions. And I've just written it down here what it is. What is the collision theory of reactions? What it is, is it says that they occur when particles collide. Why does a plus b go to c? Well, because things collided into each other and they got close enough or they maybe had the right orientation so that new bonds could form or others could break. And the theory also says that not all collisions result in the formation of product and that there are two factors that matter, the energy of the collision, and the orientation of the particles. I want to highlight those two factors. So energy of the collision. We're going to start with the energy of the collision and we're going to talk about that in terms of where you get that energy. You get it from heat. You get it from thermal energy, which means these things are vibrating and moving, maybe if it's a gas, they're bouncing off the walls of the container, that's temperature. How does that matter? Well, in order to understand how that matters, we have to understand, we have to think about this-- and this is what collision theory for reactions tells us to do is to think about a reaction in terms of an energy landscape and an energy barrier that you have to overcome. So here I have this Ea, which is the energy that it's going to take. I need to put that energy in for the collision to be strong enough. Think about it as like, well I just kind of didn't collide, I had a very low energy, so I'm below Ea and I can't get over this hill. It really is if you think about it like pushing a car, did I push it hard enough so it can go over the hill, or is it just pushed a little bit, that's the kinetic energy we're talking about, could it get over that hill. Because what we're seeing is that for this reaction to happen there's some amount of energy that's needed, that's the activation energy, that's the activation energy. OK. So what does that mean? Well, first of all, you should all be feeling something right now. And you know what I'm talking about, I'm talking about Svante, Svante Arrhenius. Right, somebody said it. You were feeling it. Why, because I'm talking about a process that has a barrier that is thermally activated, a process that has a barrier that is thermally-- Svante Arrhenius, crickets. Crickets and then intrinsic carriers. Activated processes, reactions, right, they have a barrier. They have barrier. Now here's the thing though, Ea is typically like maybe an electron volt, and we know that KbT is 0.025 eV. We've talked about this before and how is that possible. It's the same as what we talked about before because this is an average energy. If you put the Boltzmann constant in to room temperature, This is at room temperature-- If you put the Boltzmann constant in at room temperature you're going to get that small energy, but we know that thermal energy is a distribution. This is something we've talked about, so here it is again. So this is the Boltzmann distribution is what this distribution is called. And if I have my energy for activating a reaction, my barrier for the reaction to happen, shown there with the vertical line, it means that I have two temperatures. So yeah, kt may be small, but there's some tails out there where there is enough thermal energy, there's enough kinetic energy for this collision to lead to the reaction. So those tails grow larger and larger, the amount of molecules that have enough energy grows larger as you increase the temperature. That is Arrhenius. The probability for this to happen grows, that is what Arrhenius gives us. And so here it is mapped. This is something that is in the textbook that is a nice diagram because it maps the two together. You've got the reactant going to the products, there's an activated complex, it needs that activation energy and over there we've turned on the side these kinetic energy distribution plots. So you can see, when do I have enough? Well, when I'm over that activation energy and there is going to be some fraction of molecules that have enough and that fraction depends on temperature and that is Arrhenius. And so what we get is that the rate, the rate k, k the rate is equal to some constant A times e to the minus Ea over KbT. And remember, we use Kb for one collision, for a molecule or an atom. We would use R or r for a mole. Here I'm talking about a single event, one molecule colliding with another to make the reaction happen. Now, this is called the frequency factor and it's a constant. Constant, constant, and that's something that you could understand how this might depend on a combination of things. But it's assumed, incorrectly, but it's a good enough approximation for us, but it's assumed to be independent of temperature so we pull it out and call it a frequency factor. And it has in it information about whether these things really made the right collision or not, that's what A has in it. This exponent, Arrhenius is telling us this exponent tells me about that fraction, that fraction, but A tells me about orientation. So here's an example. Here's N0 plus 03, and on the top line you can see the N0-- right so, the O's are red and N is blue-- and the N0 is coming in, but it's coming in either way whether it's N or whether it's 0, it's coming into the wrong oxygen. It's hitting that wrong oxygen and so it doesn't do anything because the one in the middle isn't going to be very reactive, not at least at this temperature. But then you've got a different case where it comes in a little differently, and in fact, the nitrogen is what's coming in and leading the way to that oxygen on the end and that leads to the reaction. So you can imagine now it's not just how fast they're moving, which is what the exponent tells us, it's more complicated. Right? And this is called the frequency factor because it's all these complicated effects rolled into the frequency of the collision being right, not just the temperature. OK. And so this gives us the temperature dependence of the rate constant, this gives us the temperature and dependence of the rate constant, which is very important. This is not time, this is temperature. All right? This is a different thing than everything else we talked about here, which involved time dependence, that's temperature dependent. And finally, the last thing and just to show you this is my very last point, is that just like before, if I wanted to run a reaction and make it go faster, one way, just like with semiconductors is to increase the temperature, but I don't want to run my phone at 600 Kelvin. So what do we do, we use chemistry. I may have reactions that I want to run a lot faster. In fact, I very often do, and what do I do, I use chemistry and those are called catalysts. What a catalyst does, and this picture shows it in the context of collision theory, I've got these barriers I'm trying to get over, here it's flipped, it doesn't matter. There's a barrier here, so the reactant is higher than the product, it's OK, there's still a barrier that I have to overcome. What a catalyst does is without being consumed, what a catalyst does is it's another material in there where it allows that reaction to happen with a lower barrier. That's what a catalyst is and you better believe this is important. So a catalyst lowers that without changing the temperature. And in just 20 seconds, here's my why this matters. Catalytic converters is one of the most important technologies for pollution that has been invented in over the last 50 years. It's changed the game in terms of what comes out of that car. Why does it work? It wouldn't work, you wouldn't get rid of these things, these toxic things, you wouldn't get rid of them unless you ran it at 1,000 degrees and you're not doing that underneath your car next to the gas tank. But if you put the right catalyst in, all of those reactions to get rid of those toxic chemicals can happen much more quickly and efficiently. Right? OK. Have a great weekend. See you guys on Monday. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 24_Point_and_Line_Defects_II_Intro_to_SolidState_Chemistry.txt | So today, we're going on from zero dimensions to one dimensions, and we're talking about line defects in order to understand what they mean. We're going to contextualize it in terms of stress/strain curve. So we'll talk about that. And all of this is called the dislocation. So that's our goal for today. Now, on Monday, after lecture actually, a student came up to me and asked a really good question. And it was related to the Hume-Rothery rule. Remember, on Monday, we talked about point defects, zero dimensional. So you have a localized disturbance in the lattice-- of vacancy. You take it out. You can have that occur in a material that's ionic. Not a metal, but maybe an ionic material. And then you got to think about charge neutrality, Schottky and Frenkel. And then we talked about substitutional defects, which would be where you take an atom out of the lattice and you put another one in. And Hume-Rothery, I mentioned, had come up with some empirical observations and so I listed these. So he studied a class of metals and came up with these general guidelines. And as with so many things that we've learned so far, these are general rules that get broken sometimes. So the question the student asked was a very good one. Said, well, if the valence of something you're trying to substitute into something else has to be the same or higher, how does p-doping work? How does a p-type semiconductor work? That's a really good question, right? Because the valence is lower, by definition, that's how you make it p-type. A-ha. So my response was, well, because Hume-Rothery was designed around metal, I'm not sure how much of these apply to semiconductors. Although for semiconductors, you do want to make sure the sizes are similar, at the very least. But that got me thinking. And so I went and I read some papers last night. Here's one on the valence effects and relative stabilities-- valency effects. So this is from the '80s and I just wanted to cite something here. So here they say almost a half century ago-- so that's Hume-Rothery, so it's the '80s or the '30s-- pioneered the study of the effect of valency on metal alloy phase equilibria. They found that a lower valence metal, like silver, had only a very small solubility in a higher valence element, like antimony, but that the solubility of the reverse was large. So that's where that rule came from. And here's the next part of this paper I want to share. "Although the relative valency works well for that select group of alloys, it does not apply to the periodic table as a whole." So this other guy looked at 607 different systems and found that 22% of the time, the vacancy rule works, and 27% of the time, it violated the rule. And honestly, I'm not sure we can call it a rule anymore if more times than not, it's violated. This is a new thing for me. So that is just to say that this rule probably is really conditional on the specific elements you're talking about. I think the much more important ones are these three here. The valency rule certainly depends on the system. And all of the Hume-Rothery observations were empirical-- it's important to keep in mind. General guidelines meant to be broken more than half the time, apparently. [CHUCKLES] So where were we? But thank you for that question. It was a great question. It gave me some great reading last night-- materials. Zero dimensional, Monday. One dimensional, today. There we are. So that's the focus. Now, the story today starts at MIT. It starts at MIT in the 1940s. Because there were these two students-- there they are, Harold Hindman and George Burr. MIT students back then were a little bit older, I guess. So those guys, they were in an MIT lab and they were trying to study how parachutes break. How does a parachute break? Because why is that an important question? Because you don't want it to break. Right? And it's one of these things where you want it to never break. Like, you really want almost 100-- so they were studying why it breaks. So they were taking a parachute and pulling it apart, and there it broke. Take another kind of material, pull that apart. It broke. This one didn't, but it's too heavy. They were studying the relationship of the mechanical breaking to the material of the parachute. I love that logo there. That's the two of them in the lab at MIT pulling on a parachute. They just have a tug of war with a parachute. But the problem was that there was no instrument at the time that could give them the sensitivity to the force that they were pulling. There was no instrument that could capture what they needed to make a better parachute. So being MIT students, they built their own. They built their own and they added a whole bunch of electronics into this instrument. And that is where the name comes from. It's an instrument that had a lot of electronics for the time. Instron is the name of their company. Instron is now synonymous with stress/strain curves. So you go into any university, any company, you look at how-- if they need to measure something, like whether an egg breaks or not at what force, it's an Instron machine. It's an Instron machine. There it is-- same logo. And because those machines give you the most accurate and well-calibrated description of the force that you're applying on this material and you can apply it very, very slowly-- so these are very common machines to look at the force you apply, to take it how much distance. So we're going to talk about that today. Now, we're not going to talk about it in terms of eggs. We're going talk about it in terms of a wire. So there's a wire. It's a little hard to see there. There it is. There's another Instron machine-- this is a red one. And here they are and they're looking at this wire. And this wire is some material-- maybe it's copper, maybe it's aluminum. And they're saying, what happens to this thing as I just pull it in small increments? What is the force that I have to put on this to pull it a certain amount? Now, that is an important graph, and it's so important that I really want you guys to understand this graph. Because it is the way that mechanical properties of materials are first often analyzed and it's a plot of the stress, which is equal to the force per area versus the strain, which is equal to the change in the length divided by the original length. Now, what I mean by that? Well, let's take a wire. Here's the wire. There's that wire in there. OK. So here's a wire. And I'm going to apply a force this way. That's what the Instron is doing. So there's some force being applied to it and there's some area of the wire and there was some original length. OK. This is called a stress/strain curve. It's the force divided by the area that you apply to get the thing to stretch a certain amount. Now, those MIT students were also very smart. They knew that many materials, as you first start stretching them, the bonds between the atoms are kind of elastic. What does that mean? Well, it means that if I stretch it a little bit, it goes back. Might stretch it a little bit more-- it goes back. They're like springs. So when you think about the deformation of that wire and many other materials, you've got this elastic regime. And that means that it's reversible, as I just said. Reversible. And the displacement-- so it's reversible displacement. If I stretch it, it goes back. Reversible displacement only occurring under some applied force. Applied force. This is an elastic stretch. If I've stretched this material elastically and I let it go and it goes back, it's like a spring. Hooke's law is going to apply. So Hooke's law applies. So F equals kx or minus kx. You can press it or-- OK. Hooke's law. So that means that if I start pulling on this and I start here-- so there's no stress and there's no strain, and if I start pulling on this, then I'm going to get a straight line, like that. So that's going to be the elastic regime. But these MIT students-- the thing is, they knew something else. They knew that for a lot of materials, it didn't keep going like this. It didn't keep going linearly in this nice, elastic spring way. They knew that either the thing broke completely or it deformed-- or it started deforming. And that is called the plastic regime. We're coming to plastics later. And this is a permanent shape change. Permanent shape change. And you can imagine, if I'm going to permanently change the shape of a material, it's got to be ductile. So if it's something that's going to break-- remember, we talked about ionic versus metallic. The sea of electrons allows those metals to be ductile. And that has to do with the bonding with the electronic structure. If you want a material to be able to change shape permanently and not crack, you need that ability to be ductile. So there's a plastic regime. And that makes this change dramatically. So what happens now is, if I keep stretching it beyond this elastic regime, the stress/strain curve is no longer linear. And in fact, it might look something like this until, at some point-- so this would be the plastic part where it's literally deforming, it's changing its shape. And then finally, I've stretched it to the point where it fractures. So that's called the fracture point-- you just can't pull it anymore. You just can't pull it anymore. And they're studying parachutes, but they knew that there were a lot of surprises out there, there's a lot of complexity in this. And they needed to understand this regime and this regime in order to really engineer the parachute to make it better. They had to get that data and there were no instruments that could move it and measure it and calibrate it carefully enough. By the way, this plastic deformation that happens-- this permanent shape change, but without breaking, that happens very early on. So there were people working, for example, with aluminum, saying, well, given the bond strength of aluminum, I predict a certain mechanical strength that aluminum could go to. And then they find that it starts deforming 100 times lower than that. 100 times lower. Why? Because it enters into this plastic regime. It doesn't stay elastic. So what I want to talk about today is how-- I hope I've convinced you how important this is, and I'll have a why this matters on that as well. But what I want to talk about today is how these one dimensional defects fit in with this. How these one dimensional defects fit in. And in fact, they explain it all. They explain it all. This is what the wire looks like. When you stretch a wire and you look at it carefully, it's not just a little bit longer. It has these very set features that occur. It's not just deforming in any random way. It's deforming in a very specific way. It's deforming in a very specific way. And so what I want to talk about is that way. The plastic deformation mechanism is called slip. The mechanism is called slip that leads to plastic deformation. And it leads to that picture and it leads to the data that they got from the Instron machine. So where does it come from? This is a 2D picture of a lattice. This is a 2D picture of a lattice. So you can imagine what I've done-- oh, I'm going to hand this out. I love this. I can just sit and look at this-- this is so cool. So this is a cubic-- you can think about it as a simple cubic lattice. Here's a plane, here's another plane, there's another one. Look at that yellow one there. So many planes. I got lost there just thinking about it. I'm going to the other side. You could get lost, too. That's one of the planes. That's one of the planes. Imagine it. Now, you notice here, I've got hanging out-- because why not? Oh, why not? Because there's no other possibility. Because vacancies always exist, as we learned Monday. They cannot not exist. So there is one right there. Cool. There's an interstitial point defect, there's a substitutional point defect. That's all from Monday. They're just hanging out. But here's what we're talking about now. Look at this. I want to take this little symbol-- this T, and I want to look at the lines that come down from it. So there they are. So these are now planes. These are planes. These are crystallographic planes. And now, look at the other side of it and notice those same planes are following-- see to the left there? You see how that's now a plane, a plane? So this is actually lined up from the outside in. It's lined up, it's lined up. Ah! Right here, I put an extra one in. You see that? That's an extra plane that doesn't go all the way through. If it went all the way through, it wouldn't be an extra plane. So it's sandwiched in here. And then here, it continues over there. That one kind of goes like that, and that one goes like that, and then they go back to the normal planes. I have inserted a half-- if you want to think about it-- of a plane into-- right there. Look at that. That is called a dislocation. That is a one dimensional defect. It's a dislocation that is caused by putting an extra plane that stopped somewhere, and where it stops, you draw a little T in the crystal. Oh, let's go with the mean. Come on. There we go. We'll go back to this. So that is called a dislocation. That is called a dislocation. That is the 1D defect. It's called a dislocation. It's a line defect formed by what is essentially a misregistry of atoms that gives you an extra plane. Now, you can already see that if I put a line defect in there, it's messing with the bonds at this place here. Here, the bonds just go along, they go along. And then all of a sudden, I've got a misregistry now because of that dislocation-- because of that dislocation. One way to look at it is with the model. Another way is just get ears of corn. I did this last night. I actually-- I got ears of corn. That's really true. And I looked inside. This isn't from there. This is from the internet. But there it is. The corn needed to grow another row and so-- it's not going to grow a half kernel. Well, actually, it could. It does sometimes. But here, it grew another full row of kernels. And there it is. There's a dislocation. It's very similar to what we're looking at. Now, there are two types of dislocations. This one, where you insert an extra plane in this way is called an edge dislocation. There's another type of dislocation that we won't talk about in this class called the screw dislocation. You can look it up. It's a different type of line defect. But the one that we're concerned with is called an edge dislocation and it's described exactly in this way. And the way you note it in an atomic scale drawing like this-- by putting that upside down T where the edge is. An edge dislocation. Edge dislocations in corn. Now, you can actually see these things. And on these models, the dislocations look very nice and uniform and you can hold it and look at it. But in reality, here's an actual video of dislocations. These are groups of dislocations. Notice, these are 1D line defects. They're places where you've got this extra plane. Notice, that they're all over the place. They're not just straight, they bend around in the crystal and they move. And the movement is critical. The movement is critical. That's what's going to get us back to this, the movement. This is the defect, and then the movement is what gets us this. So let's take a look. So here's an here's an example. This is a piece of material that under the microscope is being pulled. Watch what happens. Well, those things are moving. They're forming, they're ending, they're reforming, they're interacting with each other. That one didn't, but whatever. And they're spreading out, they're growing. Those dislocations and their movement is why this doesn't crack. It's why this can move, this can bend. It's why this point is 100 times lower than you might have thought. And that can be understood by thinking about the bonds. That can be understood by thinking about the bonds. Here's a sequence of pictures. So this is the dislocation. You see it there. Now, what I'm doing is, I'm pushing-- or let's not push yet. We'll do that on the next slide. For now, I'm just watching it move. Here you see it moving. There it is. How does it move? Well, here's a model. There is a dislocation and you can see that if I take this bonding area around here and I connect this atom to there and then you start pushing-- there's a strain field around there-- and then you push it over, you might be able to move it over to there. And maybe you could keep on doing that and keep on moving it over and over until it gets all the way to that end-- all the way to the end of the material. Here is a super polished, high detail animation of that process. Here it is. I'm starting over here and watch the dislocation. There it is. There it is! And it's moving and it's moving, and those bonds are breaking. Let's watch that again. Here it is. There, it formed. But why did it form? Because, you see, what happened is, I took this material and I applied a force to it. I went back to the wire and I did this or maybe I did this. And the material said, OK, hold on, how can I respond to this thing? Well, either I can respond elastically and all my bonds could stretch, but I'm getting to this uncomfortable place here. Or a dislocation could come in and allow me to translate a whole set of bonds over by one. So I'm applying a force to this material. You can think about it as a force maybe on the bottom g-- here, let's watch what happens. Look at this. I've got a little extra room here. Now watch what happens. There it is, there it is. And I have moved the whole top row of atoms over-- the whole top three rows over by one. So I have transformed this so that the force-- if you think about it, the force on the bottom could be applied that way and the force on the top that way. The atoms get to a certain point where it's like, no, my elasticnesss-- I'm not sure I want to go anymore. But a dislocation comes in and saves the day and it allows that to happen. Because here's the alternative. if I want that same translation to happen, that same slip, I would have had to break all the bonds at once. If I didn't have a dislocation, this is what would-- ah, there's the dislocation. One bond at a time, one bond at a time. Or the much, much harder ask, especially since there's 10 to the 20 something of these, is to break them all at once. There it is. Bam. That's why that doesn't happen. It just takes too much energy. But if I can just call up a dislocation to the rescue, I can translate an entire set of atoms with much, much, much less energy. Did you see that? So the dislocation moving is what allows plastic deformation, it's what allows these atoms to slide over one another. It is what enables this region to be there and it's what defines it. A very important point here-- aw, you got a goody bag. That's OK. Was he in the class? It doesn't matter. We share. [LAUGHTER] If you guys have friends-- just came in and took one. If you guys have friends that need goody bags, we are here to help. I will never turn down a goody bag. There is a plane here that's moving on another plane. There is a plane that's moving on another plane and it's doing it only because of this location, that one dimensional defect. Now, that is slipping, so it's called a slip plane. It's called a slip plane. And it's the plane along which the dislocation moves. [PHONE RINGING] OK. We got a phone going on there. Now, I want to make a really important point here. Because what the dislocation is allowing, what this defect is allowing you to do is resolve forces. That's what you're doing. I'm putting this force on-- it's again, the Instron. I'm pulling the wire and the material is like, hold on, I got to respond to this. How can I do-- yeah? [INAUDIBLE] Yes. And I will come back to that. It's this plane. Where's the model? Who's got the model? Is that the dislocation model? It's the plane where the dislocation comes in and can move. Why? Because that's the plane that is-- by motion of the dislocation, that's the plane that's slipping along another one by exactly this animation. I'll talk about this slip plane in a minute also and more about it. So what this motion, what this slip plane, what these movements of the planes allow is is it allows you to resolve an applied force. So I come at this material with a force and I resolve it. It's resolved at the atomic level-- and I'll finish this-- at the atomic level along that slip plane. That's what it does. The Instron pulls or I pull on the parachute, like the logo, and then either the material is going to break or maybe it's just in that elastic region and then it will go all the way back or it's going to deform. And if it deforms, what's it doing? It's feeling the force and it's trying to resolve it. And it resolves it by those dislocations moving. The planes move on top of each other. Well, so let's go back to this plane. What plane is it? If I look at the planes, which are somewhere out there-- you can imagine that if I've got all those planes and they're looking at the Instron and the force being applied to me, which one is going to do the slipping? Well, you can imagine this with a very simple analogy. If I push on something-- if I push on a rope, what does it do? Well, it kind of curls up, it doesn't do too much-- it doesn't move. But if I really want to try to resolve force, I got to move those atoms. But if I push on a stick and not a rope, then the stick just all moves. So the plane that is going to move across another plane is going to be the most dense-- the closest packed. It's going to be the closest packed. So the slip plane-- let's write it here. Slip plane-- and I'll give you an example in a minute. Slip plane, closest packed or highest planar density. Because then I'm pushing on the stick. In the material, I've got stick and ropes defined by the plane. You can see it over there. Some of those planes, they're beautiful-- the yellow one, but it doesn't have that many atoms per area. So if I try to move that one, it's not as strong of a plane to move and to resolve my forces at the atomic scale with. So I'm always going to go for that strongest plane. So that's a slip plane. Now, a great analogy for this is the rug. So if you're helping people move or you're setting up your own rug, and you get it and you put down the rug base like that, and then you put it down, and you're, ah! I missed it by a foot. I got it wrong. This rug weighs-- rugs are really heavy. You pick it up, you're like, oh, I'll just move it. You call a friend and you both pick it up, and you're like, this is 1,000 pounds. Who knew? It's really a pain. And so then you think about dislocations and you think about how a material would handle this situation. And you say, well, if I just make a little dislocation, like a little crinkle there, and I move that, think about how much effort it takes to move that instead. Not much. You make a little crinkle, and then you move it, and then you move it. It doesn't take much force to do that. It takes a lot less than picking up the whole rug. That's the motion of a dislocation. We just talked about the plane that it is, but which direction? Which way should you move it? And so this is the second part. So you have the slip plane, the closest packed or highest planar density, and then you also have the slip direction. The slip direction is the other part. Which way does it go? I've got the closest packed plane. Which way should it go? Well, for this, you can see from the experiments, and when you look at the wire-- again, if I take a copper wire and I pull it and I look at it with my eyes, it just looks like the uniform wire's been stretched. But now I magnify it. This is what you see. So you can see that there are very definite directions. This is actually all the same direction. So that's a slip plane and there's a direction that it slides. I think this can be understood-- I love this picture, because I love ping pong. These are ping pong balls and you guys can do these experiments yourself. You put a bunch of ping pong balls in a row, and then measure the force that it takes to slide it depending on whether they're closely packed or not. This is a little counterintuitive sometimes when you first see it, but it makes sense. The more densely packed the plane is, the easier it is for one set of atoms to slide across the other. The easier it is. So you can feel it. If I have to let this one-- and they showed it just by angle at which it would slide or roll. I like the idea of two people going in there-- and here, you'd have to apply more force to make that one slide than this one. You can see it, again, with this super precise animation. There it is. One of them and there's the other one. Look at how much more work you got to do to get that thing to go over and over. No. If you've got a high density of atoms, it's actually easier to slide. It's easier to slide because you've got more bonds. And so I'm not having to fully break a bond before I get to the next one. So the slip direction then is going to be the highest density direction, so it's also the close-packed-- I've got an example there-- close-packed direction. So notice, this would be like a direction-- a vector and this is a plane. Well, that's jogging some memories of understanding crystals by their Miller planes and directions. We say, well, what would this be for, say, FCC? There's an FCC crystal and there's an FCC lattice. So if I had, say, FCC as an example-- well, OK, so let's see if I can really quickly-- oh, boy, here we go. There it is. And here are the face atoms. There we go. OK. That's the outer part of FCC-- three of the faces. And now, OK, but if I take a 100 plane-- so let's take the 100 plane. Then I've got that. And you'll remember, I've got these atoms on the corner. Why did I draw that so much bigger? Let's not do that. That would be like the 100 plane. And this one might be the usual lattice constant A. You could also look at the 110 plane. Here it is. The 110 plane. That would look like this and this. That would be the 110 This is just bringing back memories-- I know good ones. Here is the 111 plane. So here, we've got this one, this one, and this one. In an FCC metal or crystal, which plane is going to slip and along which direction? You can get that now. Because you can calculate the planar densities. In which case is the packing of atoms-- in which of these planes is it the highest? Well, you've got to know the number of atoms. So how many atoms are in this? 2. How many atoms are in this? Oh, why? 1/4, 1/4-- I'm just counting in the unit cell of the plane. So 1/4, 1/4, 1/4, 1/4-- 1. It only counts in the plane if the plane goes right through the atom. So here in the 110 I've got 1/4, 1/4, 1/4, 1/4. OK. That's 1, plus 1/2, 1/2. Two. I've got two atoms in that plane-- effectively, two atoms in that plane. But what about here? Oh, boy. OK. 1/6, 1/6, 1/6. You see it from the geometry it's a triangle. So that's 1/2 from those corners, plus 1/2, 1/2, 1/2. 2. I've got the same number of atoms in those planes of the unit cell-- of those planes drawn within these unit cell boxes. But they're different planar densities. They're different planar densities. Because the area here is equal to a squared in this case, it's equal to root 2 a squared in this case. So it's larger. Same atoms, larger area. That's not going to have a higher packing than that. And now I get to have really a lot of fun with triangles and thinking about the height here and stuff like that. And the area of this one is equal to root 3 over 2 a squared less than 1. That's going to have the highest planar density. You know it. That's going to have the highest planar density. So that's a connection now. So you can connect now what we've learned about Miller planes, about crystallography to this simple calculations of density, to this incredibly important behavior of materials-- plastic deformation. Because now you know how that's going to plastically deform. It's going to deform along its closest packed plane. And you can also look at the directions. And it's shown here that the slip direction will be along one of those-- that's the highest density direction, the 110. So it's going to slip along one of those directions. Good. Slip plane plus this together combined-- these are called slip systems. The slip plane and the direction. That's called the slip system. OK. Now, there's another thing that happens with this location that's so cool and so important to materials and to the things we want to do with materials. It has to do with the fact that, like I said, the dislocations come in and they can be a total mess and entanglement. By the way, how did it come in the first place? If I started with a perfect crystal, where did it come from? How could there be a dislocation? It can come in from the edge. You've got an edge there. There can be a mismatch and I call it up-- I need a dislocation or I'm going to break. Boom. Comes in from the edge, comes in from the edge. They come to the rescue. This is a simulation of exactly that done by the chair of the Department of Civil Engineering, Marcus Beuhler-- a simulation he did some years ago. But he's modeling the dislocations-- that's what you're going to see here-- as a piece of metal is cracked. Watch this. Watch them come in. It's saying, I need you, help me. And there they go. Now, look at what happen-- there's so many of them forming and forming, and now tangling and tangling and tangling. Then he's going to zoom in. It's very cool. Those are dislocations like we saw in the experiments, but this is a computer simulation of them. They're still kind of tangling. Here they go. They come in, they get called in to relieve the atomic force and let the material slip. But there's something that happens now. There's something that happens. Because now we go back to the rug-- we go back to the rug. And look at what's happened. I'm going along and I did the calculations and I had no one to lift it, it's too heavy. But I'm going to create a little dislocation and move it along. And I come across someone else's dislocation here. Look at this. I'm trying to roll mine and there's another one there. What's going to happen? I can't go anymore. I'm blocked. I'm blocked. So when dislocations tangle up, you can imagine now it prevents dislocations from moving. But dislocations moving is what gives me the plastic regime. So as I introduce dislocations, I make it so that it's harder for dislocations to move. There's actually a name for that. There's a name for that because it really changes the mechanical properties. It's called cold hardening or work hardening or strain hardening-- sorry, cold work. Cold working. So work hardening or sometimes strain hardening. Why is it called hardening? Because you're literally making this material harder. What's happened is, I've taken away ductility, so I've made it more brittle. I've taken away the plastic deformation. It can't do as much plastic deformation, because that only happens by motion of dislocations and I'm locking them in. But I'm letting it maybe go a little further in how much it can elastically deform. Maybe now, once it gets to here, it says, you know what, I've got all these dislocations in there, It's not letting me plastically deform, so I'll keep elastically deforming. But that gets you up to much higher stresses. It gets you up to much higher stresses with elastic deformation. And so this is a plot-- look at that. That's called the-- oh, I should have put that down there. Let me label that because it's a pretty important point. I labeled the fracture point. This is the yield point. There it is, the yield strength. That's called the yield because it's where it yields to plastic deformation. So here you go. You've got some steel, you've got some brass, and you've got some copper. This is plotting-- look at that. This is the percent cold work. Cold work is work hardening. Work hardening is adding dislocations on purpose-- on purpose. So that what? So that I increase the yield strength so that this material can elastically deform now up to higher and higher and higher strengths. But the ductility goes down. When you bend a paperclip, don't you go and tell anybody it's because of heat that it breaks. That's not why a paperclip breaks. It gets hot-- or warm. You bend it back and forth, it breaks because you are putting dislocations into it and you're making it more brittle. In fact, if you heated it up, you would anneal those out and make it more ductile again. That's how you get rid of the dislocations-- you've got to heat it back up. It makes it brittle, the ductility goes down. Why does this matter? Well, this is one reason. If you don't plan your cold work carefully, you might make the material too brittle. You wanted it to be so hard because this was such an important ship and it was going to be a big deal and the launch was really exciting. And then it cracks in half, the entire ship. Why? Because you didn't look up what dislocations mean. You didn't take 3.091, that's why. That's a pretty big crack. The main why this matters-- oh, I couldn't help it. I am a big fan of wind. Wind energy is growing and growing and it's such a great national resource. Here's the global capacity. This is install capacity for wind turbines. But see, this is a mechanical materials problem that you are now equipped to think about more deeply. Because, you see, you can do a lot of different experiments on those turbines. So the blades here are critical. You can imagine that you want them to be light, but if they're too light, they may not be strong enough. And then how do they need to be strong? Because you've got huge amounts of wind coming at them. And it turns out, you need to hit just the right balance of elastic deformation before it goes into some plastic regime. You need to hit just the right balance of ductility. So here's, for example-- these are some simulations. Here are some experiments on a new material for a wind turbine blade. And then you put it out there and look at what happens-- ice. By the way, this ice comes off at hundreds of miles an hour in chunks. These farmers are not happy about that. Seriously. And those are bugs. Actually, bugs in wind turbine blades is a serious problem. How do you clean bugs off of it? Because it dramatically changes the aerodynamics and the efficiency. It also can damage the blade itself. So there's all sorts of work going on. How do you make bug-proof wind turbine blades? OK. Well, now just spray it with something. Ah, but then does it have the right plastic deform-- does it have the right yield point or is it just going to crack? And by the way, it's got to have 5 times 10 to the 9 cycles before it can fail. That's the metric. So that's a pretty big ask of a material. It all comes down to understanding this curve. And in the broader sense of materials, this to me is a very exciting ask. Why? Because if you look at a plot of the density of materials-- heavy, light. Good. Kilograms per meter cubed. And the Young's modulus-- now, this is a measure of the strength of the material. It's a measure of how much strain you could put on the material before it breaks or goes through deformation. But look at this. Different materials are here-- rubbers, foams. OK, foams have relatively low Young's modulus, but they're really light. That could be good. Up here you've got metals and alloys, ceramics, you've got polymers in here. But notice, I've got so many different applications and needs in the applications and I've got this plot where I've got nothing here and nothing here, even though, if I could fill this out and dial up any stress/strain curve for any density or Young's modulus, you can make a big difference in a lot of different applications. So I think this is a great challenge. Have a good night. See you guys on Friday. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 29_Acids_and_Bases_I_Intro_to_SolidState_Chemistry.txt | We're going to pick up where we left off on Monday. And where we left off was we were dissolving stuff. We talked about how things dissolve. Why does something dissolve maybe in this solvent versus that? And how does it dissolve? And what makes them-- and then we say, well, what if we had something and we wanted it to dissolve all the way until it couldn't dissolve anymore? Well, that's called saturation, right? And we talked about that. We talked about how if you had a general reaction, maybe we had something like this, aA plus bB goes to and comes back from cC plus dD, then this reaction can happen in both ways, right? And so the general concept that we called a reaction quotient, reaction Q, is equal to-- remember, it's equal to the concentrations raised to those stoichiometric coefficients. And it's the products over the reactants. OK, so that's what we-- now then, we say, well OK, that's a thing that has to do with maybe-- where's that reaction going? What's dominating it? And remember, these are the stoichiometric coefficients, and they're related to literally the probability that that reaction can happen. OK, and so if you want an intuitive reason for the exponents, you can think about this as, well, if these all have to get close enough in a given volume to react, then that's where those exponents come from. But unlike the rate law-- and I said this Monday, and I'm saying it again today-- they are not things that are experimentally measured. Instead, they just come from the reaction itself, OK? So we're just getting back into the right mood here. But then we said, well, but look. OK, this thing is reacting. It's going back and forth, whatever. But what about once it reaches equilibrium, all right, equilibrium? And when it reaches equilibrium, it's the same thing, but we call it Keq, in equilibrium, eq'm, equilibrium, right? So that reaction quotient has a very specific value, a constant, a constant, once this way is happening in the same amount as that way. Right, that's the saturation point. Well, if you're dissolving something, that's the saturation point. OK, so it's where the precipitation-- remember, we wrote this as dissolution and precipitation when we were putting stuff in solution. And that's where we got to. And we went a little bit further, and we had a specific example. And that's where I want to start today, and that's silver chloride. OK, so here we go. So we have silver chloride. And if I write this all out for silver chloride, I've got AgCl. And I've got a little pinch of it, like a little speck of silver chloride. And I'm putting it into a beaker. So it starts out as a solid, and I add some water to it. And the water is l, because it's everywhere. It's the liquid. It's the solvent. OK, it's the liquid that I'm dissolving it in. And we say, well OK, that's going to go like this. So that reaction, it's going to give us silver ions-- remember salts, we did salts-- and chlorine ions, and back to some more water. OK, that's H2O. Now, the aq means aqueous. Because it's a salt, right, these are now dissolved ions, because it's a salt. So we write them like that. And we know we got water everywhere. So oftentimes, we leave the water out. Oftentimes we don't really write the water, because it's on both sides. And anyway, the water is just surrounding stuff. It's not necessarily being consumed. The concentration of the water isn't changing. The water is the water in the liquid phase. Well, OK, then we went even further and said, well, if you write a Keq for this, then that would look something like the concentration of silver plus in solution times the concentration of chlorine minus in solution. And that would be divided by the concentration of silver chloride as a solid. And then we said-- but hold on. But hold on. In the solid phase, the concentration isn't changing, right? As a solid, the concentration of silver chloride is silver chloride. It's a constant. And so then we said, well OK, if this is a constant, this is a constant. Then we can also absorb it into the equilibrium constant. And I somewhat confusingly wrote it as Ksp in both this form and the form that we know and love for Ksp, which is simply literally the solubilities of these times each other. Right, and that is Ksp. So I wanted just to clarify this, because a student asked a very good question, which is, which is it? Is this in Ksp or is this in Keq? Well, the big full picture for Keq, you'd write it all in. But the solubility product is focusing in on the dissolution of these ions. And so you wrap this into the equilibrium constant, because it's another constant. So that's the solubility product. And that's where we got to on Monday. The solubility product is a special equilibrium constant, where what we're talking about is-- what are the concentrations when you've reached saturation of this dissolution precipitation reaction, OK? That's the solubility product. That's Ksp. So you wrap this in, and you get Ksp. You can think about it as, well, maybe you could have called this some Ksp in a pre-Ksp. And then you wrap this in, and then it's still just [INAUDIBLE],, because it's still just a constant representing this reaction. OK, I just wanted to make this super clear, because this part leads to confusion often, that this gets left out, because it's constant. It doesn't change. OK, good. Now we get to that, because this is a constant. OK, so what does that mean? Well, it means that if I dissolve this thing in water, then the equilibrium, the equilibrium is a fixed number. This equilibrium product is fixed. And that has a very important meaning. It's the green curve. You see, it says equilibrium. And so what that means is that if I'm at point B, well, there I go. I've got my-- we said Ksp. Ksp for this one equals, what was it, 1.7 times 10 to the minus 10th. Units are coming. And so that's the Ksp for silver chloride and H20. Oh, and you have to say at some temperature. It will depend on temperature. You often will just be at room temperature, 25C, for example. So often these get quoted just at room temperature. OK, so once I know that, then I know that that's equilibrium no matter what, right? And so that means that if-- and then we solve this. You said, well OK, if I dissolve silver chloride, and I had nothing else, then the Ksp would equal the concentration of silver times the concentration of chlorine ions. But those are equal. And so we let each one of them be x, then it's equal to x squared. And so we know that the concentration of each of them is 10 to the minus-- let me write it exactly-- 1.3 times 10 to the minus fifth. Now, it's a concentration, so you know that it's got to be capital M, moles per liter. And since that's the case, since that's where you know what it is, then now you can back up. Because this is the concentration of Ag ions and it's also the concentration of chlorine ions, then you know that the units of Ksp must be M squared, all right? Units in this case, moles per liter squared. Must be, because otherwise the concentration isn't right, the units of concentration. OK, now we're going to play with this. OK, so I'm at B. I'm at point B. You see that? There I am. And I've got my concentration of chlorine, concentration of silver. And I'm here. And so that's 1.3 times 10 to the minus fifth. OK, now if I didn't have as much, if I weren't at the saturation point, then I could add more. I'd be here. And I could add more solid silver chloride. And eventually, I'd reach this equilibrium. But if I added more than that, so now I'm adding way more than I can, then it's just going to be a solid. And that's kind of boring. So now I'm up here. But maybe I add ions in. So these are the ions. So maybe I add a whole bunch of ions. Well, they're just going to precipitate. I can't be here and be in equilibrium. That's the point of the green curve. This K doesn't change. This K is this K. That's what the green curve is. It is that being constant. That's very important. That is equilibrium. Now, there is a way though to move around on it, because if I did, imagine now-- look at point C. Imagine now that I've found some source of chlorine ions and I add a whole bunch of them, right, a whole bunch of them into the solution. I don't want to add silver and chlorine together. I just add chlorine. Well, you can see what's going to happen. I have to get to equilibrium. And so in order to stay in equilibrium for silver chloride in solution, then I must lower my concentration of silver, because I added all this chlorine. The only way to do that is to eat up some silver and precipitate. That's the only way to do that. And that has a name. And that's what I want to talk about next. That is called the common ion effect. Now, this is an example. This is an example of a broader principle which I mentioned Monday, which is Le Chatelier's principle. OK. And that is that position of equilibrium will move. It's still equilibrium, but it's moving on that green curve to counteract change. This is a very general principle. It applies to many, many things, not just concentrations. It applies to changes in pressure and other changes that you make in a system. Here we care about concentration. And in this particular case, we're going to see it for what happens when you add ions of one particular type and not the other. That's the common ion effect, but it's a very general effect, that it resists the change. Now, let's see how that works. If I want a source of ions, let's say I want a source of chlorine, one way would be to add a whole bunch of sodium chloride. So let's suppose that I have-- OK, so as an example, I'm going to add to this nice equilibrium that I reached here-- so I'm at point B, OK? And I'm going to add 0.1 moles per liter of sodium chloride. And this nearly-- and so we're going to say it does-- it nearly fully disassociates. And we'll be talking about that later today. And so that means that NaCl basically goes to Na+ plus Cl-, both in solution. So if I put 0.1 moles per liter of sodium chloride into this container of the silver chloride, I'm basically going to get 0.1, because of the stoichiometric coefficient here, 1, 1, 1, right? So if I put 0.1 moles per liter of this, I'm going to have 0.1 moles per liter of chlorine ions that I just dumped into the container. But the sodium is not going to do anything. But look at the chlorine. That's going to mess with my equilibrium. But K is a constant. So let's see what happens. We can do it visually. But we can also do it back when we use the ICE table, right? So now we're going to go ICE again, AgCl. And here's Ag+. And here's Cl-. So now I've added the-- OK, so my initial is that this is solid. And remember, this is the solid that I'm putting in there. And then my initial is that I started in that nice equilibrium at point B. So I've got 1.3 times 10 to the minus fifth. And over here, I've got the same. But now the change, now the change. OK, so we're going to add some chlorine ions, 0.1. So I've added 0.1 moles per liter of chlorine ions from the salt, from sodium chloride. Yeah, but now some of that is going to react, because I just said, Le Chatelier's principle tells me that now it's got to counter that. And you can see it right here too. I'm adding a whole bunch of chlorine. Look at how far I'm going out. 0.1, that's literally all the way out to here. And so the amount of silver has got to go down. And what's going to happen is it's going to consume some of the silver and form precipitate to counter all this addition. It's going to run the reaction the other way. So it's going to lose a little bit there from the 0.1. That's going to react with the silver ions. So that's going to also lose, and this is going to gain. That's how much that's going to gain. So those are the variables, right? This is the ICE table. That's how much it's going to lo-- these are going to react to give me some precipitate. And so the equilibrium is more solid forms. That's not a good E. More solid forms. And over here, it's 1.3 times 10 to the minus fifth minus x. And over here, it's 0.1 minus x. That's the equilibrium condition. I've added in a common ion. This is the common ion effect. Yeah, so now if you do the math, then what happens? Well again, Ksp is the same. It's the equilibrium constant. So it's the constant. So that is going to be the same value. It's going to be 1.7 times 10 to the minus 10th. But you see, Ksp is also equal to the concentration in equilibrium of the silver ions times the concentration of the chlorine ions. So it's going to be 0.1 minus x times 1.3 times 10 to the fifth minus x. And then we don't like doing all this math, and so we simplify. Then we can simplify, because look, you started with silver ions in a very small concentration. You can't take away something you don't have. And so x has got to be somehow this or less, right? And so x is a small number. And this is a really big number in comparison. And so we like simplifying our lives, right? And so we like saying that that's like 0.1. We know that. That'll make the math a lot easier. It'll make the math easier. And so now we can say that this is equal to 0.1 times 1.3 times 10 to the minus fifth minus x. And then we get that x equals 1.7 times 10 to the minus ninth. That is the common ion effect. What I've done, I've taken this thing in equilibrium, and I've added one of the ions. And it changes the solubility. Remember, the solubility is what we care about. That's what the solubility constant helps us determine. And I've literally just changed it by orders of magnitude, because now the amount that this-- so this thing precipitates like crazy. I added in a little bit of some other salt that had chlorine ions. And all of a sudden, silver chloride precipitates out. So here's a video of this. This is kind of cool. So watch. This is an equilibrium container. And what I'm adding is salt. And look at that. What is that? That's silver chloride precipitate, right, because the thing has to reach equilibrium. And so in order to do so, what you do is you suppress the solubility. Literally, by adding chlorine, the common ion effect, you suppress the solubility of the silver chloride. It's pretty cool stuff. That is the common ion effect. Now, why does this matter? Let me give you one-- no, I'll get to why this matters in a second. One more question. Now without even doing the math, check this one out. This is cool. Without even doing the math, I can take barium sulfate, and from the common ion effect-- so barium sulfate is going to go. So what does that one look like? Well, it's going to go to Ba 2+ plus SO4 2-, both in solution, in solution. Dissolved ions, this is a solid. Now, here's the question. Which of the following will be required in the least amount to dissolve the same amount of BaSO4? I don't need any constants or math. I can use this same principle, because if I had some 0.1 molar or 0.01 moles per liter of either of these, what are they doing? They're serving up ions, right? In the one case, you're serving up-- so in case A, you're going to give me-- what do I have there? H2SO4, so let's do A first. H2SO4 is going to give me a whole bunch of H+ but also SO4 2-. And in case B, you have BaCl-- OK, so you have BaCl2. And if I put that in, I've got this in water, right? But if this dissolves in water a little bit, then it's going to serve up some Ba 2+ plus 2Cl-. Either way, now you know what's going to happen, because if I'm trying to dissolve something and I add any of the ions that it's dissolving into from some external source, it's going to drive it this way. It's going to precipitate. So the answer has to be just water. Stick with pure water in this case. Otherwise you're going to have more trouble dissolving, just like we just showed. You're going to have more trouble dissolving, not less. That's the common ion effect. OK, so now why does this matter? Why does this matter? We go back to the pteropod. And by the way, I didn't have this link, and I should have when I showed you the-- and this is your goody bag, et cetera. There's some really nice articles here that you can find related to these experiments and other things about ocean acidity in case you're interested. But see, what I did was that this was my why this matters on Monday. And I wanted to tell you about the goody bag and about how things dissolve, because Monday was about dissolving and finding a saturation point. Now we can get to the next place, which is, why does that matter for the pteropod's shell? What is the chemistry that matters there? OK, so I made the ocean a little more acidic. Why does that matter? But you see, now we're armed with the knowledge we need to answer that question. Now we're armed with it. All we need to do, as always with everything in life, is look at the chemistry. That's it. Say that at the Thanksgiving table. You'll be very popular. We said CO2 plus H2O. This goes-- oh, find some equilibrium to H2CO3. Now, this is called carbonic acid. This is called carbonic acid. There's the pteropod up there. And there's the reaction that's really relevant that you'll see from the ones we're about to write down. Why? Because the thing is that, what happens to carbonic acid? Well, carbonic acid also goes through a dissolution reaction. So this is CO2 dissolving in water. So carbonic acid goes like this. OK, it goes into HCO3- plus H+. Now, here's the thing. OK, what is the shell made of? Well, the shell that's dissolving, the core material is calcium carbonate. So that's CaCO3. That's the shell. And the shell also has an equilibrium reaction that happens. The shell of a sea creature is in dynamic equilibrium with the ocean. And so it's going like this. It's going to, well, OK, Ca2+ and CO3 2-. OK, but the thing is, it has an equilibrium constant. All these have equilibrium constants. So for example, for this one, the Ksp, the solubility product constant-- because this is a solid. This is the solid shell, solid, going to ions in aqueous solution. So the Ksp for that is somewhere around 5 times 10 to the minus ninth, 5 times 10 to the minus ninth. Yeah, but here's the thing. We just went through this. If I consume one of these or change the concentration of one of these and not the other-- we just did this. If I could change the concentration, if I consume any of these, then I might drive the reaction-- consume or produce. If I change any of these independently, I'm going to drive the reaction, because that's how we keep to our K. That's what Le Chatelier's principle tells us. And so what ends up happening is you've got the extra H+ ions. So these are ions in solution. Where did they come from? They came from the CO2 giving us carbonic acid, which then gave us H+. Those are what I'm talking about. Well, they react with the CO3 2-. This lowers the CO3 2- concentration near the shell-- [STUDENT SNEEZES] --right? Gesundheit. And if I lower this, because I've taken some of this now and I've reacted it, so now I've got less of it. And because of what we just saw, if I've got less of this, you're going to drive this way, which is going to dissolve more of the shell. That's why this works. Well, that's why this happens. Works sounds like a positive thing. So this lowers the concentration of CO3 2-, and that drives more dissolution. That is what's happening. And we now can understand it in terms of the concepts that we've just learned. Historically, I mentioned 50 million years. Actually, by some accounts, it's 300 million. It depends on which studies you read. But for at least 50 and maybe as much as 300 million years, the ocean has had a pH of 8.18. Now, where's the-- oh. So let's just say last, oh, 50 to 300 millionish years, the ocean pH was 8.18. And today, it's 8.07. And the prediction is that in 2100, it will be 7.8. Now, as we will see in a little bit-- and you say, what's pH? And many of you probably already know, but I will tell you what it is in a little bit. But because this is a logarithmic thing, this is a lot, right? Today the ocean is 25% more acidic than it's been in 300 millionish years. And in 2100, it will be 126% more acidic. That's why they use 7.8 in the experiments of the pteropods. OK, right. Why does this happen? Why is this molecule an acid? We're talking about acidification of oceans. Why is this an acid in the first place? What does it mean to be an acid? And that is the next topic. That is what I want to talk about next. What is an acid? And an acid is something that is very specific. It has a very definite meaning to it. And I want to talk about that today, and then we'll continue after the break. So this molecule, carbonic acid, is called an acid because of that proton. And you could feel it. You could feel it in this whole thing. The proton is the thing that caused the problem, right? An acid, this is an acid because of the proton. So an acid, it's a dissolution reaction, dissolution. It's what we've been talking about. It's a dissolution reaction that gives an H+. Now if I think about it as a generic case, generic acid A, then the reaction looks something like this. I'm going to call out the acid and the proton that it gives separately. OK, so this is HA. So we do this because it highlights that it's an acid, H+ in solution-- and these can all be in solution-- plus A- in solution. All right, H+, A-. Right, so I've taken a proton off. You see it's right there. That's HA, HA, where A is HCO3 and H is H. And then what I've done is I've transferred the proton into solution. That's an acid. Now the thing is, you will see a lot of places and people writing H+. And we will do that too, because so many textbooks use H+. But what I want you to know is that H+ is never a thing. We just write it that way. H+ is not stable in water. So the proper-- ah! The proper way to write this general reaction would be HA plus H2O is going to go to-- and now I'm going to write the full, so the full goes both ways-- is going to go to H3O+ plus A-. [STUDENT SNEEZES] Gesundheit. This is what happens, because H+ is not stable in water. But you will see H+ written all over the place. When you see H+, and if it's in water, know that it's H3O+. That is what is stable. OK, H+ in water is not. Now, there's something here that's important in terms of terminology. And that is that these are-- notice that these are related. The HA and the A- seem very related. And the H2O and the H3O+ seem very related. Well, they are, because a proton is the only difference between them. And so those are called conjugate pairs. Conjugate, related, conjugate pairs. OK, those are called conjugate pairs. Now, back in the day, there was a lot of interest. People knew about these kind of liquids, mostly liquids, for a long time. And there was a lot of experimentation before this. Let's see, well, there's these general classes of materials. We're going to call them A and B. Well, A seems to always taste kind of sour. B is bitter. A reacts with carbonates to make CO2. B reacts with fats and soap to make soaps. One reacts with metals. The other doesn't. And by the way, if we mix them together, we seem to always get salt and water. This had been going on for centuries, this investigation into properties of acids and bases. But who was it that came along? Who was it that came along? Svante! Is there anything you can't do, Svante? I ask you again. In the late 1800s, it was Svante to the rescue who first proposed that what is happening here is a dissolution reaction. That is what an acid or a base is. It is that on the one hand, you're given an H+, which is what I just said. But on the other hand, for a base you're given a OH-. And it was Svante Arrhenius who first put that down and first conceived of acids and ba-- are we snapping? Are we quiet snapping for Svante? I love it. I've just recently learned what that means, which makes me happy. Quiet snapping. It's instead of clapping. But you don't make noise, but you still want to give props. I got it. I'm there. I'm right there. Now, OK. OK, now here's the thing. So we'll talk about it. So Arrhenius says bases give OH ions in solution, and oh, acids give H+. But we know that that's H3O+. OK, we know that. Yeah, but the thing is there's disillusion reactions. These things lead to really small concentrations. And so Soren Sorensen. That is a beer bottle. By the way, his research was funded by a beer company, and it was all about the taste of beer. He called it looking at proteins, but it was about beer. And what he realized, he said, I've been doing all these dissolution reactions of ions. I think Arrhenius was right. Ions are really important, and we're trying to measure these things. But I'm sick of all these zeros. I don't like writing 0.0001 or whatever all the time. And oh, look at that, 1.3 times 10 to the minus fifth. That's not efficient. And so what Sorensen did is he talked about how you've got the power. Now, what do I mean by power? So many things, right? But if I look at H+, OK, H3O+, and I say it-- let's suppose the concentration is 0.00, oh boy, 00001, that's 10 to the minus 7. But Sorensen wasn't happy with that. You simplified it. Wait, two, four-- no, it's not. OK, maybe it is now. He said, well, 10 to the minus seventh is still four characters I got to write, four. So he went to the math department at his place and he said, what else can be-- I want to talk about the power of these ions. And they're like, power, well, that's the logarithm. And so if you take the log, it's the power of hydrogen. Power of hydrogen is the pH. And that's simply minus the log. They didn't want negative numbers either. Oh, but you can have negative pH. You can. Why? Because this is the definition of pH. Or you could do pOH, if you're going with Arrhenius. So the power of hydrogen is the pH. The pOH would be minus log of the concentration of OH-. That's the pOH. But pH is the one that we most often see to describe whether something is acidic or basic. And so now you see, well, OK, well, this is a pH. Now this is easy, right? It's a pH of 7. And if the concentration were 0.1, it would be a pH of 1, minus log, the power of hydrogen, the log, literally the log. OK, now if you look at scales of pH, and it's a really fun thing to do, then what you find is that usually, almost always actually, you'll see the scale ranging from 0 to 14. That's what Sorensen originally decided. I'm going to talk about these ions in solution. I don't want to write it out every time. I'm just going to have a simple scale. We'll take the minus log, and there you go. Yeah, but you can see that if the concentration is more than one here, then you could have negative pHs. There's nothing special about zero here. It just so happens that most things, certainly that he was playing with at the time, were in this scale. And here is a few things that we know. There it is, right? OK, don't play with that one, but play with lemon juice, vinegar. By the way, vinegar-- OK, wine is around 3 and 1/2. Vinegar is around 2. The word vinegar is [NON-ENGLISH SPEECH].. It's eager wine. That's it, right? But it's because of the acidity, the acidity. Well, it's more than that. But it's the acidity is changing by orders of magnitude, power. And then you can go back and you see, but OK, there's coffee. I like that a lot. And blood. By the way, by the way, talk about ocean acidity, blood has a very narrow range of pH. And if you change the pH of your blood by more than 0.2, it is very likely to lead to death. So just think about that. That's what we're doing to the ocean. OK, anyway, that was an aside. Seawater, oh, pH 8.07 currently and dropping. OK, baking soda, ammonia. So this is very powerful, no pun intended actually. Now, these are dissociation reactions. These are dissolution reactions. I'm taking an acid, and I'm dissociating it in water. And so you have the equilibrium constant for that dissociation, right? And so if you go back to that, that's still going to matter here. And that's going to be important in thinking about what an acid is. Because, again, why is this it? Well, because there's these protons. Well, how many of them are there? It depends on how it dissolved. Where did it find its equilibrium? And so for some of them, like I just wrote, we just talked about HCl plus H2O. That went to H3O+ plus Cl-. OK, now the thing is, if I had 0.1 moles per liter of HCl, then it's going to lead to a concentration of H3O+ of around 0.1 moles per liter, because it's nearly full, nearly full dissociation. What do I mean by that? Well, I mean that-- yeah, OK, so you'll often see for an acid like that, one arrow. Well, we know that in reality, there's another arrow there. But see, here the acid dissociation constant is huge. So now we have a thing for the equilibrium of an acid, which forms by dissociating the acid into its ions. And this is-- well, in this case, it's 10 to the sixth. It's enormous, which means that the equilibrium lies very, very, very, very far over. And so oftentimes you'll see if it's going to strongly dissociate it, like sodium chloride did, then you'll sometimes see it written as just one arrow. But there's a little bit going back as well. That is another equilibrium constant. And we will talk about Ka a little later and probably pick up on it next week. OK, so near full dissociation. Now, the other thing that can happen that's important is water can do both. Let's see, where should I go? I'll go back over here. So water, see that picture there? They got it wrong, but it's OK. They got it right, because we all get along. If you want to put H+, fine. But you know that it's actually H3O+. OK, but you can put H+, no problem. OK, now there it is. And you can see that what water can do is it can do both. And that turns out to be extremely important. So if I take water and I mix it with water, ah ha, then you can get a combination of these ions, oh, plus a whole bunch of water. So this would be like dissociation. Now, the word for this, water can turn into either. It can be basic or acidic. It can deliver OH- or H3O+. That's called autoionization, and it also has a special word. Water is called amphoteric, amphoteric. Now, that means that the same thing can be both, both acid or a base. It can act as either. So if you take pure water-- so let's see. If I take pure water and [GROANS] and I look at the concentrations of these, so H3O+, it's all happening from the water. There's nothing in it. I didn't add anything to it. It's just pure water, H3O+. So that means that if that reaction happens, it's going to generate the same number. So the H3O+ would equal the concentration of the OH-. And in pure water, that is a value that is equal to 1 times 10 to the minus seventh moles per liter at 25C, at 25C. And so now we have yet another. That's neutral. You can understand that that's neutral, because you have the same-- if an acid is H+ or H3O+ in solution and a base is OH-, and I've made the same number, then the acidity and basicity are neutralizing one other. So that's neutral. And in fact, you can see that that pH is 7. Oh, and we can even write the equilibrium constant for water, which is going to be those guys, H3O+ times OH-, which is equal to 10 to the negative 14th, because that's the dissociation reaction for water. Water plus water goes to those ions in solution. We ignore the water, and then we're left with this same thing we've been doing, 10 to the minus 14th. Now, that takes us-- those are neutral. And so speaking of neutral, that's where we're going to go next. And I think what I want to do now is-- so we're going to go next into neutralizing things, so like if you add an acid to a base or if you add a base to an acid. And we already said on Monday, you get salt and water. But see, to understand that, you need a broader definition. Svante is amazing, but Svante missed something. And so we need a different definition of acids and bases that's more general. And that's what we're going to start with on Monday. But wait, because this is a great place for me to work on my arm. I got to hit-- OK, we're going to go all the way up there and all the way up there and right there and right there and, well, there and there and there. [YELLING] Oh, that was excitement. Let's go back there and there. Well, OK, and there. Oh, that was the same direction there. And there and there in the middle and there and there. Oh, that was there and there and there and there. You guys are seeing I need a stronger arm here. [GRUNTS] I can't really get too far. And [GRUNTS]. Oh, that side is all-- and there and there and there. And one more, I'm going deep. Have a great Thanksgiving, everyone. [APPLAUSE] See you guys on Monday. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | Goodie_Bag_1_Atoms_and_Reactions_Intro_to_SolidState_Chemistry.txt | [SQUEAKING] [RUSTLING] [CLICKING] CLAIRE HALLORAN: Today, we're going to be doing goody bag 1, atoms and reactions. Our objectives today are to identify material based on its properties, observe a simple chemical reaction, and identify the limiting reagent, and estimate the number of atoms in a real object. To do this goody bag, you'll need vinegar, a plastic cup, a ruler, metal samples of aluminum, copper, iron, magnesium, and tin, but if you can only get copper and magnesium, it will work, and a pipette. Some conceptual questions to think about while you're doing this goody bag are, how can we tell different metals apart, and how can we count atoms that are too small to see? First, we have five samples of metal that we would like to identify using their properties. The easiest to identify is copper by its orange-ish color. Next, we're going to try to find out which sample is iron by testing how easy they are to bend. Iron will be the most difficult to bend. So this sample bends very easily just with my fingers, so it must not be iron. This sample is very difficult. I can hardly get it to bend at all with my fingers, and this sample also bends fairly easily just using my fingers. So we know that this sample, the hardest to bend, must be iron. Next, we're going to try to compare the density of these different metals. So this metal feels the heaviest and most dense. This one's feels lighter, and this sample feels light as well. So we know that tin is the most dense metal, so this sample must be tin. To identify which of these samples is magnesium and which of them is aluminum, we're going to test which one of them reacts with vinegar. So we're going to place each of these samples on the lid of our plastic cup to make sure we don't spill any of the vinegar and use our pipette to put a drop of vinegar on each of the samples, and the one that reacts will be magnesium, because we know magnesium reacts with vinegar watch closely. It doesn't look like we're getting any reaction from this metal, so it must be the aluminum now, let's try the other. You can see that this material's forming tiny bubbles after we put the vinegar on it, so it must be magnesium. Those bubbles are hydrogen gas. Now, that the bubbling has stopped on our magnesium, we note that the reaction is over. There's still plenty of magnesium left, so this indicates that the acetic acid inside of the vinegar must be the limiting reagent. Finally, we want to estimate how many atoms are in our sample of copper. To do so, we're going to use a ruler. Using our ruler, we can measure the width and the height of this copper sample. So using the metric side of our ruler, we align the 0 with the edge of our material and see that it is about 13 millimeter wide and 25 millimeter long. Today, we did goody bag 1. We identified our metal samples based on their material properties, we observed a simple reaction and identified the limiting reagent, and we measured our copper to estimate the number of atoms in our sample. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 5_Shell_Models_and_Quantum_Numbers_Intro_to_SolidState_Chemistry.txt | So we're going to talk a little bit about Bohr. And then we're going to talk about ionization energy, which is a theme that's going to come back a couple more times. Very important. And then we're going quantum. We're going quantum. And that's why this is a very exciting day for me. And I'm hoping it will be for you, too. It's why I have t-shirts. It's also a goody bag day. There's just all sorts of things happening today, so I'm very excited. This is how we ended our lecture on Wednesday. I was showing you this as part of my why this matters, which started with refrigerators and then we talked about a chemistry problem, CFCs. And we talked about how those were destroying the ozone layer of this planet. Now, why is that important? And I had this picture up there to help explain it. Just to make sure we all see what this is, I'm showing you here, again, this is the energy-- watts per meter squared-- that we get from the sun. And you can measure that up on top of the atmosphere, so there is no atmosphere there. And this is what it looks like. That's the energy as a function of the wavelength or frequency, or energy of the photon. You now know how to go back and forth between those. But see, on the planet here, once you have the atmosphere, you get a different spectrum. And the reason is-- and it's shown here-- that some of the molecules in the atmosphere are absorbing parts of that spectrum. So what we see on the planet is different. And we talked about this in terms of why this matters and how ozone-- so I mentioned, oh, if we can only see out here-- the visible is here. That's the visible. There's the frequencies of the visible. But what if it were here? The planet would be totally dark, because no light would get through the absorption that happens from those molecules. On the other hand, over here, we've got ozone that's protecting us. It is absorbing light in the higher frequency, which means lower wavelength. That's the UV. And that's why that was such an important problem is to not destroy those precious ozone molecules. So I wanted to make sure that we-- and we understood this in terms of energy transition, because how do these things absorb light? Well, they do just what we said, which is they take the energy from a kind of light and an electron then takes that energy, absorbs it to be excited, and it goes from one energy level to another. And we did that in terms of the Bohr levels, so I wanted to do that. And that is what you have right here. You have a way to see that visible spectrum. So we'll talk about that as well. Let's talk about it right now, because have a slide on it. So you have what is known as a spectroscope, or the most powerful way to see light ever designed. And if you look through this-- OK, don't look at the UV please. Don't look at the sun. Please don't do that. But we gave you light sources to look at. Like that, right? Or you can look at that. Or you can look at things around you with it. And what are you seeing? You're seeing this. You're seeing the spectrum in the visible. You're seeing it in this part. But from this, you are also-- what you are seeing are literally electron transitions. You are literally seeing-- and that's where I want you to think about. That's why I wanted you to have this power, this power, which means responsibility, as we know. But now, you can not only see light. You can see electron transitions. I want you to think about that as you're seeing light. And I want you to think about it in terms of what we learned on Wednesday, which is Bohr. And so let's just put that here. So we had Bohr. And in the Bohr model, we said the energy is going to be quantized minus 13.6 electron volts times the atomic number squared divided by n squared. And so for hydrogen, for example, this is equal to minus 13.6 ev, because e is 1 over n squared. So that was for hydrogen. z equals 1. And so that's it. And that's it. n is equal to 1, 2, 3, et cetera. It's quantized. This was the whole kind of discussion and theme of Wednesday's lecture. And that got us to these levels, these Bohr levels, which are places where-- energetic places, energies of the electron. So we'd have these levels that kept on going until you're at somehow-- so this is energy. And this would be like n equals 1, n equals 2. These are the energies right there. I'm just plugging in. And this would be n equals infinity, so the energy here would be 0. And for the hydrogen atom, the energy here would be minus 13.6 electron volts. That's hydrogen. So this is-- now, I'm leaving this here and I want to use this. That's why I'm putting it back up. Because look, now you can use this to start understanding light. And so you use this, and you look at light sources, and you can start thinking about things in terms of energy transition. So let's take a look. So what if I just look at some light sources, all right? OK, so that's what you see. That's the sun, all right? So that's daylight. And now we're getting into the mode of seeing light like this, all right? It's an intensity, salute versus wavelength, right? And there you go. And now if you look at sort of an incandescent light bulb, it will look something like this. So it's warm. There's a little more red, all right? You can't get up to the sun's temperature, but you can it pretty hot. So you see this nice kind of looking spectrum. And, oh, here's a halogen. All right, that looks kind of closer maybe. Look at what happens when we go fluorescent. Look at that. Why does fluorescent light give us headaches? Well, that might be other reasons. But look at this spectrum. That doesn't look like that. That's why we all have so much trouble with fluor-- and here's LEDs. And you can buy your cool white LEDs and your warm white LEDs. But look, they're still not. They're not that, right? There's been a lot of work on this. Can we get this to be that, all right? And so now we are armed with great power. And so you can look at things. You can say, oh, I see that it has this large peak because I'm looking at a fluorescent light within it. And I see that it's got this stronger peak in the blue. Maybe I could just absorb a little of that. What if I came up with a coating, all right, that I could put on that cool LED bulb to make it just a little bit more look like that? Well, now we know how to we might be able to use a board to do this, all right? Because what if-- so let's say, OK, the wavelength of that is around 450. So if the wavelength is 450 nanometers, then I know-- remember we did this-- waves, light waves, we can go back and forth between frequency-- hello. How's it going? All right, welcome to class. And energy, energy equals hc over lambda. Oh, Planck, thank you. And this is roughly 2.75 electron volts for that wavelength 2.75 electron volts, right? OK, well, but see now you say, but I want to absorb some of those photons. I'll pose a problem. What if I have an atom-- so question, right? So I have an atom, an atom that has an n equal 4 to n equal 5 transition that absorbs. So I'm telling you now this energy. So I'm telling you now I've got some atom, and I know there's a transition in it. It has an electron transition. Because remember, we're going from electron transitions to light energy, back and forth, back and forth. Look at this right here. I'm just saying, look, OK, n equals-- just keep going. n equals 3, n equals 4, n equals 5. So what I just asked you is, I said. Well look, some atom has a transition between n equals 4 and n equals 5. That means that if an electron happens to be here, it can absorb energy to promote it up to here of exactly 2.75 vb. Uh-huh, well, I can use Bohr. I can use Bohr now, right? Because so I can actually answer this problem, answer this question. If I know that, then I can say, well, OK, if the initial state is 4, and the final state is 5, right-- I've gone from 4 to 5. I've promoted an electron up in energy, lifting it up like a ball against gravity, all right? Then if then I can use Bohr to say that 2.75 eV is equal to minus 13.6 z squared times 1 over 5 squared minus 1 over 4 squared, that would be the transition energy, this, dictated by Bohr. And, oh, I have everything except z, which I can solve for. z equals 3. z equals 3, OK? So now all I need to do is come up with a way to code all the cool LEDs with lithium, right? Because this is lithium. And we'll be fine, except there's a problem. And this is part of the problem that we need to address and solve today before the weekend. And that is that, see, Bohr only works for one electron. Now, it's OK. z can be higher. z is 3, and it works here. But I assumed only one electron, which means this really would be like li 2 plus. Oh, OK, we're going to come back to that. But Bohr only for one electron, all right? So if I'm going to get something like z is 3, that's fine. But it can only have one electron there. It can only have one electron there or the model doesn't work. It was one electron with some z of protons, all right? So this could be li 2 plus. That sounds harder to put into a film, OK? We will be coming back to this. That is a serious limitation of the Bohr model. And we can't do a lot if we can only do one electron, right? You can do a lot of ions, but that's it. OK, so I just said the word ion. What happens now if you take an electron not from one level up in energy, but up, up, up, up, up, all the way until it's free? What happens if you do that? Well, that is called ionization. That's ionization. And, in fact, when you look out into the beautiful world, the universe-- remember I said on Wednesday, 75% of this is hydrogen. A whole lot of it is ionized. It's getting ionized from energy that's knocking electrons out. And then electrons are coming back and cascading through these levels, emitting discrete frequencies of light. And this is what you see. It's beautiful, right? And what you're looking at is ionized hydrogen getting electrons coming in and cascading through the Bohr levels, all right? Ah, but what does ionized hydrogen mean, right? OK, we can now understand it. We can now understand it because now, instead of going from 4 to 5, I'm going from anything to infinity. It's like escape velocity, right? It's like, how much velocity do I need to give something? So it just gets away from the earth. And it's not going to feel the earth's gravitational pull. Well, here, I don't want this electron to feel the potential of the atom anymore, the z, the positive charge. I want it to just be free. That's here. That's infinity. That's n equals infinity, all right? So if I think about this, that's actually quite easy to put into Bohr, ionization energy. So to save time, we'll write IE, right? And this is equal to the energy required-- oh, I'm going to save time here. Look at that-- req'd to remove the electron from the atom. OK, good. And if we think about this in terms of Bohr, it would be minus 13.6-- let's say for hydrogen, because I've got hydrogen up there. So for hydrogen, meaning z equals 1, it would be minus 13.6 times 1 squared-- that's the z squared, all right-- times 1 over nf squared minus 1 over ni squared. That's that delta e, which equals minus 13.6 times 1 over infinity minus 1 over ni squared, all right? And the last I checked with my colleagues in the math department, that's 0. That's 0, all right? So the ionization energy, you literally can just read off here. If an electron is in the ground state, if there's a happy electron right here in this orbital, the amount of energy I need to input into that atom to knock the electron fully out, the ionization energy, is 13.6 electron volts, right? It's just the energy of that orbital. I can get a little fancier and say, well, OK, what about this question. What if I had light with 100 nanometers? So now I'm telling you my source, all right? Before I didn't tell you. Say something goes into it and knocks it all out? No, now I know what's going in. The wavelength, or the energy of the photons, is 100 nanometers. And I'm shining that on hydrogen. Well, now, OK, let's see here. So a little example. Oh, I'm asking a question. What's the question? What is the lowest energy level that the electron could be in such that a photon from this light source ionizes the atom? OK, let's see, because I now know the pieces, and I know the definition, all right? So now if I have the energy of 100 nanometer photon, well, that-- I'm not going to go through it-- is equal to 12.4 eV. Uh-huh. But see, if I now want to use this to ionize hydrogen, then I get something like this. I get the delta E equals minus 13.6 times 1 oer infinity squared-- OK, fine, we'll keep that for a sec-- minus 1 over ni squared, right? This is how much. This is how much delta I need to ionize from any given place, any given energy level ni, all right? OK? But I have 12. So what if I set this equal to 12, all right? If I set this equal to 12.4, that's what I have. Then what I get is ni equals 1.05. OK, good, that's my answer. No. Panic. Everybody should be panicking. No. That can't be my answer. I can't have-- I can't have something that's not an integer, because I said it over there that n is a quantum number. 1, 2, 3, nothing is allowed in between. Mm-mm. So what you can see is that I got so close. I almost had enough energy to get n to be 1. But I don't. So I cannot use this light to ionize an electron in n equals 1. I can't. But I could use it to ionize an electron in n equals 2, right? Because there, if n is 2, well, then I've got actually plenty of energy, all right? So cannot ionize from ni equals 1, but ni equals 2 is OK, all right? So that's the outcome of this. Now, I could also use that same 12.4 eV photon to ionize from n equals 3, or 4, or any of these. But I can't get to 1. And there's one other concept that's important. Because look, I gave it 12.4 eV. What did I actually need? Well, you can even see it here. You know what these levels are from the Bohr model. Here, this is 13.6. This is minus 3.4, OK? That's telling me this is definitely not drawn to scale. But still, you get the point, 3.4. And I just said, well, I could probably get it out of 3.4. And I can because I've got 12.4. And so if I did this-- so if ni equals 2, then I would have excess energy. Excess energy would be nine electron volts. And that is kinetic, right? So I've shot this thing out of the atom with light that's got way more energy than I needed to ionize it. But that energy, it's still in the electron. I transferred that photon energy to the electron energy. And so that is going to be in the kinetic energy of the electron that's now free, the electron free. I should have said free electron, right? Excess kinetic energy comes after ionizing, right? So these things are the things we can do with Bohr. So I wanted to kind of take what we learned Wednesday and apply it to a few problems and go back and forth with this idea of photon energy to Bohr level changes. And that's what I want you to do in this week's goody bag as well. OK, now ionization is an important concept. Like I said, we'll be coming back to it. And you can actually plot-- and many people have measured, and this is very important-- the ionization energy of elements, like as a function of atomic number. And the first ionization energy is important, right? The first line ionization energy is the energy required to remove the outermost electron, right? All right, so if I did have a whole bunch of electrons, so if I had a whole bunch of electrons. the first ionization, the first ionization energy is the energy required to remove the outermost one. So that's an important number to know. If you want to take an electron off of an atom-- and will be doing that because that's how we're going to make our first bond next week-- if you want to take an electron off an atom, this is one of the first things you ask. How much energy does it take, all right? How much energy does it take? So that's an important concept that we'll be coming back to. Now, OK, I alluded to the fact that we needed a new theory. I already alluded to that. And I said that's where we're going today. The Bohr model works for one electron, as I said. All right, that means it's good for hydrogen. Oh, we just ionized hydrogen, and that was a lot of fun. And it's good for He plus, or Li2 plus, or osmium 75 plus. Yeah, that's real. But what about two electrons? What about two, just two? Forget about 10, 20, 30. 2, the Bohr model can't do it. It doesn't. It doesn't work for anything more than one electron. It was derived for one electron, and that's where it stays. So we're at this point now. This is the picture I started to draw this week on Monday. And we're at a point where we need something deeper. And some of the work that went into this was really critical. And I'm not going to go into too much detail, but there's one thing I want you to know. So remember, Planck and Einstein, they were playing with quantization of light. And Einstein's photoelectric effect found the particle nature of light, with Planck's-- you know, with together, their energy, equals h nu, that quantized energy of the photon. And they saw it was a particle, but it also was a wave, all right? There's a lot of discussion, and interest, and attempt to understand what it was. Was it a wave or a particle? And Compton did some beautiful experiments to show, again, evidence that it was a particle because he saw how light interacted with electrons. And it was like this collision that particles do. And then De Broglie, or as [INAUDIBLE].. De Bro-- I don't know-- would say he went further. De Broglie went further. He said, everything is both. Light, you guys don't know if light is a particle or a wave. Well, I'm saying everything is a wave, everything. And everything has this duality. So he wrote something very important, which is the relationship between anything's momentum and its wavelength. So we'll come back to this. So I want to write it down. So De Brog-- he said that the wavelength of anything can be written as, oh, that same quantization number, constant from Planck divided by its momentum, literally, just its momentum. Anything has a wave nature, anything. And light has a particle nature. What's going on? And it led Einstein to say at the time-- he said, quote, "It seems as though we must use sometimes the one theory and sometimes the other while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality. Separately, neither of them fully explains the phenomena of light. But together, they do. This was the beginning of quantum mechanics. This is where they were. And later, once they really got the quantum mechanics kind of going, he said this. The more success the quantum mechanics has, the sillier it looks. Why? Well, there's only one way to really learn about quantum, especially if it's the first time. And that's from Dr. Quantum himself. And so I do have a video of Dr. Quantum. And I'm hoping you will enjoy this with me. I enjoy it every time I see it, most Friday nights. So let's watch Dr. Quantum explain. Because there was one really important experiment that happened that he's going to explain right now called the double slit experiment. And it changed everything. And I don't care if you see it in another class. We're seeing it here because this sets up what happens next in chemistry. Let's watch Dr. Quantum. --are. The granddaddy of all quantum weirdness, the infamous double slit experiment. To understand this experiment, we first need to see how particles, or little balls of matter act. If we randomly shoot a small object, say a small marble at the screen, we see a pattern on the back wall where they went through the slit and hit. Now, if we add a second slit, we would expect to see a second band duplicated to the right. Now, let's look at waves. The waves hit the slit and radiate out, striking the back wall with the most intensity directly in line with the slit. The line of brightness on the back screen shows that intensity. This is similar to the line the marbles make. But when we add the second slit, something different happens. If the top of one wave meets the bottom of another wave, they cancel each other out. So now there is an interference pattern on the back wall. Places where the two tops meet are the highest intensity, the bright lines. And where they cancel, there is nothing. So when we throw things, that is matter, through two slits, we get this, two bands of hits. And with waves, we get an interference pattern of many bands. Good so far. Now, let's go quantum. It's my favorite line. I love that line. An electron is a tiny, tiny bit of matter, like a tiny marble. Let's fire a stream through one slit. It behaves just like the marble, a single band. So if we shoot these tiny bits through two slits, we should get, like the marbles, two bands. What? An interference pattern. We fired electrons, tiny bits of matter through. But we get a pattern like waves, not like little marbles. How? How could pieces of matter create an interference pattern like a wave? It doesn't make sense. But physicists are clever. They thought, maybe those little balls are bouncing off each other and creating that pattern. So they decide to shoot electrons through one at a time. There is no way they could interfere with each other. But after an hour of this, the same interference pattern seemed to emerge. The conclusion is inescapable. The single electron leaves as a particle, becomes a wave of potentials, goes through both slits, and interferes with itself to hit the wall like a particle. But mathematically, it's even stranger. It goes through both slits, and it goes through either, and it goes through just once, and it goes through just the other. All of these possibilities are in superposition with each other. But physicists were completely baffled by this. So they decided to peek and see which slit it actually goes through. They put a measuring device by one slit to see which one it went through, and let it fly. But the quantum world is far more mysterious than they could have imagined. When they observed, the electron went back to behaving like a little marble. It produced a pattern of two bans, not an interference pattern of many. The very act of measuring, or observing which slit it went through meant it only went through one, not both. The electron decided to act differently as though it was aware it was being watched. And it was here that physicists stepped forever into the strange neverworld of quantum events. What is matter, marbles or waves? And waves of what? And what does an observer have to do with any of this? The observer collapsed the wave function simply by observing. You can see why I show this. We do movie night at home a lot. It's Friday night, guys. Seriously, you know that you might have a little free time. You get together with some friends. There's no better way. But look, this is why-- I hope that's as mindblowing to you as it is to me. It was mindblowing to them at the time. This led Bohr to famously say, anyone who is not shocked by quantum theory has not understood it. And here's the thing. There are some things in life where you can ask questions, and you can understand them. You can answer them. I can ask the question why do-- or somebody might say, why do I wear a t-shirt under a sportcoat? That's weird, stylistically. I mean, stylistically, that's-- and I say, well, OK, here's the answer. Because I put comfort over stylistic norms. Whatever, that's an answer. I could ask a harder question. I could ask a harder question. Why did Justin and Selena break up? Because Jalina was totally hashtag couple goals. Just saying. OK, anyway, now I'm just-- why? Oh, but still that's a harder question. Why? Because they were so good. But there's probably an answer to that too. That's a harder question. This was mindblowing. How can something be both? How can something that they know is a part of-- How can everything be both? This was really difficult to answer. And, in fact, the real answer came with a mathematical solution, which we'll learn next, but not quite understanding. We still don't really understand quantum mechanics, even after 100 years. We don't. And so this is just, I think, a beautiful part of our story, of our detective story. And just to show, people are still working on this, right? How much quantum can we see? How much quantum can we see? Well, look, here's somebody who, about 15 years ago-- and it's published in this paper-- and a little more, almost 20 now. They actually took-- forget about electrons. Those are teeny, tiny things. This is a whole group of 60 carbon atoms. That's called the fullerene. And they shot this through-- they did the double slit experiment with this. And they showed that with the grading-- wait, without, you get the signal peak, just like we saw. And with you get interference. You get interference. So they were able to show all the way up to a large molecule, 60 carbon atoms, that these things are still-- they have wavelength. They have these incredibly strange properties that you just saw in the double slit experiment. All, everything has this duality, this duality between waves and particles. Now, OK, so we needed a new model, right? Bohr can only do one electron. And meanwhile, there's all this stuff going on at the time showing that you need a new theory. How do we explain waves? And to kind of put a little bit more of the nail in the coffin for Bohr, you had Heisenberg. So I'm just going to write this on the board before I go into why this matters. And what Heisenberg said is that you cannot measure both position and momentum exactly. You can't get them both, OK? And that's called the Heisenberg uncertainty principle. But that was it, because-- look, you look at that and you say, well, he-- saying this. But in Bohr, you got them both. You've got them both exactly. Remember, I put energy down there. But we also put r on Wednesday, exactly. Gesundheit. You can't get them both. And if you can't get them both, that's another problem with Bohr because Bohr says you can get them exactly, right? So we had these pointers, double slit experiment, Heisenberg's theories, all right, saying we need something more. We need something more. And I'll get to that. I want to do my why this matters. And then I'll get to the more that came, which we'll then expand on Monday. Before I do that, let's do why this matters. So why is the fact that an electron is a wave, why is that so important? Well, first of all, because it sets up this detective story of the electron set up this new theory that's coming, quantum mechanics, right? But also, it immediately, just like we started painting with them, right-- remember that was one of my why this matters-- we also realized that we could see with them. We could see with electrons. Because if electrons are waves, then I can shine them just like I shine light and see what it shows me, right? It can illuminate matter. So if you look at the frequency here of light, this is an electromagnetic spectrum-- radio, microwave, infrared, visible, UV. Now, here's the thing. If you want to see something, some feature size, you're limited by the wavelength of the light. It can't be bigger than the features you're looking at, roughlyish, OK? That's what you'll-- so if you're trying to see something tiny, but the wavelength of light is really big, you won't see it. So we need-- let's say we want to see atoms. Let's say we want to see atoms, or even more, nuclei. Look at how tiny. Those are 10 to the minus n, 10 to the minus 12 meters. Those are x-rays or gamma rays. But the problem is, if we shine x-rays on things-- and we will do that when we look at crystals-- but if we shine x-rays, it's very hard to then collect them and make a photographic image, OK, at least one that gets you that resolution. And gamma rays are even harder to catch, all right? But see, here's the thing. The electrons give you exactly what you need. Because if we do this math for an electron-- so bring this one back down-- if we do this math for an electron, well, I'm going to use-- oh, I thought I had the middle one. So if I have an electron-- let's suppose I have an electron that is-- electron, I'm going to say I accelerate it over 100 volts. I'm going to take an electron, and I'm going to put it over 100 volts. I'm going to give it some kinetic energy, right? So it's kinetic energy is then going to equal 100 eV, right? OK, so now I've got an electron moving with a kinetic energy that's 100 eV. Now you can convert this to joules, and you can set this equal to 1/2 mv squared, right, mass of the electron times its velocity squared. And then once you have the velocity, so you get the V. And then once you have that, you get the momentum, the p, right? And then once you have that, you get the wavelength, right? So I can go now from something that's easy to do. 100 volts is a lot, but in a lab safe, not in your dorm. You could apply 100 volts to an electron, get it going at this speed. And once you know the speed, you know the momentum. And if you know mv, then you know it's wavelength. In that case from this relationship, you would get that it's about 0.12 nanometers. But look at this. The wavelength of a simply accelerated electron is right where I need it. It's right where I need it. It's an Angstrom, right? So now, if I take advantage of the wavelengths of the wave nature of electrons and I shine them on materials, then I can see materials that way. And I can see them at that scale. And we do that all the time, all the time in many, many different areas of technology and research today. We use electrons to image. In fact, the best images you can get are made with electrons, all right? Here's an example of using what's called a scanning electron microscope, all right? So you see a butterfly, but you want to really see a butterfly, or we can go even further. And instead of just drawing pictures of this these beautiful materials made of carbon-- those are called nanotubes. This is called graphene. Gesundheit. It's a single sheet of carbon atoms, one atom thick material. Notice with these materials every single atom is a surface atom. That's pretty cool. They also have lots of other cool properties. And I'll give you examples throughout other lectures of how these kinds of materials can be used. But for now, I'm talking about seeing them. And this is what happens when you actually look at them with an electron. That's a nanotube. And here is a picture of graphene. The only reason we can see graphene is because we have electrons. And we take advantage of the wave nature of those electrons. Well, you say, well, OK. But why does that matter? Well, that matters tremendously, because one of the first experiments that really did what Feynman, what Richard Feynman wanted-- Richard Feynman predicted the field of nanotechnology 50 years ago. He gave a famous speech at Caltex called There's Plenty of Room at the Bottom. He's also an amazing teacher, and he taught actually the double slit experiment. I highly recommend you Googling that lecture. And he said that someday, you can put the atom where you want, all right? And the first time that was done was in 1989 by IBM. They had 35 xenon atoms. They moved them around. But the point is, you couldn't realize nanotechnology. You couldn't realize the ability to actually move atoms if you can't see them, right? And this, the ability to see what you were doing changed everything. It changed everything. And nature had been doing this. And I love these examples. So nature has been doing nanotech for a long time, all right? So you have the inner ear of the frog. It's a cantilever that is sensitive to a few nanometers of movement. The frog can actually hear that. You've got features in the ant's eye. I love the silk moth. The male silk moth has a single molecule detection system onboard that can sense a single molecule pheromone. It can detect a female silk moth two miles away, two miles away. We have nothing like that. We have no technologies like that. I can't even tell if someone's in the next room. I have to look at my phone or something. This is because of nanotechnology, that kind of detection system, all right? But it was the ability to see atoms and molecules with electrons that kind of blew open this entire field, all right? And it's made it so that we can now try at least to rival nature. Here is one example. You're not just seeing graphene, but check this out. You're seeing a single atom of graphene, and you're seeing what happens under a certain kind of irradiation. And you're seeing this hole. And you're seeing the hole grow. And that's really important, because something I care a lot about are membranes, right? And another one of these matters, I'll tell you about membranes. But here, I'm actually making the thinnest possible membrane that you could make because it's only one atom thick. And I'm controlling how I make that. But I would never know what my controls do if I couldn't see it in real time, all right? So this is my why this matters. It's seeing things at this scale. OK, so back to-- so there's Heisenberg, right? So in every moment, the electron is only inaccurate position and an accurate velocity. This was, again, saying Bohr, you got to think harder. And we had all of these experiments showing the particle wave duality of nature and of atoms. And so the question then became-- all these pointers from all these different places, the question became, how do we describe this wave nature of matter? How do we describe an electron as a wave? And I like this, because not only do I think if Schrodinger had a swimming pool this is what he would do with it, but also I thought maybe this would be cool for an MIT kind of swim test thing in the future. Because you can see what happens. This, I want to do this experiment. But this looks really fun. They're making waves. They're literally making waves, right? And so the question now-- look at that. See, but the question is, how do you describe it? Well, there are wave equations. There are wave equations that we know about that relate the position of a wave to it's time change in velocity, all right? And the thing is, this is it. This is how we need to describe an electron. Now, think about it for a second. An electron-- by the way, it just occurred to me that you should take this on that same day. Because if they don't have real candles, and they only have these, you should have this handy because you might need to pull it out and see what kind of LED they gave you. What does it look like? Anyway, OK, that was a side point. Bohr says electrons can only be in certain places. Well, that's mindblowing because-- by the way, think about that. An electron can only be here or here. Oh, and by the way, it can transition between them, but it never be between them. Did anyone catch that? What's up with that? It can't be between them. I am telling you that. But it can transition. Where did it go? Oh, OK, that's a weekend kind of exercise. And now we're saying they're not even in one place. They're waves. They're like the water. That's cra-- that really was at the time-- they thought how is this possible, right? But Schrodinger came along. And he said, I'm going to write down the equation anyway. And that was the big contribution, the really critical contribution that Schrodinger did, is he said, I'm going to write down kind of like a Newton's equation, like nf equals ma, but for quantum mechanics and for waves, like electrons. And I'm going to describe them. So he developed the equation that we saw, that on Monday we'll solve, to describe the wave nature of quantum mechanics. And just to end-- so on Monday we'll pick up with Schrodinger's equation for the atom. The question is, what does this all mean for chemistry? What does this mean? We're not-- I'm not teaching you quantum. I'm not teaching you quantum just so you know quantum in isolation. I'm telling you about it because it completely revolutionized chemistry. And that is where we will answer that question on Monday. Have a really good weekend. [SIDE CONVERSATIONS] |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 2_The_Periodic_Table_Intro_to_SolidState_Chemistry.txt | Now, today is lecture 2. And we're going to be talking about counting. And we're going to be talking about the mul which is a way to count to help us count. And then-- and this is really exciting-- we're going to be building this. And that is super exciting. Now, everybody should have gotten one of these on the first day when we talk about this later in the lecture. If you don't have this, and for that matter-- oh, there's Laura. Laura has like thousands of these. And if you need one, please ask Laura. And also if you need a goodie bag, which we gave out on the first day, please get in touch with Laura. And she'll give you one. We have extras here. You can stop by our office later, get one please today. All right, now, last lecture, we left off with-- remember, those guys like Boyle and Priestley and Lavoisier, conservation of mass, right? And what these guys were doing is they were smashing and burning and pushing on things and trying to figure out what it was that was that sort of fundamental object that you couldn't break apart anymore. And remember, it was Democritus that said there is such a thing, right? He said there was a such a thing called atoms, "atomos". And so we come to Dalton, who said, look, these things we're trying, we're calling them elements right now. Let's, in honor of that indivisibility idea, let's call them atoms. And Dalton did a lot of really important experiments. And he was also trying to smash and break and measure and figure out what are these indivisible-- and what are these things that you can't break apart any more? What are these atoms? How many of them are there? What do they do? And he came up with a pretty important observation which was that he called it the law of multiple proportions. And in this, so what Dalton did is he said, look, I'm going to take something and react it with something else. And it's going to react fully. OK, so there's no limiting reagent. There's no excess. It just everything reacted here and everything and want to measure how much stuff reacts. And then how much I get. And so if you take like an example of this law of multiple proportions, if you take an example of how that came about, what he observed is that if you had sort of, for example, maybe you had sort of case one. And you react some amount. So he's going to react some amount. So let's say he reacted-- OK, I don't know, some amount. Let's write it. Yeah, some amount of something, maybe, carbon. And some amount of something else. How about oxygen? And he would get some amount, right? OK, we know he's doing all these measurements. I'm trying to get you back in that mood. How much do I have? I'm going to weigh it, measure it. That's what he's doing over and over. But he had some amount of something. Aha. Now, he would do this over and over again. OK, I'm just taking these two elements as an example. He did this for many, many elements, right? So some amount of carbon with salmonella. And in case one, OK, everything reacts. And I get this carbon to oxygen mass ratio of let's say 1 to 1.33. I measure it. That's how much of the sum amount, right? 1 to 1.33 reacts fully. Remember, no excess, because they add a little more, and it's excess. No, this reacts fully. And then you do it, again, case 2. And he'd find that this same ratio was 1 to 2.66. Ha, ha. And he noticed a pattern. And he noticed a pattern. And he called it the law multiple portions, because what he found is that you could keep on going. And in some cases, not all cases. But in some cases, like carbon and oxygen, for example, these were common ratios, these series. 2.66 divided by 1.33 is an integer. These things would happen in integer ratios, integer amount. So I could keep going, and maybe I'd get three or four. Right? And what that means is that well, over here, this sum amount of something is it said something about the something, right, because it says maybe these somethings are like CO and CO2, right, for example. Maybe, aha, but not definitely. And that frustrated Dalton. That frustrated him. We'll come back to that in a sec. Right? But he did this. And he observed these ratios, these integer ratios of what would react with what-- he's trying to figure out what is he making? All right, what is he making? Now, he also, as with Lavoisier, remember, I showed you Lavoisier's tableau. Well, he had his own table. And here it is. So here in 1827, he had identified 36 elements. Now, the thing is, though, OK, 0.1, if you're going to try to brand something, which he clearly tried with these symbols, you can't make them this complicated, right? I'm pretty happy that that's not how we write our elements. You can see that this would be very hard. But he did come up with these symbols. And you can see it would be very hard to keep going without getting [inaudible]. So I think we're pretty lucky that [? brazilious ?] came along and figured out how to name these elements in a consistent way. But that wasn't his real problem. His real problem was what I just alluded to. See, he knew that he had these different elements. But he didn't know what the compositions were. He was so close. He was so close. He could see that these-- ah, but is it CO and CO2? Or is it CO2 and CO4? That he didn't know. That he didn't know, because he didn't have an accurate way to measure the mass of an atom, not at that time. But he did have this nice ability to see these series. OK. But in comes other measurements. And you had Gay-Lussac who was looking at volumes, right? And so Gay-Lussac was saying, well, look, if I take a certain amount of volume of a gas, one gas, and I reacted again completely, no access, OK? And I reacted with another gas. And then I see I get something else over here. I can measure the volumes. And I do this for the cases where there's nothing left over, nothing limited it. And so he would find, for example, what's shown up there. He would find that you've got for every two volumes of hydrogen gas-- so you've got two volumes of hydrogen that he would react with one volume of oxygen. And he would notice that he would get two volumes of something that had-- let's not put a dash there-- it had hydrogen and oxygen in it. He knew that. Now, he knew this because he was very careful. So nothing leaked. He said no. Everything is accounted for. Lavoisier holds-- conservation of mass. I'm not losing anything here. And as Lavoisier said, in a chemical reaction, mass is neither created nor destroyed. So I know that whatever is over here must consist of whatever I have here-- oxygen and hydrogen. And he kept doing it for other lots of different types of gases. But it was very strange, because what happens is sometimes, the volumes are the same. Like he did it for hydrogen and chlorine. One volume of hydrogen, and one volume of chlorine would give you two volumes of hydrogen and chlorine-- something that had hydrogen and chlorine in it. And so there, you'd mix one and add another one. And you'd get two. It seems it would kind of makes sense. You add one volume. But here, you'd mix two volumes, add another volume. And it would feel like it was reduced. Something in the reaction happened that gave you a lower volume. What could it possibly be? Right, what was going on. This is what they were grappling with. This is where they were. And then along comes [sighs] Avogadro. Now, I wish we all called him Lorenzo Romano Amedeo Carlo Avogadro which is his full name. But we shall call him Avogadro. And he made a critical proposal. He said, look, the way to explain what's going on in Gay-Lussac's volume measurements is to make the following assumption-- oh, I have another board over here-- is to make the following assumption. And I think if you've had high school chemistry, you know what this is. But I'm going to put it here, because of how important it is. So Avogadro said that at constant temperature and pressure equal volume equals same number of particles. So if I have the [? efomethesine ?] temperature [inaudible],, if I have the same volume, I have the same number of [? particles. ?] Now, what are those particles? Oh, he was flexible. This is important. He was flexible. He said, those particles could be atoms, or they could be combinations of atoms. They could be molecules. Remember, at the time, they hadn't all settled on the atom as a name, even. What were these things? Equal volume, same number of particles. And this really helped, because you see, if you make that assumption, and you believe these volume measurements, then it becomes very clear what you have there. Right? If I have the same number of particles in here, and I believe in Lavoisier, who tells me nothing comes in or out in terms of the math, I didn't create or destroy atoms, then you know that if I have two volumes of water, let's say I started with o and h. It doesn't work. I can't make enough water. I just can't make enough particles. The number of particles on the right has to be equal to the number of particles in those two volumes of hydrogen. It's got to add up. Counting has to work. That's what Avogadro gave us is this idea of counting. And that's why we've named a constant after him. So I'll get to-- Gesundheit or in honor of Avogadro, "salu-tay". [laughter] He was Italian. And so if you believe Avogadro, if you believe Avogadro, this becomes clear, because the only way for these things to work then is if here I'm counting how much volume's here. So here I got HCL only if this is a two, and that's a two-- diatomic gases. That's the only way the numbers work. And here, I've got to have two and two. And this must be H2O. Oh, or at least it's one simple and very plausible explanation if Avogadro's hypothesis was right. Dalton didn't buy it. Dalton said, uh, uh, no way. You've got leeks, dude. And he really was harsh. Everybody thought water was OH at the time. And so Dalton said, you must have made a mistake. I block you on Instagram. [laughter] I block you. And it was pretty tough, because Avogadro put a lot of really brilliant work out there that was really ignored for 50 years until Cannizzaro came along and others. 50 years later, two years after he died was when we had the first evidence that the scientific community was willing to believe that his hypothesis was right. What a big deal his work was. OK. Now, I mentioned that this is-- I'm mentioning all this because we are going to use his constant a lot, the Avogadro constant. And it's a way-- and so I want to talk about that constant-- and I want to do it in the context of our candle. And so because this constant, this number, this idea of the number of particles you have in chemistry is the link between the atomic world and the macroscopic world. That's what his constant does for us. OK? Now, let's put that into context. If I were to ask you a question like this, I have that candle. Remember, the candle, the C25H52. And I say, well, OK, you've lit the candle. How many candle molecules burn in a minute? I want to know that. Don't we all? Right? That's a pretty important question. Why are people nodding? Thank you for that agreement. And I could say, well, I assume the 12-hour candle of the mass and 90 grams, let's get that off of a candle on Amazon. Right? How many molecules burn in a minute? What I need to know is, well, how many are we sort of talking about? How do we know how to even answer this question? And it comes from the link. But see, to answer this kind of question, which I'm sure so many people wanted to know back then, you must have a way to talk about big numbers. You must. Why? Because there's a lot of these molecules in a very tiny amount of volume-- a lot. Drop of water for perspective has 10 trillion molecules-- 10 trillion molecules. Anybody know how many cells you have in your body? 600 trillion? Let's call it 100 trillion just to be nice. Right? It's a 100 trillion cells. And so if you're going to talk about how many cells you have, or how many molecules are in a drop of water, you need to have a big number, a big scale. And that is what a mole is. Avogadro didn't come up with this number. This was gotten much later, right? But we name it after him in honor of the work he did on counting atoms and molecules. And that's the number. This is the way we talk about it. And just to be complete, Avogadro's constant, which is a mole, one mole is equal to 6.022 times 10 to the 23rd. It's just a number. All right? It's just a number. But it's a really important one because it provides the link. OK, if I take like how many eggs are there in a dozen eggs. It's 12. How many bags of sugar are there in a dozen bags of sugar? It's 12. All right? How many atoms are there in a mole of atoms? That number. That's all it is. It's a number. OK? Oh, but a dozen eggs weighs differently than a dozen bags of sugar. So a mole of one atom will have a different amount of mass, a different weight than a mole of another atom. And that's the link that he gave us. That's the link that this number gave us. And, in particular, again, going back to this idea of counting, we can count now with this link. If I said, how many pennies are there? And I gave you a mass of pennies, right? If I had 2.5 kilograms of pennies, and I say how many pennies do I have? Then all I need to know is how much each penny weighs. I'm not going to count them. Right, you don't want to count them. You want to do something a little easier. So if I have one penny, weighs 2.5 grams. Then I know that I can just get this really easily. I have 1,000 pennies. OK. What a mole does-- a mole gives us the link. So a mole, if I have moles, then I can go to the mass of an atom-- the mass of an atom. And from that, I can go all the way up to grams. And this is critical. OK. The mass of the atom and the grams per mole are the same number. That's what the mole gives us. Right. So in the case of carbon, carbon has a mass per atom of 12.011 AMU. These are atomic mass units. It's just a way to measure things that have very little mass. But the link here is that if I have a mole of these things, if I have a mole, one mole of carbon, then I have exactly 12.011 grams. Back and forth, grams, atoms. Numbers of atoms, mass per atom. All right, so one carbon atom has a mass of 12 atomic mass units. We don't need to know. This comes from weighing neutrons and protons. All right, we'll get to that next week. We don't even know that yet. But this is a mass of things that the very tiny scale. And if I have a mole of those atoms, I get to grams. OK? So it's my link. Now, where is this amount? Where is this critical number that gives me the counting per gram? It's, of course, here in your periodic table. This is where you get that information. OK, so we'll be building that up as we go. Well, let's say why do I need to count? Well, let's go back to answer our question now. That's the link. That number, Avogadro's constant links these atomic masses to grams. All right? Now I can answer the question. How many candle molecules burn in one minute? Well, let's do that. Because now I know that one mole of C25H52 is something like 352.7 grams. How do I know that? Because I can look up what a mole weighs of carbon in here, how many grams. And I can look up how many grams of mole hydrogen weighs. And then I can multiply 25 and 52. And that's how many grams I get. One mole of it is that. Aha, the link is coming. The link is coming. It's very exciting. Because in a 12-hour candle, if it's 90 grams, so if I have a 90 gram candle, as I've said in the problem, then I know now that that is about 0.25 moles. Right, because of this. See that? 90 over that. So now, I know how many moles of candle I've got. I know how many moles of candle I've got. But remember a mole is just a number. It's just a number. So this is how many moles of candle molecules I have. And OK. In one minute. It says in one minute. So in one minute-- well, OK, there's 60 minutes in an hour, and it's 12 hours, just a 720-minute candle. Right? So in one minute, 1/720 of that is what burns, if it lasts 12 hours. I'm just taking the information I was given. And I'm using the math of minutes and hours. 1/720, OK. All right, good. So 1/720th of the candle burns. In one minute, 1/720 of the candle burns. And if the candle is that many moles, that's this many moles-- 0.003. No, 0347 moles. Why did I do all that? Because I can now answer the question, cause of Avogadro's constant, because a mole is just a number. A mole is just a number. So if I've got that many moles, then I know that I've got 2 times 10 to the 20th molecules roughly. So I have been able to answer this question by going back and forth. [inaudible] I look here to get the grams per mole. I figured out how much candle I had. In grams, that gave me how many moles I have of candle. And then gave me the number-- back and forth, back and forth. That's what we do. Avogadro's constant gets us in and out of the atomic world. OK? And so we're going to practice that a little bit. And this gets me back to what we started on Wednesday. And I always like to give you guys advice if I can. And last time I mentioned, if you're on a date at a restaurant, ask them to give you, not a candle, but some C25H52 as a test for the quality of the restaurant. [laughter] Now, I'm asking you to do something else. It's noisy. You want to talk. Right? You go out on a first date. You're a little nervous. You want a quieter room. Ask them if there's a room where you can kind of shut the door and get that noise out. And it's quiet so you can talk. But now, you are armed with very important knowledge, because now, you know how much volume you have of air in the room and therefore, how much oxygen you have. [laughter] And if you light a candle, and you're taken 10 to the 2 times into 20 the molecules of candle out of the year, by the way, how many molecules of oxygen are you burning? Ah, how many molecules of oxygen? Ha, for at 38, balance-- 38 times as many molecules. For each one of these that burns, you need 38 molecules of oxygen. So 38 times as many O2. Now, I don't know about you, but if I'm on this date, I'm looking it up. The human-- [laughter] --needs 1.9 grams of oxygen per minute. You need that. I mean, maybe it's variable. But you need roughly-- and there's two of you. 0.4 grams of O2 is burned per minute. 0 point-- who's going to win-- you or the candle? So you got to be just aware of this. And now look, This is kind of thing we can do now. And I'm not saying that everyone's going to understand this. You might not be with an MIT student. You might be with a student from some other school down the street. [laughter] All I want to say, be nice, be nice. It may not be their fault that they're where they are. [laughter] It may not be their fault. Go through this with them carefully, slowly. [laughter] Explain these concepts. And this gets me to the other topic of conversation that you can have. And this gets me to my why this matters moment for today. Why does this matter? Why does counting and thinking about how much of stuff you're using by using chemistry and moles? Why does that matter? Well, let's take an example. Let's take an example of nothing less than the number of humans on this planet. This is the population of humans. And you can see that it wasn't all that much until lately. But let's focus in on this part here. Let's focus in on that part there. Right. So that's billions of people. So if I zoom in on this, there's a change. Look at that. It's kind of going a little bit up. But now, it really kicks up. What happened during that time? What happened is a very important chemical reaction, became easy to do or a lot easier. OK? And the process that enabled that is called the Haber-Bosch process. But it is what allowed us to feed billions of people in a sustainable way. It's arguable whether we're doing that sustainably but in a way where we could actually produce enough-- enough what? Enough ammonia. And here's the deal you see, because plants need nitrogen to grow. Right? And there's 70% of the atmosphere is nitrogen. But it's useless, because it's N2. And N2 is one of the strongest bonds in nature. And plants can't break it. Now, we knew how to break it before. We knew how to do this before, right? This is called fixing-- fixing nitrogen. And the way it works is you go N2 plus h. This is a reaction, goes to NH3. What is wrong with this? [inaudible] It's not balanced. Thank you. Did anybody say, balancing is the same as mole ratio. Balancing and mole ratio's the same-- counting. Remember, we talk about that. But balancing is counting. So let's see-- oh, 2 here. OK, that helps me with the nitrogen. Oh, but this is H2. I meant to put that in the first place. And then so this is a 3. That reaction was known and doable. But it took tremendous amounts of energy. So it's very difficult to scale. And Haber-Bosch came up with a way using catalysis to do it at much lower temperatures. Catalysis is something we'll learn later in the semester. But how do we answer this question? What's my question? My question is how long can we keep going? 50% of every protein you put in your body-- 50% come from this-- comes from some plant that was grown using this process. That's how important it is. It's 500 million tons of nitrogen is made this way every year. Well, it's just counting. One mole of ammonia is 17 grams. Oh, we're doing the same thing. How did I know that? Periodic table. That's how I knew that. One mole-- if that many molecules of ammonia, I've got 17 grams of it. Let's say I just need to make the same amount of-- it says so times 10 to the 6 tons. I need that much tons of NH3 per year. That's our-- yeah, per year-- per year. OK, so that sets up my problem. I know how much I need. I know how many atoms we're talking about in one mole. Now, I can actually understand how many moles-- so I'm just going to not do the detailed math but how many moles I have. So I've got 30 times 10 to the 12th moles. This is how many moles I needed per year. How much do I have? Well, we know how much the atmosphere weighs-- it turns out. We know how much the atmosphere weighs. The mass of the atmosphere is something like 5 times 10 to the 21st grams. And if I take 78% of that as my N2, so 78% is N2, then I can tell you that I've got-- let's see, 1.4 times 10 to the 20th moles of N2. That's how much I have available. OK? So if I keep on taking N2 out of the atmosphere, then I can now answer the question just like the candle. All right, how long do you have? I can answer the question, how long can we keep taking N2 out of the atmosphere and using Haber-Bosch, right? Oh, well, you would use the balanced reaction. So for every mole event do I take, I get two moles of ammonia. That's good. And you can work backwards. And you can learn that you're good. We're good for now. We have roughly 100 million years. OK? We have 100 million years that we could keep consuming. And then we'll run out of N2 in the atmosphere. How fun was that? OK, I love doing this. Now, when I get excited-- wow. Wait, before I get excited, what's the limiting reagent? [? add ?] enough N2. Is it N2? How do you know? How could you show? How can you prove? Limiting range, it means what runs out first? Well, oh, I'll just take H2 from the atmosphere, right? Na, uh. There's no H2 in the atmosphere. Where's the H2? Di da, it's over here-- H2O. It's in the oceans. How much H2O do I have? Well-- [laughter] It turns out you've got a lot. 10 to the 23rd-ish moles of H2O, right? 10 to the 23rd-ish moles. And so because of those coefficients of the reaction, you know exactly which one is going to run out first. You know it's N2, so your answer over there was correct. N2 is the limiting-- I love that little dance. I'm going to try to learn that. I'm going to learn that later. If you try this, you go 100 million years. You run out of N2-- plenty of H2 left. Let me read you. When I get excited, I didn't miss this. I like to throw stuff like T-shirts. Now, the thing is these T-shirts are not-- [laughter] Now, and I go like this. [laughter] And then I go all the way-- ahh, to the back, all the way up there, all the way up there. All right. OK, now look-- why do I throw T-shirts? I have no idea. [laughter] Sometimes things are just better left unknown. But please, I'll throw T-shirts 34 more times. Every lecture, I'm going to throw some T-shirts when I get a little excited. And please, if you get a T-shirt a second time, share. Right, don't take it-- give it to somebody-- trade. Right? Sharing is caring. Now, where were we. OK, now, here we are. We got to build this? Here's the thing-- in 1669, there are only 12 elements known. But yeah, I hope I've given you a sense. All these people and many more, I haven't even named. We're working on discovering new ones. By 1799, we had 34. Dalton and I mentioned had a table of 36, and they kept going. And as we got more and more elements, a lot of people were wondering, how do we organize these? How do we organize these? And are there patterns? Are there patterns, right? That became a hot question. Yes, they still wanted to smash and break and burn and make new, discover new things. But they also wanted to know what it all means. It wasn't just like a few-- it was a lot. And so they started looking for patterns. And one of the first people was Dobereiner, who came up with what he called the Law of Triads, which was a law that worked three times. [laughs] But what he found-- [inaudible],, everybody was arranging these elements in different ways. [inaudible] arrange it by mass. No, arrange it by something else. And he found that if you listed him kind of by mass-ish, then the middle atom of the three would weigh exactly half or roughly half of these two. That was cool for three times. OK, Newlands did something really important. Well, first of all, he loved music. And he arranged elements like this and again, kind of by mass. And he noticed that there were repeating patterns every eight. Now, unlike a lot of triads, this was really important, because the Law of Triads, you said, look, I can take these three. I can take those three. That's kind of cool, but it wasn't periodic. It didn't happen these three then, the next three. And whereas the Newlands, it was periodic. And it sort of also only worked for 2 and 1/2 runs. The Law of Octaves-- he called it the Law of Octaves. Oh, this was the time to look for patterns. And many people took a stab at it. Julius Meyer made some really important contributions in measuring the molar volume by knowing the densities and finding patterns. they were looking for periodicity. But as many of you may already know, it was Mendeleev who figured out what needed to be done. And what he figured out was absolutely critical. Let's go to the middle. And notice, he's listing them-- OK, he didn't know whether to put it this way or sideways or what. This is oriented differently than what we do today. But it's OK. He's just listing them and looking for patterns. But what Mendeleev did that were so critical-- what he did that was so important is that he organized them. So Mendeleev-- Mendeleev-- he had, at the time, this was-- let's see 1869. He had 63 elements. And he called this a periodic system. And the critical piece is he arranged them by mass. And this is so important-- properties. He really believed that the periodicity was the driver, and that the properties were critical in this organization-- the properties and the mass. Right? He was so certain that you had to take them both into account that he actually left gaps. That was new. At least for this many elements and arranging it this way, that was different. And so like, for example, look at this. Over here, so here you have aluminum, silicon. And you know what else? Uncover this, so you can it. Oh, zinc-- OK, zinc. Right, arsenic. But he said, no, don't just put this next to zinc because of the properties. This doesn't have the right properties. This should be more like phosphorus. So we're going to leave gaps. And this is really important-- he predicted the properties of those missing elements. So if you call this, for example, with silicon, we know this is germanium. We know that now. But back then, no one had smashed and burned and broken enough stuff to find germanium. So he said, well, we can predict it. Because it comes here in this table in this organization, therefore, it must have properties similar to other elements in that column. And he called it eco-silicon. Eco just means one next, right? And so these are predictions by Mendeleev-- predictions of the properties of this missing element. And here's what you get when you actually find germanium years and years later. And you measure those properties. And look at that-- it's incredible. Right? Thank you for that wow because I felt it too. Any more T-shirts? Aw, I felt the wow, and it made me excited. [laughter] All right. This is a big deal. You go back to here-- iodine and tellurium-- no, I'm not going to give you the mass ordering, because it doesn't make sense in terms of property. So I'm going to keep the mass the other way, even though this has a higher mass, look at that question mark. I'm still going to put it before because it makes more sense. That's what Mendeleev did, and it was absolutely critical. It really brings us to the current system that we use. So this is now the periodic table. Our periodic table today is based Mendeleev's organization. And we're going to be spending a lot of time with it. So we're not going to go through everything. But a couple of things. First of all, there's a key. That's the key to the table. If a table doesn't have the key, don't use it. It should have a key. And so, for example, oh, look at that number. That looks nice. Right? That's the AMU or the grams per mole which we now know are equivalent-- not equivalent but relatable. You have a mole, that's how many grams you have. Or if you have one atom, that's how many AMU's you have, right there. The key tells you what things are. Right? Now, if you have a more complicated one, don't panic. Here's yours. This is your periodic table. Ha, look at this. It's all in the key. There's a lot more information in here then in the basic ones. But it's all in the key. It's all described in the key. And the key may be different on the front and the back. There may be different properties included. But it's all in the key. And look at it. Some of these we know like the weight-- atomic weight-- that's what we've been talking about. Melting point sounds kind of obvious-- boiling point density. And then you get to these other things-- electronegativity, first ionization. All these things, what are they? We will find out literally next week, all right, as we build up our understanding of the atom. But it's all on the key. Now, this is so important that I really ask you with passion to carry it with you everywhere you go. I really do, because you never know. Remember the restaurant example. You never know when you need it. You never know. And, in fact, you can send me pictures. Students do this. I love it. You don't have to send me pictures. You can just post it on your feed or whatever it's called. [inaudible] What's it called? I don't know-- a thing. So they took it all the way up to Dartmouth, because there were some people up there apparently that needed help with their chemistry. [laughter] Here we are at a restaurant, looking at the periodic table. Here's another. It's a good thing to have at a restaurant. You never know. Here it is again. Look at how happy they are. Look at that, right? They're really getting into it. They're really getting into it here, teaching family members about this or being proud of it with the whole family. [laughter] Take it with you. It'll come in handy no matter what. Here it is under a Christmas tree. Right? It's in here somewhere. And there it is. And look, even if you're dressing up for a theater, it's going to be important. OK, now, this is the one I want to get to-- OK, even if you're a little tired. It's going to be what comforts you. I think it's being used as a pillow. That's OK, it's comfortable. Look at this. [laughter] Right? Right? You know even at the dance club, even at the dance club, oh, I see that. You might need this. You don't know. You don't know. Look at how happy he is. [laughter] Look at that, because he has this. Take it with you everywhere you go. Now, there's a bunch of things about this table that we'll be talking about. We just built it up. The first things is just some nomenclature. Right, just how do we name things? We name things that the rose, we call them periods, and there's seven of them. The cones, we call them groups, and there's 18 of them in the way we organize, in the way we present it, right? But there's something kind of important that happens here which is that these rows down here are actually 14 elements longer. So right, so this is 18 across. But these guys fit in right here. They go right there. Right? These are called the lanthanides because that's lanthanum-- innovative naming. These are called the actinides. OK. They go in there in between. Why don't we just put it all there? Well, you can. But then you got the bottom rows are 34. Right? And then the next one's up. There's only two rows in the periodic table that are actually 18. Then because then you got 18 here. And then these are eight. And then you got two up top. Now, why? That we will understand when we understand the atom a little better next week. OK? But for now, I just want to call your attention that these are the names that we use. And those top ones are called mean group. These ones are transition elements. You have alkalines here. You have lanthanides, actinides. These are the names that we give these regions that get more clear when we know what electrons are. And I think I'll tell you just the last bits about just a couple more observations about the periodic table, I think, when we start our lecture on Monday since it's 11:54. And as promised, I want to let you guys out right on time every lecture. Have a fantastic weekend and see you Monday. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 12_Molecular_Orbitals_Intro_to_SolidState_Chemistry.txt | And speaking of VSEPR, that's what we did last lecture. We're going to move on today. But I want to just clarify something about VSEPR. When I was talking about this I tried to put two different concepts in the same place. And I put something in the table that shouldn't have been there. And people corrected me. I want to be sure that there is no confusion around this. So what we had was a case of formaldehyde. And we had the case of SO2. And I want to be really clear about this. Both of these molecules, when you're thinking about VSEPR, the number of electron domain count-- Right. Remember, that's how we start. The number of electron domains is the same for both of them. And we did this on Wednesday. Right. And if the number of electron domains is the same, then the electron domain geometry is trigonal planar. Right. OK. But I was also telling you about repulsion order at the same time. And so I put the wrong thing in the table. And then I erased it. And I just want to be sure we're clear and we don't get bent out of shape about it. [LAUGHTER] Stop. [CLAPPING] Get out of here. Stop. Stop it. All right. All right. I'm going to still be here. I'm going to be here. All right. So like here, right, the shape. Remember, what the table was about was the number of lone pairs. OK. The number of lone pairs. And so if I've got the three binding pairs and no lone pairs, then it's going to be a trigonal planar shape. So that is trigonal planar. And over here I've got two bonding pairs and one lone pair. And so this shape is called bent. Now, what I did is I used that word. Aw, why? I don't know. But you make mistakes. When you're in the arena, you keep going. And if you make a mistake, you admit it. You leave it all there. You do your best. Forget the rest. That's all you can ever do. Now, here's the thing. I said bent. I said the word bent. And what I was talking about was how that bends a little bit because of the repulsion order. Yeah. But that shape is not bent. I want to be real clear about that. That shape is trigonal planer. And it's a little bit distorted. OK. This shape is bent. That shape is bent, because it's got a lone pair. Remember, the shape is about where the atoms are. But it's also a little bit distorted. Right? Because the lone pair pushes on those guys a little bit more. All right. OK. So those are the two concepts that I was saying all at once. And I want to make sure we understand them. There's trigonal planar. there's bent. And then there's repulsion order, which talks about little changes in the angle from those parent shapes. OK. Good. All right. Now, onto today. Now, here's the thing. Now, in all of this stuff with Lewis, it's a really good way to think about molecules. But this tells me that there's electrons on the bond. It tells me that there's electrons really localized in the region between the atoms. But see, the thing is, how much is it localized? How much is it really? Are these electrons-- if I were to draw the two electrons, are those dots right there? It doesn't really tell me more information than that. But if we remember the picture-- let's see if I can pull. Yep. There it is. There's a picture I showed you when we first talked about covalent bonds. Sharing is caring. Remember what happens. I've got the two protons. And we talked about these. A repulsion. And then the two electrons. Repulsion. And then I drew for you the fact that, but they can come together. And they can be like, you know what? You can use my proton for a little of your attraction. And hey, can I use your-- And they share it. And each electron sees both of them. And this is where those electrons are. But look at that. The proton is there and there. And the electron is in this probability cloud around both of them. That's the molecular orbital. And Lewis can't tell us about that. So we need something else. And that's what today's topic is about. Today we're going to talk about something called molecular orbital theory that gives us a sense of how these atomic orbitals overlap. It gives us a much better and a deeper description with more details about these covalent bonds. OK. All right. Now, so let's write a few things about molecular orbital theory. Molecular orbital. And what we're going to do is say, MO, and Molecular Orbital theory. So we'll keep a list of some important things. All right. Well, first of all, the first thing is that atomic orbitals, or, as we may call them, AOs, are the basis for-- watch this efficiency, the MOs. I didn't have to write it again. Right? OK. So what does that mean? Well, it means that I'm going to construct the MOs from the AOs. Now, the AOs are the things we've been playing with and making. Right? We solved the Schrodinger equation. So the MOs are going to be linear combinations. All right. Combinations of the AOs. And that's why you might see this, in fact, called LCAO theory. LCAO. Linear Combination of Atomic Orbitals. OK. And in particular, sum and difference of orbitals is what we're about. Because this is like what we talked about before. These are waves. Right. These are waves. Waves can either constructively or destructively interfere. And that's what we're going to explore in MO theory. By adding or subtracting these wave functions, these orbitals, these atomic orbitals. So that's enough to start. Let's see. So look, basis. A basis is nothing more than the things that I'm going to use to represent the things that I want. Right. And the things that I want are these molecular orbitals. And the things that I'm going to use to represent those are these, these beautiful atomic orbitals that we spent a lot of time understanding and developing. We'll start with the simplest case, which is-- I think I have-- oh, I even highlighted it. And I said, let's first make molecular orbitals using combinations of the S orbital. That's the S orbital. That's the 1s orbital up there. Remember, by the way, blue and orange is just the sign of the wave. Right. This is not a charge. Please don't make that confusion. These are just waves. Right. Waves. And let's go back and see it. The p orbital has a plus and a minus. It's a wave. All right. And the 2s orbital has that too. And then you square it, and you get the probabilities. Right? OK. So we'll start with 1s where it's all just one sign. And I can add it. I can add an S and another S. Or I can subtract it. So let's do that. Right. So we'll do that over here. All right. So if I take a 1s orbital, and I add it to another 1s orbital, well, you can see, as I bring these close together, if they have the same sign, these are waves. So if the waves are both positive, and I bring them together, they can constructively interfere. So as I bring them closer and closer, well, I might actually get the picture that I showed you, right? I might actually get something that looks like this. OK. I might actually get something like this. Now, if I subtracted them, then as I bring these orbitals closer together, well, they're going to cancel each other. And you can see right here, if I bring these orbitals-- by the way, which are coming from two different atoms. Right. The whole point is we're talking about molecules, molecular. So if I bring them together now like this, but I'm subtracting these orbitals, then you can see that right at the interface, right between them, it's always zero. Because I've just subtracted the same amount. No matter how close they get, it's zero. That's a node. Right. And, in fact, what you're going to get is something that looks like this. Now, so what I've done is I have added and subtracted the same AO. And I've created two MOs. So these are my MOs. And these are my starting AOs. That's about the simplest thing you can do in MO theory. So I'm using these as a basis. Now, there's something very important, which is that-- I kind of already alluded to this-- these are two atoms coming together to make a bond. This is the bonding axis. This is the bonding axis. OK. Now, if an orbital, if an MO is symmetric, so if it has cylindrical symmetry about that bonding axis, then it's called a sigma orbital. It's called a sigma orbital. That's the MO. Sigma is a notation. A sigma orbital is one that has symmetry around the internuclear axis, the two-- Does Pauli exclusion not apply to that? We're coming to that very soon. Because these are just the orbitals. I got to do something with them. It's just like over here. I made all these orbitals. But then I had to fill them. That got me excited. But you can see right here. When I add them up there's charge in between. So this looks like it's going to have a bond. That's a bonding MO. But if I subtracted them, this doesn't look very happy in terms of electrons being in between and the caring is sharing principle of a covalent bond. Right. So if I were to put an electron in this MO, it doesn't look bonding. In fact, it looks-- Non-bonding. Or antibonding. It's actually not quite non-binding. We'll get to that later. This would be antibonding. Why? Because there's not even a chance of electron density in between. That's your starting principle of the relationship. Let's try to share. Oh, by the way, right half way, no chance. That's not a very good way to start. And you can see why this is antibonding. It's the opposite. You're actually pushing charge away from the bond if you're an electron in that orbital. So that's an antibonding orbital. Now we get to the question. What do you do with these orbitals? Well, you make them. I made these with 1s. And the way that we typically draw this is we go back to the same picture that we've been working with, 1s here and 1s here. And I brought those orbitals together, and I formed one that lowered the energy of the system, and another that actually raised the energy of the system, because the electrons are repelling. They're actually less bonding. They're antibonding. All right. So these are my MOs. Those are my AOs. Right. AO, AO. And now, this is called a sigma orbital. And the way we refer to the antibonding orbital is with a little star. They're both sigma because they both satisfy that. Right. They're both sigma cause they both have symmetry around the inter nuclear bonding axis. But now, see, OK. So that's good. But now let's fill them. Let's fill them. Because now let's write down the rules. And then we'll fill them for some examples. OK. So I'm going to keep going on my MO theory list. Right. As you can see, when I do it this way, the number of MOs equals the number of AOs used to create them. So that's one thing that we can see right away. Right. I used two AOs. I added and subtracted them. I got two MOs. That's good. Well, another thing is, each MO will be just like an AO. It makes a lot of sense. I had room for two electrons here. I had room for two electrons here. I got to have room for four. Two here. Two here. Right. So each MO has a max of two electrons. And they must obey Pauli, just like in AOs. They must obey Pauli. You can't mess with quantum mechanics. Uh-uh. They're still electrons. They're still quantum mechanical. All right. You're just making their wave function more sophisticated. But they still have to follow the Schrodinger equation and the principles of quantum mechanics. And so, finally, what we're going to do is we're going to fill them. So we're going to fill the MOs with electrons, starting with the lowest energy, just like we did for atoms, lowest energy first. OK. And one more thing. Just like we did with atoms is we're going to obey Hund. Hund. Hund's rule, which tells you about how to fill degenerate orbitals. Well, we're going to have degenerate orbitals here too. We don't yet. But we will. So we're going to fill them. So let's fill them for the very simplest cases. OK. So we'll do some cases together. And we'll see how that goes. All right. So what I'm going to do are four cases just so we can compare. So first, we're going to do H2 plus. OK. That's the first one. And here, what we have is the same graph. But now, over here I actually can label it now. This was just an orbital. But now, gazuntite. Now, I'll say, OK. Well, this would be like H and H plus. OK. These are my MOs, sigma, sigma star, 1s, 1s. Oh, boy. So if that's H and that's H plus, and they're coming together to make H2 plus-- that's what's in the middle there. That's the molecule. All right. Well, then I also can now fill in the starting AOs. So let's do that. That's it. Because somebody had to lose an electron to start with. So now, when these come together, you fill. You obey the Pauli exclusion. You fill it from the lowest energy first. You say, well, I only got one electron. There it is. That's my MO diagram populated by electrons. OK. Now, there's a principle that comes out of this. There's a way of understanding bond strength now. There's a way of understanding bond strength. And that is a very important concept. Now, let's put it here in our list. OK. And it's a concept called bond order. Now, this one I really want to write BO. But I don't know if that's a good idea. I got Laura on that one. OK. And there's a definition for this. It's going to equal 1/2 of the number of electrons in a bonding orbital, minus the number of electrons in an antibonding orbital. Sorry. This is a little bit small here. But we'll do a bunch of them so you can see what this means. Right. So now, the bond order is important because the bond order of a molecule, if it's higher than a stronger bond, and it has to be greater than zero for stability. Let's see how that works. And we'll see this intuitively as well as we fill these up. Right. So if I count the bond order here-- oh, I'm going to do it. I don't feel like writing it out. The BO on this one is 1/2 times 1 minus 0, which equals 1/2. OK. Good. It's 1/2. Well, it's greater than 0. So right away I know that the H2 plus molecule is probably stable. That's good. But let's go to some more examples now. Let's do a few more. All right. Now we're going to work our way to H2. And in this case, energy is always going up in these diagrams. Now, each of these H atoms is bringing an electron. There's no charge to the system. And so now different quantum numbers. Pauli. But same sigma. Right. And so now that's the H2. Now, the bond order here is equal to 1. Right? I didn't have any electrons in an antibonding orbital. All right. But you know what's coming. I'm going to do He2 plus next. And let's just put it here for comparison in real time. If I were He2, each of these would be the AO occupation of an He atom. Right? But since I'm He2 plus, one of them is missing an electron. So this is what you're going to see. These are both He atoms still, but one of them is plus in this picture. But see now, OK. I've got three electrons. They're going to go here. And I've got to put one up here. I have to. Cause I had three. I got to keep on filling, just like in an atom. But now I'm filling bonding and antibonding orbitals. All right. And here I've got this and two up there. So if you look at the bond order, here the bond order is still positive. Here the bond order is zero. And, in fact, that's why He2 is not a stable molecule. Because you're putting just as much charge on this thing that wants to bond it as you are on this thing that doesn't want to bond it. And at the end of the day, it leads to a molecule that is not stable. So that would be He2. But He2 plus has a bond order that's similar. Did I get something wrong? He2 plus 1/2. OK. Now, because this weakened it. Right. This weakened the molecule, compared to this, because now I added charge to an antibonding orbital. So you would expect from the bond order that this one is more stable than this. And, in fact, that's what you find. Right. The H2 molecule has a stronger bonding energy. Now, we can write the molecular orbital configurations just like we did with the atomic orbital configurations. OK. And so like for here, so for this case-- I'm going to write it up top if I can fit it. So here we would have sigma 1s. And we're populating it with-- Ah. That's not my example. [GRUNTING] This is my example. Sigma 1s is the orbital. And we're populating it with one electron. Right. So that's how we would write that. Now, if we go to the next examples, we would have this one is this molecule you would write as sigma 1s 2. So this would be sigma 1s 2. And oh, now I've got sigma 1s 2. Oh, this is getting interesting. Because now it's sigma star 1s 1. So everybody see that? I populate. This is just like SPD. But now I'm just filling those MOs up. And I'm showing you what the molecular configuration is with those molecular orbital names. Sigma 1s. Sigma 1s star. OK. Good. Now, we can keep going. We could do this all day. Actually, we kind of will. But we're going to move on-- don't worry-- from sigma. Let's see. Let me go over here. There's another concept that I want you to know about MO theory. So if I kept going, I could have, for example, let's do lithium. OK. So now, lithium 1s. Oh-ho. 2s. Lithium brings this. What does it bring? It brings three electrons. And over here, 1s and 2s, three electrons. So this would be the lithium dimer. But I want to make an important point. When you would draw this one, here's how I would draw it. Whereas when I would draw this one, I would draw it like this. OK. Now, those are supposed to be the same spacing. I'll tell you in a second. Notice that the distance here between the bonding and antibonding orbital is larger than it is here. The reason is because it's related to the overlap. So there's another point, which is that the overlap of AOs is related. So the greater the overlap of AOs, the greater the change in energy between bonding and antibonding orbitals. And let's be clear. Those are molecular. But see, you could see like, if I'm lithium, those 1ses aren't going to overlap too much. Right. They're kind of close to the core. So you do get MOs there. But the difference that you get depends on how much these electrons are near in energy and overlapping. All right. So the 2s can overlap a lot more. OK. Now, if I were to fill this up, we would get something like this. These sigma 1ses-- oh, sigma 1s. What do you think these are called? 2s. Sigma 2s. Sigma 2s. Sigma 2s. OK. This is getting a little bit-- I'm going to draw this out here. That's the sigma 1s star. Right. And this would be the sigma 2s star. Well, what am I going to occupy? Well, how many electrons do I have to pour into the molecular orbitals? [GRUNTING] Lithium 2 is stable, because its bond order is-- you see. Bond order. Number of electrons in bonding orbitals, one, two, three, four, minus the number of electrons in antibonding, minus 1, 2, divided by 2. So the lithium dimer bonding order is 1. All right. And then you can go on, and you could say, well, if I had beryllium, I'd have another electron here and here. And I'd fill those antibonding orbitals up again, like in the He2 dimer. And I would have, again, an unstable molecule, which is, in fact, true. OK. This is sort of the simplest way you can think about molecular orbital theory because it's the simplest orbital. But you could spend the whole night. And it is a Friday. What else do you have-- this is what you do on a Friday. After I watch Dr. Quantum I'm going to just add every single one of these with every single one. It's finite. It's not going to take forever. Let's just do p. OK. Now, with p orbitals there's something interesting that happens, because now that sign-- you see how the p has a plus and a minus? Right. And so you've got to kind of think about that a little bit. OK. If I take a p orbital, and I'm going to do it like this. Minus, plus. And I'm going to add it to a p orbital that looks like this. And notice, I'm taking these and I'm adding them along an axis where they're kind of both along the axis. Right. They're both along the axis. And by convention, we'll do the pz as the one along the axis. Remember, there's px, py, and pz. When we solve for the p orbitals, those are the Ms. Those are the Ms. Right. One. Zero. And minus 1. Now, by convention we put the pz along the bonding axis. And you can see right away, so if I do this, well, I'm going to start adding these wave functions constructively. And what you're going to get is-- let's see if I can draw this-- something that looks like that. And where this is actually going to be plus. And this is minus. And notice that I'm going to have nodes in there. Right. I had nodes in the original orbitals. But I got a lot of bonding in between. I got a lot of bonding density in between. That is a bonding orbital. And the other thing we see about this is that it is symmetric around that bonding axis. So it's actually a sigma pz orbital. It's a sigma. We call it a sigma, because it's symmetric around the bonding axis. Now, if I were to take a px-- oh, well, let's actually subtract these. OK. So minus, plus, minus. Now, what you see is something very different, right. So now, oh, boy. Can I draw this? Maybe. I don't know. OK. I'll take that. Right. And so now what you see-- so I'm subtracting these. And so the density goes down. And now, you're going to get minus, minus. Right. Plus, plus. But I've reduced the electron density. And even worse, I'm back to that situation where right in between where I want to share the most, I'm saying no. No probability density there. And that's an antibonding sigma p orbital. So this would be sigma star pz. OK. Sigma star pz. Now, the other thing that you can do is look at the other directions. Right. And so I'll just take one of those really quick. And if you do that, you see something different. All right. So now, let's combine these like this. By the way, some textbooks. We're adding and subtracting. We're constructively interfering and destructively interfering. So sometimes you'll see a textbook add like this, or add the other way, which is subtracting like this. But it's the same thing. Right? You can decide how you want to orient it. But we're adding and subtracting constructively and destructively. That's what we're doing. So I'm going to choose to add it like this. Because then you just see, if I do this, then those are going to have some kind of overlap. Right. Those are going to have some overlap of plus appear and some overlap of minus down there. And that is called, oh, a pi orbital. Yeah. Yeah. But if I subtracted them, then what you would see is something that looks more like this, where you got that node in between again. Right. And this would be a pi star orbital. This would be for like px orbitals. Now, notice, these cannot be sigmas, because the pi orbital is not symmetric around the bonding axis. OK. These cannot be sigmas. But that's why we have another symbol for them. Luckily, we've got a lot of symbols, and the chemists are geniuses at naming things. And so these are called pi orbitals. And as you can see, OK. I have three orbitals. One, I put it along the axis. And then I've got two going perpendicular in the other plane. And this is one of them. And then the other one would be the other one, py. So as you can see, I'm going to have a sigma star, two pi orbitals, and two pi star orbitals. Right. And then what we got to do is we got to put them on our energy scale. And instead of going through drawing it all, I'll save myself a few minutes here. I'll show it to you. All right. So there is that energy scale. And oh, they're pointing out the nodes. Isn't that beautiful? You can see right here. They're not saying what molecule this is yet. This is a 2p orbital. And it's pointing along the same bond axis as the other one. So this is going to be a sigma p orbital. This is a sigma p orbital. And you can see that. And there's a sigma star. Right. Good. Now, if we go to that, there's the next one. There's a pi orbital. And you get the same exact effect. You've got a lowering of the energy, cause you're putting electrons on the bond, and a raising of the energy cause you're taking them away. Right. And that's what it looks like for the pi orbital. And we can put it all together. And if you put it all together, what I want to do is put it together for O2 and then show you a video, and then do my why this matters. Now, if you put this together, I'm going to show you the system for oxygen and nitrogen. OK. And then we'll do a couple other cases too. But we're going to do oxygen first. So if I take oxygen, then I'm not even going to write 1s anymore. That's way down in energy. It's not really involved in the bonding. I'm going to leave it out. But I've got my oxygen 1s, oxygen 1s. OK. Good. So those are going to come in and form molecular orbitals like that. And we know that they are all filled. OK. Now, up here I've got my oxygen. Sorry. I just said I wasn't going to do 1s. And I'm not. Those are 2s. Those are 2s. Sigma 2s. Sigma star 2s. OK. Good. All right. Oh. Let's see. Now, over here, I start with my ps. And I start with px, py, pz. And in oxygen, how many electrons do I have in here? So it's like this. [INAUDIBLE] Everybody should be shouting. [SIGHS] That's so much better. That was close. And then I've got to-- well, I'm not going to have enough room. I've got these over here. px, py, pz. Ha, ha, ha, ha. And there is oxygen. But you see, now, they come together. And they form these MOs. And we just went over what types they are. Right. There's a sigma pz orbital. And there's a sigma M. But the ordering is about the same thing that we learned, which is that it has to do-- this delta is up there. This delta is up there. And so in that pz for oxygen, you can overlap more. It pushes those apart more. And so what you get is that the pis are inside like that. The pis are inside. Right. So if I write in here, it's going to get too small. So I'm going to do this. This would be a sigma 2pz. This would be a sigma 2pz star. And over here, these would be pi orbitals. Pi px, pi py. And these almost can fit pi star px and pi star py. All looking like the shapes that we've been drawing. Right. And then the filling part comes from the filling of the AOs. Right. I've got my filling of the AOs here. I've got the valence filling for oxygen. There is a 1s down there. Right. And so I've got to put four electrons from there and four electrons from here into the middle. So let's do that. One, two, three, four, five, six, seven, eight. Pauli. Pauli. Right. So for oxygen, the bond order, if you add it up, the BO is two. It's a double bond. We know that. Oh, but now we know so much more. And this is what I want to make a point. Lewis could have gotten us this. But Lewis can't get us magnetism. Lewis can't get us magnetism. Molecular orbital theory can. OK. And let me show you magnetism right up close. Because if I take liquid nitrogen. That's not me though. And I pour it through a huge magnet. I want that magnet so badly. But if I did that, look at that. Liquid nitrogen just goes right on through. And it's really fun to do actually. But there you go. That's liquid nitrogen. But look at liquid oxygen. Liquid oxygen. Now you pour it through, and the magnet holds it in place. All right. Whoa is right. There's four more things we've got to cover. One is paramagnetism. Another is something called mixing. Another is what happens when you go hetero nuclear. Oh, yeah. You'll see what that means in a second. And finally, what I said I would get to, which is non-bonding, which are not the same as antibonding. What I'm covering right now, there, is paramagnetism. Because as you saw on the exam in one of the questions, we explained it. We said if you got unpaired electrons, then you are-- what does paramagnetism mean? It means that if you put a huge magnet on it, you'll respond. You will feel a force from that magnetic field. And so we did that in the exam. We had a silicon atom, because we hadn't done molecules. But now we got molecules. So I can tell you why that experiment happened. What's happening? I can tell you why that experiment happens. Really? I didn't even know there was more. [LAUGHTER] It happens because of Hund's rule, molecular orbitals, and the fact that I've got these two unpaired electrons in the O2 molecule. And you know that in nitrogen those two electrons are gone, cause nitrogen is missing one more here and here. So you've got two less. Everything's filled. Everything's filled. Paramagnetism. Now, OK. Speaking of magnetism, I couldn't help but show you this. This is diamagnetism. Now, diamagnetism, everything is filled. But you still feel a little repulsion to a magnetic field. And some of you may know there is this thing called the Ig Noble Prize. Only one person in history has won both the Ig Noble and the Nobel Prize, Andre Geim who discovered graphene with scotch tape. But before that he won the Ig Noble Prize because he did this to frogs. Because water is diamagnetic. And so it repels a magnetic field. It's just got to be really, really, really high. I hope that frog was OK. It looked sort of OK. So the frog was floating. And it was like a study about levitation using magnetism. Why am I showing that to you? No particular reason. [LAUGHTER] But this does get me to the why this matters, which has to do with how you cook pasta. And, of course, since we're talking about O2, when I have finished cooking pasta, what do I do? I pour it. There it is. It's like it's sophisticated. I pour it through a colander. That's a membrane. That's a membrane. You did use a membrane. You did a filter. Now, but I could also have done that separation, that same separation, I could have left it on the stove. I could have. And it would have boiled out all the water and left me still with the pasta separated from the water. I've accomplished the same exact thing. I have separated the pasta from the water. Test done. Pasta may not taste as good. [LAUGHTER] But you've done it. But see, the thing is that if you separate things this way versus that way, you can just feel how much less energy it's going to take. In fact, you can save over 80% of the energy if you do a membrane based separation as opposed to a thermal one. Gazuntite. So those are two ways to separate pasta. But see, there's two ways to separate lots of things. Like how about 1 to 10 nanometer particles? How about chemistry? How about molecules? How do you separate those? Well, you got the same two ways. And if you count up all the things we separate this way, it's a lot. It goes on and on. And it will go all the way down the Infinite Corridor. And this is how we do chemistry. In fact, has anyone seen this on the side of the road? That's a distillation column. It's a big pasta cooker. That's all you're doing is boiling one molecule off of another over a long time with a whole lot of fossil fuel. In fact, if you look at the US energy consumption, roughly a third of it goes into industry. 40% of that is for this one thing. It's boiling pasta. But the pasta is 1 nanometer to 10 nanometer particles. 40% goes into boiling one chemical species off of another. Separation. Separation. That's 12% of all the energy. That's the same as every single drop of gasoline in every single car truck and bus. Just to give you a sense of how much energy that is. You'll say, well, why aren't we using a colander? Why don't we just pour it through a colander like we do our boiling pasta? Well, we do that for one field. Desalination. I got a [INAUDIBLE] on that another time. But we don't do it for all those other things. And there's a really simple reason. We don't have the right pasta colander. We don't. There's no option for that size scale that can withstand the conditions that are in all of those chemical separations. But if we did-- we take so much O2 out of the air. We need O2. But we don't want the N2. So we have to separate it. How do we do it? We go cryogenic. We go to very, very cold temperatures, which is the same as boiling. Right. But you're still spending all this fossil fuel to lower the temperature. That's how much O2 we generate each year. And this is how much energy it takes. It's like 1/2 a percent of all US energy, just to get-- But what if you could use something like O2's paramagnetism? What if you could use something about the chemistry or O2 to do this separation more efficiently, lower energy, or maybe make a new colander that does that? And if any of you have ideas, come talk to me. This is a problem I care a lot about. OK. Ah. But I had some other-- paramagnetism. Unpaired electrons. We got that one. Ha. [SIGHS] Why is chemistry not-- why can't they follow the rules? Why? But they always got to break them. And what we see, this was oxygen. Sigma. Sigma S. Sigma. Sigma star. Pi. Pi star. But look at what happens for nitrogen. Why? Because of something some people like to call mixing. Remember, I said that-- where did it-- somewhere I said that the closer in energy, or the closer in symmetry orbitals are, the more overlap they can have, and the more they interact and can mix together in the ways that I've been talking about. Yeah. Well, it turns out that if you go below oxygen, in what are called homonuclear dimers, which is where the atoms are the same, then you can get mixing even between this sigma and that sigma. And so what happens is, instead of them being kind of separate like this for N2, there is an interaction. You see. You can think about it the same way. I'm throwing more orbitals into the mix. And so because they can contribute to overlapping, you're changing that delta E even more. That's effectively what's happening. And so you can see this one for nitrogen goes down, but that one goes up. Because that's the delta E. Because it's able to mix in. It's able to mix in because they have the same symmetry. And they're closer. For those smaller atoms, they're a lot closer in energy. And so if you look at this, what happens is those switch. They switch. It's real. They switch. Now, it doesn't change the thing I just talked about, which is the magnetic properties of N2, because they are still all filled. But it is important. Because if I were to pull an electron out of N2, it would come from a sigma orbital, not a pi orbital, because of those interactions. And those happen below O2. So this is from your textbook. And you see, it says 2s 2pz interaction. Remember, 2pz is sigma. Sigma. Same symmetry. Able to mix. Don't mix very well here. But here they're strong enough to change the ordering of that orbital and that orbital. Right. You see that? That's an important effect. OK. All right. But it only happens below O2. It happens below O2, because that's where those energies and orbitals can line up in that way. OK. So we just covered that. Now, there's two more things. And then that's all of MO theory. One is I've been giving you the same atoms. But what if we go from a homonuclear dimer to a heteronuclear dimer, which just means one is one and one is another type of atom. How do we draw an MO diagram for that? And the second thing is what happens? The second thing is what happens in this case? In HCl, H is bringing only one S electron to the party. But Cl is bringing all of the-- it's bringing S. It's bringing P. What does it do? Right. How does the MO diagram look in that case, where I've got all these extra electrons coming in from one of the atoms. Now, I will not do this in 30 seconds. But I will, next week, give you a nice sort of thorough explanation for each of these two last MO cases. In the meantime, have a very good weekend. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | Goodie_Bag_2_Electronic_Transitions_Intro_to_SolidState_Chemistry.txt | [SQUEAKING] [RUSTLING] [CLICKING] VIVIAN SONG: In this video, we will be going over a Goodie Bag number two which is about electronic transitions. What you'll need are four LEDs-- white, blue, green, and red-- and one spectrometer. The objectives in this video are to understand photon absorption and emission, apply the Bohr Model, and use the spectrometer to see sample spectra. The conceptual questions you should keep in mind are, how do electrons transition to different energy levels? And why is spectroscopy a method of material characterization? As a reminder, the Bohr Model can be used to model these electronic transitions, assuming that there's only one electron in the atom. So right now, I'm going to draw my energy axis and three different energy levels that my electron can hop into. So this is one, and two, and three. So let's say that our electron starts in the ground state. And it absorbs some energy and can hop to a higher energy level. But in doing so, at this higher energy level, it is in a more unstable state. Eventually, it'll want to return to that more stable state. And when it does so, it emits energy in the form of a photon. So emit photon and absorb energy. These different electronic transitions can be tracked with this equation. So the change in energy is equal to minus 13.6z squared times 1 over nf squared minus 1 over ni squared. Where z is the atomic number, and f is the final state of the electron. And i is the initial state of the electron. And this change in energy is in electron force. So remember that these electronic transitions are quantized since these energy levels are not continuous, but they're integers. This means that the photon that the electron emits is going to have a certain set of different wavelengths. And we can see that with this equation energy equals hc over lambda, where energy is equal to this change in energy. h is Planck's constant. c is the speed of light. And lambda is the wavelength of our emitted photon. After looking at the red and white LEDs through the spectrometer, you should see something like this. Notice how for the red LED, mostly the red and orange bands are visible. Whereas for the white LED, almost all the colors are visible. Another thing that you could explore is using your spectrometer to look at the ceiling lights and seeing how that spectra would differ from the spectra that you observed from your LEDs. In summary, electrons absorb and emit photons of different wavelengths because electronic transitions are quantized. By capturing this information, the spectrometer becomes a very useful tool for characterizing our materials. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 13_Hybridization_Intro_to_SolidState_Chemistry.txt | But back to MO Theory, which is where what we talked about last Friday. And there are sort of two last things I wanted to mention now related to MO Theory before moving on to the topic of hybridization. Now, before I do that, you know, I'm going to use a couple of videos today and I'm-- I'm not going to show you this one, but I just want to mention there's lots of videos that help, you know, animate this stuff, right? So like this is what-- so like-- Oh, OK. [VIDEO PLAYBACK] Bonding molecular orbital. [END PLAYBACK] Antibonding, we talked about this last Friday. But look at this animation. I couldn't do that on the board. Watch this. [VIDEO PLAYBACK] Antibonding molecular orbital formed by the combination of two 1s orbitals are called sigma 1s and sigma star 1s respectively. [END PLAYBACK] Kind of remembering stuff now, right, from Friday? [VIDEO PLAYBACK] The bonding molecular-- Oh, look at that. -- is lower in energy than the original [INAUDIBLE].. [END PLAYBACK] That's that kind of animation that you just can't-- you know, I can't animate the bonding and antibonding. Well, I kind of can with my hands, but there it is. And there's lots and lots of those kinds of videos. I'll show you a couple today related to hybridization, because it really does help see these 3D orbitals, right, in their full shape and what's happening. So I highly recommend it, especially if you're having some conceptual questions related to stuff we talked about Friday and today. Take a look at some of those videos. Now, where we left off on Friday, before I get to nitrous nitric oxide, which has all those different uses in the world, and many more. We left off with the homonuclear dimers. That's what we were doing on Friday, right? How do the atomic orbitals come together to make orbitals that are molecular MOs, right? And we did it for dimers that have the same atom, right? And so here-- so we did nitrogen and 2 and we did O2, right? And so let's do that on the board and that-- because now we're going to start with one that has N and O, heteronuclear dimer, right? Two different atoms. But if I had the same one, remember, this is what it would look like, right? So here-- here would be, say, N2. Would-- you know, I'm going to just stick with the valence. So the 2s for our nitrogen atom, and the 2p would have, remember, Hund's rule. All right. OK. So that's what the 2p would look like. And then over here, we would have the other atom. This is how we roll when we talk about MOs. And we'd have the 2s over here. And this is how we start. These are my AOs for the 2 nitrogen atoms. Right. This is what we did, we're just getting back in the mood because we had the long weekend. And when we make the MOs, remember, we have all those sort of things that we talked about in terms of how these-- this difference between the bonding and the antibonding orbital, right, is related to the overlap in the same symmetry of the two AOs that come together. [INAUDIBLE] we talked about that, the more overlap, the biggest this delta. Gesundheit. All right. And then we also-- so that would be filled, this would be filled, right? Sigma 2 s, sigma a. Sigma star 2s, right? And then for nitrogen, we also talked about how-- oh, boy, these are reversed, right? See, I'm now already... ...backwards twice, which is not what I want. OK. And this was-- but this is the weird thing about nitrogen, because if we go to oxygen, then the way it goes is, you know, you've got your 2s in oxygen. OK. And in oxygen, you've got your 6 valence electrons in the 2p. And over here, you've also got the same. OK. And you've also got that 2s. And-- but for oxygen-- OK. So those are the sigma, sigma star. But for oxygen, we go to the ordering that we usually have, which is that-- remember, those pi orbitals have-- and this isn't necessarily on a scale. But those pi orbitals-- all right. So this was sigma, sigma star, and these are the pi orbitals. Pi and pi star, right? Sigma p. Sigma p. All right. 2p, 2s, sigma. This would be sigma s, sigma 2s, sigma 2s star, sigma 2p, if you want to be exact about it. Sigma 2p star. Pi x, pi y, pi x star, pi y start. We talked about all this on Friday, I'm just getting us back in the mood. But the way we left off was that these are-- OK. So for-- if I fill these all up now-- right. Remember, we do the MO filling, then we'd have this, and this, and-- ooh. Now, hold on. Yep. This and there and there for oxygen. And then over here we're filling them up. We got-- those are filled, those are filled, and this is filled for nitrogen. Remember, I showed the movie pouring liquid oxygen, pouring liquid nitrogen. Oxygen has unpaired electrons, which means it's paramagnetic. It has unpaired. Electrons, the O2 dimer. This is O2. These are the MOs of O2. These are the atoms. These are the MOs of N2. N2 doesn't have any unpaired electrons so it's not paramagnetic. But the thing that we left off on-- that's what I'm saying is, that these two switch order for those dimers, for lithium through nitrogen. Those two switch. All right. We talked about that on Friday. The reason is because for those atoms, these sigma-- these sigma p orbitals can mix in with the sigma s orbitals. And so you get more. Because, remember, the more overlap, the greater the energy, rate? The greater that delta between bonding and antibonding, right? So the more mixing you have, the more spread you get. And it's so much that for those dimers you push the orbital up. OK. That's all what we talked about Friday. Now, I got on of each. I got one of these on the board so that you can see them as we now take one atom from one side and one from the other. It's actually kind of what you expect. You just have to have one thing in mind when you think about No-- so if this is N, OK, over here, and we're going to put 2s there and 2p there. Here, here, here, here. There's my N atom. And now I've got O over here. So I've got my-- but, now, see, the thing is always more electronegative. And the way this works for a heteronuclear dimer is-- let's write this on the board. So for a heteronuclear, that means that the two atoms are different. OK. But for a heteronuclear dimer, the bonding MOs are closer in energy to the more electronegative atom. Electroneg atom. Atom. So what that means is that if I draw-- so if I draw my oxygen levels-- but you know this, right? You know this, also, from all the work we did on atoms, right? You know a little something about-- remember, this axis is energy. This axis is energy. And so if this is 2s for oxygen, and then the 2p's are going to be sort of like here. OK. And we'll fill them up thinking about Hund's rule. Well, now, what we're going to have is that those-- the MOs for the No molecule are going to come closer, they're going to come further down from that side, right? Then they come from that side. Do you see that? There's an asymmetry. I made it kind of subtle. I made it kind-- and why is this curved? That's not good. What's going on here? There's a little bit of an asymmetry here, right? So-- so this is lower down-- these are lower down, right? So when you draw your orbitals now-- oh, man. I'm going to draw this a little bit lower. Right. So now when you draw then, you see that bonding orbital is going to be closer. You see, this is a bonding orbital still. It's still bonding, right? It's going to be a sigma p, sigma 2p. But it's closer in energy to the more electronegative levels. You see that? Right. So it's not symmetric anymore. OK. And then the same what happened with these guys, right, they'd be closer in energy to the-- yeah. They would be closer. And I'm not leaving myself enough room here. That's why I've got some videos. They do it better. All right. OK. We'll leave that one there. It's a little bit better. Not much, but a little bit. So the bite-- now-- oh, by the way, for No-- let's talk about MO for a second, because there's another thing that's important that we talk about, which is bond order, and, you know, the bond-- oh, electron configuration is another thing, right? So like the configuration, we can write this out, right? So the configuration of, say, 02 would be sigma 2s, right, you're just filling them up with two electrons. Sigma star 2s with two electrons. Sigma 2p with two electrons. Pi, 2p with four electrons, and then pi 2p star with two electrons. That's how that looks. It's a little bit messy, but I hope you can read it. That's just reading up, right. It's the electron configuration of the MOs, just like we would write the s, p, d electron configuration of that, but this is a molecule. Now, sometimes you might, you know-- say there's a convention we use, which is that the pz is the sigma. Nobody can agree on it, so different textbooks say px but we're saying p-- if pz is sigma, then this would be like a sigma 2pz, right, and you might, if you really want to write this, you know, you might write that that's pi 2px if you really want to keep track of it. There's 2 orbitals in there. You didn't fill 1 orbital with 4 electrons, right? So this would be pi 2px and pi 2py 4, right? That's what's happening in there. OK. But the other thing you can see from the bot-- from this filling is, also, the bond order. And so for the bond order-- bond order, here you would have 1/2 times, and it's all of the-- remember, the bonder is all of the bonding electrons minus antibonding electrons divided by 2, right. And if it's zero, it's not a stable molecule. If it's greater than zero, it's stable. And the higher the bond order, the higher the bond strength. So that's also from Friday. And so this is 2 minus 2 plus 4 minus 2, and so you get 2. Nope. I missed a 2. Plus 2, plus 4 minus 2. 2 minus 2, that kind of cancels out, right, plus 2, plus 4, minus 2, all divided by 2. So the bonder for oxy-- now, the bonder for No-- bond order for No is going to be 2.5. Well, that's kind of interesting, now, you can see-- OK. So No is stronger-- more strongly bound than O2, less strongly bound than N2. And it's kind of interesting. How can I make No a stronger-- a more strongly bound molecule? Well, I can take an electron away. It's a little bit counterintuitive, right? Because for No plus, the bond order goes up because the electron that I'd taken away was in an antibonding orbital. Right. So No plus actually has a stronger bond than No, even though it has one less electron. You get this all from just looking at which electrons are occupying which orbitals. In this case, you had in this very much not drawn to scale-- oh, boy-- figure, right. What you had was one up there. So if I take that out, my bond order increases. Right. OK. All right. And the last point I wanted to make about MO theory before we talk about hybridization is with one other example, and that's HTL. OK. So for HTL, you got something else happening that's important. So we just talked about electronegativity driving where the-- you know, that this is tilted. OK. Fine. And now we got the fact that chlorine comes into this situation. You know, it's like chlorine is like, hey, I got 7 electrons. And hydrogen is like, hey, I got 1. So how's this work, right. So this gets to what was brought up Friday, which is, there's another type of MO. And it's not bonding, and it's not antibonding, it's non-bonding. That is not the same as antibonding. Non-bonding doesn't take sides, it just is. Because the way this works is hydrogen comes at this-- OK. So hydrogen's up here with this 1s. Chlorine is, you know, let's say over here. It's got 3s filled, and it's got-- let's draw these here. OK. 3p. And chlorine's got 7 electrons in the 3p. Do they-- do they all come together and form MOs? No, they can't, because there's only 1 on this side, right. So this 1 orbital will form-- let me make myself enough space so I'm going to exaggerate this a little bit. So this 1 orbital will form a sigma bond. Sigma 3p, sigma 3p bond. This would be a sigma 3p star. Now, that's exactly what we've already done, right, the sigma. But-- but-- oh, there's something a little interesting about this one, right, because it's an s. It's an s. But that's OK because the s can talk to a sigma p. This is -- we talked about this. The s can talk to this. That's OK. Because it's aligned, right. Now, but these guys don't have anyone to talk to. They don't. So they can't make their pi orbitals. They can't do that. And so they just stay there. These stay there exactly at the same place. They're not doing anything different. They're just hanging out. Those electrons are non-bonding. So these are not pi orbitals. Sigma and pi are special names. They refer to these orbitals of the molecule that involve the LCAO, the linear combination of the atomic orbitals. These do not. Nb. Nb. That's what we'll call them. Just to be very clear. Just to be very clear. Now, they're filled. They're filled, and they do not count in the bond order. By the way, this is, also, a non-bonding orbital. That's a non-bonding orbital. Non-bonding orbitals. These are not going to count. So in the bond order, you only have-- you have 1/2 times 2 minus 0 equals one. These do not participate in bonding. I got to fill this. 2 electrons are in that one. OK. So that's another example of something that happens in these MO diagrams. One of them brought too many electrons to the party. And so there's no one to talk to. But it's OK. They're just going to stay there and wait. Maybe something else will come along. Right. OK. Now, that is, right there, what I just drew, look at that. 2p 2s, and you get this thing, right. Oh, it's a beautiful molecular orbital between-- OK. Fine. It was a 1s and a 3p, but you get the point. It was a p oriented along the axis with sigma symmetry. That means if I look along the axis-- sorry, cylindrical symmetry, sigma means if I look around the axis, everything is the same, in a circle, right? Right. So-- but-- and so that's a nice sigma orbital between two different atoms. MO theory. But see the other thing that can happen that is something that I want you to know today, is that orbitals on the exact same atom can also do this. They can also do this. They can combine to form a new orbital. sp, same atom forms an sp orbital. It's called a hybrid orbital. So we're going to give a couple of examples. This is hybridization. Hybridization. And I can give a couple of examples, then my [INAUDIBLE] matters. So this is when two or more ao's that are similar in energy. They're similar, they're not the same but they're close. You know, you can't hybridize if one level's way up here and the other level's way down there. But if they're close, 2s 2p, then those orbitals can combine. And what they do is they combine to form, and this is critical, sets of equivalent orbitals properly oriented. This is really important. That's why I'm writing it all out. Properly, not property. Properly oriented to form bonds. Oh, this is such a big deal, to form bonds. It's the last word there, but I did it all caps to emphasize. Because that's what this is all about. You see, it-- now, let's take an example. I'm going to take an example of methane. Who brings what to the party? OK. I got H-- I've got 4 H's. H1s, H1s, H1s, OK. This is definitely getting repetitive. H1s. But I did it. There they are. And they come, and they're like, hey, carbon, what you got? And the carbon is like, OK, hold up. I got my 2s, that's here, and I got my-- all my 2p's, and what are we to do? What are we to do? How can these four hydrogens come and bond with the carbon given that it looks, at least, like the carbon isn't bringing the kind-- you know, how's it going to do this? And, also-- and this is critical. This is why I wrote it here. Properly oriented to form bonds. You see, because this taps into what we learned with VSEPR It does, because VSEPR says you got to not repel. Or, no, you're repelling, but try to minimize it. I mean, that's what's at the core of VSEPR. So what this system sees is a way out. And by way out, I mean a way to happiness. And by a way to happiness, I mean a way to lower energy where all 4 of these can bond to the carbon atom if it has 4 same bonds. If it had 4 same bonds, then it can orient them tetrahedrally and make VSEPR happy, minimize repulsions, right. And you get your methane molecule. So what ends up happening is this carbon atoms says, hold up. You 4 hydrogens, you wait there, I'm just going to go into the back and do a little reorganization, and I'm going to come in it with-- I'm going to say, well, if I had-- if my carbon atom comes at-- comes at it with 4 equal orbitals that look like this, those are called sp3, because I took all 3 p orbitals and I mix them up with my s orbital and I formed equivalent orbitals. Now, carbon comes out of the back room, and it's like, hydrogens, I'm ready, I can make a minimum energy, and I can follow VSEPR and we can have-- all have very, very low energy, compared to whatever else I could've done, if I stuck like that. What I'm really talking about when I'm talk-- OK. You might know carbon, you know, it doesn't go into the back room. It doesn't. But what is it doing? It's nothing more than solving its Schrodinger equation. That's what this is all about. You just change the boundary conditions. And so what happened-- that's all hybridization is. You're solving the Schrodinger equation all the time. We are always solving the Schrodinger equation all the time. And in this case, carbon said, well, how can I solve it and minimize my energy with the 4 hydrogens? This gives you the lowest energy. Because, now-- oh, I got a movie. Now, I can make those, see. So I have them-- I have a picture here. s and those three. And it says, how can I do it and minimize repulsions? This is how. Three equivalent-- 4 equivalent orbitals to make it sp3. And this is where I think a movie helps. So I'm going to play this. It's like a minute long. [VIDEO PLAYBACK] Each 2s orbital is a two lobe shape converging at the nucleus. - There's the 2s. So there are the three 2p orbitals. However, when hybridization occurs, the s and p orbitals cease to exist, They don't cease to exist, they just didn't-- it's a new boundary condition. You're solving the equation. Right. So just to keep that straight, right? These are the orbitals-- and the two sp3 orbitals have an entirely different shape. - That's true. We can see that orbital hybridization explains the VSEPR placement of carbons for VLANs electrons. Since all four 2sp3 orbitals are equivalent, each 2sp3 orbital repels the others with equal force, resulting in identical bond angles. The carbon atom only hybridizes when it is in a bonding situation. Here, four hydrogen atoms bond to carbon by overlapping their orbitals with carbon's hybrid orbitals. So what would be the reason for this? If we go back and see that both carbon and hydrogen have unpaired electrons, the overlap allows the electrons to pair, and thus go to a lower potential energy. [END PLAYBACK] Oh, that's such a beautiful place to stop. Lower potential energy. Happier. That's what this is all about. OK. So I-- you know, like I said, you can see now the difference, right, if you just watch-- it's one minute long, and I think it really brings it to life. So if you're, again, want to see this a little bit more conceptually, these movies are good. I have one more movie I'll show you that's like a minute long. OK. Now, the thing is-- so what we just did is we did carbon hybridizes to sp3. It's called sp3 because you're bringing 3p orbitals and mixing them with s to form 4. And that allowed it to do this. But, you see, if I had ethane, which is C2H6, it's the same thing. Now-- because, basically, what's happening? A carbon atom comes into this system of ethane, and it says, I need 4 bonds that are kind of equivalent-ish. All right. I got to give one to each hydrogen and I've got to give one to that other carbon over there. And by doing that, it can minimize its energy, it can minimize repulsions, it can be the happiest molecule it can be. But to do that, carbon must have these four equivalent sigma like bonds. It must be able to form those four equivalency. So that's very similar to methane, except that in this-- so, also, ethane will be sp3 hybridized. It will have sp3 hybridized orbitals. But then, and you know it's coming, right, so ethane has C2H6. But what if now I've got ethylene. I think I have-- maybe I have a picture of it. Do I have a picture of it? Yeah. Oh, there it is. Ethylene. Ethylene is used. This tiny molecule is 2 carbons and 4 hydrogen instead of 6. This is used in so many applications I can't possibly list them. If polyethylene is a polymer made out of this as a base, we will polymerize this a little bit later in the semester when we talk about polymers. It's also used in the back of your supermarket in case you didn't know that. Very soon-- actually, kind of almost now-ish, most of the fruit we get will be super green when it gets here. And then they use ethylene to actually induce, sadly, the brightness. Doesn't taste as good. But, anyway-- OK. Now, but for ethylene, you've got a different situation. Because for ethylene, each carbon atom needs 3 bonds. Each carbon atom needs 3 equivalent bonds. And it can do that, too. So, for C2H4, each carbon atom needs 3 equivalent bonds, and it can do that if it hybridizes in the sp2 hybridization. So let's draw that, right. So these are the original orbitals of carbon. And they will go to 3 sp2 orbitals. Notice, I can put one electron in each ready for action. Ready for bonding. One electron in each of these equivalent hybrid orbitals, sp2. And I got one left over. Ho, ho, oh. One left over. But the one left over is a p orbital that didn't get in on the hybridization. But that's OK. That's OK. These are p, right. So this is s and this is p2s 2p. And that's a p orbital right there. Those are my hybrids, and that's a 2p orbital. There's another 2p orbital on the other carbon. Also, got left out. Now, I only needed 3 because, you see, I formed this nice spatially maximize to minimize repulsion, right, separation, maximize to minimize repulsion so it's a planar-- oh, but it's a planar for another reason. Because each of these carbon atoms has a p orbital, and we know that 2p orbitals can form a pi bond, right? And I have a picture of that. That's exactly what they do. So this is the sigma. The hybrid orbitals are sigma, right, because they're along the axis, and look at them. If I look down this axis, I can draw a circle, and there's full cylindrical symmetry, right. But I've got this left over p orbital here on each carbon atom. That's that one there. And I didn't need it to do this but it's still there. And so it forms a bond. It forms a bond. Right. It forms a pi bond. So that p with the p from the other one forms a pi and a pi star, right, in the MOs, and you occupy the pi with the two electrons. And that's a bond. So in this you've got-- you've got a bond from this that's above or below-- in this orientation, it's above or below the plane. And you've got a bond from this. So you've got a double bond. That's exactly what we know already, but we know so much more now. We know so much more. Because-- because, now, we know that it's not just two-- you know, two bonds and four electrons, we know that they have shape to them. We know that they have structure to them, and we know that those bonds are different. These are pi bonds, and that's a sigma bond. Right. And you, also, know why this molecule stays planar. Because if I try to twist it, then those p orbitals won't align, right. It's like a little spring. If I-- if I try to twist it, it's going to go right back. Because it's minimum in energy is when these things can maximize their overlap. All right. So that's going to stay planar. OK. Now-- and so for this case, you've got a double bond. So this is double bond, and I think I have a little carbon, carbon, double bond. And I think I've got a video to show this, because this is another case where, you know, again, I just want to make sure you guys get the intuition here, and I thought this is a pretty cool animation of sp2. So I'll play it for a minute. [VIDEO PLAYBACK] [INAUDIBLE] bonds. The one 2s and two of the 2p orbitals hybridize. Consequently-- - That's cool animation. --this hybridization is termed as sp2 hybridization. - Look at that. The hybridization leads to the formation of 3 [INAUDIBLE] sp2 hybrid orbitals. As you can see, each sp2 hybrid orbital is bilobed. One lobe bigger than the other. The half-filled p orbital, which was not involved in hybridization, flies at right angles to the plane of the equilateral triangle. Now, let us understand how this hybridized state results in the formation of a double bond. For this, imagine a similar sp2 hybridized carbon atom approaching this carbon atom. As these atoms come closer, an orbital overlap takes place along the internuclear axes. This bond is called a sigma bond. At this stage, the unhybridized p orbitals which lie above and below the plane of the sigma bond, also come very close to each other and overlap laterally, resulting in the formation of a pi bond between the two carbon atoms. [END PLAYBACK] All right. OK. So I thought about it. I watched that movie, of course, many times over the weekend. And I thought about can I animate, you know, the electrons like that and do the-- and I-- I think I could. But I don't know that it would be a good use of our time. So I thought that they did a pretty good job of bringing it to life. Now, this is a great time for me to tell you about my why this matters. And, of course, the sp2 carbon atom matters because of drinking water, of course. And, you know, this is a well-- in a district in India where more than 50% of the wells exceed the WHO limits for arsenic by around a factor of 5. 1.8 billion people in the world drink fecally contaminated water on a daily basis. 600 million people boil their water in the world. Boiling doesn't help with arsenic, though, arsenic just stays in the boiled water. It kills bacteria, but it doesn't help with toxic-- toxic elements. If you look at the world as a whole, and you look at sort of where there are water-- water crisis, where the water crisis is at this level, it's a little over 3 and 1/2 billion people. And in those countries, almost 185 or so countries, where this is a serious problem, if you look at the cost of disease, and the cause of death in those countries, 70% to 80% of all disease and almost 30% of all death can be attributed to the water quality. Right. So when I talk about water as a problem, I really mean it's a serious problem. It's a problem of life or death. Now, this is-- to get fresh drinking water, if you look at the planet, and you say, well, where is the water on this planet? Where is the water on this planet? I like this picture because we consume water volumetrically, not in an area, right? We use water as a volume, and this is the water we have on this precious planet as a volume. This bubble here is all of the ocean water. It's about 70-- it's about 97% of all the water. That bubble there is freshwater that is inaccessible. It's frozen. And this one here, you can't really see it, there is another dot there. That's less than 1% of the world's water. That is our drinking water ecosystem that is what we are destroying. And-- so what-- OK. So there's a lot of things we can try to do about this. One of the things we can try to do is to see if we can tap into this in a way that's more affordable and more efficient. Because there's a lot of water here, right, and use that. But the problem is, if you look at the cost of desalination today as opposed to the cost-- that's desalination. If you look at the cost as opposed to just digging water out of a well, like the one I just showed you in India, it's over a factor of 10. Still. And, in fact, one of the big issues with desalination isn't just the total cost or the operating costs, but the capital cost. How do you build this plant in the first place? And so it takes too much money. If you look at the thing that's at the heart of desalination, it's a filter, right. It's a filter. And I talked about this in my last, why this matters, when I talk about separations, and I promised that in my next why does this matter, I'd talk about water. Right. Which is what I'm doing. If you look at a desal plant, this is one of the world's largest, this is a plant called the Hadera plant in Israel, and you look at what their costs are that-- remember, the costs digits-- half of that cost-- almost half of the cost isn't just energy. And almost all of that energy is in pushing salt water through a filter. Through a membrane. That's called a reverse osmosis membrane, because you're going against the osmotic pressure. And if you look at the membrane itself-- this is a picture of it, it's actually a very small layer on top of this active layer that does all the work-- it's not a very good design. In fact, membranes for desalination are pretty bad. Right. They kind of do everything worse than they should, except that they work, which is good, and they're cheap, $1.00 a square foot. But, see, they're very-- they take much more energy than you need. They foul up very easily. And then that means stuff gets kind of, you know, stuck in them. And then you can't take it-- you can't clean them because these polyamide membranes, which are the same polymer used in these membranes for 50 years, haven't changed, the material. Those polyamide membranes are destroyed by chlorine. So even in a desal plant-- in a desal plant, if you have drinking water in your feed stream, drinking water has six parts per million chlorine. That's not much. They still will go through the cost to remove it. They will remove it from the feed. Why? Because if you leave that little amount of chlorine in the feed, these membranes get destroyed. By the way, there's 40,000 of these membranes in this plant. Each one is 2 meters long and 40 square meters of area. So-- so these are so delicate that you can't really clean them well. And that's part of what the cost of a plant involves. That's part of what the cost of a plant involves. Now, this would be like a plant, right, you have sea water coming in and, you know-- and then you've got your membrane module that's taking the salt out of the water. And then the product water. But because this filter is so delicate-- there's the picture-- because that filter is so delicate, you actually have to add a whole lot more to the plant. So much of a desal plant is built around essentially protecting this membrane. So that's cost. That's cost. Right. A better membrane could change this. But like I said, the membrane-- oh, there it is. You can't Snapchat on that. Snapchat? What is it that people do today? Not Snapchat. Yeah, Snapchat. Yeah. OK. Thank you. You can't Snapchat on that. That's what I meant. I know it's not My Space, but I don't know-- but, anyway, this is like even-- you know, when I look at-- at membranes today as a material scientist and materials chemists, all of your membranes, I think, that's what it looks like. Why? Doesn't have to be the case. We can do so much better. And so-- OK. Energy costs goes into pretreatment, secondary treatment, and this is where this comes back. This beautiful material that we have now understood more than we did before. Because, remember, before I showed you graphene and I showed you-- no, I showed you benzene. And we talked about it as a Lewis resonant structure, remember? Right. A Lewis resonant structure to help explain how it looked. You know, gra-- graphene does not have alternating bonds. Remember, that's what we talked about before. It's a Lewis resonant structure. So it lowers its energy by having two sets of structures that have sort of alternating bonds. We talked about it in the context of benzene. But, see, now you know so much more. You know so much more. Because, now, you know why graphene is so special. You know why graphene is so special. It actually is the hybridization. It's the hybridization. Because on top, and on the bottom, of a single atom of a single sheet of graphene, you've got pi bonding all the way. Those pi bands are going across the whole surface. And those electrons are critical. Those electrons that occupy those [INAUDIBLE] are critical to the very special properties that graphene has. So you now know the secret. It's sp2 hybridization. And, you know, this has allowed us-- this has-- you might tell I'm kind of passionate about this problem. Partly impassioned passion about a lot of problems but, also, we happen to work on this, and we have been developing this. This is like the ultimate membrane. I can soak this in chlorine overnight. I can put it in negative pH solutions. I can heat it up. It doesn't degrade. And we have figured out how to poke holes in it at just the right size, or to stitch it together so that maybe you can filter particles by the flow in between these sheets. Right. And this material, as a membrane, is so promising that we've actually started to commercialize it as of the last year. And I think this is really going to make a big difference in a lot of areas. Why? Because of sp2 hybridization. That is why. That is why this is such a special material. OK. That's my why this matters. Now-- oh, we did see 2H6. There it is. We did see 2H4. There it is. Look at that extra pi bond, not animated this time. And, of course, you can go on and you could do see C2H2, and you can imagine what happens in this case, is what you would expect. Is what you would expect. Because, now, I've only got-- see, each of these carbon atoms only needs two sigma bonds. One to the hydrogen and one to the other carbon. So it's only going to grab one-- it's only going to hybridize-- to make those two equivalent signal bonds, it's only going to hybridize with one of the p electrons. So that's sp. That's sp hybridization. But, notice, that in C2H2, it-- you know, now, I've got 2p orbitals left over on each carbon atom. And so you can imagine that with these 2p orbitals, well, you know this already, right. When-- when we talked about p orbitals, we talked about how, you know, you might-- you know, you've got this, right. Awe. OK. That's supposed to come like out of the plane and into the plane. They're orthogonal. They're orthogonal. Right. And so-- so two of them are coming at the other atom. One is like this, and the other's like that. And then the other atom is one like this and one-- well, it might not at first, but then when it sees those, it's like, hey, wait a second. If I do this, we can pi bond. Right. The other ones are like this, and they pi bond, and so you get a carbon carbon triple bond. You get a carbon carbon triple bond for a C2H2 with sp hybridization. Now, it's not-- oh, there it is. Right. OK. Forms pipe bond network, forms sigma bond network. It's not just carbon that hybridizes. It's not-- carbon is a classic example. But, you know, if you look at BeH2, it's the same thing. BeH2. Well, Be, Be starts with an un-- can I raise this up? Be starts with orbitals that don't look like they're going to give me the equivalent two sigma bonds. Be starts with these as electrons filled. Right. But if Be had something more like this, if it had an sp electron in each of those orbitals, well, now, when I see the two h orbitals, the two 1 SH orbitals coming at me, I can form those equivalent bonds. So BeH2 will also hybridize [INAUDIBLE] to make that molecule. And all sorts of other things. The deal orbitals can get in on the action. All right. So here's SF6, which we talked about. We drew this Lewis structure. Took a long time to draw those lone pairs. But, now, you know that the way this actually works because these are equivalent bonds. So the only way this can work is if I form hybrid orbitals. Right. Right. But they didn't have to be equivalent bonds until we learned about VSEPR. And it said, no, I want to maximize my separation to minimize repulsions. And so then we know that if they're equivalent bonds, the energy of the system will be lower. Right. And so that-- that's-- you know, that's going to have to take some d orbitals and create these hybrid orbitals. Look, they're beautiful. Because, now, this thing from the sulfur, this thing can say, hey, I can take six of you, and I can be equivalent bonding to all of you. That's what hybridization is. OK. It's written right up there. We're just keeping on saying the same thing. OK. Now, I think that I'm sorry-- I got very excited about the water so I have to throw some T-shirts out. And, now, where do I-- where do I-- OK. I don't go there. I don't go there. I got to go back there, and back there. Oh, that's very loud. OK. And right there. And right there. See you guys on Friday. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 34_Introduction_to_Organic_Chemistry_Intro_to_SolidState_Chemistry.txt | It's our last Friday. Did I hear an, "Oh?" Thank you for that one "oh." It's OK. I'll take what I can get. It's our last Friday, and so next week is our last week. We have class on Monday, and Wednesday will be the last class. And we've got one more smallish topic to cover after this, which is diffusion, and we'll wrap up on Wednesday. But today, I want to keep talking about polymers. So this is our polymer week. And I want to go back to the properties I've put here, the properties we've talked about. Right? So on Monday, we focused on what a polymer is, and we focused on the ways you make it. All right? The radical initiation, chain, addition, and the condensation polymers that you know, the condensation where you have two different mers. And then on Wednesday, we talked about things that matter, kind of like engineering polymers. Like what are the properties of the polymers that you can change, that we can change? All right? And so we talked about, well, you could pick a different monomer. That will change the properties. Right? We've been through that. You could try to grow longer or shorter strands, so that's going to change properties. The interactions between the strands, which will, of course, depend on what you pick there. Right? And then we talked about the density and the crystallinity and how that could depend on things like cooling rate, or maybe the physical structure itself of the chain. So what do I mean by that? Well, when I say physical structure, of course I mean the chemistry that makes that physical structure, so I could branch the chain with the same monomers, but now instead of being one linear chain it's a bunch of branches. Or maybe you could have a functional group that you can control which side of the chain it's on. And all those things would lead to differences in density an crystallinity. And then the last thing we talked about is cross-linking. And what I'm going to do is I'm going to add one more thing to this list today, but before I do that I want to pick up on the last slide of Wednesday, which is this. So we tend-- so all this kind of goes in together to give you these solids called polymers. Right? But we talked about how sometimes they're not fully solid, and that's what's going to come in the middle. But you can make a solid thermoplastic, which is these strands. Right? So what is a thermal? Ah, what's the? Anyway, so with thermoplastic we'll look over here. There is no cross-linking, so you don't have any of this. But what do I have? Well, I have these really long strands that are bonded together with these IMFs, typically in a thermoplastic you would have something like maybe van der Waals, and maybe you have some H bonds. And these materials, because you didn't cross-link them, and you didn't cross-link them strongly with some kind of strong bond, because of that, you can reheat them and reprocess them. So these are actually often the plastics that we recycle that we can recycle. And like I said, they're mostly linear and maybe slightly branched polymers, and they're used in a whole lot of applications, so we call them thermoplastics. Thermal-- you can heat them. Right? You can heat them, and you can see what happens is if I heat these back up, these links can start moving. They wiggle, and they wiggle until they can just kind of slide past each other and become a vicious liquid and then a melt. On the other hand, if I've got a thermoset, I heated it up, and then as it set I crossed linked it. And there are lots of ways you can cross-link. We talked about some of those. Right? And now I've got a pretty high density of cross-links, so that's drawn here. So I've got these long strands. Remember, they're super, super long, right? But now, every so often I have something linking them together. And if I put a lot of cross-links in, and if they're strong cross-links, then you can get really hard plastics. Right? And so these are high-- the thermoset, they're set. They're set. Thermoset, good. It solidifies, and it can't be reheated. Why? Think about it. If I heat that up, then, OK, I might melt the polymers, so the chains are kind of wiggling around, but those cross-links aren't going anywhere. They're really strong. And so what happens with thermosets is they're very difficult to recycle. They're very difficult to recycle because what'll happen is before those things melt the whole thing catches fire. The whole thing catches fire. It burns often. Or maybe it doesn't burn, but when you melt it, you've got this weird chemistry now of the cross-linker mixed into the polymer, and it's not useful anymore. So these kinds of plastics-- the cross-link is really great. You make, you know, like I said, over 30% of all toys. It's used in many, many applications, these harder plastics, but they're not recyclable. And then, we talked about this intermediate region. And this is where we ended. We talked about elastomers, and elastomers are in between. And so in an elastomer, you've got light cross-linking. And the cross-linking, like in slime, that cross-linking has a weaker hydrogen bond. All right? And remember, we talked about how that could lead to viscoelasticity, really cool stuff. In other cases, like in the case of the vulcanized rubber-- remember that, Charles Goodyear? That story from Wednesday? In that case, you put a covalent bond of sulfur between the strands, and that gave you a much stronger rubber that still had some elasticity. But now, the best analogy here is still the spaghetti. I keep coming back to it. I'm not letting go. Think about it. If I've got a bowl of spaghetti, and I pull on it, that's my plastic bag on the left. All right? That's my plastic bag. I pull on it, and those chains are all crumpled up and tangled, and I'm untangling them, and then I'm sliding them past each other, and pulling them until the whole thing rips. That's the bowl spaghetti. But now I've got this spaghetti that attaches every so often, one strand to another. Think about it. I've got a bowl spaghetti, but in each strand of spaghetti, there's like five places where it's attached to another strand of spaghetti. Now what happens? I can pull that as well. I can pull it because those spaghetti these are all curled up, and so there's some amount of pulling that I can do while they uncurl until those links take over, and I can't pull past them. All right? That's what the light cross-linking will do. And you say, well, if I had heavy cross-linking in the spaghetti, then the whole bowl of spaghetti wouldn't be very easy to move because I'm always up against all the links between them. All right? And so that's why these elastomers are so interesting, because they fall-- it's all interesting. But the elastomers fall in between, where there's some of that uncurling of the polymers until you get to a point where, depending on how much you've cross-linked, and the bonding of the cross-link, you're going to go up against the cross-linking. All right? And so these are used, so they're free to move. Now, must be above its glass transition temperature. I want to talk about that for a second because I did mention this, and I want to be sure that we all have a good sense of what that means. And so I wanted to-- taking this elastomer, say, what happens when you melt it? All right? And we drew this curve, but I want to go very carefully. This is the temperature. Oh, we love this curve, the molar volume. Right? And now up here, you've got your liquid. Right? We know that. And then here is the place where it would crystallize. Can a polymer crystallize? Sure. Right. If a polymer crystallized, then it would just be these strands in a crystal. What is a crystal? We know it's a repeating ordered structure. Yes, polymers can crystallize. But as we've talked about so often, what happens is because, like we talked about with silica, you know, you've got this very viscous, amorphous, can't find the crystal sites-- same with polymers. And so you get a glass. And then this was the Tg. Now, why am I showing this? Because with these strands-- and I want to show you. You know, we've drawn this before where often what you get in polymers is some degree. It's right here, some degree of crystallinity. Right? So what do I do? How do I do this? Well, you literally have both. All right. So I've drawn this already. So if I'm here on my polymer, then it might look like what I've drawn already on the board. Oh, I love doing this. And then there's the crystal, and then there's more. I've drawn this already. Right? You've got this crystalline region, so you've got the crystal, and here, because I'm below the glass transition temperature, this part here is an amorphous-- oh boy, it's kind of small, but this word says "amorphous"-- solid. It's a solid. That part is a solid because it's below the glass transition temperature. All right? OK, but now let's start heating this up. And as I get to the-- so now I'm going to get here. And what happens is it doesn't actually go straight to the full liquid. I mean, it's got this crystalline stuff in there, right? It's got these regions. This crystal has one melting point. Remember, the melting point of the crystal is the melting point of the crystal. It doesn't change. Glass transition temperature can be tuned. But this is a melting-- Now, so what happens. Well, what happens is there's more of a curvyness to this, and you might kind of have the shape look more like this. So what's happening? This just gets a little bit more at the detail of how a polymer would melt if it's got any crystalline regions. And as I've told you, very often it does. Not always-- it could have no crystalline regions. It could be totally amorphous, but I've drawn this enough that I wanted to explain this curve with that as our starting point. All right? And I've also talked about how the degree of crystallinity is so important for properties, and it can be controlled. So what's happening? Well, in this region here-- let's draw that picture. So now I'm in here. And what happens is I'm below Tm, but I'm above Tg. So that means that, literally in the same strand-- this is so cool. That looks the same. It's not supposed to be the same because now this is a liquid. This is a viscous liquid, and this is still a crystal. That's an xtol. I'm below Tm. So the crystalline region has not melted, but the rest of the polymer is above its glass transition. This is so cool. This one strand is both a solid and a liquid, one strand. That's so cool. That's in here. All right? And then I get over here, and now here-- OK, well, let's see. So now this is all liquid, and the crystalline part is starting to come apart. So crystal melts, xtol melts. All right, well, that sort of was supposed to be the remnants of a crystal. And then above Tm, the whole thing's a liquid, right? So that's kind of zooming in. I wanted to make sure that we understood this plot in the context of these polymers. And that also gives you a sense-- OK, now, that's this one strand. Now, I've got a bunch of strands. Did I connect them or not? And that adds the layer of what an elastomer is. Right? Because now you can imagine, is the elastomer above or below Tg? Where's is Tg? Are we above or below Tg when we bounce a ball? Right? How does that influence the properties of the ball? A rubber ball is likely cross-linked. And you can imagine that if you're below Tg, then all that amorphous part of the ball is a solid, so there's an elastic response. Sure, it could still bounce. If you're in this region, then that part of the ball is a liquid, but it's likely cross-linked so the ball doesn't fall apart. Right? So it's going to bounce differently. Right? It's going to bounce differently. OK, so anyway, I wanted to just go into detail of that to make sure that we feel our oneness. And I mean, you could add here. This is, because it's so important in just-- like I was saying, just in terms of what this material, how this material behaves, you could add the glass transition temperature, which, as we've already talked about, has a number of ways that it can be tuned. On the other hand, it may also be tuned by some of these other things that you do, but it's a very important part of the polymer property ecosystem. OK. Good. So that's the elastomer. Now, there's one more thing that we can do that I wanted to talk about. Oh, that's the picture. Yeah, I thought I'd show you another picture. So I'm not the only one that draws things like this. All right? OK, there's a nice one from Encyclopedia Britannica, and you can imagine this region being a solid down here, then a liquid. But that's still solid, and then that region melts all at once at Tm. The last thing that I want to add is another way that we have to control polymer properties, and it's another bullet here, and it'll be our last one. And it's really the composition and sequence. So what do I mean by that? Well, it turns out that-- remember when we did condensation. Right? Polymerization, a poly condensation it's called. The condensation polymerization. We had two different mers. And we could have picked the box. Remember, the box inside the mer could be different things. And then when we've done the radical initiation, we've just had one mer with the double bond. Right? Now, if you have one mer, then that's called a homopolymer, Homopoly. And if you have two, it's called a copolymer. The thing is that we actually can even make copolymers with both approaches. And not only that, but we're learning more and more how to control the sequence, and that's what this shows here. So I could have a polymer A. Let's say I have a-- yeah, I'll put it below this. Let's say I have a, you know, polymer, polyethylene, PE, which is-- oh, we know now how to draw these-- C, C, N. And then I've got the other one, which is PVC polyvinyl chloride, and that looks like this-- C-- all I've done here is swap out a hydrogen for a chlorine, and I've got a totally different polymer. All right? Some other end, OK? And an N, whatever. I can now alternate these. I can alternate them, and I could alternate them in this way. So PE might be my A, and PVC might be my B. And if I alternate them, we might-- you know, if it's like regular alternating, we might write PE-- this is how we'd write the copolymer-- a, PVC. All right, but you could also have it be random, so then you'd write PE-- you can take a guess-- r, PVC. Right? You could write a grafted version-- PE, g-- well, you get the point-- PVC, and so forth. What I mean when I write this is that I've taken these two polymers, and I've copolymerized. I mean, sorry, these two. Well yeah, these are the polymers. I'm taking the monomers, and I've made one polymer out of the two of them, but I've controlled the sequence. It's not-- Now, maybe it's not controlled. Maybe it's random right there. Or maybe I've figured out how to do this in a way that I have the backbone all one type, but I can have these side chains, the branches, another type. That's really powerful, it turns out. Or maybe you can control that I have a certain number of them, and then a certain number of the other, and this is called a block copolymer. And some of you may have heard of block copolymer chemistry, which is growing, is a very growing, very powerful field. Because when you make these blocks and you control their properties, you can imagine that you control all sorts of things about how that polymer behaves. Imagine you make something that bonds to A but not B. Like dissolves like. All right, we've been there. B is something that bonds to B but not A. OK, what's going to happen? They're stuck, but they're going to try to come together and stay apart in the same strand. All sorts of interesting things can happen. It can be engineered when you can control these blocks or these graphs. All right? So that's the last thing that we-- Here's an example. This is-- I was trying to look for a good exam-- here's something called-- actually, what is it called? Surlyn. Who comes up with that? I don't know why that's the name, but that's the name. It's the Surlyn resin, which is the ingredient that goes into the polymer, OK? And look at what it does. Oh, it provides clarity, toughness, versatility. Surlyn is a leading choice for food, cosmetic, medical device, skin, stretch, packaging, as well as golf balls. The tunability is enor-- why? What is Surlyn? Well, we don't know exactly because they won't tell us, but Surlyn is actually really cool. It's using-- OK, it's using-- where? Dah! Where did I put it? Composition, sequence-- it's using sequence to make a graft polymer where one type is going along the chain and the other type is coming out. But really importantly, the other type is made specifically to form ionic bonds. So here B is made so that it can form ionic bonds with other parts. Ionic, remember, that's basically by grafting, by creating a copolymer, where one type wants a form ionic bonds. I've cross-linked. I've made a cross-link built into the polymer itself. The cross-link is going to be linking the strands together, maybe within itself, maybe with other strands but with an ionic bond. Right? But I've done it all just built into the polymer. And that's why you can see here. Look at this it's an ethylene copolymer. All right? So ethylene is the backbone. The co is what's coming out of the side, and that's an ionic bond, so it's also called an ionomer. What's an ionomer? A polymer that has ionic cross-links, ionic bonding cross-links. And what's cool about that? Well, you get the whole world of ionic bonding now in the mix. And so what does that mean? Well, you could have it be very strong, golf ball. You can have it be thermally responsive. Right? You can tune at what temperature these bonds break. There's all sorts of flexibility, and you can see it right here in how they're marketing it. It can do almost anything. Another example of the copolymer-- I'll give you a couple examples of copolymers. This is one that you may have seen, maybe many of you experienced, you just don't remember. This is the diaper. Right? And so what what a diaper is is it's got this copolymer made out of acrylic acid. These are the monomers. Notice the double bonds? Oh, we're going to take advantage of those, and we're going to make a polymer, but we're going to do it in a controllable way. So we're going to take these two mers, and we're going to control how they come together. And very importantly in these kinds of materials-- they're broadly called hydrogels. These are materials that can absorb hundreds of times by weight, hundreds of times water into them. Why? Because in one of the copolymers you've got this sodium atom, and what happens is there's the dry state. It's all curled up and the chains want to be all together, but as soon as you introduce water, those sodiums like going into the water. And what happens? They leave behind this oxygen that really wants the water because it's a negative charge. And so you get a sodium ion going in the solution. You get the solution coming to the anion . And even more than that, now if the sodium leaves, then what happens? Well, what happens is all those negative charges are left and they repel each other. That's going to help this whole thing want to expand. Right? So that's a diaper, 2% of all landfill by the way. Now, so that's another example of a copolymer and a really interesting way of tuning the properties. In here it has to do with, what did we do? We tuned the ionic character of the backbone. We tuned the charge. We made it responsive it in terms of its charge to something, in this case water, the presence of water. That's cool. OK. Last one, and here I want to talk about mechanical strength. So this is the tensile strength. So how far could you pull this thing, this material before it breaks? And then here's the elongation until it breaks, so it gives you a sense. Now the tensile strength, this is how far can you pull it in the elastic regime? You can see that for steel it's really high, but you can't pull it very far before it breaks. We know that, right? So steel is really strong, but it's not very flexible. There is nylon. We did nylon, so it's kind of not nearly as strong, but it has a lot of elongation. There's fiberglass. This is plastic reinforced glass, where you can get a lot of strength, very little flexibility, more than steel, a little bit more than steel. Well actually, it was a lot more, but still it's only 3% or 4% elongation, and here's cellophane. That can be a naturally occurring polymer. You can also make it synthetically in the mix. And I want to point out nitrile. Nitrile rubber is a really interesting copolymer, and many of you may have seen or used-- if you're in a lab, you might use nitrile rubber gloves. You might have them at home at the sink. And one of the advantages of nitrile rubber is, again, this enormous tunability by just picking how you copolymerized. How much of one did you put in the other? And this is-- maybe I'll use this board here because it's a really cool copolymer, where you're putting in acrylonitrile. So here's what this looks like. So you put your CH2, your CH, your double bond your CH, your CH2, and there it is. And that's going to repeat some number of units that you control because it's a block copolymer before you put this one in, and here's the kick. It's got a triple bond nitrogen on it. And that's going to go a certain m. All right? Now, that comes from acrylonitrile and butadiene, but again, it's the synthesis to make the copolymer. It's the sequence. I keep on forgetting where. It's up there. It's the composition, the architecture. How did I put those copoly-- if you do this in the right way, it's the amount of the acrylonitrile that you put in will control the strength. It'll make it stronger and stronger, because you've got this really nice, strong bond in here, and this will give the material a lot of strength. So that's why you can go up to, you know, fairly reasonable. But then, how much of this I put in is going to determine how far you can stretch it, and so you balance those. And you can make a whole bunch of kinds of nitrile rubber. OK, so I hope this has given you some examples of the tunability of polymers. And you know, if you take the bigger picture-- and this is a little hard to read. You know, it will be in the slides, so you can look up this paper here. It was published in Science a few years ago-- and you look at the fracture toughness with the yield strength. So what is that? Well yield strength is-- Remember, we know that because it's when you're in the elastic region, and you're pulling, and then you yield to plastic deformation. Right? That's the yield strength. Now what about fracture toughness? Well, that's if you start a crack, does the thing tend to crack more or not? That's the fracture toughness basically, right? And so you could put materials down on this plot. You've got ceramics. You've got concrete down here. You've got-- so you're not going to be. So the polymers sit here. All right? Here's metals, and alloys, metallic gases. You can look at this on your own, and look at each one of those, but I just want to point out, polymers have a fairly wide range, but there's so much interest in going beyond. There's a lot of interest in using polymers in many other applications. We can't get there yet because we can't push it out here or maybe out here. We still need to figure out how to tune it more. We've got all these ways to control the properties, and we're still only at the very beginning of understanding how to engineer polymers. Now, there is no better way to make that point clear than to look at nature. All right? And I already showed you the tree and you know, the examples of nature as a polymer engineer. I want to talk about that a little more because nature is not just a polymer engineer. Nature-- I'm going to write this down nature. Nature is a polymer engineer gone wild, and I'll show you why. Polymer engineer. Humans, what can humans do? Well, we are a natural polymer engineer material, but what can we make with all of what I've just shown you? What can we make? I can put like one, or maybe two, or if you really go into the research, you get three mers, trying to control where they are in the chain. All right? You've got three, maybe two. We have really two in most materials. Copolymer-- we're so proud of this nitrile. Two mers, and we control them, and we make rubber sheets. But nature-- but then they're everywhere. They're everywhere is just that. All right? So nature can have-- so humans have the same functional group every stop. All right? OK, that's one mer. Maybe I've got two, maybe two or three. Nature can have different groups, and here's the key. It can have them everywhere. OK? So what I have is I have many, many more possibilities. I mean, we have these possibilities too, we just can't control it. All right? So if I look at-- like let's go back to condensation polymerization because this is what nature does. So this was what I drew for you on Monday. There's a dicarboxylic acid. Here we're making nylon. And a diamine, and we're making polyamide. And remember, the box-- in nylon 66, the box is 6 carbon atoms. I call that boring. It's at six carbon atoms from hydrogen, but that's kind of boring. But in nature, this box is an amino acid. That's much more interesting. That's much more interesting because if you look at an amino acid, and this is an amino acid-- why is it an amino acid? Well, it's got an amine group here and a carboxylic acid there, so it's an amino acid. But see, here's the thing, R. This is nature's "the box." We can put six carbons in. We're really proud of ourselves. Nature can put almost anything it wants for R. All right? So R, just to spell this out, is nature's choice, and I'll show you what it chooses. Nature's choice-- because there are hundreds of amino acids, and this R group can do many things, but just 20 is all we need to make proteins. Most proteins are made out of just essentially 20. But if you do the math-- I think I have the math here-- and you take-- let's see. OK, if I take two amino acids. Let's say I take two amino acids, so I've got two different R's. Two amino acids, and I've got a length is 2, and this is called the dipeptide. I'll tell you why in a sec. OK, but let's compare this now. Now I've got 20 amino acids, and I just told you that 20 is what nature makes most proteins out of, 20 different R's. There's an amino acid. R is nature's choice. But see, if I've got 20, then I've got 20 squared equals 400 dipeptides. I've made a two-unit polymer. It's not a polymer. It's a peptide. But now, what if I take the 20, and I've got-- but now I've got, let's say, 1,000 units long, then I would have 20 to the 1,000 possibilities, which is 10 to the 1,300 combinations. So 10 to the 1,300 possibilities-- that's because there's 1,000 units, and I've got 20 amino acids, right? But now you think, how do you possibly think about what to do? That is nature. It's had 1 billion years, and it's given us the world that we live in, that we know. It's messed with these combinations in a way that's impossible for us to even understand. That's why I say, nature is a polymer engineer gone wild. It's got almost limitless flexibility, and it's used it. Right? Now, how does nature make its polymer? Well, its condensation polymerization. This is just what we saw. Look at that. This is what an amino acid is. It's got this carboxylic, so there's the OH group that can condense with the H here and form the link. There it is. That's called a peptide bond. When two amino acids come together, that's a CN. That's a peptide bond. Right? But look, this had R1. That had R2, and I've got 20 different amino acids to choose from. The possibilities are endless. So the protein synthesis is condensation pol-- you knew that. You knew it couldn't be anything else because there is no double bond. Where would I have done a radical initiation with these amino acids? If I'm nature, I've got to make it with condensation polymerization. That's how we are made. OK, now, so just to give you a sense, so the R-- I'm not going to go through this, obviously. I'm just giving this to you as if you want to read more about it. There's some wonderful charts here, 20 amino acids, the 20 that are most common and that nature puts together. And what they've done that's really nice here is they've grouped them, right? Because what did nature choose to do? How is it most utilizing the properties, the tunability in these amino acids. I said there were 20, but most proteins are-- I'd say there are hundreds, but most proteins are made from these 20. It gives you the tunability in having maybe non-polar groups. Right? So it might be then hydrophobic. Polar groups, hydrophilic-- maybe you can put, nature can put, R groups in there that maybe have a charge, or that lose or take an ion. All right, so it can play with the charge, and it has done all of that. It can have groups that become acids, that make something acidic, and so forth. OK, so that's cool. What has nature done? Well, I got to show you this spider because-- and if you're interested, Professor Bueller does some wonderful work on spider silk-- because I think this is a great example of something that nature can do that gives us a sense of how far away we are. Like I said, we're very proud of this, and we want to do more. And so we take our two mers, and we mix them together, and we make branches, and maybe we're going to try to add a third. Meanwhile, nature has had a lot more flexibility and a lot more time. What can it do? Well, here's spider silk. Now, this is a spider. Spider is an incredible polymer synthesis machine. It's an incredible polymer synthesis machine. And here it is weaving a web, and I love this-- [MUSIC PLAYING] --because I think it's so cool. OK, there's music, I guess. I forgot about that, and there it is. Now watch. Out of here, this is the back of the spider. There it is right there. It's making protein. That's called spider silk, but these are proteins. These are polymers. It is doing condensation polymerization right there, and then there's all sorts of structural stuff that it does. Right? It's got a specialized hook. It knows how to step. It creates-- look at that. There's a branch place where it knows to put it. Right? It's already created the glue. So not only is it putting this spider silk out there, but it can put other types of polymer depending on what it needs. Does it need something really sticky, less sticky? Right? And so there it is weaving its web, and it's generating this polymer on the fly. It's doing condensation polymerization. Now a couple properties about spider silk. OK, so here we go. Let's see. So spider silk, this is just one example of what nature can do. It's five times stronger than steel. Remember the mechanical strength chart, nothing was stronger than steel. Spider silk is five times stronger. Just to give you sense, the example, I found that I like. If you had a spider silk that was a pencil width, and you made a strand, it would stop a Boeing 747 in midair. That's how strong it is. Oh, it keeps strength-- here's another thing-- below 40 degrees C. We can't do that. Just take a rubber ball and put it at that low of a temperature and try to bounce it. It's going to shatter. Right? Spider silk can keep that strength and not break. Its elastic, so it's got-- throughout all of this, it's got an elastic property of 4x, so it can be stretched to four times its original strength. Compare that with nitrile. Nitrile rubber could also go to very, very high elastic elongation. All right? So if you go back to nitrile-- here it is in the table. We're very happy with this, but this only had two monomers to play with. Right? We played with two monomers, and we got to here, and look at the sacrifice in the strength. Look at how much we had to sacrifice strength. Spiders don't have to do that. OK, and oh, here's the last one. Fully recycles-- now the thing is that this is actually kind of incredible. Spider webs get dusty. They lose their stickiness. So many spiders know this and just simply have to weave a new web every day, but they don't leave the old web there. They actually eat it, and they fully recycle it, fully. Right? They eat the web, fully recycle it, process it, have this condensation polymerization work, and make a new one the next day. All right? So we don't come close to this spider. We don't even come anywhere near it in terms of where we are. Even though I gave you all these wonderful things that we're doing with engineering, we still have so far that we could go if we could just figure out how nature works. OK? And this gets me to what we do, and so I've got my "why this matters" now. And so a spider eats its web, fully recycles it, spins a new web the next day. Here's what we do with our polymers. I already talked about the oceans. Here's what we do on land. These are tires. Now, tires are very difficult to recycle. Why? Because they're too vulcanized. They've got too much cross-link. Remember, if the cross-link density is too strong, which you need to make a tire, then you can't recycle it. And in fact, what happens is-- and there's a tire mound. Here's it is from a satellite picture. Those are tires. Those are tire mounds. Here's what happens when one catches fire. Here's what happens when many catch fire. All right? It's actually a very hard fire to control once they catch. But we don't really know how to recycle them well yet. What can we do? On the science and engineering side, the fact of the matter is there is a lot of work to do, but there are promising directions. So I wanted to leave you with a little bit of that, and I'm not going to go into great detail. I just want to show you, and there's references here you can look. This was published a couple of years ago in Nature. One direction is in self-healing polymers. This is a very exciting direction. What can we do? Well, we can go away from this whole single-use idea and make stuff last longer. All right? So that would be beneficial. Maybe not if it ends up in the ocean, but in terms of just how long we can use these materials. So one direction is, well, you've got these polymer networks, and you incorporate little gels, little beads in here, but the beads don't open up until a crack comes along. So they're sensitive to a crack. And when they feel a crack in the material, they open up, and they pour a healing liquid that then solidifies. That's a self-healing kind of approach. You could do that at different scales, all the way down to the strand. You can do that on larger scales. And here's a whole system where there's actually this healing material being flowed through a polymer structure, just like arteries in our body. Again, always there to try to heal the material. Right? Another direction is in fully recovering. And again, I don't want to go into full detail here. You can look up some of this stuff. This was published last year. Can we take the polymer and chemically decompose it all the way back to the monomer? Can we go back to where we started? Can we do what the spider does? Right? The answer is no, not today. But if we could, that would open up a whole lot of doors for recycling that are closed today. Can we do this in a way that is efficient? Right? And then, another direction of work that I think is very important is in making thermosets so heavily cross-linked so you get all the hardness and all the properties you need of the polymer that's heavily cross-linked, but easily breakable in the cross link. And there is good work going on in this direction. Can we make degradable thermosets? That's another really important direction. And last, we should be encouraging people, if you can't do any of this, at least take the polymer out of the landfill and make something with it. And there are projects-- you can look at here, Waste Management Journal, in making bricks out of polymers, incorporating them into construction materials. Right? These are important directions, and we need a lot more hopefully in the very near term. hopefully in the very near term. No more just talking about alignment. OK, have a good weekend. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | Goodie_Bag_8_Reactions_Intro_to_SolidState_Chemistry.txt | [SQUEAKING] [RUSTLING] [CLICKING] VIVIAN SONG: In this video, I will be going over Goodie Bag number eight, which is about reactions and reaction rates. What you'll need are pH strips, small measuring cups, stirrers, gloves, citric powder, cups, a scale, and seashells. No sea animals were harmed in the making of this Goodie bag. The objectives are to understand how climate change has caused ocean acidification. Assess the order of the reaction from empirical data. Learn how temperature can affect the reaction rate. And measure the pH of a reaction using pH strips. Some conceptual questions to think about are, what is the reaction order? And how does the reaction rate change if the temperature increases or if we have outside pH effects? Now, in this Goodie bag, we will be exploring the application of reaction rates in ocean acidification due to climate change and what that means for many sea animals. As you can see from this graph, there is a strong positive correlation between increasing carbon dioxide levels in the atmosphere, which is marked by the red, and decreasing pH in the ocean, which corresponds to increasing acidity, as seen in the green. What exactly is going on in terms of the chemistry? First, carbon dioxide from the air dissolves into the oceans and reacts with water to form carbonic acid. Then the carbonic acid donates a proton. Finally, calcium carbonate from seashells reacts with excess protons in solution to formed bicarbonate ions. What does this all mean? Seashells are dissolving in a more acidic ocean, which is very sad. One final note about the slide. I've only drawn the forward reaction arrows. But these reactions are actually reversible. So there should also be backward arrows. However, the forward reactions are more dominant to the reverse reactions. So the net product is that of the forward reaction. Now, let's dive into the experiment and see this happen in real life. First, take out one of your seashells and weigh it with your scale. Next, put on your elegant nitrile gloves. And take a large plastic cup and have it filled with tap water about halfway. Measure out two scoops of the citric powder using the small but accurate measuring cup. And put it into the water. Then stir while using the stirrer until all of the citric powder is fully dissolved. It might take a minute of stirring, but the water should look pretty clear. After that, you can use your pH strip to measure the pH of the solution. Have a timer ready. And when you're ready, place the shells that you weighed into the low pH solution. And record what you observe. After 20 minutes, take out the shells with your gloves on and record the exact time. Let the shells dry for maybe half an hour or so so that all the water has evaporated. And then measure the weight. Repeat this step two more times so that you have data points of the mass of the shell at zero, 20, and 40 minutes. The pH that I measured was a pH of 2, as you can see using this pH tester. While doing the experiment, you should see that the sea shells are dissolving in the acidic solution. And here, I've captured a hyper-lapse of this happening. The focus of our experiment is to figure out the reaction rate of the dissolution of calcium carbonate. As a reminder, the reaction rate is equal to some rate constant times the concentration of the calcium ions to the n power times the concentration of carbonate ions to the n power, where n plus n is equal to the reaction order. And that is determined empirically from the data that we've gathered. As you can see from the data, this reaction is 0th order. I plotted the concentration of carbonate ions on the y-axis, which you can figure out, assuming that you poured 0.2 liters of water in your experiment, and time on the x-axis. Because this plot looks linear, the rate is constant over time and does not depend on the concentration of carbonate ions already in solution. Thus, this is a 0th order reaction. You could also imagine the same plot with calcium ions since we know that the calcium ions and carbonate ions exist in a one-to-one molarity ratio. In summary, the reaction rate order is determined empirically. And the reaction rate can depend on temperature and pH. Now, we didn't have time to do this pH and temperature dependence experiment in the video. But you can imagine how temperature and pH might affect your reaction rate. And finally, ocean acidification linked to climate change is causing our seashells to dissolve, which is very sad. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | Goodie_Bag_3_Ionic_Solids_Intro_to_SolidState_Chemistry.txt | BABATUNDE OGUNLADE: Hello, everyone. Today we'll be working through Goodie Bag 3, which is on ionic solids. In order to work through this Goodie Bag, you'll need a conductivity meter, some small measuring cups, some medium sized cups, a small scale, a writing utensil, some nitrile gloves, some stirrers, and four unique solids. Today, I'll be using sodium chloride, magnesium sulfate, magnesium oxide, and sucrose. Our main objective today is to use solubility and conductivity measurements to determine if a given solid is an ionic or covalent compound. And as we do this, I'd like you to think about two questions. First, how do solids dissolve in water? And second, what factors may influence how conductive a given solid is in water? So first we're going to look at the solubility of our solids in water. And to do this, we're going to dissolve 1 gram of each solid in 30 milliliters of water. I've already weighed out my solids here. And I've also already weighed out my water. Notice how I'm wearing a pair of nitrile gloves. The compounds that I'm working with today aren't that dangerous, but just as general safety, it's good to wear gloves when handling compounds. So I'm going to pour each solute into each respective cup of water and stir and mix for about a minute to two minutes and see if the compound dissolves. So after one to two minutes of stirring and dissolving, you can see that our magnesium oxide right here has not dissolved in our water. You can see a very clear boundary between the solute, which has settled at the bottom of the cup, and the water, which is on top. If you compare that to our sodium chloride, though, if you look very carefully, you can see that our solute has dissolved very well in our water. So the next thing we'll be doing is measuring the conductivity of our compounds when dissolved in water. And to do that, we're going to be using a handy dandy conductivity meter that I have right here. So I have some cups of water prepared here to be used as both a baseline correction when we're measuring our conductivity of our compounds and as a water bath to clean our probe in between measurements. OK, so first I'm going to turn on my conductivity measure. And I'm going to make sure that my units are set to microsiemens per centimeter, or some equivalent unit. And first I'm going to measure a baseline of just regular water here. I have tap water here. And so when I do that, I see that I'm measuring around 900 microsiemens per centimeter. So that is the connectivity of my water as is. And I'm going to use that as a baseline when I'm measuring the connectivity of my other compounds. So since this is water, it's already clean, and now I'm going to measure my sodium chloride, which is right here. And so right now I'm reading around 4,700 microsiemens per centimeter for my sodium chloride. I'm going to take note of that. And eventually, when I'm collecting my data, I'm going to subtract my water baseline from this value here. Make sure in between each measurement that you dip the probe in the water bath to clean it, and you can continue on with the rest of your measurements. So the last experiment we're going to do today is to measure the solubility of our compounds in water, except with 2 grams instead of 1 gram. So I've measured out an additional 1 gram of each solid here, and I'm just going to pour each one into each respective cup, mix and stir. And after one to two minutes, I'm going to see if each compound is dissolved. So again, we have our magnesium oxide after round two of solubility. Again, we can see that our solute has not dissolved in water. There's a very clear boundary between the water on top and the solute on the bottom. So in this case, round one solubility and round two solubility look identical, but it may not be the case for our other three solutes. All right, so now that we've finished our last solubility test, let's put all our data-- our first solubility test, our connectivity test, and our second solubility test-- on the board. So here we have a chart of all the data from our experiments. I have the compounds that we use going down right here. And I have solubility here at 1 gram and 2 grams, and conductivity after water subtraction here. So before going into this data, first thing I would like us to notice is that for sodium chloride and for magnesium sulfate, these conductivity values are much lower than I expected. This should be close to the 10,000 microsiemens, and this should be closer to 8,000 microsiemens per centimeter, but the fact that we're able to measure conductivity for these respective compounds gives us insight into what type of compounds they are. So first, if we look at sodium chloride, we could see that at both 1 gram and 2 grams, our sodium chloride was able to dissolve in water. And we were able to measure conductivity readings for this. This means our sodium chloride was able to dissociate into ions when dissolved in water. And this is an extra check for us to determine that sodium chloride is indeed an ionic compound. Next we can look at magnesium oxide. And we see that for magnesium oxide, at 1 gram and 2 grams, we weren't able to dissolve the magnesium oxide in water. And not only that, because of that, we weren't able to measure any conductivity. But we know that magnesium oxide is an ionic compound just from our understanding of bonding. So we had to think more closely about why this is the case. So magnesium oxide and sodium chloride are both ionic compounds, but magnesium oxide has a much higher lattice energy than sodium chloride. This means that when magnesium oxide is placed in water, it's more energetically favorable for it to stay as a solid than it is to dissociate into ions of magnesium and oxygen. So magnesium oxide is still an ionic compound, but it's lattice energy is high enough such that it does not dissolve and does not give us conductivity measurements. So it's still ionic. Now we can look at magnesium sulfate. And we see that for 1 gram, we do get some solubility. And at 2 grams, we get no solubility. It was not able to dissolve. And we're able to measure some conductivity. This means that magnesium sulfate dissociated into respective ions and we're able to measure some conductivity while in water. But magnesium sulfate has a limited solubility in water, which is why at 2 grams it wasn't able to dissolve completely. So magnesium sulfate is still an ionic compound. Finally, if we look at sucrose, we see that we were able to get 1 gram of sucrose to dissolve in water, and 2 grams of sucrose was able to dissolve in water. But we were not able to get any conductivity. This means that sucrose dissolved in water, but did not dissociate into respective ions. So if you think about the structure of sucrose, sucrose has a bunch of polar groups on it, which make it able to dissolve in water. But sucrose does not dissociate into ions, which is why we were not able to measure any conductivity. So sucrose is a covalent compound. So by using our knowledge of bonding and our solubility and conductivity data, we're able to affirm the identity of our compounds. That is, whether or not they're ionic or covalent. So today we looked at how the type and strength of bonding of a solid influences its properties. Specifically, we looked at how these aspects of bonding influence these solubility and conductivity of these solids in water. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | Goodie_Bag_5_Electronic_Materials_Intro_to_SolidState_Chemistry.txt | BABATUNDE OGUNLADE: Hello, everyone. Today we're going to be working through Goodie Bag 5, which is on electronic materials, namely semiconductors. In order to follow along, you'll need a multimeter, a pair of alligator clips, some large LEDs-- I'll be using red, white, blue, and green-- and some regular sized LEDs. I'll be using red, blue, and purple. Our main objectives today are to identify differently colored LEDs by irradiating them with light across the visible light spectrum. And in the process, to explore the relationship between the bandgap of a semiconductor and its critical absorption wavelength. As we work through the Goodie Bag, I'd like you to think about how the size of the bandgap of material may influence which wavelengths of light you can absorb. Let's get started. First, connect your black or negative lead to the 10 amp port on the multimeter, and then your red or positive lead to the voltage ohm milliamp port. Then connect the probe end of each respective lead to one end of an alligator clip. After you do this, you should still have two free ends of the alligator clips. Grab one of your small LEDs and connect the free end of the alligator clip connected to the red lead probe to the positive end of the LED. This is the longer leg of the LED. And then connect the other free end that's connected to the black lead probe to the negative end of the LED. That's the short leg of the LED. Then turn on your multimeter and set the dial to 2,000 milli DC volts. This is the order of magnitude of voltage we'll measure across our small LEDs when we irradiate them with light from the larger LEDs. Finally, make sure to keep track of which small LED you're testing so that which LED to refer to when we start comparing data between small LEDs. So now we have our small LED hooked up to our multimeter. And now we're going to shine each one of our large LEDs onto the small LED and record whether or not we have some non-zero voltage reading. If the large LED has enough energy to excite electrons across the bandgap of the small LED, we should see some non-zero voltage across the small LED. So first I'm going to try and shine this blue light onto my small LED, and I'm going to look at the multimeter and try to see if there's some voltage drop. As you can see, our volt meter is now measuring some non-zero value, which means that this blue light has enough energy to excite electrons across its bandgap. After you've done this, repeat this for the other large LEDs, and then repeat the entire experiment for the other two small LEDs. This should generate a set of 12 data points. And using our knowledge of bandgaps and wavelength absorption, as well as the data that you've acquired, we're going to determine which small LED is what color. So after running through your experiments, you should have some chart that looks like mine, where I have my small LEDs going down the side here and the large LEDs going across. So each square represents a time when a small LED he was irradiated by a large LED. And a check represents when a non-zero voltage was measured across the multimeter, and an x represents when there was a zero or very low voltage reading across the LED. So in order to dig into what this means it's important to understand the balance of the visible light spectrum in terms of energy and wavelength. So if you remember the visible light spectrum, known wonderfully by the acronym ROYGBIV, where you have red on one end and you have violet on the other end, we know that red corresponds to around 700 nanometers in wavelength. And we know that violet or purple corresponds to around 40 nanometers of wavelength. And if we think back to the relationship between the energy of light and its wavelength, we know that the formula is E equals hc over lambda. So as the wavelength of light decreases, the energy of light increases. So going across, as we go from red, green, blue, and white, we have increasing energy because we have decreasing wavelength. And just as a reminder, white is a composite of red, blue, and green. So it has all three of these combined in it. So now let's start to identify which LED is what color. So if you look at small LED 1, it was irradiated by red, green, blue, and white light. And there was a voltage drop across the LED for all of them. That means that at the very least, red had enough energy to excite electrons across the bandgap of that small LED. Because that's the case, and we know that we're dealing with LEDs which emit like in the visible light spectrum, we know that this small LED then must be red. So if we look at small LED 2, we can see that both red and green light did not have enough energy to excite electrons across the bandgap of the second LED. But when we shined blue light onto the LED, we got a non-zero voltage drop. And the same thing when we shined white light, which also has blue light in it. Because, like I said at the beginning, we knew that our small these were red, blue, and purple, and we've already found our red LED, that means that this LED must be the blue. So finally, if we look at small LED number 3, we can see that red, green, blue, and white light did not have sufficient energy to cause some voltage drop across the LED. If we look back to ROYGBIV, and we think about the energy of red to blue, that means that they didn't have enough energy to excite anything after it. And that means that our final LED must be violent or purple. So by shining light across the visible light spectrum onto our small LEDs, we're able to identify which one of our small LEDs are what color. So right here I have small LED 1 hooked up to the multimeter. And we determined it was red. And so we can check that by just turning on the LED. So as you can see, this LED is shining red, which means it's the red LED, and we're able to confirm that by using our data as well. You can check that the other two LEDs are the colors that we determined by doing the same thing. So today we looked at a direct application of semiconductors in LEDs. We looked at the relationship between the bandgap of an LED, of a semiconductor, and the range of light that can absorb across its bandgap. We found that for semiconductors, there are some critical absorption wavelengths, some maximum wavelength, below which the LED will absorb all other wavelengths of light. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 26_Engineering_Glass_Properties_Intro_to_SolidState_Chemistry.txt | This is a great day. We're having our second lecture on glass, and we've got a demo-- where did Peter go? So we've got a demo coming up for you from the director of the glass blowing lab here at MIT. And now before we start, I just want to mention, so next Monday-- so we're going to talk today-- we're going to continue what we started, which is the chemistry of glass, and we'll talk about that cooling curve or the effects of different things on the glass formation and then how to engineer the properties of the glass. Before I start, I just want to mention that next Monday we have something called the Wulff Lecture. And this is a lecture that is open to the public. It's geared towards freshmen. It's put on by the Department of Materials Science and Engineering every year, and it's usually some sort of really interesting, exciting topic and usually a very dynamic lecture. And in this case it's one of our very own, Sam Shames, who graduated in 2014, and he'll be talking about his adventures as a material scientist and how he's used that knowledge in his life and also in the company that he founded which is now selling products in over 70 countries around the world. So I think that will be a lot of fun. If you can make it, it's next Monday from 4:00 to 5:00 PM. All right, now the second point I want to make is that there's a mistake on tomorrow's quiz, just in the schedule. So quiz number eight will be given at the beginning of class. Quiz eight is at beginning of class, not the end. I think it might mistakenly say the end. So you might notice sometimes the quiz-- so tomorrow the quiz is going to cover defects and then whatever was covered up until Monday's lecture and Tuesday's recitation, which includes the beginning of what we talked about-- [DINGING] OK, let's not do that. Let's turn that off. And we had an exam on a Monday, and so we didn't want to give you a quiz that week. So we're doing defects tomorrow even though we covered defects last week. And then we didn't want to give you a quiz right before or on Thanksgiving. That didn't seem right. So for the next quiz what we're doing is we're still making a quiz for you to have, and we're going to give it to you next week but we're not going to count it. So next week you will still get a quiz that you have for your studying because I know you'll miss it. But in case you're not here or you don't want to do the quiz over your turkey, we're not going to count that one. So quiz nine won't be counted. All right, so that's the quizzes. Now, where were we? Here we are. Speaking of quiz material and thinking about glass as a potential problem on tomorrow's quiz, yes. You might want to know about some of the things we talked about Monday. Here's an example. OK, here's a question. I have two glasses. They form from liquids, and they're cooled to some temperature I call T1. And I tell you that T1 is greater than Tm, which is the melting point of the material. Now remember, the melting point is the same. It's a number. The melting point of silica, that's a number. But when the material forms a glass can vary. Why does a glass form? Well, a glass forms-- this is what we covered on Monday-- it can form because of three different things, right? One is the crystal complexity. That's important. Another is the viscosity. And the third is the cooling rate. Those are the three things that we covered on Monday for reasons why a glass might form, and then we started kind of talking about the cooling rate. That's what we need to understand to answer this question. So, OK, two glasses forming liquids. One glass is formed. Plot the molar volume versus temperature. All right, so now, let's see. How about this? Could this be the volume per mole versus temperature? So I'm going to try it out here. I'm going to go like this. I'm going to go like this, and that looks like the liquid, maybe. And then I'm going to go like this and maybe this, and we'll call that T1. Does that look right? No, this can't be right. This can't be right because, you see, I said that T1, this is a glass transition. That's a glass transition. That's a Tg. That's another Tg. But I've said in the question T1 is greater than Tm. T1 is greater than Tm. Well, so this can't be true because Tm has to be out here. Tm is out here where there's a different transition that occurs, and it turns into a crystal. So this couldn't be right, T1 greater than Tm. So what about another one? Let's try another one. So how about if I-- OK, volume per mole. Now I'm going to go like this. So I'm going to go like this. So now here comes one, and here comes the other. And OK, so I'm going to put now T1 maybe-- let's try T1 is going to be out here, and Tm would be-- actually, let me do it differently. Sorry. I'm going to go like this. I'll just connect-- how do I want to do this? I want it to join these up there. Yeah, that's another good mistake. I'm making mistakes on purpose. And let's make T1 be out here. So T1 is here, and then Tm could still be there. This could be Tm. And then maybe is that right? Is that a possible curve? Well, that fixed the T1-Tm problem because T1 is now greater than Tm like I said in the question, but what is going on here? What is going on? You say, well, but I cooled one faster than the other. Isn't that kind of what's happening here? Maybe I started changing the curve? No. That's not what this is, so don't make that mistake. You see, remember, the slope is equal to the thermal expansion coefficient-- thermal expansion coefficient. And those slopes are the same for the different glasses. Just because I've cooled one differently, they're not going to have different slopes. And this is the fundamental challenge. This is the fundamental confusion point that happens which is that this is not time. This axis is not time. So this is wrong to. What we're doing is we're talking about something involving time, cooling rate. But we're plotting it on something that doesn't involve time, and that can lead to confusion. So I wanted to point that out. You say, well, shouldn't the slope be different because you're cooling it slower or faster? No, that's not what this slope is. This slope is a thermal expansion coefficient. So the right answer to this must be that you have your nice liquid phase and you've made your two glasses. So here's one. This has cooled faster. This is slower. This would be Tg. This would be another Tg, maybe Tg1, Tg2. And the Tm would come down and have-- well, I've run out of room here, but Tm might come down like this. And so T1 must be somewhere up here. That would be a correct plot for that question. So does everybody see? I just want you to get a feeling for this plot. We introduced it Monday, and I really want you to feel your oneness with this plot because it's a very important plot to understand why glasses form. All right, now moving on, there's a lot of properties of glasses, and here I'm talking about oxide glasses, so silica. Remember, silicon oxide, oxide, silica, alumina. These are, oh, oxide. We talked about that. Chemically inert, electrically insulating, mechanically brittle, optically transparent, visually arresting-- I like that one-- these are the properties that we know about these kinds of glasses. We already know them. But what I want to talk about today is how they can be engineered. And this plot is a basis for understanding that. And so is spaghetti, and that's what we're going to talk about. So the very first one-- which we'll set up our demo in just a few minutes. The very first one that I want to talk about is the mechanical properties. We know glass to be very brittle, but could we change its mechanical properties? Could we do something about the mechanical properties? And by the way, if those are the things that dictate whether a glass forms or not, then those are the things that you might think we might try to engineer if we want to change the properties of glass. You can imagine I'm a silica-- oh, you got to pass this out. I'm a big silicate, an SiO4, and I'm connected to these other silicate groups through this shared bridged oxygen. And I'm a bulky thing, and I'm trying to move around in the liquid, crystal complexity. I'm cooling down, and I'm trying to move around. And [INAUDIBLE] told me to go that way, but it's actually over there, or there's no building there and I don't know where to go. Where is the lattice site? It's complicated. Glass could form. Glass could form. Or maybe I'm a silicate group and I'm like, hey, could you please get out of my way? You're really slowing things down, you other silicate group that I'm attached to. Viscous, get out of my way. No, I can't, viscous. Or maybe you're a silicate group and you're cooling down and you're like, that was very sudden. I didn't even realize that happened. All of a sudden I'm supposed to be a solid? This is what's happening. And so instead of getting into this beautiful quartz crystal it doesn't get there. It doesn't get there. A supercooled liquid wants to be a solid. It wants to, but it can't for some reason, for some reason that has to do with those. So let's just have this in our hands while we're listening and thinking about things, and let me just tell you about this one way-- if I take a piece of glass-- if I take a piece of glass, and now imagine that I take some part of it, like this part here. And in this part I'm going to cool faster, and in this part here I'm going to cool slower. So imagine it's a slab of glass, and it's like the edges are cooled at a different rate. That's actually exactly why a car window, if you look at it through polarized glasses, has spots on it. Those spots, some of you may have seen these. Those spots are changes, purposeful changes in the mechanical properties of the glass to keep you safe. Why? If you call it slower, then you might have some volume that you want. The volume here, let's call this Vb and let's call this Va. That goes along with the graph we used before, b and a. So it wants V-- what did I do-- b. This wants Va. But look at what happens. If I purposely cool the top really fast or maybe I make spots that I cool fast, then what happens? Well, this wants to have volume a inside. This wants to have volume-- well, this cools first. So this cools first. So inside, so what happens? As I cool it now, I have this. This is now a glass. This is now a solid. And I should say glass solid, so it's solidified. And this wants to have a lower volume-- lower volume. It wants a lower volume, but that's been frozen at a higher volume. Why? I cooled it faster. That's it. I cooled it faster, and now the inside is like, wait a second, I'm not cooling so fast. I don't need such a high volume. Can I please get a smaller volume out of this? And the top is like, well, I'm already stuck. I'm already a solid. And what happens is it pulls. And so what you get is wants lower volume pulls on the surface, and that means that this is going to be the top, if I'll just draw it one more time, will be under compression-- under compression, and the inside will be under tensile stress-- under tensile stress. You can kind of imagine this, right? You can see it. I've frozen the top because of cooling rate. It's right here. I just cooled the top faster. The inside isn't cold yet. It starts to cool, and it tries to solidify at a different volume. It pulls it in. But see, that changes the mechanical properties of the surface. That's called tempering. That's called tempered glass. This is tempered glass, and there is no better example of temporal glass than the Prince Rupert's drop, which is the demo that Peter has come. Here's Peter. There's Peter blowing some glass. This is what he does in his free time. He just throws these big glass-bowling parties. He also happens to be the director of the glass lab, like I said. He also teaches the classes, the glass-blowing classes. There's one for freshmen during IAP, and there's another one in the spring. They're very popular. But because you're in 3091, he'll get you in. He doesn't like it when I say that because it's not true. [LAUGHTER] But come talk to him. He's right here, and I think you've got a demo for us. So I'm going to let Peter take the stage and show you the best example of tempered glass you can do. And if you want to use the mic-- if you want to talk. --a little kit out And one year Professor Cima, when he was teaching this class, told the class that they had to come to the lottery for the glassblowing class, and it's a big class. And so dutifully everyone came to the lottery, and we already have about 150 students plus 3091. And the lottery was in room 6120, so we overflowed it by a lot. So I told Michael, don't ever do that again, please. Which is why I do it every year. But we like 3091 students. We'll try to give them preference. [LAUGHTER] Trying to work it for you. He says that. How many of you have seen Prince Rupert drops before? Some people, yeah. There's a lot of online stuff about them, but you get to see a live one today. So what's the difference between a Prince Rupert drop and tempered glass like the type [INAUDIBLE]?? Anybody want to take a guess at that? They're both under stress, but the stress is generated in a different way. So the dots that were on that windshield are formed by air coolant. That's how tempered glass is made. It goes through what's called a [? lear ?] or a rolling conveyor, and it's heated up very, very quickly, and then it's cooled very quickly. And it's cooled by this array of compressed air jets on both sides that take it from almost the transition temperature-- it just about wants to move but doesn't quite, down to room temperature in three minutes or so. And that super-rapid cooling is what generates that stress on the-- the difference between the inside and the outside temperature in the glass. And so it's air cooled, but Prince Rupert drop is cooled by-- anyone want to guess? Water. Yeah, by water. So water is a better coolant, right? So as a result, you get a lot more stress embodied with the glass, a lot more stress. And so that's relevant because you don't want to make a window that breaks with that kind of energy and so you get explosive energy all over the place. So tempered glass is engineered to fail. It's very strong, but it's engineered to fail so that when it breaks, it breaks into small pieces so that if you're walking along the sidewalk and a skyscraper blows out a window, you don't get huge shards of glass falling down and hurting people. You get little pellets gently raining down, which isn't such a bad thing. [LAUGHTER] So I've got two kinds of Prince Rupert drops that I made here and a vessel in which to demonstrate here. So first I'm going to show you-- they both cool rather quickly, but one was cooled in air. One was cooled in water. This is the one that's cooled in air. And by the way, I used to give a demo in the Glass Lab where I would take the Prince Rupert drop, put it on a sealed table that we call the marver, and hit it with a hammer as hard as I could. I was never able to shatter this head with a hammer. In fact, I dented the marver so many times that I stopped doing that demo. So I have a different way of doing it now which is-- I won't do that today, but usually I'll have a student come and try to break the Prince Rupert drop in their hands. It's really hard to do. Most people can't do it. But the air-cooled Prince Rupert drop doesn't have the same kind of strength. So watch what happens. Not much. Well, let's try down farther. Maybe there's more stress there. Nope, not [INAUDIBLE]. Let me try one more time. Maybe where it's thicker maybe there will be more stress. Maybe a little. So, here's the one that's water cooled. OK, don't blink. Five, four, three, two, one. That's the speed of sound in glass, which is, I think, about three times faster than the speed of sound in air. So you can't see it. But there is some great high-speed video-- maybe some of you have seen that online-- of Prince Rupert drops exploding. And actually up in the Edgerton Center they have a camera that can catch that, and they've got some really cool video. I'm going to leave this. This is a very cool fragment that shows the fracture, the failure pattern, which is really interesting. It fails from the center of the drop radially out at an angle that looks to be about a 45-degree angle. You can probably explain that. That will be on the quiz tomorrow. [LAUGHTER] 45 degree [INAUDIBLE]? Was there a question? I'm going to do one more just in case anybody blinked, and then I'm going to be on my way. By the way, Jeff mentioned that we had an IAP pass that starts during IAP for freshmen and a spring class that's open to everybody at the institute. So if you're not a freshman, you can come to the lottery. The lottery is for all of our classes, are shown on the Glass Lab website. And I think we're going to be doing the lottery for the spring for the IAP class in November late this month, and then the spring class will probably be sometime in January, and it starts in March. OK, ready? Five, four, three, two, one. [INAUDIBLE] will even have some more energy. Thanks, Peter. [APPLAUSE] That was awesome, and I actually do have a high-speed video of it, so I'll show that to you now. And again, so if you have questions about glass, if you want to watch-- Peter can make animals for you, I think, on request. Oh, this guy. But anyway, thanks a lot, Peter, for coming by. As you know, I'm a big believer in hands on, seeing things. Can we pass-- is this-- no. Well, your call. Just don't eat it. So now you know on parent's day, when we have parents here, we had the two sheets of glass. One is called annealed. It just means it's not tempered, and the baseball was supposed to break it. It didn't. Peter's got very good annealed glass. He's the one that gave us those big sheets. And it breaks into these huge, dangerous shards. Thanks, Peter. Take care. Oh yeah, thank you. [APPLAUSE] If you notice, the tempered glass, we tried to break it. It also doesn't break. It was supposed to be the only one that didn't break. But then you take a little hammer at the edge and you just lightly tap the edge and the whole thing breaks. Why? Because it's got so much built-in stress. It's got all this compressive stress, and the edge isn't protecting that. The edge of that tempered glass isn't protecting the stress. So when I just dig in a little bit, it's able to all crack and release that stress. So that's what happened on family day. Here's a high-speed video. These are two guys from Corning. And we can listen to this. [VIDEO PLAYBACK] They're actually pretty cool. - So there's a few things going on here. The cold water rapidly cools the exterior surface of the glass, hardening it almost immediately. The interior, still molten, cools more slowly. As it cools, it contracts and attempts to pull the surface in with it, but it can't-- well, not very much. The surface has already hardened, so it gets pulled in only a little, compressing it, while also creating an internal layer that remains forever under tension. It is this action that gives the glass its uncharacteristic strength. We call it compressive strength. - It sounds like the same principle as how an arch provides strength in structural engineering. - Yes, kind of. Now, Jamie, I'm going to ask for your help. We're going to attempt to destroy this Prince Rupert drop. I just want you to tip that hammer past its center point. Go ahead. That's what Peter was talking about. So you can hit it really hard. - Now I feel like we've been swindled. - Swindled not. We have just experienced-- That's good writing. That's good writing. - --the power of compressive strength. It does, however, have an Achilles heel. Take those nippers right there and nip the backside of the tail of this Prince Rupert drop and watch what happens. Wait, wait, wait, cue the high-speed camera. OK, here we go. - Oh! Here it comes. - That was even cooler than I thought it would be. [END PLAYBACK] So that's what you just saw live. And what that is is it's a relief of-- thank you, or not. [APPLAUSE] Yeah, sure. As Peter said, tempered glass like in your car, it's done with air cooling. That's literally what those circles are. If you cool it even faster-- bam, faster cooling. You now know, what does faster cooling mean? Well, if it's going to form a glass you might be all the way near the melting point, and you're going to have an even-higher volume per mole which is going to cause even more stress within the glass which is going to make the surface very strong because it's all compressed, but it's also going to have a lot of energy in it that it wants to release. Let's see. We were here on Monday, and because I was talking about other ways to change the properties-- in fact, remember, I showed you a couple of different examples of glass, right, a glass cup and a glass bottle, and it turns out there is not just silica in it. There's all these other things in it, and there's many, many kinds of glass. And I listed a few here. This was from Monday, so I'm showing it again. And going all the way back to glass made in ancient Rome you'll find that it's got these other ingredients. Why? So that's what I want to talk about next. Well, let's use this. So if you notice-- and this is where we left off Monday-- if you notice something like calcium oxide, well, it's going to go to Ca2+ and O2-. Let's look at Na2O. Na2O is going to go 2Na+ plus O2-. How about alumina? That's on there, Al2O3. Well, that's going to go to 2Al3+ plus 3O2-. Notice a pattern. The pattern is you get O2-. We can understand why O2- minus has been used, why chemistry has been used to change the properties of glass for millennia. And the reason goes back to the pictures that we drew on Monday where we started by talking about the chemistry of the silicate group. So remember Si? Remember this? An SiO4 is happy if it's got four minus charges. Remember that? We started with Lewis and it satisfies the octet. But then in glass what happens is you bridge. And so you have this. And I won't draw all of them. I'll just draw it the way we drew it before. And you have these bridges, and all of these oxygens are bridges in glass. So now I'll say, well, what if I wanted-- and remember, these are forming these long, viscous chains. What if I wanted to make them flow a little more easily? What if I wanted them to form shorter chains? What if I wanted to cut the pasta? Don't do it in Italy. They'll get really upset. But Americans are OK with it. If you need to cut the pasta, it's OK in this country. But if I need to cut the pasta in chemistry and it's glass that I'm cutting, O2- is my knife. It's my chemical scissors. Chemical scissors, I like that-- chemical scissors. Why? Well, you can see because I go back. What did the oxygen do? By bridging it didn't need that extra electron. But if I bring now something like Na2O-- so I've got an O2-. Now I'm going to add an O2-. Then what will happen is you're going to cut this. So now I'll just draw the-- I'll see if I can-- OK, so now I've added not just an oxygen atom but the two charges that each of them needed before they bridged, you see, to be happy? And so now this can have one of those charges, and over here you can have another one, and so on and so on. But see, now I've cut. I've cut the glass pasta, and I've done it chemically and I've done it from understanding of octet and Lewis, from basic chemistry principles that we've learned. Where's the sodium go? Well, they'll be like a sodium. They'll want to hang out close by these. There will be these ions in there. These sodium ions will be in there still, right? If it was Na2O, if it's something else and it's a different kind of ion that might be leftover, but the main thing is I've delivered the chemistry needed to cut this. And this is called chain scission. There's a term for it, chain scission, the technical term for cutting glass pasta. Well, this is powerful. This is very powerful because if I can cut this-- those are the silicate groups, right? And if I can cut them, then I can change another property. Look, I can change the viscosity. I've hit another one of those three reasons why glass forms. And so I'll show you a slide in a minute that shows you the change in the viscosity, but just to give you a sense of what can happen, how does that impact? How does that impact? Well, for silica crystal, the melting point is greater than 1,200 C. For Tg of soda-lime glass, it's 500 C. So you can imagine now I've got another mechanism. I've got another way, not just processing, not just cooling rate, but chemistry to mess with this curve. Because you can imagine now that I might be able to get to lower and lower transition points. You see this? I might be able to get to lower and lower transition points if these things can move faster and not get locked into a solid as easily if I add more and more of these chemical scissors. So maybe this one, now instead of being cooled slower, maybe this one had more soda than this one. Maybe now they're cooled at the same rate but I've changed the amount of impurities, the amount of oxygen, O2- very specifically, that I've put in. So it's another way now to change this curve. Now, these are called-- the thing that forms the glass-- and we're really only talking about silica as our glass. You can make other types of glass not made from silica. We're talking about silica in this class. So silicon is the atom that is making those long spaghetti strands, and it's called the network former. Silicon is called the network former. And then the thing that I put in there-- the thing that I put in there that modified the network is called the network modifier. So you can see on this graph that what you've got is your bridging, the bridging there that's going between two network formers. And then you've got these atoms that are hanging out near a place where you've broken a bridge. So those would be modifiers. Those are called network modifiers. These are just terms that you will hear and see when you think about glass-- network formers, network modifiers. The chemistry that you introduce is modifying the network, and it's what is done in many, many different ways with many, many different [INAUDIBLE] to change the properties of glass. And so here's the curve that shows you another kind of important way of looking at this. And I don't need you to remember all of these terms, but I want to show you because this is how glass is engineered. Remember, today the topic is how do we change the properties? We talked about the cooling rate. Now we're talking about adding chemicals. And so now you're going to work with the glass to do something with it. If you're going to do that, this is one of the first plots you'll look up. And it shows you-- it plots the temperature. This really doesn't work at all-- the temperature on one axis-- so that's like the x-axis-- and on the y-axis is the viscosity because now that's the property that we're changing, right, the viscosity? So there's silica up on the upper right. Can you see that? So fused silica, that's basically almost quartz. That's basically quartz. So you are really close to making a nice crystal, but notice how high of viscosity you have versus temperature. Notice that you're at this very, very high value for viscosity. It's a really thick material. And people care a lot about viscosity when they work with glass. Why? Because you need to know, for example, is the viscosity at a point where I can work with it? Can I shape it? That's the bottom. That's the region that says working range. I can still shape it. I can make a window. There it flows under its own weight. So now I can shape it and it will lose the shape. That's kind of important to know if I'm making a glass animal. There its stress can still be relieved. Remaining stress can still be relieved, and up there you can rapidly cool it without introducing new defects. But look at the tradeoff between the temperature that you have to go to and the viscosity and the viscosity that you have. So if you go now from this almost quartz all the way down-- OK, there you take a little bit of stuff and you put it in, but it's still 96% silica. And now you go to boiled silicate. Remember, it's like 75%, 80% silica plus other additives. And look at how much you've changed the temperature. So I can now work with this material below 1,000 C. That has huge consequences for industry, for actually making different things out of glass, which we now do regularly. And the Romans did it too, but they didn't know why. They said, well, if you add a little bit of this dirt, something different happens. Let's try a little bit of that one. Now we know. We know why. And so this curve now is something that we understand and we understand how to control it. And if you go back to the example of making glass tough, I was just showing you these ions, these sodium ions. They're left in there. Well, it turns out that you can put ions into glass, and maybe you're doing it to change the viscosity so the thing stays liquid for longer, and you can maybe work with it in different regimes. But maybe you're putting the ions in there to do something else. And so actually there's a whole another thing you can do once you have ions in the glass like sodium. You can substitute them out. Imagine that I have-- that's what this next picture is showing. Imagine that I've got a glass in here, and those red circles are sodium. But now I've introduced a different ion on the surface. That's potassium. What happens? Well, if I have a high concentration of potassium and I put it on a surface of glass that has a bunch of sodium in it, so now I've got a glass with sodium ions in it. And I expose it to a potassium bath. So now I've got potassium up here. And what happens is if the concentration is high enough and the conditions are right, I can get potassium to go in for the sodium. And so actually now I can substitute. So these were here. I can now substitute those for potassium. Well, but this is a big deal because potassium is bigger than sodium. And so if I am able to on the surface, let's say, put something bigger in there, then you can imagine that it's pushing the other stuff into each other. That's causing the same stress. K+ plus bigger than sodium, and this causes this huge surface tension. Same principle as tempering, but now I've done it chemically. This is Gorilla Glass. This is all phone screens. This is what they do because you can get really, really hard glass this way. And if you go back to these guys, they have a little-- and this is just a very short video. They've got this sort of longer videos on the Corning website that talk about this, and they shoot bullets through Gorilla Glass. You can make glass really strong with just chemical substitutions, and all you're thinking about is atom size. Because by putting this larger atom in, you cause this same kind of compressive stress at the surface. [VIDEO PLAYBACK] - Gorilla Glass, it's been refined over time, but like all Gorilla Glass variants that came before it, it is compressive-strength glass. But it's not made in the same way as we just demonstrated, the rapid-cooling method. No, instead Corning uses an ion-exchange process. To break it down simply, the surface ion particles that naturally form during the manufacture are replaced with larger ion particles. Once exchanged, the larger ion particles create the same sort of inward pressure that we see on the Prince Rupert drop. And with this method they are able to control and manage the resulting-- [END PLAYBACK] The resulting what? What was he going to say? I don't know actually. I don't remember. I have to go watch it now. So that's just a video showing you what I just showed you, which is that that's yet another way of engineering the properties of glass, and there are so many more. One of the points I want to make today about how to engineer glass-- so we're talking about how to change this curve. Maybe we change the viscosity by adding O2 which changes the curve here. Maybe we change the cooling rate. Maybe I can use additives, ions in different ways. But it turns out that we have come to a place where we can engineer every single one of these properties. We can make conducting glass, even though it's usually been and usually is insulating. We can make chemically active glass. As you just saw, we can change the mechanical properties. There are now flexible rolls of glass that look literally like Saran wrap almost, but they're glass. And so it all comes down to the pasta. And so where we are, if we use the pasta analogy-- which as you, I think, can understand I really like because I really like pasta. This is a great analogy for glass because it's like in the Roman time, all we had was one sauce, everything. Just mix it all in-- meatballs, pesto, red sauce, carbonara. I can go on. And that's all they did. They just didn't know why things were happening. But now that we know about the electronic structure of atoms, we can make glass into any delicious pasta dish we want, right? OK, don't eat it, but you know what I mean. And so I'll give you a couple of examples. So the point is the last 20 centuries, we were really just getting warmed up when it comes to this material. What the recent understanding of how to engineer this material has led to are real breakthrough ideas. And so I'll show you just a couple in my why this matters. Because glass is, after all, made of-- there it is up there in case you can't see it. This is the abundance of elements versus atomic number, and you notice the very top two elements are silicon and oxygen. And so it would be really nice if we could make a lot of stuff out of these really abundant cheap elements. Think sand. Could I take sand and make a lot of stuff out of it? Well, not if I can't control. Not if I have to heat it up to 3,000 degrees or its viscosity isn't what I want or it doesn't give me the properties I want, but that has changed. That has fundamentally changed. So we can now look at materials like this, these super-abundant materials, and we can completely rethink them. And I'll give you one example. It's already a few years old, but I think it's just a really cool idea, which is called the Solar Sinter. And here he's developed this machine that is entirely solar powered. They're solar cells for the electricity. And it's a focused beam of light from the sun that gets hot enough to engineer the glass. So he takes sand from the desert that he's in. He puts it in a container, and he's got a 3D printer that he's made. It's entirely solar powered. There's no fossil fuels, but he can take sand and turn it into something structural. And so there's a vase that he's made, and you can take that out. And again, that's just the beginning of rethinking what we could do with these super-abundant materials. Here's an example from also a few years ago from an MIT lab. This is Neri Oxman's lab, and she's in the Mediated Matter lab here at MIT, and she's developed a 3D printer. [VIDEO PLAYBACK] [MUSIC PLAYBACK] And so there it is printing something with glass. And she can print lots of different designs now using glass. She has a cool video. And there is the printhead. And again, 1,900 Fahrenheit. I'm not sure that I want a 1,900 Fahrenheit printer on my desktop. But as you can imagine, what they had to do-- I'll just give you one more example and then we'll stop. What they had to do is understand everything we've just talked about. How do you engineer the viscosity, the melting point to make it all work in a 3D printer? The last point I'll give you is this. I love this. This is from a few years ago. It's a group in Japan that made a glass that is as strong as steel. And what they talk about-- this is what I like. They say just think of a world where your smartphone wouldn't shatter. OK, cool. Buildings could be bolstered against natural disasters, even cooler. And then somehow they bring it down. Wine glasses are reassuringly safe. [LAUGHTER] Was that really a problem? I don't know. And what they did, fabrication was conducted using an aerodynamic levitation furnace where ingredients were floated in the air using oxygen gas and melted together using CO2 lasers, and they get a transparent superglass with 50% alumina. That was so hard to do because the aluminum didn't want to be a glass. It wanted to go and become a metal, a crystal. But by doing it this way, they were able to capture it in the disorder. All right, have a great night. See you guys on Friday. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 16_Doping_Intro_to_SolidState_Chemistry.txt | We are talking about semiconductors today. Yeah. I finished our exam 2 concept math that I thought maybe you guys would find useful. This is much like the last time, when I gave you guys the concept map. I'm just giving it a little earlier this time. And here, you can see all the connections between the lectures, the recitations, the problem sets, the quizzes over here, the goody bags, and the exam topics. Speaking of the quizzes, tomorrow, there's a quiz. And so you can look up here and say, OK, well, what's going to be on that quiz? Might cover some molecular orbital theory. Might cover the band gap stuff that we did on Monday and that you covered in recitation yesterday. Oh, and intermolecular forces we also did. And you're asked to bring to the quiz the voltmeter and these large LEDs. Now, I'll talk about that, a little bit more about this, in a few minutes. So remember to bring that. Laura will also send a reminder email about that. OK, you have voltmeters. You and voltmeters. How many of you, this is the first time you're using a voltmeter? How many of you haven't used one but you've never seen one-- you've used it before, but you've never had one this accurate as the 3091, right? Yeah. You never know when you need a voltmeter. You should always carry it with you with the periodic table. Now, on Monday, this is what we had. This was a conduction band. This was a valance band. And then we had a conduction band and valence band that were a little bit closer together. And then, we had this case, where maybe it's filled up somewhere in the middle of the band. And remember, the valance band is filled with electrons, and the conduction band is not, right, in the simplest picture. And this difference in energy between this VBN-- that's the VBN, the maximum level of the valence band-- and the CBN-- the minimum level of the conduction band-- that energy difference is the gap of the material. Here, you can see energy gap equals zero. Well, in this case, it's a metal. Or if you want, it's a conductor. And we'll talk about why it's a conductor more on Friday, but a little bit today, why metals are conductors. Today, I really want to focus on this case. All right, so that's a metal when there is no gap in the material. If the gap is large, then that is an insulator. What do I mean by large? Well, what I mean is it's not in the range of this special case, which would be, say, zero less than the gap less than around 3.5 electron volts. This is the case that I want to talk about today. I want to spend today talking about semiconductors. Those are called semiconductors. Let's write that here. Semiconductors. Sometimes you'll see them called metaloids. Sometimes you'll see them called insulators with small gaps. They're all kind of right. Metaloid, thank you. Now, the reason why we classify these and why this range of gap is so important is what I want to convey to you today. These are really special materials, these semiconductors. Now, before I do that, I want to just clarify one thing. And I'm going to use diamond and graphite as an example. On Monday, what we did is we talked about the concept of bands with a very simple 1D model, right? This was a 1D model of s orbitals, 1s orbitals. I said, well, if I have a mole of 1s orbitals, then that goes into a band. And I called it a 1s band. I always say, well, if you fill it in different ways, that is what determines the properties. But I want to be clear-- and I stressed this on Monday, and I want to stress it again. Whether this forms this and then the next set-- you know, whether you get a 1s that forms a band like this and then a 2s that forms a band like this and then another one, maybe 2p, like this. Or whether there's breaks in them, or maybe these kind of overlap-- so you get overlap, and you get sort of something like that. All that depends on more than just those atomic orbitals. It depends on more. It depends on how these things are bonded. This is such a great example. I've got the same element here. Its carbon. There's nothing else in this system. Carbon, pure carbon. Everywhere is carbon. But on the one hand, I've arranged it in a certain way. And on the other hand, I've arranged it in another way, and their properties couldn't be more different. So just to take this a little further, if I think about diamond on the left, diamond is an insulator. Diamond has a gap of 5 and 1/2 electronvolts. 5 and 1/2 electronvolts. It's an insulator. But OK, that makes me think of methane. Remember, methane, you had these SP3 states-- SP3, SP3, SP3. And the reason it formed SP3 is because all these H1s states came along-- H, H, H, H. These are 1s's, 1s. And so carbon came along, and it said, well, if I arrange-- if I got four of you here like this and I arrange myself-- if I arrange my orbitals in such a way that then I can arrange so that they're minimizing repulsions, that gives me methane. That gives me the ground state of methane. But the only way to do that is to have four equivalent orbitals. Those are hybrid orbitals. We talked about this when we talked about hybridization. The idea being that if I took one of these and one of those-- so now you can imagine how it's going to look. You know all this now, right? There's a 1s orbital of hydrogen. You know that those will form a sigma and a sigma star and occupy that bonding orbital. That's the idea of methane. That's the idea of molecular orbital theory with some hybridization. And it's going to form those bonds along each direction. But now I'm talking about something different. I'm not talking about such a simple case. I'm talking about one mole of SP3. That's what diamond is. Look, the carbon atoms have done the same thing. They said, hey, wait a second, OK, it's not four hydrogens. It's like 10 to the 23 carbons all in one small space. How can we organize to lower our energy? And so they form-- well, we know this, that it's going to form a band. But it doesn't just form a band and then half fill it. That's not what diamond does. No, actually, diamond breaks this apart so that you have a huge gap. And those are filled, and then those are not. The SP3 band in diamond breaks apart into a valence and a conduction band. The valance band is full. The conduction band is empty. Why? Well, you don't need to learn the specifics of how to get from there to there. But I wanted to show you how complicated this is. I will ask you whether the SP3 band splits or not. I just wanted to show you how dependent this is. Because now, look, you can understand this now, right? So I've got four electrons per atom going here. There's a huge gap, and then nothing in there. That's diamond. 5.5 eV gap. One mole of SP3 in diamond. Now, it depends on the structure. If I had just put them together in some amorphous soup with no-- oh, amorphous. That's a word that's coming later. But if I just put them together with no order, I wouldn't get this, this splitting. You take graphene, which is one of those sheets, and make stacks and you've got graphite. It's another form of pure carbon. In that case, you've got your SP2 bonds to form in the plane, and those also split and they fill. But now you've got your pi bonds, and those don't split. So in graphite, what you get is you've got your SP2's. You've got a mole of those, and you've got your P electron. Remember those? Now you do a mole of that, and you've got the same thing. But see now, you've got-- these are the SP2 bands. It also split. Remember, I've got my electrons here, my SP2 electrons and those there. But now, I've got the pi bands that didn't split. Graphite is a metal. You can understand it by looking at the bands of these materials. Now, again, let me emphasize, I am not going to ask you to know when something splits like this or not. Why didn't the pi band split into two like these did? When does an S and a P become a single band versus an S band separated by a P band? I'm not going to ask you to know that, but I wanted you to see it. It doesn't just depend on the initial states. It depends on the structure of those atoms. It depends on how those atoms are bonded together. Now, what about this semi-conductor? What about this semiconductor? Well, if we take a semiconductor and we look at it-- I told you this on Monday. There are two things that you can do in your goody bag, especially. One is you can take an electron, and you can shine a light on it, and it's going to excite this electron up, just like in the Bohr atom, right? You can absorb photons if the photon energy is the energy between one state and another. We did a lot of this earlier. But see, in a semiconductor, remember, I've got an almost infinite number of states here. And so what I need is for that photon-- there's not an exact value here. There's a continuum. But here, it's forbidden, just like in the Bohr atom. There's no states. There's no electron. So here, the energy of the photon absorbed-- let's write this out-- has to be greater than the energy of the gap. This is the energy of the gap. This is the energy of the gap. Now, by the same token, I can take the same semiconductor and run it the other way. I can put a current on it and feed electrons, electrons from current. So I can attach it to a battery, and there you go. This is a semiconductor where I'm literally-- all I'm doing is I'm feeding electrons into its conduction band. But now, when those electrons fall down into the valence band, they emit a photon. Photon emitted. And just so I have room, with energy of the photon emitted is going to be exactly equal to the energy of the gap. You can see that because these electrons, I might inject them into the conduction band anywhere, but they're going to wind up very, very, very quickly going to this minimum conduction band state. And that gets me to another important point, which is on the absorption side, I can overshoot this. Like I said, I can absorb higher than the gap. I can't absorb lower, but I can absorb higher. And if I do, if I absorb higher than the gap-- so let's say this is my state. So if my photon comes in and it kicks in electron up really high, it's a high-energy photon, then that electron very quickly will go back down. And that's called thermalization. So what happens is that electron loses energy to heat to get to the CBM. We'll come back to this later. That's thermalization. So the electron can be kicked up by light to any level, but it's going to very, very quickly go to the CBM. That's why when I inject it-- that's why these are the same color. These are high-quality LEDs, clearly. Well, they work. That's a good thing. So the batteries might be slightly different, and the currents might be different, a little bit. I might be injecting electrons at different parts of these conduction bands, but this is always green, and this is always red. And that's because those electrons very quickly get down to the CBM and the VBM. And the CBM and the VBM are determined by the semiconductor inside. Well, you've also got-- so this is a photon emitter right here. I've made a light emitter, right? But I've also got a detector, which here's what I said on Monday. So now, I take an LED, and I hold it-- gesundheit. And this is what's inside of the emitter. It's the same thing. It's a semiconductor. But these two semiconductors have different band gaps. That's why the light is different. They have different band gaps. Now I've got one with some band gap, and I hook it up to my voltmeter, and I just look. Is there a current or not? If this thing conducts through it, it means that electrons have been promoted from here to here. And that's something we're going to talk about today. Electrons cannot move around easily if they're in the valence band. But once they're in that conduction band, they're free. How do I get them there? Well, you've got power. Shine red light on it. Does it conduct? Check the voltmeter. I'm literally holding two semiconductors and doing these two processes right here in each hand. It's a power that I have. I have semiconductor making light, semiconductor absorbing light, or maybe not. If I shine right on this and I get no current, well, it means that the energy of the photon from a red light source is too low. Maybe it's able to excite it up to here but not up to here, given whatever the gap in here is. You see that? So the semiconductor has this powerful-- maybe I take a different semiconductor and red excites it. Or maybe it doesn't, so I go to a higher energy photon. With this goody bag, you're touching and feeling semiconductor physics and chemistry. Because band gaps in materials are rare. You see, if you look at the periodic table and you classify it by metals, metals and nonmetals-- so insulators, conductors, and then these weird things in between-- there aren't that many of them. There aren't that many of them. But as you can see, these are really, really important technological materials. Why? Because this is visible light. This semiconductor range here is in the range of light we care about-- visible, UV. So I've got an electronic material where the electrons in it can interact with currents, small currents even, and wavelengths of light that are things we can see. You can imagine how important this could be, for example, for making LEDs. But I need a lot more flexibility in my material set, and that's where chemistry comes in. So this is, one, why this matters for today. So, look, I mentioned that getting red was hard back in the day for color TV, because the red phosphorus were difficult to make. For LEDs, it was blue. In fact, there was a lot of work around gallium nitride. Here's gallium nitride in one structure. Notice this. There's gallium nitride. It's not on here. Darn it. Gallium nitride in other structures have different gaps. Oh, there it is. There's one. But the point was they wanted blue. They wanted blue, and they needed to make a material that was both cheap to make and lasted, so it didn't degrade, and gave you a blue light. And that was really hard. And that's what the Nobel Prize was given for in 2014. Because without blue, you can't make white light. So there's all sorts of technologies you simply can't go into with LEDs until you get blue. How did it happen? It happened because of chemistry because people figured out how to change the how to take one element another element and then maybe alloy them together. Maybe you take gallium nitride and you alloy a little bit of aluminum in here. That means mixing it in. Now, all of a sudden, you've got a different band gap. Band gap engineering is really the centerpiece of the semiconductor revolution. Now, the connectivity. We've been talking about connectivity. And we'll talk about metals on Friday. Here we are. Insulator's really low. Least conducting, 10 to the minus 25. Hook your voltmeter up to Teflon. 10 to the minus-- wood. Low-conducting materials. But we're interested in these, and these are kind of in-between. They're not very good conductors. That's why they're called semiconductors. But the fact of the matter goes back to what I've been saying. And this is the next point I want to talk about, which is I cannot have-- so in a semiconductor, there's no electrons. If there's no electrons in the conduction band, then there's no electron conduction. Now, I've said this before. I said this on Monday. I said it already. Why not? On Monday, I said, well, these electrons are stuck. What does that mean, they're stuck? Stuck in the bonds, stuck in the anti-bonds. They're stuck in states. The point is, here, that this valance band is filled. Now, if I want an electron to move through a material, which is what conductivity is, after all-- by the way, siemens, conductance, one over ohms. If I want electrons to move in a material, they need freedom. They need freedom. What does that mean? Well, it means that I'm trying to move through this wire. And for an electron, freedom means I can go to any state that's nearby. State, a wave function, a state of probability that I can move to. And then to that one. And then over here. And then there's is something there. I don't want to be. I've got to go this way. There's a bad thing over there. It's scattering me. I'll go this way. But if electrons don't have the ability to move into a free state, they're not going to conduct. That's why you've got to get the electron up here. Why? Because the conduction band-- I shouldn't really fill this in. I'm just showing you empty states there. The conduction band has 10 to the 24 states. Freedom. Freedom. So as soon as an electron gets up there, it's like, oh, I can go anywhere I want. I can be there. I can be there. I can move in response to this field that's pushing at me. I can conduct. I can't do that unless I have conduction, unless I get electrons into the conduction band. This is OK. Now, this is why, you can see right there, metals are clearly good conductors, based on our band picture. Because those electrons near that line there, you fill them up, and they're halfway in a band. But right up above there, an almost infinitesimally small energy away, is an empty state and another empty state. These ones you can see, it's going to be hard. I'm going to have to shine really high energy light to even get-- or really high temperatures. Again, semiconductors are special, because their gaps are right in that range where I can get electrons to go from the valence band to the conduction band in different ways. And there's two different ways that I want to talk about. The first way is simply with heat, and this is why I mentioned this on Monday. Actually, no matter what, there's some probability for an electron to have a high enough thermal energy. Yeah, electrons get hot, too, right? Everybody feels the heat. So that the thermal energy can be enough to get the electron to knock it up above. So with thermal energy, you can get-- let's see. At room temperature for silicon. So this is the case of silicon, which has a 1.1 electronvolt gap. Silicon, 1.1 electronvolt band gap. It's a really important semiconductor. And for silicon, at room temperature, I got about 10 to the 10 electrons in the conduction band per centimeter cubed of material. This is how we think about it. We think about how many of these do I have per volume. It's a good way to measure it. These are also an electron in a conduction band also is called a carrier. So for silicon at room temperature, I will have-- just because I can't help it, because there's that many-- about 10 to the 10 carriers in the conduction band. Now, carrier is a carrier of electricity. That's why it's called a carrier, right? It carries electricity. So those are the ones that are carriers, the ones that made it up. But if I go to 600-- so now I'm going to go to 600 kelvin. Oh, we only use kelvin. Kelvin is the thermodynamic energy scale, temperature scale. 10 to the 15 carriers per centimeter cubed. This seems like a big number, but is it a big number? What's a big number we know and like in this class? The mole. So like roughly-ish, is this all a big number per centimeter cubed or not? It's actually very small. But I need them to be there if I want better connectivity. If I want conduction in these semiconductors, I've got to put electrons up in the conduction band. Do you remember, just as an important point, I mentioned the energy of room temperature, the thermal energy of room temperature kBT is about 0.025 electronvolts. That seems way smaller than 1.1 eV. It is. But that's because that's not what temperature really looks like. Temperature is a distribution. There's a distribution. Now, you don't need to know that for the band gaps. We're going to come back to distributions, Boltzmann distributions, when we talk about reactions, reaction kinetics, later. But that's why, at room temperature, the tails-- hot means a range of energies. Hotter means a wider range of energies. And it's those tails that have enough energy to get over the gap and into the conduction band. It's those tails. That's why I've got a little bit of electrons that get up there at room temperature. Now, if you look at a plot-- here's a great way. There's the log connectivity of silicon. It's from [INAUDIBLE]. And there's the log connectivity on this side of tungsten. That's a metal. Totally different behavior. How is that possible? Well, we're going to explain the metal on Friday. Today, we're interested in this. This one, now we understand. Thermal energy excites carriers into the place where they can be free and conduct. So as I increase the temperature hotter and hotter-- there's the temperature in kelvin. I just gave you examples from down here, and now we're going higher and higher. I can get the connectivity up orders of magnitude higher for silicon. You know why now. I'm populating. I'm getting over this gap, and I'm populating the conduction band. So the answer is we just need to run our semiconductor industry and our phones and our lights at around 1,000 kelvin, right? No problem. All our devices. No, that's obviously not a good idea. How do we get around this? The answer, such a good word, it's chemistry. Chemistry gets us. Chemistry saves us, as always. You've got to know your chemistry. And in particular, this is not the doping that you may read about in the news related to Olympic athletes. This is the introduction of some small amount of an impurity. An impurity is something that wasn't there normally. It's like something I mixed in that's not normally there. If I put a small amount of something in it, then I can tune the properties. And in this case, I'm talking about the connectivity. So how does this work? If I look at this case-- I'll do one case here. I've got my silicon atoms arranged in the solid that gave me a band gap. And again, that is due to the arrangement of them. If I didn't arrange them in this particular way, I might not have that band gap. I might not have the same band structures. It depends. But this is what I get for the silicon arrangement when the band gap is 1.1 eV, which is the one we use in most of our electronics. But the thing is, I want-- so this would be silicon. This is just silicon by itself, valance band, and here's the conduction band, CB, VB. Now, I want to add electrons into this material. And so you can just kind of imagine how this would work, right? If I just look at a Lewis dot diagram for silicon-- let's just forget about this structure for a second. Silicon has these four valence electrons. So you can imagine silicon might look something like this. And OK, I'm going along, and I'm going along. It's tetrahedral and stuff. But now, I'm going to take one out. I'm going to take this out, and I'm going to make it phosphorus. What is phosphorus? It's, wait, where did it go? Panic, and then-- I'm recreating what happens for you guys-- I found it. That was close. But phosphorus is right there. It's right next to silicon. It's group 15 instead of 14. But it just has one extra electron, right? So maybe it could do the same thing, but then have this one extra electron. That's doping. Where does it go? It's all about freedom. It looks down and says, wait a second. There's no states here I can go to. No states. I can't go anywhere here. So I will have a state, because I have picked phosphorus carefully. And because of the way the chemistry and the bonding worked with phosphorus in silicon, this will introduce a state near the conduction band, and that's the key. So now we've got the valence band, and we've got the conduction band. And what's happened is there's a new level here that is called the donor level. That level is where the extra, extra key electron goes. It can't go in here, remember. This is all filled. So now, I've doped with P. But look, if I've got an electron here, now this difference in energy is really small. It's really small. I've engineered it this way. I've thought about it. So if I put phosphorus in here, that extra electron isn't sitting far away anymore from the conduction band. In fact, now it's so close that room temperature just kicks it right on up. So effectively, it's going to go up here, because that's a small delta e between the donor level and the conduction band. This is how doping works. This is how doping works. So I put a phosphorus atom in. There's an extra electron that comes out of that. The extra electron needs a state. It gets a state. That's called a donor state. Donor makes sense, right, because it's donating an electron into the conduction band. Once I get the electron into the conduction band, I'm home free. Because now, I've done what temperature was doing before, but I did it with chemistry. And so you could ask questions like this. How much phosphorus do you need to substitutionally dope? Substitution meant just what you-- substitution was I took a silicon atom and I substituted it. That's substitutional doping. To substitutionally dope a mole of silicon and get a free electron density of 5 times 10 to the 17. I'm going big here. Because look, temperature-- gesundheit. OK, maybe I'm willing to run my semiconductor in a really hot room, my phone at 600 kelvin. I'm still only at 10 to the 15. I want 10 to the 17 now. How much phosphorus do I need? You can solve that problem. I'll just write down a couple of the steps. So the volume of one mole of silicon is one mole times 28.1 grams per mole. Oh, and I got the density on there, too, right? Divided by 2.33 grams per centimeter cubed. How did I get the density? I looked it up in the periodic table. And so one mole of silicon is going to give me 12 centimeters cubed of silicon. One mole of silicon is 12 centimeters cubed. I need 5 times 10 to the 17 electrons in the conduction band per centimeter cubed. You see why per centimeter cubed is a good measure here. So that means I need 12 times that total. So I need 12 times that. So I need 5 times 10 to the 17 P atoms. Why do I need that many P atoms, times 12? Let me just write my units here to be very careful. P atoms per centimeter cubed times 12 centimeters cubed. That's how P atoms I need in this mole of silicon to get what? It's one to one. Because each P atom had one extra electron. Those other electrons are in the bonds. They're not going into the conduction band. It had one for one. One electron went to this donor state, went into the conduction band per phosphorus atom. So if I want that many carriers, if I want that many free electrons, I need that many atoms. Well, you guys can look up the phosphorus grams per mole, and you can get that this is 0.0003 grams of phosphorus. That's really not much. You see why it's called an impurity, right? You see why it's called an impurity. That's not much. I have all this silicon, and I only need 0.0003 grams of phosphorus, and I've bumped up the number of carriers by orders of magnitude. Not only that, I've done something really important, as well. And you can see this, because now, if I plot something like temperature versus number of carriers-- well, you saw the graph for temperature before, right? It was like this. That's the graph from a couple sides ago. But now, if I do it with chemistry, it's flat. Until the temperature gets really hot, I could-- give myself more room here, because that's kind of the point, right? Number of carriers. I can make this now flat. From doping, from temperature. And so you can see, OK, at a certain point, the temperature goes up, up, up. And at a certain point, the temperature is going to be so hot that I'm adding electrons, even beyond this. But you saw the temperatures there, 1,000 degrees kelvin. No, this allows me to have a predictable, flat number of carriers in the conduction band no matter the temperature. Just think about how important that is. Semiconductors are kind of discovered in the 1800s. But it wasn't until physicists and chemists were able to control the band gap and how to put carriers in and out of it until they could really work on the semiconductor revolution, which is what led to Snapchat. Did I get it? Snapchat is still used. I knew that. Slack. Now, you can do the same thing going the other way. You can do the same thing going the other way. If I now put gallium, so if I dope with gallium, well, you see, gallium has one too few electrons. And so if I dope with gallium, what I have-- so the gallium is there. Where is it? It's on the other side. Where's silicon? Silicon. Phosphorus is over there, group 15. Silicon is group 14. Gallium is group 13. One fewer electrons, but it really wants to make those four bonds. The material isn't happy if you break one. And so what does it do it? It steals an electron. It takes it. It says, I need an electron to be happy, so it takes one out. And that's called an accepter level, where what's happened is the gallium has actually pulled an electron out of there. And that leaves behind a positive charge. This is called a hole. That's called a hole. Think about it. If I take an electron off of an atom, it's positively charged. If I take an electron out of the VBM, there's a positive charge in there. That's really important, because these positive charges effectively are like electrons. They're like negative charges, but they're positive. So they also can conduct. They also can conduct, right? And that's really important. So these are two different types. And you can imagine the brilliance of the naming. These are P type, because it's a positive charge. And these are N type, because it's a negative charge. Those are N type semiconductors. What does that mean? I've used a little chemistry, sprinkled a little something with extra electrons. It's got negative charge in the conduction band. P type, P means positive. It means that I have introduced states here, because I needed an electron. I say, I need an extra one. I'm going to grab it from somebody. I don't care who. And so you create a positive charge in the valence band that can conduct electricity, just like a negative charge in the conduction band can. This is what the entire semiconductor revolution has been based on. And really, so much of the focus was on simply taking these elements-- in particular, silicon and germanium. But then, many, many more, as I showed you in the LED graph, many, many more have been used to make, to engineer, these simple-- what are seemingly simple properties, the band gap and the doping. And so I'm not going to go through this in detail. I'm just leaving it here so you have it. But you get charts like this. Here, you have, OK, carbon is up there. Well, if you dope it with boron, you can see that it's P type. You see that? Because I've got one fewer electron than carbon. So I'm going to grab that from the valence band, create a positive charge P type. I could also dope silicon and germanium. I'd say carbon, silicon, germanium. So I'm looking at this middle column, doping it with either side. Why can't I put aluminum in carbon? Why didn't they list that? Well, because if I try to put an aluminum inside the carbon network, the network gets all messed up. Doesn't fit. So that's also important. Can it fit? But it's through this that the semiconductor revolution started. It's only through this, this kind of control and this knowledge of atomic orbitals, really, if you think about it, and the structure of the atom. Now, I'm not going to talk about transistors, but this did lead to the development of the transistor. There's a really nice video there. It's, like, eight, nine minutes. I suggest you wait until your Friday night social. But if I take a P type and I take an N type and I take a P type and I put them together, some really important things happen. So important that it led to this, which is the very first transistor. That's a transistor. It's a dope semiconductor, one type, next to a dope semiconductor, another type. Now, those are the three people-- Bardeen, Shockley, and Brattain-- who won the Nobel Prize for this work. And that's the first transistor. Does anybody know how many transistors we make today? Per second. That's the only time metric we can use. Yeah, it's something around 10 trillion. We make 10 trillion transistors per second today. This is a big deal. This started it all, and it's all about chemistry and the physics of this device. This led to diodes and photodiodes. This led to the whole revolution. And what I want to point out here is that this also led-- it called on chemistry. So I love this chart. Unfortunately, they didn't update it, but this is an Intel chart that they used to show. And what they're showing is how are they making chips. How much of the periodic table do they need to make the current chip? Well, in the '80s, there's almost nothing lit up here. There's a little bit over here that you can't see, stuff we just talked about. But look at the '90s and look at the 2000s. And today, it's 75% of this. It's literally 75% of this is in your phone. Yeah. Why? The reason is exactly what we've talked about today, because they need more and more and more flexibility and ways to tune the band gap and the semiconducting properties of these materials and the doping. Why do they need new ways? Because they're making them so darn small. They're so small, it gets harder to figure out how to do it right, how to do it in a way that's also stable over time, for example. So they need new chemistry. This is a call to action to the field of chemistry, and chemistry responded. And there's the data on the cost. I thought that was kind of cool. So these are orders of magnitude of cost and number of transistors. And I'll just say one more why this matters. Because the semiconductor is also the same thing we use to take electricity from the sun, to generate electricity from the sun. And you can imagine, you are doing that in your goody bag. You're literally using this semiconductor to generate electricity by shining light on it. Why is this important? Well, I like this comparison. All the coal, oil, and gas known to humans is what you get from the sun in about 20 days. This is the sun. Now, we've seen the sun before. This is how we see it in our class. The sun is a spectrum of different wavelengths and intensities. And this is already on planet Earth, because you can see those chemistries that are in the atmosphere have already absorbed. Remember ozone down here, helping us with UV radiation? And out here, you've got a lot of water and other things, CO2. Those are absorbing. That's why it doesn't look smooth. But the point is, I want to semiconductor to absorb as much of this as possible. If I want a good solar cell, I should absorb as much of this as possible. But the problem is that it's a constrained optimization problem. So let's see, if I-- I'll just use this. This goes back to Bohr, by the way. Remember, Bohr, I'm absorbing light. It's just now, I've got an actual solid. I don't have an atom. I've got a solid, which is what you need if you want to generate a current. I've got to hook leads up to it. But my solid is a semiconductor. And you can imagine, say, well, if my gap were really, really tiny, then any amount-- almost any amount of this spectrum would excite electrons. And I might grab most of that spectrum. I might absorb most of it. But the problem with that is that, then, all of these will thermalize, like I said in the beginning. Loss to heat. Loss to heat. So if my band gap is really small, I absorb a lot of the light, but I lose most of it as heat. If my band gap is really big, then there's so much of this light that I cannot absorb. By the way, also, the voltage that you get out of your solar cell is essentially this band gap. So if I get to really, really small band gaps, I have almost no voltage. That's bad, too. But if my band gap is too big, you think, oh, I'll get a high voltage. No. You won't absorb any photons. So it's a constraint problem. It's a constraint problem. You can actually solve this, and you can plot-- in fact, here's a little animation. Energy comes in from the sun. It excites an electron. This is how PowerPoint sees it. Leaves behind a hole, and you get them out. That's a solar cell. And this is the chart I want to show you. Because you see, if I take this constraint into account, that I don't absorb light if the gap is too big, but I do absorb light and it all goes to heat if the gap is too small, it means there's some sweet spot. There's some sweet spot. And that's what is plotted here. This is the band gap of a semiconductor, and this is the maximum that that solar cell efficiency could be. That's the maximum that it could be for that gap. So you can see the sweet spot is right there. That's a thermodynamic derivation called Shockley-Queisser of the maximum efficiency you could ever get out of a single material. And notice, if you put the materials here, silicon is 90% of the solar cells you buy today. And notice that it's not quite at its maximum potential. It's also not quite in the middle, where you'd want. It should have a slightly higher gap, and you'd get to higher efficiency. But these materials out of gallium arsenide, they do have the-- what is GS? That's not an element. Gallium arsenide, it does have a better gap, but it's much more expensive. And so we go. And a lot of solar cell research is on getting as close to this point as you can. See you guys on Friday. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 10_Lewis_Structures_II_Intro_to_SolidState_Chemistry.txt | OK now, exam one. I mentioned this before. I thought this would be helpful to everyone. What I did is put together what I call a concept map. And I will post this, here it is, oh, there is a periodic table, isn't that lovely? This is what I put together for you guys. And I call it exam one concept map. Now, I'm not going to go through it now. But I wanted you guys to have the big picture of what we've done. This really is a celebration, especially when you look at all of the things that we've learned and how they're all connected together. And that's what I wanted you guys to see. OK so, look at this. So here are on the left here are-- this is all not graded stuff, here's what's been graded. Oh, and this is what's graded on Monday, exam one problem topics. Look at all those things. Now over here, we have the three goody bags that you guys have had. And there have been problems in there. And here we have the problems that you've had from the textbooks and the topics there. Here we've had your seven recitations. Here you've had-- it's even color coded. It's even color coded. So here you've got the lectures. So these the nine lectures and all the topics that we've talked about. Today is number 10. And as I said, I will tell you something today about Lewis structures that will not be on the exam, and that is resonant Lewis structures. So we'll teach that. And then those are the three quizzes. Now you notice, there's some things, like here, OK, we've covered this stuff, like Lewis dots, and electronegativity, Lewis structures, stuff like that we've covered that has not been on a quiz. But it could be on the exam. So this is a concept map that I hope gives you a sense of how this all connects together. And how this connects together to give you the knowledge that you need to answer questions about these topics. Now, the exam is not-- these exams walk the balance. We want you to learn how to solve these problems. What I want you to do on Monday is run out of knowledge, not run out of time. Did I get a-- thank you. Now, you might run out of both, but listen-- [LAUGHTER] But these are not meant to break you in a speed test. These are meant to see if you have learned the concepts. So what I ask you to do on this exam is show us what you know. As you may know already, I am a big fan of partial credit. Lots of partial credit. If you show me you know something, you will get credit. And so that gets into, also, exam strategies, but a lot of you already know about this stuff. But please remember, don't just go in order, and then run out of time, and not even see the last two questions. Take a look. Tell me what you know. Show us what you know. We're not about the last decimal on some calculation. I'm much more about do you understand the concept? Do you know how to solve this problem? So show us that, please, on the exam. OK, now was that another loud snap in favor. I think that means it's like positive reinforcement. Topics, OK so, the other thing I want to say about the exam is we're going to connect concepts together sometimes. That's the point. Exams aren't just going to be another three quizzes plus one more question on Lewis. No, we're going to see if you can also weave together the knowledge. So an example would be, if I were to ask you, OK, I'm going to give you a PES. So let's see, here would be like a question about a PES question. So here's a PES diagram. And here I'm going to draw, I'm going to say, OK, I give you this PES. Here we go, let's label it, three, four, five, six. We don't need more than that. OK so, we're going to go, oh, and a break in the axis. Now you know all about that. And we're going to go all the way up to six here. And we'll do another break. We'll go up to two there. And now this is going to go up to here. And you can read off of here. Remember, with the PES, that's a relative electron count. I'm blasting this atom with enough photon energy to get all the electrons out. And then I'm counting their relative-- I measure the kinetic energy, I can get the energy. Oh by the way, this would be like decreasing, the way I plotted it before. And the way we usually plot PES diagrams is the x-axis is going down in energy. Because those are the 1S electrons there. These would be the 2S. And these would be the 2P. And I can keep on going. So I've got 3S. Now what do we got here? We've got 3P, How many? Four. OK, good. OK, a question you might be asked, what atom is that? Sulfur. OK, good. OK that's sulfur. So I can answer that kind of question. But now, let's take another concept and weave it in here. OK, if that's sulfur and I say, well, what if I went up to chlorine. Right so, sulfur has 16. Sulfur has 16. And chlorine is 17. What would this look like for chlorine? I gave you this. Well, you might say, well, I know, it's going to just be like this. But let's go to the right height. And now I filled that with the one more electron. And so it must be chlorine. But that's not taking everything you know into account. Because you know that if I go up to chlorine, I've added a proton. But I haven't really done much more to shield those outer electrons. And so everything wants to be closer to that more positive charge. So you've got a reflect that in the PES. So if I were really going to say, well now, you know. You might not even know quantitatively. But you know qualitatively, if I'm going to draw this peak for chlorine, it better be to the left of that peak. It's got to move in just like the orbitals and all those things we've talked about. That's the kind of thing that I mean. I want you to be able to synthesize concepts as we think about exam questions. Everything is going to shift over, if you go to chlorine. Now, on Wednesday, we talked about Lewis structures. So let's talk about Lewis structures more. Let's suppose I've got acetaldehyde, a wonderful toxic compound, H4O. OK, here is a possible Lewis structure for this. All right, let's see, I'm going to draw the two carbons here. Here's my oxygen. It's got some lone pairs, put another one there, why not. Hydrogen here, and we got one more hydrogen there. What is wrong with this? Double bond on the hydrogen. Question really is, what is not wrong with this? I can count three things, right now, that are wrong with this. What are they? Double bond-- this has to have, remember, these things need to be happy and follow their octet. Octet means for hydrogen, two. But otherwise, so that's good. That's good. That's not good. OK so, we're not-- is this good? No, no. Is that good? Now, hold on, because OK, but there's something else that's fundamentally wrong with this. Anybody notice? How many electrons? This is how you start a Lewis problem. How many electrons do I have? How many electrons do I have? Remember Lewis is all about the valence. So I've got 4 times 2, that's 8. Eight electrons, four electrons here, and six electrons here, how many do I have? So I've got 18 electrons total. Now what did I put into this diagram? So each of these bonds, remember, a line is two electrons. It's a bonding pair. So I've got 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. I didn't even get the electron count right. I didn't say it's acetaldehyde two minus. Wrong number of electrons. And octets not formed. There's something else. There's another way you can see how this Lewis structure is wrong. How is that? What else did we learn on Wednesday? Really important concept. Formal charge. Formal charges are all over the place. Lowest not lowest formal charge. Now, when I say lowest, I mean closest to zero, because that is what gives you the most stable structure, as we talked about on Wednesday. You can see this, formal charges here are all over the place. So the formal charge on this carbon is-- OK so, you go the number of valence electrons of the free atom minus dots minus 6. So 4 minus 1, 2, 3, so that's plus 1. Oh, 6 minus 1, 2, 3, 4, 5, 6, 7. So that would be minus 1. So this is not a good Lewis structure for all of these reasons. The right Lewis structure would look something like this. Very easy fix. And C, and here you go. And now, you can check all of these things that we just talked about. You can check the formal charge. You can check the total charge. I could've taken two electrons off of here. And it still would have been wrong. But this is the right Lewis structure, because we satisfy all the things that we talked about on Wednesday. OK, good, now, as I mentioned, there is one other concept oh-- The last concept that will not be on the exam, oh, but it could be on exam two, I don't know, is the last concept about Lewis structures that's very important. And it's what happens when you have structures that look similar. If I look at all these things, number of electrons, octets forming, formal charges, these look good. These both look good. And so the question is, which one of them is right? And this gets into this concept that I want to teach you, which is called resonance. And just to define resonance, I'll put that on the board. These are called resonance structures. OK now, resonance, when referring to Lewis structures, the way I want you to think about it is that it describes delocalization. I'm going all caps, delocalization, why not? I'm not shouting it. I'm just saying it's an important concept. Describes delocalization of electrons in molecules. So in the case of ozone, which is something we broke apart and talked about, we talked about ozone already, in the case of ozone, now, what's happening? Well if I draw the ozone molecule the way-- if I draw it the way one of those is, so I'll draw it the way the left hand picture is, like this. And O and, I'm not talking about shape. No shape yet. [INAUDIBLE] I don't know. Next week, shape. This week, no shape. OK, hold on. Now, if I look at this molecule, this is a double bond. There's more electrons on this bond than here. And what we know about that is that that's going to make that bond stronger. And those atoms are going to be closer. So this length would actually be around 1.2 angstroms. And this one would be around 1.5ish. But see, and now, you say, well, maybe it's the other one. 1.2 over here 1.5 over there. But now, it take ozone, and I can measure it. I can measure those bond lengths. And I do that. And when I do that, what I find is that it's neither of these. But in experiment, the bond length is 1.3. And it's the same. It's not different. So what is happening? Physically, what is happening is delocalization. And remember, electrons do things to be happy. Happiness means lowering the energy of the whole system. And so what these electrons realize, what these extra two electrons here realize is well, OK, I couldn't make this bond short and that bond long. But maybe if I delocalize across the whole molecule, maybe if I do that, I can be happier. Meaning I can be in a lower energy state. And that is what resonance talks about. That is what it tells us about. It tells us when I have these different states that look equivalent, that the molecule essentially, pictorially, is going back and forth between them so fast that it's an average. That it's a delocalization. Now, the way that we think about this in terms of our Lewis structures is by having one resonant form and another resonant form. OK now, hold on. So, here OK, let's do this, and this, and double bond on this side, good. Chemists are very particular about how you use arrows. And you have to be very careful in chemistry. But here's a case where they really like the arrow. And they really like curvy arrows in particular. And what the curvy arrow shows you is what happened to the electrons. So it's really just a representation of where the electrons went to get from one resonance structure to another, how they changed. But it can be, actually, helpful to understand whether you really have a resonance structure and what's happened. So for example, in this case, these are the two resonance structures for ozone. OK so, what happened? Well, to get from this structure to this structure, you can see, what's happened is that you can imagine this lone pair has gone onto that bond. And this bond, where am I? This lone pair has gone onto this bond. OK, so I've made a double bond to get to there. And I've made the double bond here to get to there. Now what's missing? Well, I've got to make the other lone pair appear here from this bond. So you can look at your resonance structures and realize that that also happened. And you can look over here, and I've got three lone pairs on that one. So the only way to get that would have been, let's try draw these over here to make room, if that happened. Those curvy arrows help us just see how the changes in the electrons happened in these resonance structures. But the way to really think about this, and this is why I wanted to capitalize this word, is that the actual structure is a combination of these. So let's see, I'll do this. And the way you would write the actual structure would be like this. OK, this, and this, and this, and dashed lines, oh. And this is called the resonant hybrid structure, Resonant hybrid. It's a mixture of the structures. And it shows, with the dashed lines, the delocalization. And if you want to be complete about this resonance structure concept, if you want to be complete, then you would also count formal charges. And so you would see that the formal charge-- let me make sure I get this right. So over here, we've got what? 6 minus 1, 2, 3, 4, 5, 6, 7. So the formal charge here was minus 1. The formal charge here, 6 minus 1, 2, 3, 4, 5. So that's plus one in the middle. And the formal charge here is 6 minus 1, 2, 3, 4, 5, 6. So that's zero. And over here, I'm not going to do the counting again, it's the same. But the zeros and the minus ones have switched, as you can see. Now, in the resonant hybrid structure, what's happened-- notice these are the two Lewis structures. So you do have formal charge. Formal charges are not zero everywhere, but that's OK. This is the best you can do. But you've got the two that are equivalent. And they go back and forth. And in this case, the way you would draw that is that you would have the plus is still there, but now you're sort of sharing the formal charge on the ends. So if you want to think about this in terms of formal charge-- notice something about formal charge. I did not add or take away charge, in this case. I got it added up. The addition of all formal charges, by the definition of formal charge, must be equal to the charge on the molecule, must be. So if the molecule is neutral, then the formal charge must add up to zero, which it does, does, and it does. Good way to check your work. OK so, we can go further. So that's the concept of resonance. And I did it, as I like to do, kind of slowly. And now I will give you another example. But I'm not going to go through it as slowly. And but oh, let's ask this question, so if I had-- OK so, here's two other cases. Now, this one we're going to get through quickly. Is this a resonance structure? And I'm going to give you the answer, the answer is no. And the reason is that resonance structures only involve the curvy arrow motion of electrons. The curvy arrows showing us how electrons might change in this, but not of atoms. Notice I cannot get from here to here without moving an atom. That's not a resonance structure. That is not a residence structure. So we're done with that. And we can put, well, let's put a little thing here. Only change location of electrons, not atoms. Let's keep this separate. Not atoms. But now, we get to another one, where we didn't change the location of atoms, but we did change the location of electrons. And you can see the structure up there. And I won't do this, because I just did it for ozone, but you can go through this yourself and show, with curvy arrows, what's happening between these structures. These are three resonance structures. These are three resonance structures And you can see how they go. So you can see, like here, I had that extra lone pair on the oxygen, that might have a curvy arrow to here to give you the double bond there. But as I did that, in order for carbon to keep its formal charge, to keep its formal charge, I had to do something with this double bond. And so this is how you think about resonance structures. And just to give you the final for that one, just show you what it would look like. Once you've got the hybrid structure, it would look something like this. So there, there, and oxygen with its two lone pairs, and oxygen with its two lone pairs, and the delocalized electrons. And then you've got your charge. And this would be minus 2/3, minus 2/3, minus 2/3. Notice on these three oxygen atoms, notice that, again, the formal charge adds up to the charge on the molecule. There it is. Charge on the molecule, that's a 2 minus. This is CO3 2 minus. Formal charge adds up to the charge on the molecule, good. OK, one more example, because now we're going to combine this and learn one other really important concept about Lewis structures. Let's see if this, OK. So here's an example-- do I have this up here? No. Here's an example, what if I had the thiocyanate ion? This is a favorite. So this is CNS minus. Might as well write the name here, thiocyanate. This is a favorite for demonstrating concepts about Lewis. And you'll see why in a minute. Because first I'm going to draw a Lewis structure. This is a minus, so it's an ion. First I'm going to draw a Lewis structure that seems to make sense. But then as you'll see, there are resonant Lewis structures that also seem to make sense. So I need to use some of the concepts we've learned to figure it out. So if I draw this, let's see, I'm going to draw three. So N, triple bond, C, sulfur, lone pairs, lone pair. Or I could have N double bond C, double bond sulfur, lone pair, lone pair. Or try to squeeze this in underneath, so that you can, put it here. N, three lone pairs, single bond C, triple bond sulfur. Now, these are resonance structures. I can get them by drawing my curvy arrows and moving electrons around. And so you might say, well, is this not just going to be an average. Are these electrons delocalized according to their resonance structures. But see, once we start thinking about formal charge, we get more insight. So the formal charge here is minus 2. 5 minus 1, 2, 3, 4, 5, 6, 7 minus 2. Here it's zero. And here it's plus 1. Sulfur 6 minus 1, 2, 3, 4, 5. So that looks pretty bad, a large formal charge minus two. OK over here, oh, I didn't draw the lone pairs, sorry about that. Here's two lone pairs here. We will not mess up the electron count again today. And so here, the formal charge, let's see, this one would be minus one. And in here, it's zero. And over here, it's zero. That looks better. According to my rules on Wednesday, this looks like a more stable structure than that. So even though it's resonant, I'm going to say that's probably not going to be part of the most stable ground state structure. But let's look at this last one. Zero, and over here, I've got zero, and over here, I've got minus one. What do we do? There's one last point that we need to learn. And it has to do with the formal charges. And it's a general rule in writing Lewis structures. And that is that, let's see if I can fit it here, because it's very relevant to here, that atoms with the negative formal charge should be-- I'm not going to fit it here-- should be on the more electronegative ion, electroneg atom, not ion, atom. So let's take a look and see what this means. Let's understand it, first. I just wrote something down. Let's make sure we understand it. So atoms with the negative formal charge should be on more electronegative atoms. That kind of makes sense, right? Because remember what our definition of electronegativity is. Electronegativity is the desire of an atom to bring bonded pairs towards itself. Remember, and it makes a molecule polar, polar covalent, for example, maybe ionic. But anyway, it's that desire to bring bonded pairs towards itself. So if an atom has a negative formal charge, then you think, well, that should be on the more electronegative atom. Because the negative formal charge means it's got a little bit extra charge compared to when it was free. And so if you look at this, oh, now you need to know electronegativities, something that we know. The electronegativity of nitrogen, so chi, remember, of nitrogen is around 3.0. And chi for sulfur is around 2.5. And that tells us that this is the ground state structure. All from the concepts we already know. All from the concepts we already know. Now, at the same time, the atoms that are more electropositive would want to have positive formal charges, in general. So the same rule applies the other way. OK now. So that's the last part of Lewis structures that I want you to know about. And it all ties together. You can see resonance, this is why this is such a great example, you can think about resonance. And could you move electrons around and get different structures? Yes, but then those aren't actually equivalent, because they have very different formal charges. And then two of them had kind of similarish formal charges, but one is more stable because of that. Now why does this matter? And I'm going to tell you an example of why this matters. I'm talking about delocalization to stabilize. And I'm talking about how the actual structure of a molecule isn't this rigid two bonds here, one bond there, that's not what that says, but it was in ozone. But it's this shared equivalent bonds where the electron is shared and the whole system is lowered. Now, there is one molecule where this is very much important and dictates all of its chemical behavior. And that's benzene. And back in the day, in the 1930s, oh, by the way, Pauling, electronegativity scale. we talked about him on Wednesday. But back in the day, they didn't know what benzene was. They could kind of measure some things. But none of it worked. And Pauling wrote a number of structural formulae have been proposed for benzene, but none of them is free from very serious objections. The way they wrote stuff. Look at these shapes for benzene. If we don't know the right shape and why you have the right shape for benzene, then we cannot understand all of the mega amounts of organic chemistry that we do with benzene. So this is what's happening with benzene. You have these two structures that benzene actually averages over. These are resonance structures. Back in the day, they called them Kekulé structures because Kekulé, as you can see, was kind of close and thought about this as well. These are resonance structures. It totally dictates the chemistry of benzene, which then leads to massive amounts of things that we do with benzene. Because with benzene, and this is just, we don't need to look at this in detail, this is an example of a few out of almost unlimited variations of benzene that we can make. Because we can take one of those hydrogen atoms and put something on it, or two places. And each time we do that, we get a different material, a different molecule that gives us different properties and different uses. But all of this, even though we draw it, still with those lines, we know that inside, it's delocalized. It's delocalized. Well, it might change once you add stuff to it. But by itself here, by itself here, it's delocalized. And that sets up the properties of benzene. Now OK, let's go a little farther. So I like these last two here. Naphthalene, now, you can keep on going, and keep adding benzenes, and we're going to get somewhere that I really love. This is getting me excited. Oh, look I'm going to add one there. I'm just going to keep adding benzene. Now I could stay here, oh, don't get benzene wrong. I can keep going. And if I kept going, I might get sheets of benzene that are now called graphene. And if I stack those up, I've got graphite. I've got graphite. That's what graphite is. It's these very large sheets of benzene. Now, there are two very well-known phases of pure carbon. One is diamond. And the other is graphite. Does anybody know which one is more stable? How many people think graphite? How many people think diamonds are forever? I'm giving you guys some advice, especially like, you're out, and maybe you're out on a date. And it's the weekend. And there's a candle. And you've talked about combustion. And you've written that down. And you talked about how long you have in that room, because you know about oxygen and the limiting reagent that it's either you or the candle. We talked about that. We also talked about how you need your periodic table on that date. And how you may need your spectroscope, because they might give you an LED candle instead. You need to talk about LEDs. And I'm not saying that all of these conversations involving knowledge you gained in this class will lead to this place of seriousness, but it could lead to engagement. It could. And now, some people when they get engaged, one person buys the other a ring. And sometimes that involves this material. My point to you right now is when you go to the store, maybe it's Tiffany's, maybe it's somewhere else, and you buy that ring, ask for a warranty. And make sure it's about 100,000 years, because after that time, that ring is going to turn into graphite, because graphite is the lowest most energetically stable form of carbon. So I'm just, again, always trying to help. But this is the most stable form of carbon. And it's a bunch of benzene, but repeated. Not benzene, the hydrogens are gone. It's just pure graphene, but repeated in these beautiful rings. And when we come back to hybridisation and molecular orbitals, which we'll start after exam one, you'll see other beautiful properties of structures like this. By the way, speaking of seeing, you can see this material. What are you throwing at this material to see it like that? Not light, not photons, you're throwing electrons. I just came across this the other day in Wired, New microscope shows the quantum world in crazy detail. They took electrons and they shine it on these little particles of platinum and iron. And they're able to literally see every single atom as they break apart the particle. By just throwing electrons at it, they say, the transmission electron microscope was designed to break records. Using its beam of electrons, scientists have glimpsed many types of viruses, et cetera, et cetera. They got down to 0.4 angstrom resolution in that work. It's a beautiful thing. And being able to see these materials has revolutionized what we can do with them. Diamonds and graphite makes me, when I talk about graphite, I get excited. I get excited. Now I'm working on my arm here. So I got to go up that way. And I got to go this way. All right, hold on, hold on, let's go that way. Now, wait a second. I'm very excited. Oh wait, there was a lot of noise. I tried. I got to work on my arm. It's such a beautiful material. Too excited. We'll have more. Now I don't mean to bring us down after why this matters and excitement. But here's the thing, if there's one thing you know about chemistry by now, it's that chemistry lives in the fast lane. And what I mean is, they know about the rules. And they also know they're meant to be broken. And Lewis and octets are no exception. And so there are exceptions to Lewis's octet rule. And I'll give you a couple of examples. And it has to do with the concepts that we've been talking about, but in particular, like boron. Boron trifluoride, well, that one, you could draw it. If you wanted to, you could draw it like this. And let's see, here, F, and you can do this, F. But the problem with this is then, when you look at the formal charges, you've got a minus 1 there and a minus 1 there. And so actually, what this structure does, is boron is like, you know what, I'm flexible. I don't need my eight. I can be OK with six. As long as you keep sharing, fluorine atoms, as long as you keep sharing those electrons so we can make these three covalent bonds all equivalent, I'm OK. And so this is actually-- and you can see, the formal charges on all the atoms here is zero. This is actually the stable structure, even though boron does not, this does not satisfy the octet rule. Chemistry, breaking the rules. So boron is said to be happy. Boron can be, what is called in this world, electron deficient. And by the same token, you can go the other way. And sulfur likes to go the other way, because sometimes sulfur, so there we have sulfur tetrafluoride. So I'm using the same external atom, but sulfur also prefers to look like this and to have one extra lone pair there. And now we can go like this, and like that, and like that. And so what sulfur will do, oh boy, lots of dots, and so what sulfur will do is it will form what's called an expanded octet, expanded octet. And this will become also, a little bit-- we'll sort of understand this also a little bit more once we look at shapes. Shapes are going to play an important role in molecular stability. But for now, I just wanted you to know that there are exceptions to the rules. And in particular, as you go beyond the first couple of rows, so you go down in the period table, you do find atoms more willing, like sulfur, to have an expanded octet in order to reach the lowest energy state. Now, in the last two minutes, I come back to this. Let's just bring it all back home around Lewis, and around electronegativity, and around all these covalent bonding concepts. And what I want to leave you with is what was kind of the point, part of the point of your goody bag. Which is, I wanted you to get your hands on these materials so you could touch and feel the differences and see that actually it's the properties at the end that are also very important for determining what a material is. So you can make rules like electronegativity difference less than 1.6 is going to be polar covalent or non-polar covalent, or greater than two, it's probably ionic. But at the end of the day, you've got to come back to properties. I'll put up a slide showing this, but ionic solids are hard, brittle, and solid at room temperature. Covalent solids can be gases, liquids, or solids. Covalent solids have a low melting point. Ionic solids are typically high. Covalent solids are poor conductors. Ionic ones are usually insulators as a solid. This is what you need to also do. You can't just go by one metric. You got to look at the properties. It's what Mendeleev did. And it's what chemists have always done to make progress. OK so, on that note, have a great weekend. And see you at the exam. [CROWD NOISE] |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 23_Point_and_Line_Defects_I_Intro_to_SolidState_Chemistry.txt | Happy Monday everyone! How you all doing? You know it's good because it's a goodie bag day and so I'll get to what you have in your hands in this lecture. Today-- So we've been talking about these things for a while now, right? We've built up an understanding of the stamp and the basis, and then we understood how to talk about it like in terms of the planes and the symmetries, and then we shined x-rays on it, and now we're going to start taking them all apart. And so don't be sad, don't be sad, because actually that's more real-- that's more real. And that's what we'll talk about today. So what our plan is-- you know-- So today we're going to start creating defects in the perfect order that we've had so far. And in your goody bag you have a defect generation machine, which we'll talk about. And then on-- And so today we're gonna talk about one kind of defect, that's a point defect, and then on Wednesday we'll talk about line defects, and then we'll just mess the whole thing up, and we're going to make it amorphous. So that's where we're going. And beyond that if you want to know more about where we're going, there's the concept map up there for exam 3. Which again, you know we don't pick these dates, so this date came later in the semester that we wanted but it's that's what it is. I think it's all the way up in early December. But-- Because of the fact that there's a Thanksgiving holiday in between and there's one topic here we have to start before the exam. It's not here because it won't be on the exam but we're going to start polymers but won't be on the exam. I'm going to spend the whole lecture before exam three just reviewing for exam three, because we're gonna have that Thanksgiving break and all that. So that's what's not listed here these are just the topics that are covered on exam three. Okay. Good. If anyone has any questions please do let me know. The concept map. Now the thesis of 3.091 you have seen in multiple different ways. I have tried to convey to you that the electronic structure of atoms is the key to life. It's the key to chemistry. It's the key to understanding and within that you get things like this which is what we've been talking about recently like composition and arrangement. Right? That's these crystals that's these different-- BCC FCC. Okay. And then the chemistry you put inside. But you see the other thing about it is defects. Because the thing is that defects, which is the topic of today and Wednesday, they are absolutely crucial for understanding the properties. If they're there and I just told you they're always there. The question is how much are they there? And so if you don't know about defects-- if you don't know about defects then you really cannot fully understand properties. There is a very strong correlation between the two and that's why we have to talk about them. We have to understand them. Now I love this quote from Colin Humphreys he said, "Crystals are like people, it is the defects in them which tend to make them interesting." Yeah. And that's actually really true because defects, right, defects sound you know... 'I don't want to defect.' No, actually oftentimes you do want defects. Sometimes you don't. So there are some kinds that are kind of like maybe not things you want in your crystal and then there are other kinds that actually you are engineering purposefully to be there. Either way you got to understand them. Right? And that's what we're doing today because --Gesundheit-- so you know if you take this three layer-- We talk about graphene. It's a one atom thick material. Pretty cool-- every atom's on the surface. Here's another example of what's called a 2D material. Why? It's not 2D but it's called a 2D material because it gets you like... you know more publicity on your work. This is molybdenum-- this is molybdenum disulfide it's really cool material. It's three atoms thick. So it's kind of like 2D-ish, right? Yeah but see that's the model. That's the model. This is reality. This is real-- How do you see-- How do we see atoms like this? That's a real picture. Do we use x-rays? No? What do we use to see atoms that well? Electrons. Electrons are our light, right? That's what we're using. This is an electron picture, electron. And if you do that you can see individual atoms and look at that there's there's places where atoms are missing all over the place. That's reality. That's reality. And those where you have one atom missing or where you have one thing that is localized and disruptive to the regularity of the lattice... that's called a point defect. So let's write that down. That's the one that we're talking about today. So the point defect, 'point defect', is where you have a localized... localized disruption... in the regularity, 'regularity'. The periodicity the repeating of the lattice. And it could be on or between, we'll see this, on or between sites. Right? It could be a defect that is, like you see here, there's an actual atom missing. What could be something that maybe you've got in there in between sites? But either way it's a disruption that's localized to a point or almost a point. And so it is --Gesundheit-- that you can see that if we just make these disruptions in something. Alumina. We talk about alumina, right? Remember alumina's really strong lattice energies? So it's like sandpaper. Oh, it's also toothpaste. It's in a lot of stuff. Alumina is a great-- But look at this. That's alumina with almost no defects. It's still got some. You can never get rid of them all! But here it is when I purposely engineer the defects in alumina and I put a little titanium or iron. Or here's it when I put chromium. So you can-- There's one property where the localized disruption, and it's not a lot as we'll see, the localized disruption changes the properties. That is key. That's just color. Right? This applies to most properties of materials. And so this first one, the point defects that we're talking about today, you can kind of think of those as zero dimensional because they're they're localized. You know. They don't really go off in a line. They don't go off in a plane or a volume. And you can have defects that are that cover all of those possibilities. But these are points. Alright? This is what the word localized means here, okay? Okay. Now in this class, we're not going to cover all of them, but we will cover these two. Today: this one. On Wednesday: that one. And that'll give you enough of a sense of the role of defects and how to think about defects and crystals. Right? So this is how we think and classify defects. Okay. Now the four-- so now we're going to point defects-- now the four, there are different types of point defects that can happen. The one that you saw here is a certain one. That's called a vacancy. It's because there's an atom that's vacant. So you literally just lost an atom somewhere in there. Okay? Now that's a vacancy and that always exists. But you could also have taken something like one of the atoms that's already in there, maybe an aluminum or an oxygen that's in the lattice, and you could substitute something in place of that. That's another-- Why is that a point defect? Because I didn't like tear something out. But no, but you changed the regularity. Localized disruption and regularity. So if I put something else in for aluminum that's a point defect. Right? And again, now go back to here. You can see it here. So we classify these, right? There's a vacancy, which is an atom missing. There's what's called an interstitial, which is an atom that goes in between the other atoms. Here it's a self interstitial so it's the same type of atom. And these are-- This is where you have a different type of atom in between and this is where you have a different type of atom as well. Those are both called impurities and I will talk about those all today. We're gonna start and focus a lot of our attention on the vacancy. And that's what I want you to use this goodie bag to understand, is the vacancies. Okay. Vacancies. Why do vacancies happen? What are vacancies? To understand vacancies in crystals we have to talk about this guy. And oh... this guy was brilliant. This-- I'm sure you know his name is Svante. Yeah, it's written right there. Svante Arrhenius. Arrhenius studied in the late 1800s so many different things and made so many contributions that it's almost hard to catalog. He was brilliant. And you know he won the nobel prize in chemistry for his work on electrolytes. But he also worked on immunology. He was the very first person in the late 1800s to come up with a model for global warming and the role of CO2 in the temperature change of the planet. And his predictions were actually pretty darn good. He was brilliant. And one of the things that he did is, he observed what it were called activated processes, and how they depended on temperature. And so we have this equation that I need to talk about today and it's going to come back. So we're going to talk about this equation, which is the Arrhenius equation, that relates the rate of some process to the temperature. And the activation energy for that process. So we got to talk about what all that means, alright? Now we will be using this. Today we're going to use it to think about concentrations of vacancies in a crystal which follow Arrhenius-like behavior. Right? But then we're going to come back to it when we go into reaction kinetics in a couple of weeks. Where we talk about reaction rates. So we'll be using Arrhenius multiple times throughout the rest of the semester. So what is it? So the Arrhenius-- So the general Arrhenius equation can be written like this. Arrhenius, he was not in the army but this is just the general equation. Okay. So let's put equation there. And so we have k, which is some rate, equals A. I'll talk about each of these. Times e the exponential of e to the minus Ea over RT. What are these things? So this is the general equation for the rate of some process. As we're going to see this applies to many, many processes including vacancies. Okay. But this is sort of the rate. So you can think of this as-- Okay. Let's just-- That's the rate of a process rate. The rate of some process. Okay. I mean you could think about it as the number of times something happens per second. That's a rate. Right? It doesn't have to be those units, but that's like a rate. Right? How often does this happen. The rate-- But now so that's dependent on something called the pre-exponential, that I will talk about. 'Pre-exponential-- exponential'. That is a factor that is a constant. And here we have this exponent-- and this is such a beautiful expression. So this is the average thermal energy. 'average kinetic energy, average thermal energy' Right? It's taking temperature and it's making an energy out of it. Now remember we've already talked about this before. That you know you don't get what you get are distributions. So you know this would be T high-- remember I've drawn this exact thing and this might be T low and this might be-- You know, this might be the you know the probability of something happening. Probability of occurring. And then in this case we talked about this as the kinetic energy of the molecules. That's a graph we showed already. But so the-- But this RT is an average. But so I just want to make sure you don't forget. This is a number. This is a number. You've got at 300 degrees, at a thousand degrees, you calculate a number here. But whenever you're talking about temperature, you're talking about distributions. There's some average of the distribution but you're always talking about distributions. That's important for understanding Arrhenius. Okay. Now this thing here is called the activation energy. Activation energy. And that is the energy that you have to get over for something to happen. For the thing to happen. Let's take-- I love this analogy of the bookcase. I'm going to draw that now. So let's suppose that you have energy, like that. Okay. So maybe this is like potential energy. Yeah, let's go ahead and say potential energy. Then if I've got a bookcase and it's very very heavy. And I try-- and I want to push it over on this hinge. So I want to rotate it and push over. Then you could imagine that at some point it's going to look like this. And then at another point, it's going to look like this. Right? So this is like step one, and that's step two, and that's step three. And now I'm pushing this thing over. Well this is the activated state. Why? Because it's where I've gotten to the highest energy, and that's where this energy graph is important. This energy axis, this is the activated state of the bookcase. Right? So if I plot the the energies. Right? You can say one, two, three. You can think about it like that. Right? I've literally-- So because it's a potential energy, right? So it's like here's the gravitational pull, here it went up a little bit. Right? So you can think about it just like that. And here it came back down. That's why I like this analogy because it really gives you intuitive feeling for what's happening here. There's a process. The process is pushing the bookcase over. That bookcase has some energy associated with it. Here and here. Potential energy. And it's got some energy I gotta put into it to activate it to go from here to here. Right? That's why that's called the activation energy. That's what this Ea means for the general Arrhenius equation. You're pushing the bookcase over. Yeah but how are you going to push it over? Well clearly you're going to run around the room and accidentally knock into it because that's temperature. Maybe that's not a good-- But if you had a lot of people running around a bookcase maybe once in a while they'd kind of bang into it. Thermal energy. Right? And then maybe another time, you increase how fast everyone's running, and you give them more energy. So now not only are they hitting it more frequently but they're they're actually hitting it with more energy. They're able to give it more energy. Now you make everyone run really fast. And the chances that this thing goes through that process are higher. That's what that is. That's a probability, right? This is what this exponential is. That was the brilliance of what he did right. So this thermal energy is like, you know, it's how much-- It's like a probability but that's why it goes into an exponential. Because when you increase temperature, again it's not just that you're increasing how much energy over here, Right? How this is get like kinetic energy of some molecules. Maybe how much energy those molecules have but you're increasing how many of them are above some threshold. So your chances go exponentially higher. Well you think if it was harder and harder to push. But this is going to go the other way too. You could have sometimes somebody might knock it the other way. It looks kind of hard from this picture. But there's a chance. It's just that the activation energy going this way, right? Going this way, the activation energy is here. 'Ea' and going this way the activation energy would be here. So this would be 'Ea' for, let's see, three to one. And this would be 'Ea' for one to three. Now again, I am giving you the general Arrhenius. We will be coming back to this when we do reaction rates. We will be coming back to this picture, and we will be talking about reactions, and then we'll be using that, and going to equilibrium. Today, I want to give you a sense for what Arrhenius is because this is where the equation comes from. And because if you think about now defect formation. Defect formation is an activated-- it's a thermally activated process. But that's exactly the point. So that's why if we write this down-- I've said this now multiple times the vacancy... 'vacany'. No. Vacancy is always present. Always present! Why? Because it's thermally activated. So unless I can get to T equals zero-- thermally activated. There's always a chance that I push a vacancy into the material. There's also always a chance that the vacancy gets pushed out. So those are happening because of temperature. You can think about that. It kind of makes sense, right? Atoms are moving. Those are the people-- right-- there's the people in the room. And now all of a sudden, something happens somewhere that allows an atom to come out. We'll talk about that in a second. And create a vacancy. Now there's a chance that it can go back too, right? And so there's at some point there's an equilibrium concentration. Both of those are thermally activated processes that have an Arrhenius like behavior. And so you get to... the very nice expression for concentration. So you can get-- can get? Yeah, why not-- get concentration. So the concentration at equilibrium. Since vacancy formation is thermally activated and it's thermally deactivated. There's some-- you know, you can say well when it's activated and deactivated in the same rates you're in equilibrium. That's how you get a concentration. And I don't need you to know the math but I need-- This is where it comes from. The number of vacancies divided by the total number of sites is equal to e to the minus E vacancy-- I will talk about this-- vacancy divided by, let's use RT still. RT. Now here's the thing about RT. Let's write this down because this is really important, right? If-- we talked about this before-- if I'm using RT then it's per mole. Per mole use RT. This is the ideal gas constant, right? That's not just used for ideal gases. This is used for a lot of things. That's the gas constant. Gas constant. That's equal to-- let's see if i have down here-- 8.314 joules per mole Kelvin. So you see if I'm working in per mole, then I use R. This is-- the energy unit the-- those energy units have to be the same. They got to cancel. So if my activation energy is in per mole, fine, use the gas constant. If the activation energy is in per atom then we just use the Boltzmann constant: kBT. Where the Boltzmann constant is something we have seen, it is equal to the ideal gas constant divided by Avogadro's number. Remember that Avogadro's number goes in and out atomic macroscopic worlds so R and kB same thing. Okay. One thing-- notice that's per Kelvin. Whenever you have equations like this that come from thermodynamics, which is where these things come from. There is only one temperature unit. There is no other. And it is Kelvin. You have to be aware of that. If you see something in Celsius, it's not going to work. It's got to be Kelvin. All of these equations have to use Kelvin. Okay. So that now-- you can see at any time, you know, any time you see an equation with an exponential. What's the first thing you want to do? I mean it's like an-- it's almost like an instinctual reaction. You see an exponential. Take a log. Don't say anything until you take a log. Right? So that's an-- so that-- so we're talking about energy of the vacancy in a minute. But if you take a log of this. Then you get that the log of what '|n(Nv)'. Well I'll talk about that. Equals log of N. Let's do it this way. Minus log of n equals E vacancy. Okay. Divided by RT. That's just taking an exponential, a logarithm of this equation. And now I can talk about these Nv and N. So Nv is-- this is literally a concentration. This is the concentration of vacancies that are forming in my crystal. Why did i get this expression and notice the constant cancelled. This pre-exponential factor cancelled. It canceled because I'm taking a concentration. I'm looking at the rate going one way and the rate going the other. And I get to take a ratio of those. That leads me to this expression. That's how you get from Arrhenius rates to some concentration. Okay. But you still-- but notice it's still-- we call this Arrhenius-like because it's still an exponential dependence. It's not a rate, it's a concentration. It's okay. It's a concentration in equilibrium. So this would be like the number of vacancies, number of vacancies. And this one here 'n' would be like the number of sites, number of lattice sites. So it's like you know concentration. Lattice sites. Right? And what is this? This is exactly here, here we go. I have my crystal. Okay. Here's my crystal and here it is... I think I'll stop here. And now I did this. We're going to go graphical and this and this. That energy is literally the energy difference. It's literally the energy difference between having a vacancy and not having a vacancy. How much energy-- Question? [STUDENT:] Isn't there supposed to be a negative sign in front of the E vacancy? [PROFESSOR:] Yes there is. Oh yes there is. Yes there is. Thank you very much. Yes. So that's-- Okay. So now the energy difference between having a vacancy and not, is the vacancy formation energy. E vacancy. Now sometimes, and this is unfortunate, but sometimes you will see the vacancy formation energy written as the activation energy. That's fine. I mean it just it's written that way to get across the point that it's an Arrhenius-like behavior. But actually the act-- but if you're clear about it. And that's why I want to go through the bookcase example, right? The formation energy which is this energy of the vacancy forming is the difference between here and here. The activation energy is this hill that you got to get over to go back and forth. Okay? So the formation energy between having a vacancy and not is what goes into our our equation. Alright. Now we're going to use this. So we will see how this works in just a few minutes. But again there are so many processes that are thermally activated. There are so many processes that have Arrhenius-like behavior. That are Arrhenius-like. And if you go to Dartmouth then they'll give you goodie bags with live crickets. And actually I really hope not. But this is one of the labs that they have where they take crickets and they measure the number of times a cricket chirps. And they're like, well okay. Let's measure the cricket chirp over 13 seconds. We're gonna cool them down, hopefully not too cold, and then we're going to heat them up, hopefully not too hot. Because crickets are nice, right? And so then-- and they ca-- but look at that. And they count it. And then what do they do? Well they didn't know about Arrhenius yet until somebody from MIT went and visited. So the first thing they did is they plotted the data. Look at that. Chirps per 13 seconds plotted. And they're all sitting there trying to fit a straight line to it. And then someone from this class is up there visiting. They're like, you know what i think, this looks like a thermally activated process. So i think it's probably exponential. And then they fit this nice exponential and it fits the the cricket tripping beautifully. And you can go even further because you see if you got this far. Well now you see this is a line. This is a line and we're going to do that a lot when we go into reaction kinetics. If you have a exponential and you take a log, that's a line versus 1 over T. Right? That's a line versus 1 over T. And so that's another way you could look at data. They didn't do it there. But, you know, you could plot for example-- you could plot 1 over T versus the log of the number of vacancies. But the lattice-- the number of vacancies is what we want. That ratio is the concentration. That concentration is in equilibrium at some temperature. Okay. The lattice-- the number of lattice sites is simply how many lattice sites you have, in whatever volume you have, for whatever crystal structure you have, for whatever element you have. We'll see that in a few examples. So that's just a concert-- it's the number of sites you have in the chunk of material. And then instead of-- The question this equation tells you the answer to, is how many of those have a vacancy? Because it's a thermally activated process. And if you plot that log in Nv versus temperature you get this really nice linear line. And the slope of that line is equal to minus E vacancy divided by R or it could be kB. R. Let's write this again per mole. Or it could be kB if it's per atom. You will see both. You will see both. And this intercept-- intercept-- is equal to the-- let's see-- the intercept is equal-- what do i have here? The log of n. Did i write it right? Log of n. Okay. Alright. Now, okay. Oh yeah. What else can you do? Well before we go on to the defects, this explains the doping. I kept calling the doping in semiconductors a thermally activated process. But look at what happens. This is the carrier concentration in that conduction band. The thing you've been you've been learning about, right? And thinking about. But look at it now versus temperature. It's a straight line. It's a straight line. This is-- this is experimentally what you observe. And the reason is because it's a thermally activated process. And in fact, in this case, what is the activation energy? Right? The activation energy for getting an electron into the conduction band is the gap. Right? And so now you say germanium has a smaller gap than silicon, which has a smaller gap than gallium arsenide. The slopes are different. The slopes are different because the energy that it takes in that activated process is the gap. That's why the slopes are different. Right? Okay. Alright. Now on to-- oh no i didn't-- I did want to mention this because it's so cool. Where are these vacancies going? Did you actually just take an atom from the middle of a crystal and remove it? No. Because that would cost way too much energy, right? And so instead they have to-- you call out to the surface. It's a call out to the surface. Or, you know, maybe the surface calls in. It all has to happen on the surface. So what ends up happening is a surface atom may go away and then another one may take its place, right? And then the next one, and then the next one, and that's literally how you can rip an atom out from somewhere inside. But by the same-- that's literally pushing the bookcase over. Right? And now you can push it the other way and the surface atom could go in to the crystal so that some atom inside is able to fill a vacancy. It all comes to the surface. This is a beautiful paper, where they're showing how you get these rings. This was published almost 20 years ago now. But how-- they're studying how do vacancies actually pull atoms from the surface specifically. And what they did is they change the temperature and they see islands growing and shrinking. Where are those islands going and where are they coming from? Vacancies. It's all about the vacancies, right? And one of the things I loved about this, is this is the abstract of this paper. Look at this. Here we show the vacancy generation and annihilation, right? Both ways on the 1-1-0 surface of an ordered nickel aluminum inter-metallic alloy. Oh, I love reading that because you guys all are experts in this now. You know what that means. You know what that means. Where do vacancies come from? Okay. And now we got to make vacancies. Right? And so this is the kind of problem you might get. How many vacancies are in a centimeter cubed of copper? How many vacancies are in a centimeter cubed of copper? Well, now you know how to do it. You're just going to apply this Arrhenius-like behavior. You're going to apply this equation to figure that out. And i won't go through all the math but i do want to just give you a sense of the types of questions that you can now answer; that you know how to think about vacancy formation. So for example, in this one you've got step one. Step one: you would find N. Step one: you'd find N. How many sites do I have? Because remember this equation is about a concentration between the number of vacancies and the number of available sites for vacancies. And so step one-- well-- so step one you'd say N equals Avogadro's number times 8.4 grams per mole. Right? Divided by all this-- is such old-school stuff right now, right? Grams per mole. Then you look that up in the periodic table and it's 8 times 10 to the 22nd sites per centimeter cubed. Why am I using centimeter cubed? Well because that's what i was given the density in. So I'm just leaving it in those units for now. Right? And then step two you can find Nv. So now we can apply our our equation. Nv equals N, I won't repeat it, times e. Now I've got my kB instead of R. So it's 9.9 electron volts divided by and then this is kBT. Now I get-- so I'm given that it's a thousand-- wait a second. What's the term, oh a thousand degrees. So do I put T equals a thousand. No! No! You never use-- you only use Kelvin. 1273, right? T is always in Kelvin for any of these thermodynamic equations. And I won't go through the math but it goes something like 2.2 times 10 to the 19th vacancies. I'll write the units here just for completeness. Vacancies per centimeter cubed. Now that seems like a lot but it's actually not. You say it's like one in a thousand or one in ten thousand atoms are missing and I'm at a really high temperature. If I were at-- then this is the power of what Arrhenius gave us. If-- Because it's exponential. It's all about probabilities and sampling and how many times did i bump into that bookcase. If I'm down at room temperature I've got so much less energy, and so many more chances-- so many fewer chances to deliver that energy. Right? Where the surface is taking atoms in and out to create vacancies. That at room temperature the number of vacancies is 10 to the seventh instead of 10 to the 19th. Right? So that's the power of that exponential. That's the power of the exponential. You might also be asked questions like this one. Right? So --oh no like-- let me do this one. Then I'll talk about your goodie bag. Where instead of now-- Okay. What-- Example two. What is the vacancy formation energy in aluminum? And now notice, instead of giving you the vacancy formation energy per atom, I'm giving you other stuff. I'm actually giving you how many vacancies you got, at some temperature. Well that's-- This just going in a different direction but it's using the same math. So you can go in different ways, arrive at the same kinds of equations, and get all of these things, and I won't go through the details. But in this case you get that Ea equals minus kBT. I'm using kB because look at that... the formation energy I'm asking is what is the vacancy formation energy-- Okay, well actually look at that. No, I just want to do it per atom. It didn't say per atom but if I wanted to do a per atom, I'd use kBT times log Nv over N. And so you see 0.75 eV per atom. Now what's interesting, copper was-- copper was 0.9, aluminum .75. You're going to have a lot more vacancies because it's exponential. This seems like-- 0.9, .75 is kind of the same. No. It's going into an exponential, so it has a really big difference on how many vacancies you have at any given temperature. Aluminum is going to have a lot more because it's easier to get them in there. Right? Notice I did the thing that I said other people do and I don't really like. That's the energy of the vacancy. The vacancy formation. People call it activation energy all over the place. Okay. Now this is your goodie bag. Now why am I giving to you? Because first of all, this is the most sophisticated vacancy generation machine you'll ever find. There are exactly 500 beads in this. Exactly. There are not 501. There are not 499. And you know why? Because Laura has spent 7,000 hours counting beads in every single one of these. So you have exactly the right number. Then there's precision tape and polycarbonate films maybe. And what you guys can do is build your own crystal. It's a 2d crystal. Here's what's so cool about this. Try to get those vacancies out. Look at this! Dude-- None of you can see what I'm talking about. [LAUGHTER] There's a big vacancy there, I've literally made this right in front of me. This is here. And now I'm gonna say-- okay, I'll use temperature. There's so many more. There's so many more because I use temperature. This is temperature. You are temperature. And you can run around, and you can shake it, and all you bring it to the dance floor, and on. Then you're like how can I get-- Whoa, wait a second. I'll just quietly go down in temperature and I'll get rid of them. Yeah, tell that to your friends down the street. See how long they go until they stop trying because you guys know not to try because you'll never get rid of them. You'll never get rid of them. You'll always have vacancies and this is proof that you will always have vacancies. You can try to tap it like that. See if you can get rid of the defects. You can't. Luckily that's a good thing because defects are what make everything interesting. Everything. This is your goodie bag. So you can touch and feel-- Oh, but we got to cover the other case. Because-- See I could have had not just a metal, where every atom is the same, but i could have had an ionic solid. You can make point defects in ionic solids as well. Right? So yeah. So if I had a defect in an ionic solid-- well now you got to consider the charge. I can't just keep pulling-- If this is sodium chloride, I can't just keep pulling out like one type of atom. If I take a sodium atom out. Like imagine that's a plus, minus, plus, minus. Right? The size-- if I take a sodium atom out then it becomes charged. Because remember the ionic bond is one of them, grabbed it and it has it. And so if I take one out, I've charged the crystal. You can't do that. It won't let you. So if I take a sodium atom out, a chlorine is going to come too. It's got to come too. You got to keep it charged neutral, if it starts charging neutral. So these have special names because of that. And there are two types of defects in ionic point defects. We're still with point defects. Local disruption in the regularity. One is called the shocky defect, right? So in the shocky defect you got both an anion and a cation are removed. Cation. Now here's the thing- I should have put 's' because you may need to take more than one out. Alright? Removed-- and because you got to ensure-- ensure charge neutrality. Charge neutrality. So like you know-- If I had so inquiry that's fine. But what if I had you know something else. What if I had instead of sodium chloride. What if I had like calcium chloride. So now what if I had calcium chloride. Well that looks like that. Because calcium goes to two plus. We know this. Right? But chlorine is only minus one so that ionic crystal is different. And what it means is that if I remove-- if I remove a calcium it's going to come out as Ca2 plus but that means i need to remove two Cl minus atoms. That's the thing about shocking defects. You got to keep the charge balanced. So now you-- It's not just a one for one. This could actually be a way to take atoms out. Or if I want to take chlorine out of this, we'll take some calcium out. Chlorine will come out too. Right? Could be a way to engineer it. So that's a that's a shocking defect. But the simpler thing that could have happened is just that one of these atoms kind of moved over. So now I didn't remove it from the crystal but it just kind of wandered over into some site. So like you know... here's BCC. I can have like an atom-- so this doesn't look like an ionic solid but suppose-- it doesn't matter. I could have an atom leave a site and go-- Notice that there are these voids, right? They're these spaces in between other atoms. Well those spaces are places where atoms can go. They're not-- It's not part of the regular lattice but it's just-- it's like a hole, right? Remember the packing fractions. We never got above about three quarters. We never got about three-- that means there's still a bunch of volume in there that's free volume. Right? Where is it? Well this kind of defect will find it. Right? So you could have the smaller ion migrate over, create a vacancy, and it's going to find a place where it likes to go. Ah. Okay. Yeah, we'll do this. And so-- so that's called a Frenkel defect. So Frenkel defect is the other kind of special name we give to a vacancy in an ionic solid, and it's where one ion-- right-- anion or cation. Anion or cation. Moves to some open space. Open space in the lattice. And you can already kind of feel where this might be kind of common. It might be common in situations where you've got a big size difference, right? So like the silver halides are a good example. Like silver chloride, silver bromide, silver iodide. Those are all good examples because what happens there is this ionic structure has a very big mismatch between the atom sizes. And so you can see. Let's blow up a picture of this is silver iodide, a cartoon. Look at that. So those are the silver atoms and what you see is because there's such a big difference in size, you're going to have these pretty big voids. And one of the atoms is small enough that it actually doesn't mind-- it's actually pretty easy to move it. Ah now, if you think about it. Well if you have enough vacancies in there, then maybe those ions can actually move really freely. Because they're so small and the voids are all kind of nearby. And you actually can get a conductor, a good conductor, this way. You're creating vacancies on purpose so that one of the ions can move around. The smaller one, right? That's what we do. We make these solid-state ionic conductors. You need the vacancies. You need the vacancies. Okay. Now, I told you that we got this map here. We've talked about literally just one. Luckily I don't have nearly the same amount of material to talk about these others. The vacancy is clearly the one that I'm very interested in. The substi-- the self-interstitial is actually pretty easy to understand. The self-interstitial is actually pretty easy to understand because it simply doesn't really happen much. And the reason is, if you look at this. Here's a picture of is basically what i drew on the board, right? But now I've put-- so these are-- this is not a vacancy, right? This is a kind of point defect but it's not a vacancy. I've actually added an atom into the lattice and I've squeezed it in between all the others. You can just feel how much energy, how much strain, those other atoms are gonna have to make and how much energy that's going to cost. Those are high energy defect. So a self-interstitial-- a self-interstitial-- self just means it's the same atom as the rest of them, is very infrequent. Is very infrequent. The self-interstitial-- let's see-- I'll just use this one again. The self-interstitial has an energy of formation of something like you know five electron volts. And you think but-- five electron volts. Self-interstitial. Self-interstitial. It went in between, so it's not substitutional it's interstitial and itself because it's the same atom type. The energy energy of formation is 5ev. I think 5ev doesn't even sound that much higher than like the one ev you had for copper, right? 0.9 ev. But it goes into the exponential, right? So it's like one per centimeter cubed. Literally. You get like one of these happening per centimeter cubed instead of 10 to the 20th, or 10 to the 10th, or you know many many orders of magnitude higher. So the self-interstitial is actually not very interesting. It's actually not very interesting. Now because we have like half a minute left, I'm not gonna talk about these. We'll have a very brief discussion about these last two types of point defects and then we're gonna make line defects on Wednesday. And my why this matters, I'll give you on Wednesday for this. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 6_Electron_Shell_Model_Quantum_Numbers_and_PES_Intro_to_SolidState_Chemistry.txt | Last week we talked about-- we went quantum. We had Dr. Quantum show us the way. And this week, we're going to go quantum for electrons, right? That's what we're going to do this week. These are my two favorite quantum mechanicists of all time. I believe that this sign is meant to teach us something about the double slit experiment. And just as you're leaving the lot, knowing that there is some chance that your one car could actually split into two, go through both exits, and recombine. Just like an electron does through two slits. And in fact, we know that that's possible now. And if we wanted to, you could write down-- from de Broglie-- you could write down like the wavelength of a car, right? So like in a car-- I don't know, your lambda of the car would be equal to h over mv. And if you plus some numbers in, this is going to be something like oh, you know, maybe 10 to the minus 37th meters. It's small, but it's a wave. Your car is a wave, you know? The baseball is a wave. The electron-- we are all waves. And I know, and that's what-- so the electron is a wave. And I know-- hashtag, we are all waves. Because it's a thing. It's really a thing. Because it's true. I am a wave. You are a wave. We are all waves. It's just-- our wavelengths are pretty small, but they're not zero. And so the next time you get-- oh, we're on the same wavelength. You really are. It's true. We are-- right? That's deep. We are waves. All of us. And so, the wavelength-- OK, the wavelength for an electron was something like 10 to the minus 10th meters. Which made it a really nice way to interrogate images. And that was my "why this matters" last week, right? On Friday. So you can take advantage of that wavelength to do things. Like see electron microscopy, right? But everyone's awake. That's what quantum mechanics told us. But what we want to do now is know how to represent this wave. So everything's waving. We've got that. But how do I describe it? Right? And that's where we left off. And we left off by me telling you that it all led to Schrodinger, who came up with the wave equation for quantum mechanics. And the wave equation of Schrodinger is really pretty incredible, because-- so I'm not going to write out Schrodinger. That would take me forever. So I'm going to write S. Eq., right? The Schrodinger equation, right? And here-- so what he had is he had the sort of wave behavior that was captured. And it was captured automatically. And it gave us also allowed energies, you see? So, allowed energies. And this is also the quantization. Ah-- I ran out of room. Right. But now, If you go back to classical-- so if you think about classical physics, what you would have is something like maybe F equals ma. But what that gives you-- well, it doesn't give you wave behavior, but it gives you continuous energies. Continuous energies. So the powerful point here is that from that equation-- and we don't need to know this equation. We're not going to derive it. Right? Today I'm going to tell you how it's solved, but you don't need to know how to solve it. But you need to know what the solutions mean. That's what we're talking about today. But that equation-- what it does is it sets up the energies as a function of the wave. Psi-- that function, psi-- is the wave function, OK? And so you get waves by default, because that's what you're solving for-- these waves. And then you have a minus second-- oh, there's an h bar. That's an h over 2 pi. Right? But it's like the same Planck's constant. OK. But the wave behavior gives you-- because there's only certain energies that solve that equation. So quantization falls out naturally. You don't have to put it in anymore. It is a direct result of solving this equation. If I take a wave-- think about this, right. If you take a wave and you put it in a box, then it can only have certain solutions. Once you define the box, if you throw a wave in it, well, that's where those things are on the right are. Maybe it's a standing wave. And so it's going to look-- no-- it's going to be like this. One solution. And some energy. Or maybe it's going to look-- I didn't leave myself any room-- like this. And it's going to have some other energy, right? 1, 2. That's it. If you throw a wave in a box-- like, here's the first one, second one. But there are only certain ones that are going to work. All right, and you can get that sense. Because otherwise it's not going to line up and then be able to wave and keep going. It's not going to be a standing wave. Right? So you get this sense right away of quantization. Only certain waves, and only certain energies, will solve this equation. Now, here's the key, is this-- we're going to throw an electron into this equation, right? But we're going to throw it into the right potential. And that is the potential that we know it sits in, in an atom. Which is simply a Coulomb potential. So this is how we get there, right? We take Schrodinger's equation of quantum mechanics. And we solve it for an electron waving in the presence of the potential of the atom. Which is a positive potential. And you can see, this is a-- OK, V is a potential. And that's an attractive potential, right? That's an attractive potential. So what happens when we do that? So that's what we're going to do. I'm going to leave this for-- so. What happens? So if we take a hydrogen atom-- or as I've written it here, a "Hatom," then you would have-- Let's see. What do you get from this? Well, you get that the potential is going to be minus 1 over r. Yeah. It goes as minus 1-- an e and an over-permittivity of space and stuff like that. But it goes as minus 1 over r. That's the Coulomb potential. Right? A plus charge is sitting outside of a minus charge. Ah! That was a close call. The electron is sitting out here, and there's the nucleus. And the potential feels is a Coulomb potential. OK, so that's point 1. Point 2 is that we're going to apply the-- oh, I'm saving myself so much time. I love it, when you're efficient like that-- S. Eq. Schrodinger equation. The Schrodinger equation. But the way that we're going to do this is in spherical coordinates. Spherical coords. And all that means is that I'm going to write psi as a function of r, theta, and phi. And the last thing that I'm going to do is I'm going to separate the variables. And that's just a standard way to solve mathematical equations. So I'm going to separate the variables. So separate. And so what that means is that psi can be written as one function-- let's call it r-- of the radius, times another function, p, of the theta, times another function f of phi. So that's my separation of variables, OK? Again, I'm not all about having you know how to solve this equation. OK? That you can learn in quantum mechanics. But what I'm all about is what that quantum mechanics solution means for us here in chemistry. And this is deep. This is deep, because-- why? Because this whole thing is quantized, right? So just like before, there's only certain allowed values for the wave function. There's only certain allowed values for these variables. Each one of them is quantized. Each one of them. OK? And that's what we need to learn about today, is what that means for chemistry. All right. So now we're going to go through the quantum numbers and see what they mean. And we're going to start with this first one, the radial function. So radial, and then two angular functions. OK. Now the radial-- so what do we get? Well, the radial function-- I'm using a word here, I got to tell you why. Orbital of hydrogen, right? So I've solved this now. I've solved this equation, and I'm plotting this orbital for first quantization-- n equals 1, n equals 2, n equals 3. We've got to talk about this, right? But I'm just plotting what that function looks like. Well, let's compare. Because if I'm Bohr, then what I have is this. Remember this? I've got like this-- OK, there's 1, and it's going around, and it's very happy in an orbit. But see, we don't call these orbits anymore. And it's going to be clear as we talk about this picture, because these aren't distinct orbits, right? So now in quantum what you have-- I'm just going to take that n equals 1 solution. That's the first solution to the radial part, right? That's not a fixed distance. That's not a fixed distance, right? So now I'm just looking at the radial distribution of the probability. The probability of finding the electron. That's how we speak when we speak quantum. We don't say it's here, right? We say it could be here. I don't know. But there's a probability that it's there. So now it sort of has a probability cloud. Now it looks like it gets a little less-- why am I drawing spirals? It looks like it gets a little less probable as you go further out, right? So maybe the density of probability-- because look, right? It's got a peak there. But even if I just plot that first orbital, even as I go all the way out to here, it is non-0, right? So this is probability. And this would be n equals 1. So it's a totally different story. That electron isn't in one place, but it is. But it's not. It could be anywhere. It could be-- if I'm a hydrogen atom, my electron could literally be across the room. There is a finite probability that that's true. And we know that since we are all waves, and we can all be defined by the equations of quantum mechanics-- and in fact, that is the way to define the world-- that we also have probabilities. They're small, you know. But I could be over there right now. There's a chance. My weight function is not exactly 0 all the way over there. It's close. But it's not exact, right? That is a very different picture of an electron in an orbit-- in fact, it's not an orbit. It's an orbital. And that's where we say orbital. That's why we stay it. OK, but-- we're listing quantum numbers. These are the quantum numbers. Oh, oh- I'm going all caps. Didn't even know I was going to do that. So we have n, which is called the principle-- principle quantum number. And this is going to be related to the main energy, kind of e, 1, 1, right? E, 2, 2. So this is going to be related to the main energy level. Main energy level, right? And it's also related to the radial distance. Radial distance. And sometimes we call this the shell. So sometimes the principal quantum number, this is the shell, right? It's the first shell, or the second shell. So we can call this the shell. Come back to that later. And you can see right there. OK, we'll talk about n equals 2 and 3 in a second. But you know-- oh, let's talk about it now. You can see right there-- it I go from n equals 1 to n equals 2, then I'm further out. Yeah, I still got non-0. Doesn't quite go to 0 like I just said there. But it's further out, it's shifted out, right? So with increasing n-- so another thing we can say about n is that with increasing n, you get higher energies and larger distances. larger r, a [? hole. ?] But you get something else. You get something else, and you can see it there. And I've been avoiding it, but I got to address it right now. Because those higher end functions go to 0. So they have nodes. With increasing n, you get more nodes. What's a node? A node is when it goes to 0. And there's more of them. Now that's also really important. So what that means is in Bohr, my orbit was r, e. Heisenberg said no way. Can't have them both. But I had a r, which was one value. And remember-- it could only then be the r of the next value, the next n. Now I'm basically saying the opposite. I'm saying, well actually, r can be anything. There's a probability cloud, so this electron could be anywhere with certain probability, except not here. In this one place. This node. It's 0. It's literally 0. The probability of finding the electron there is 0. It cannot exist there. But how bizarre is that? The electron can exist with probability out here, and probability in here, but it can't exist in between. It's the same electron. Try explaining how it goes back and forth. It skips over a zero probability part, r. Right? But these nodes are very important. And just to give you a sense of what this means, I found a very fancy animation, because I want to make sure you guys understand. So here's the electron. It's doing this, it's doing its waving thing. But look at what happens with the nodes. Uh uh. The probability there is always 0. You see that? Can't go-- no. Not gonna exist there. Right? That is super deep. Because there's no classical analog to these things. There's no classical analog. But this is what electrons do in an atom. Now, this is what the clouds look like-- a little better drawn than mine-- for n equals 1, n equals 2, and n equals 3. So again, you can see-- this is one electron. It's colored differently, just to contrast it, but this is one electron. Right? And These are plus and minus of this, right? So it'd be like a wavy wave, and you're just showing plus and minus with the yellow and blue. But that note in between is where it can never be. But it's one electron. It's not many, it's one. And this is its probability distribution. Now I want to make one really important point, which is that-- look at what I'm plotting here. Maybe something-- that's size squared, right? So it's only the square of the wave function that we can relate any physical meaning to. The physical meaning I just related to you is the probability of being somewhere. That's what size squared gives us-- the probability of being somewhere. But ask your teacher-- not me, your quantum teacher. Ask what psi means. And they're going to have a-- it's going to be an interesting class. Force the issue. Don't let them go until they answer. We still don't know. We still don't know what the actual wave function means physically. We cannot attribute a physical meaning to it, only to its square. So we go about our business for the last 100 years squaring it and saying, OK, well it describes nature. That's great. But no one, still, knows what it means without the square. I find that incredible. And it led Bohr to say all sorts of really cool things. He said, "We must be clear that when it comes to atoms, language can be used only as in poetry. The poet, too, is not nearly so concerned with describing facts as with creating images and establishing mental connections." That's Bohr like deep in it, trying to figure out what psi means. He's like, I don't know. It's poetry. OK, that's cool. That's cool. V squared means something, v doesn't. Really? No. Classically, we've got nothing. Right? Quantum mechanics. Oh, it's a beautiful thing. Why does this matter? I'm going to give you a brief "why this matters." And we've been talking about the quantum nature of things, right? Quantum is this. Quantum are these differences. It's this probability. It's quantization of energies. What does that mean, right? Well, last time I told you that by electrons being quantum, and having wave-like character-- we're able to use them to see things, and it started the nanotechnology revolution about 30 years ago. And today I want to give you an example of that in my "why this matters." And it has to do with quantum domination, which should be the title of a movie or a book. And here's the example-- take a piece of bulk material. Take a piece of silicon, OK? And if you look at the silicon, it's kind of boring to look at optically. It's not very interesting. You know, don't tell that to your iPhone, which has a lot of it in there, but it's boring. But now, take out your little nano ice cream scooper, which you all have. And you take a little tiny piece of it. We call that a quantum dot. Why? We call it a quantum dot because it kind of looks like a dot-- its smallish. But we call it that because quantum mechanics takes over the properties. Here's an example-- shine light on this thing, or have it glow, and all of a sudden instead of being boring it can be any color you want. Why? Because if I shine light-- and here is a very sophisticated picture of a laser, which you can tell I had trouble making transparent. And so you shine a laser on this piece-- we know already, light excites charge in an atom. It does the same in solids. So here's my piece of silicon, and what that does is it sends an electron up in energy levels. But the electron is up in an energy level, and it left behind a hole. Which is something we'll talk about later when we talk about semiconductors. A hole is just a positive charge. And it turns out that that electron and that hole are attracted to each other, but not-- they can't get too close. But they want to kind of hang out at a certain distance. Where? How far? Yeah-- quantum mechanics. N 2, 3, 4 tells us. It tells us how far, right? They want to hang out-- and now all of a sudden, I've nano-scooped them out, and I've made a quantum dot. But if I try to do the same thing there, that's how far they wanted to be. And they can't be. They literally ran out of real estate. They ran out of atoms. They cannot be out here, because there's no stuff out here. So they get squeezed. I am squeezing these quantum mechanical objects, this electron and this hole. And by squeezing them, I'm changing the way they interact with light. I'm changing the color, right? Oh. It's all stuff we've learned that explains something revolutionary. By the way, if you buy a QLED TV, is it as good as OLED? No. But is it better than LED? Yeah, a little bit. That's what they-- QLED-- quantum is quantum dots. They have coated the LED emitters with quantum dots to give you better color. Now what this means, why this matters, is nothing less than the periodic table itself. Because I think I showed you this-- we're living in these different ages, which I love. We have the periodic table that has-- the ability to work with it is what's brought us to the age of materials design. But see, now I've literally just told you that by changing the size-- nothing more than the size-- I can tune a property because of quantum mechanics. Because I'm changing its quantum mechanical interactions. It would be like saying, I can take a piece of this table-- I can break a piece of this table, and it's now going to be red. That's exactly what we're doing, but at a very tiny size. That is as if we are taking this periodic table and giving it a whole other dimension. Every element can do more things than we thought possible because we can tune these properties related to quantum mechanics. So that's a big deal. That's a very big deal. OK. Now back to our solution to the Schrodinger equation. For an electron in an atom. Because now we move on and we solve for the next one, which is-- OK, so we did r. Now we're going to do theta. And what happens with theta is, you get another quantization of that variable. And that's the second quantum number. So we're going to put it here. N-- I'll number them so we don't lose track. Who knows how many they'll be? OK. Not three. OK, so the second one is l. And it's the angular quantum number. And it has also quantization 0, 1, 2, 3. OK? So l can be-- Oh, I need to put that up there. I meant to put that right here. This is just like what we learned for Bohr, right? And l can be 0, 1, 2, all the way up to l minus 1. OK? I'm going too fast, that's n minus 1. L goes from 0 to n minus 1. OK, so what that means right away-- what you can see-- if it only goes to n minus 1, then for example for n equals 1 then l equals 0. That's it. There's no other options. It stops, right? And we have a name for this. We call this s, the s orbital. And for n equals 2, l can be 0, or l can be 1. And this one is called-- this one is the p orbital. This is the p orbital, OK? Those have names. And so on and so on. d, f-- This is old notation that we cannot get rid of. This is called spectroscopic notation, because it comes from spectroscopists. And remember, I told you-- never make a spectroscopist angry. Don't do it. Don't do it. And so we use their notation still. s was sharp. Spherically symmetric. p was principle, because it sort of dominated in some experiments. d was more diffuse. f fundamental, because it looked kind of like the hydrogen atom sometimes. These shouldn't be the names, but they are because they still stay with us. Spectroscopic notation-- s, p, d, f. What they really do is just correspond-- that looks like "porbital." Let's put a little more space there. P orbital. Right? Because what they really mean is nothing more than a quantum number for the second quantum number, which is the angular quantum number. These are also sometimes called subshells, right? Because they're underneath the shell. And as you can see from this picture, it describes the angular shape. The angular shape of the orbital. The shape, right? So it gives us a shape. Now-- if we think about it, and we think about that p orbital, that has a different shape than the [INAUDIBLE].. Then you think, well, this could be aligned in different directions, right? So if I think about that p orbital-- well, I could align it in three different ways. It could go along this axis, this axis, or that one, right? And so those are p orbitals in hydrogen that have three possible orientations, and that automatically gets us to another quantum number. In fact, it's the last one, and it makes sense just physically-- that's phi, right? So the last one-- or is it the last one-- is m. And we call it m sub l, which is the magnetic quantum number. And this one can have quantization equal to minus l, right? All the way up to plus l. So the number of orientations that the orbital can have-- if I'm a-- n equals 2, l equals 1, then m sub l equals minus 1, 0, 1. Three orientations. Right? It's got three orientations-- three possible magnetic quantum numbers. Why is it called the magnetic quantum number? Because those are the experiments. We're done using magnetization-- magnetic fields-- to discover these orientations, to sort through it. To see it. OK. So if I had l equals-- all right, so if I had, for example l equals 2. Then m sub l would have five values. Five possible values, right? I should put a plus there, just to be sure we keep track of minus and plus. These are quantum numbers. That means they're integers, right? But they can have these ranges and these rules, and again-- you may think, why did this come out like this? Well, it comes out from solving this equation-- separation of variables-- with that potential. That's it, right? That's it. You get these allowed values of these three variables, quantum numbers. OK? All right. Now-- and there's the d. These are called d-- So if n equals 2-- OK. So did I mess up here? No, I didn't, because n would have to be 3 here, right? For l to be 2. Because l can't be equal to n, because I wrote it there as the list of allowed quantum numbers that l can have. And there it is for the d orbitals. And so there's five d orbitals. Five d orbitals. And you can do f on your own, and it's an extremely fun thing to do at night. Late at night. Now, this gives you the orbitals. You get them all. These are the orbitals. These are the beautiful, beautiful orbitals that we have from quantum mechanics describing an electron in an atom. Which tells you everything about where that electron can be, how it behaves, what its energies are, and therefore literally why atoms are what they are. That's what it tells us, right? Because the next thing we need to do is fill up our atoms. And know how electrons fill atoms using these quantum numbers. And then relate that to chemistry, and to the differences between one atom and another, and explaining why atoms are different. Remember, that was our goal. We needed to go quantum to answer that. So we're getting closer, but we're still not there. And one of the reasons is there is a fourth quantum number. And this came out of experiments-- I thought that I would just show you a very quick video, because it's easier to see this in action. And so these are experiments that were done by Stern and Gerlach that showed something very peculiar about atoms and their electrons. OK, so here I'm just going to play it. It's got a really nice-- oh, that's loud. Yeah? I don't mind that. That's good. So you can read the text here. OK, so they've got a magnet. A very strong magnet, right? Magnetic fields. And they're going to shoot a classical magnet. There's a classical magnet. And you see it's going to react to the magnetic field, and be pushed up. That magnetic could've been oriented differently. Like sideways, right? In which case it doesn't feel this. And you could throw a bunch of them randomly, with random orientations, and that's what you would see. [JAZZY MUSIC] It's good music, too. But now-- watch this. I love the animation here. Look at this. And the bass, right? Those are what happens to atoms with electrons. You'd shoot them through a high field, and you don't get anything. You get two things. Only two. [JAZZY MUSIC] I'm just playing it because I really want to groove out and I'm debating-- OK. All done. I was debating how embarrassing it would be if I just started grooving out. So this was Sterin and Gerlach, who discovered, essentially, a fourth quantum number that can have only two values. So they showed that this thing that the atoms were doing was because of the electrons. And that the only way you could explain what these electrons were doing is if there was this thing called spin. And if you really need to think classically-- and it's dangerous to do so, because then you'd forget that I could be all the way over there, right? Right now. But if you really need to think classically, you can think of a charge spinning. It could be spinning this way or that way. In a way, it helps our classical needs. But it's so not what's happening. But it does help our classical desires, right? And that would give you the two fields. Right? But I would advise you just to think about spin as simply a fourth quantum number. And so we call that one number 4. Not from that equation. Separate from those experiments. And that is-- we call that the spin M sub s is the spin quantum number. And it's the fourth one in our series. The fourth quantum number. And it has only two values-- up or down. Or if you like, you could write it as plus 1/2 or minus 1/2. Or you could write it as up or down arrows. So many ways to distinguish two things that are opposite. Spin up, spin down. Two values that this can have. So we're not ready yet. We're still not ready. We're so close, but we're still not there, right? Because we still don't know, well-- can't I just--? How do I--? What are the rules here going to be? For populating atoms. And this is where Pauli said something very important, and made this incredibly important statement. Which is that-- no particles in the same system-- read that as, no electrons in the same atom-- can have the same quantum numbers. No electrons in the same atom can have the same quantum numbers. And that immediately tells you-- again, this has to do with the nature of quantum mechanics-- what that immediately tells you is that two electrons-- ho-ho, this is big! Two electrons-- I'm gonna write, two electrons, and watch this-- I'm going all caps-- only two electrons can occupy an orbital. An orbital. Or- bi- tal. That called the Pauli exclusion principle. OK. So we're going to use all of this information to fill stuff up, but I don't want to do that yet. On Wednesday, we're filling up the table, and we're making our first solid. I figure it's our three week anniversary. I know you've all been looking forward to it. I've been thinking about it a lot. And I think-- that makes me want to throw t-shirts. [STUDENTS CHEERING] I got to do this. All right, all right. I got to go-- I know. I got to go backwards here. OK, OK, hold on, hold on. I'm missing there and there, I know. I got to work on my arm. Yeah. Show you love over there. OK, all right. I'll have more on Wednesday, and the reason is-- we're making our first solid. I almost wanted to leave it a surprise, but I couldn't help myself. It's extremely exciting. This is, after all, solid-state chemistry, and Wednesday we're making a solid. Literally. We are not only making a bond, but we're making a solid, and talking about solids. So that's going to be very exciting. And we're going to start by filling elements. But before we do that, I want to explain something very important, which is-- what's happening with energy? Right? We did just see something dramatic. As you go from Bohr to quantum, something very dramatic happens with where, right? And we know something else very dramatic happens, which is right here. You know, that's Bohr over there- n equals 1, [INAUDIBLE].. Bohr is 1, 2, 3, right? Now it's like-- oh no, wait, it's 1 and with 1s. 2-- I can put two electrons in there, thanks to Pauli. I know I can put two, because I have all the quantum numbers in here the same except spin, which could be up or down, right? N equals 1, l equals 0, m sub l equals 0, spin is up or down. Two electrons. But look, I'm now here at n equals 2, going from the Bohr model to quantum. I've got this and I've got three p orbitals. So now all of a sudden I've got eight electrons that can go into that quantum number. That's very different than Bohr, right? And that's called degeneracy, because you're taking something and you're saying, there are multiple variations that have the same n. But there's something else that I want to explain right now that is can be understood very physically and intuitively. And that is that the motivation-- one of the big motivations of this was that we needed more than one electron. Our atoms are not all hydrogen and ionized atoms. They are just neutral atoms going up above hydrogen. So what happens when we go multi electron? And there's two effects that I want to explain. Here's what happens to this picture. Those are the one electron levels, right? There's Bohr over there. This is what you really get in terms of energy. Right? This is energy along the vertical axis. And there's two reasons, two very important reasons, why these are so different. One is called shielding, and the other is called orbital penetration. And I want to talk about both of them in the last seven minutes of class. I'm going over here. All right, so let's talk about-- let's do shielding first. Shielding is something that you can kind of get a very good sense for it if you just look at what's happening in these atoms. So in shielding-- let's take an example. I'm going to give you an atom with 15 protons and 15 electrons. What atom is it? Anybody know? Phosphorus. OK, so I'm going to take an example of the phosphorus atom, OK? And I've got 15 positive charges in the nucleus. 15. Right? Now, 15 protons. Now I've got my levels, and it gets very difficult to draw levels as probability distributions. So I'm going to draw them as rings, even though we know they're not, right? These are my orbitals. But I've got like an electron here, OK, an electron there. All right. So I've got two electrons in the n equals 1-- n equals 1 orbital. So that's the 1s orbital, right there on the bottom. But if I go out-- OK, so I'm going to go out now. To n equals 2. Well, now I've got 1, 2, 3, 4, and-- oh boy-- put one here. 5, 6, 7, 8. Because here-- remember, I've got l equals 0 and l equals 1, and as we just said, because n sub s can be up or down, I've got eight electrons here. So here, I've got n equals 1, here I've got n equals 2. That's eight electrons. And then out here-- oh, now it gets really interesting-- because for n equals 3, how many electrons do I have left? 1, 2, 3, 4, 5. And you can already see what's happening. So remember, Rutherford told us this. All this charge is in a tiny little point here. You wouldn't even see it if we were drawing this to scale. But what are these electrons seeing? What are the electrons seeing? Well, these inner two electrons are just seeing this huge amount of charge. They are happy. Low energy. Right? The lower your energy, the happier. You're in a lower energy, ground state. Remember-- more lower energy, more happiness. More bound. Right? So those electrons down there are enjoying. But these out here-- you see? These electrons. Those electrons in here-- these shield the outer electrons, these shield outer electrons by two electrons. Right? Because they're kind of in the way. It's like you've got this negative charge that's in between me and the positive charge, right? And so if I'm out here at n equals 2, I don't see 15 positive charges there anymore. I see 13-- ish. That's shielding. But now this one-- this has the 2s and the 2p electrons. It's got eight electrons there, and those are shielding another eight. Those are shielding another eight protons. To things outside of them, right? Shielding doesn't work going in. If I'm this electron and I look out, all I see are repulsive-- I see repulsiveness. In a good way-- I'm sure they're friends. But two negative charges, right? So I'm only looking in. Looking in, shielded. Looking in, shielded. So these ones have sort of an effective-- z effective. You know, that could be something like five. This is not the z of the atom, but it's sort of like those very outer electrons don't see the whole proton cloud. They get shielded. And that's a very important part of what happens to orbitals when you put a lot of electrons in them. And there's one more thing that happens, and it goes back to this plot that I showed you before, which is that the orbitals themselves are very different than just like a straight line from Bohr. They have these shapes to them, right? So the second thing that happens is that if I have that same s-- this is s, so this would now be the 1s. And now I'm going to put-- remember that 2s looked like this. And it had the node there. Remember that? Ho-ho. That was 2s. But now we're going to put the 2p there. May not be drawn to scale. OK, here's the 2p. Something very, very interesting is happening here. And again, it has to do with what electrons are seeing. Because the 2s electron has a probability-- remember, this is the probability normalized, so we put in r-squared there, right? That probability-- all that means is if I integrate over everything, it's 1. It's got to be somewhere. But look, the 2s electron can be really deep inside the shells. It can be in there. Literally-- look, it's even inside the 1s. So it gets some of that 15-proton action, right? Orbital penetration leads to massive changes. Orbital penetration is what this is called, and it allows some of the-- but the p doesn't have that. It goes in a little, but the p doesn't have that peak, all the way in there, where it can see all that positive charge. And that is why these two things are different. You see? Because the 2s now, can be closer sometimes to those 15 protons, right? And so it lowers the energy, making it happier. It finds a lower happy place. OK? So we're going to use this, and on Wednesday, we're going to take these things and we're going to fill them up, and we're going to make our first solid. Have a very good rest of your day. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 31_Exam_Review_Intro_to_SolidState_Chemistry.txt | How is everyone doing today? [CHEERING] You know why there's a lot of excitement, and I saw the silent praise, because I know what this means, recently, and I'm doing more and more. You guys are excited, because we are celebrating something on Friday. We are celebrating knowledge, we are celebrating chemistry, and we are celebrating how much we've learned in the last month. It's called exam three. Woo. It's a midterm. Thank you for that. What I thought we'd do today instead of-- again, I mentioned this on Monday. You guys just had Thanksgiving, you're back, and now we've got a test. And I know you've got a lot of other stuff going on too at this time. So instead of throwing something in that we're not going to test you on, what I wanted to do with today was just to kind of feel our oneness with the concepts. All right? And so I want to go over the concepts that are going to be potentially on the exam on Friday, and I'll talk about those concepts, and we'll do a couple of problems here and there as we go. OK, so those are the metrics, the specs, for exam three. But on Monday, I finished with two minutes left, and I said, hey, there's one more kind of acid-base definition. You won't be tested on it, but it's the most general, and I've got to give Lewis his due. So I just wanted to just show you like that. You won't be tested on Lewis acids and bases, but I did throw this in there, and so I want to just, for one or two more minutes, talk about this. Right? These are the things you are responsible for knowing. So an Arrhenius acid-base-- OH, H plus. Bronsted-Lowry. It's proton transfer. And then there's Lewis, who broasted these guys, because he said, uh-uh, and this is the most general way to think about acids and bases. It is the most general. It is quite powerful, because you go back to say this one, and so if this is a-- what if we mix this with this? So here we have H. Now, remember, Lewis had all the dots all over the place. He loved dots, and so they're all over the place. And that's what if you had this, this is a Lewis acid, because it is ready to receive. What is it ready to receive? No more atoms, only electrons. It is the most general way to think about it. All right? And this would be a Lewis base, because it is ready to give a pair of electrons and form a bond together. And that's what they do, or that's what they can do. So a Lewis acid and a Lewis base is the most general form, and you really get rid of-- so for Lewis, it's all about electron transfer, electron pair transfer, pair transfer, or bonds if you're-- I'm just going to make it as general as possible. If you're going to mix an acid and a base, which I'll do in a second, it's a pair. Electron transfer is what it's all about, and what happens here? Well, you don't have any chemical identity, no more chemical identity. So it's full. You don't need the hydrogens anymore. You don't need the OH's. It's totally general. And this really opens it up, because now you can look at gases, liquid, solids, all forms of matter, all elements in the table, and you can think about them as whether they're going to have an acid-like or a base-like character in terms of electron transfer. So here's the general-- if you had a Lewis acid and a Lewis base, and you mix them together. Then, you can see that if you got an electron pair that could then go-- that one of them wants it, and the other can give it, and they can form a bond, then that would be how you think about mixing a Lewis acid and a Lewis base. And again, this won't be on the exam, but I just had to give Lewis his due. OK, Lewis acids and bases are all about the electrons. And then the other thing I didn't get to do on Monday was my Why This Matters, and so I want to do that. Now hair is very much affected by the softness of water. But instead of me showing you lots of pictures of hair that I found randomly on the internet, googling soft water, hard water, different forms of water, hair, I'm showing you pipes, because that's much more actually relevant in terms of like, for example, whether you can get water into your house at all. All right? And this is a real problem. If you have hard water what it means is you've got minerals in the water. Right? Things like magnesium and calcium ions are in the water. And what happens is, if you run that through pipes, they tend to react and form and deposit. And you get these deposits of very, very hard mineral deposits around the edges that eventually look like this. That's not a good thing. And so you need to soften the water, and that's something you may have heard of. But now you really know how to think about it, because you can think about it in terms of all of the things that you've learned. And just as an example, if one of those minerals that's a problem is calcium-- oh, it is, right? Magnesium is another one that can cause this in pipes. You've got to somehow take it out of the water or at least make it into a form that makes it like insoluble, so it'll just flow, and it won't have a chance to come around and deposit on the edge there of the pipe. And so what do you do? Well, you add, for example-- oh, we've done this before. Right? What if I add soda, something called soda ash? Oh, we know what that is. That's Na2CO3, and that will give me some ions. It'll give me some Na pluses, and it'll give me some CO3 2 minuses, and those are in aqueous. Now, this is all part of the stuff that we've been talking about. You're dissociating this to get ions in solution, but why do I want these ions? That's a water softener. Why? Because now you can see it. Right? Because this one here, this one here is going to remove the calcium. So if I've got these calcium, these minerals, in the water making it hard water which leads to that and wreaks havoc on your hair, then you get a little of this in there, and you're going to react with the calcium. Why? It's because of what we already saw. If I have this-- this is something we've already looked at-- if I have calcium 2 plus, and I have some bicarbonate in there, CO3 minus, then we know that I can get precipitates, Ca-- this is just what we looked at in the goody bag, right-- plus CO2 plus H2O. That's the same kind of reaction that we looked at already. We have calcium carbonate, calcium carbonate. Why? Well, because this is going to give me these. This is a source for those which then eats up some calcium. And you think, well, now this can take the calcium ions out, so that it runs through. That's what you want to do here. That's how you soften water. Well, you can add lime which gives me a source of OH minus, and that takes out the magnesium. And these are ways to change the chemistry of water, so that you don't have as many ions, mineral, these calcium and magnesium ions floating around that cause problems. So that was my Why This Matters, and now we turn to what I started with which is exam three. Oh, look at that. I look at that, and that's just happiness. Look at all those topics-- X-rays XRD, defects, glasses, reaction rates, solubility, aqueous solutions, acids/bases. It's all there. It's all there, and it's all there in you, and if not, you've got two days. So please, make it be there. Let's go through this. Let's go through this. I'm going to go through this topic right there. So I'm going to tell you the concepts that I want to make sure you know. OK, so here's the first one. So it's X-rays, and XRD. So what do I want you to know? Well, for example, I want you to know about how you have characteristic, characteristic, characteristic X-rays, and remember this? This comes from those transitions. Right? So you get like the K and the L and the M. We say et cetera. We never really did et cetera, but we've got a lot of K and L lines we looked at, alpha, beta, but there's also the continuous. Right? So this was in-- if you want-- this would be in the topic of X-ray generation, and there's that Bremsstrahlung word that means breaking. Right? So if you just want to look at this, you might have a curve that looks-- OK, so we had things like this. Right? Oh, I'm going to try to make it not look too bad, and there's another one, and so on. And remember, so this would be like intensity, intensity, and this would be like maybe a lambda going that way or maybe energy going that way or frequency. Right? Those would be like the x-axis, and these are those lines. Those are the lines, and they come from the transitions that you get from like the K level to the L level to the M level. Right? Remember, if you get the energy high enough of an incident electron onto a target, metal, then you can knock out electrons from that 1s shell. And then something cascades down, and that emits these lines here. So that'd be like K alpha, K alpha, K beta, et cetera. But then, you also, even at lower energies, even if you don't have enough energy to knock out a 1s electron, you still can slow an electron down as it goes through a metal atom. And that's where you get this continuous spectrum and that limit of how much of that type of X-ray you get. And it just depends on the incident, on the voltage, on the maximum incident electron you fire at it. So that's those first two topics. OK? Well, oh, we also did X-ray diffraction. Oh, that's fun, diffraction. OK, and this involves understanding Bragg, the Bragg condition, and also selection rules. So let's just take a look at an example of the diffraction and selection rules. OK? So what would I want? So these are like the kind of concepts that I want you to know about. How might it play out? So like, OK, so here's like an example of a spectrum. Now, that's an X-ray spectrum for MgFeO4, magnesium ferrite, and it's a kind of steel. And you put it in, you make a powder of it, and and you throw some X-rays on it. Oh no, you throw just one type of X-ray on it. All right? So for these X-ray spectra, what happened? I threw some K alpha on it, I showed right there, you got to say what you got. All right? You've got to know at least the source of X-rays has this wavelength. Good. And if I have that spectrum, and say, what's the lattice constant? What's the structure? I can get that. How would I get that? How would I get that? So I would take this spectrum, and notice something is very much pointed out to me, a peak and a number, 35.5. Right? That's the 2 theta value for the 311 peak. Well, I think I'll take that as a hint. Maybe I should think about that number, and OK, so let's see. From Bragg, so from Bragg, you know that you got n lambda equals 2d sine theta, and remember that had to do with constructive and destructive interference of waves, the waves being the X-rays. And then we said, well, n is 1 in our class, so that's just 1 in 3,091. It doesn't have to be. You could have interference off of two planes down, three planes down, but in this class, we're just doing the two planes next to each other. So n is 1. OK? Now, I'll take that angle, and I'll round it to 18 degrees. So I'm going to say for theta equals 18 degrees, then-- oh, wait, why is it 18? Because that's 2 theta. Right? Because remember, we take spectra in 2 theta, but the theta in Bragg is 1 theta. OK, that wasn't too hard. But now, I've got, let's see, sine theta is around 0.3, and oh, I got lambda. Lambda equals 1.54 nanometers, and so I can actually plug in now. I didn't even need the peak. I'm just plugging in the Bragg condition, and I've got d. d equals 2.52 Angstroms, so I'm done. No. No, I'm not done, because the question says, what's the lattice constant? What's the lattice constant? This d, I didn't say-- I did something very bad-- I didn't say what d-- d is in hkl. That's what an XRD does. XRD, that's a different than diffraction, but the d, in Bragg, is the distance between the mirror planes that are scattering. All right? And so that's some cuts in the crystal. That's some hkl, and we know from that era that there is an equation for cubic crystals that gives us hkl which is a divided by the square root of h squared plus k squared plus l squared. And that is the lattice constant of the crystal, and that would be the distance between any planes, any set of planes, in the crystal, and that's what we need. Now, I can use that, combine it with the d, and I can get that the lattice constant is 8.38 Angstroms. Good. But the second question is-- you don't even need to do any math to answer that-- what is the lattice structure? Here, we simply go to our selection rules, and we know from before if they're all even or all odd-- so here we have, let's see, all these peaks-- 220, 311, 400. You can see, they're always all even, or they're always all odd. From our selection rules, what kind of crystal is that? It's got to be FCC. All right? It's got to be FCC. So that's the answer to that question, just from looking at the peaks that you get and matching them with the selection rules that you know, that we've already talked about. You could also, if you didn't have the planes-- so this is an example, where you're given all these planes. You're given all these hkls, but what if instead it's like the one we had in class? We went through this in class. Here, I'm not given the planes. I'm given instead just two theta values. Well now, you've got to go and use the procedure of XRD, and I'm not going to do that again. We've already done that pretty thoroughly, but that's what you would have to do here, where you create your columns. Remember the columns? All right? So you would simply make your columns, you make your sign, your thetas, your 2 theta theta sine squared thetas. And then you normalize, you take the lowest one, and you normalize to 1. Clear fractions, and then find the hkl combinations, and that's how you would go from this to getting, for example, the planes. So those are some examples of things I want you to be able to do with X-rays and XRD. Let's move on. Let's go to the next topic. Let's do defects. How about this, defects, because as we said, it's just like life, crystals, defects are what make them interesting. It's the same. Everything, it's all about defects, imperfections, those are the interesting and exciting things, so we've got to know about them in chemistry, of course. So we've talked about a bunch of different kinds of defects. The first one that we talked about were vacancies, vacancies, and these could be interstitial. These fall into the category of point defects, point defects, right? You could also have interstitial point defects, interstitial or substitutional. Those are the different kinds of defects we've talked about for point defects. Also, by the way, if it's an ionic point defect, ionic point defect, you need to make sure you have charge neutrality, charge neutrality, plus two types. There's Frenkel, and there's Schottky. You can think about Schottky, you remove, but you've got to remove them in a way that's charge-balanced. Right? So in Schottky, you remove. In an ionic crystal, you've got to keep charge. That's a new thing you've got to keep track of. And so if we had-- I'll go to this picture. Well, you should know this picture. This is an important picture. We showed it a lot-- vacancy, interstitial, self-interstitial, substitutional. And then if it's ionic, now all of a sudden, you've got charges in there, and so you've just got to make sure things remain neutral. Right? In Schottky, you would remove atoms to balance the charge, and in Frenkel, you move an atom. Schottky remove, Frenkel move. That just occurred to me, that that's the same word with just re in front of it. That's exciting. Well, so if I were to ask you, for example, if I were to ask you this. You diffuse chromium into one side of the ionic crystal alumina to make it red and titanium on the other. Would you expect oxygen vacancies, aluminum vacancies, or no vacancies in each case? Just follow the charge, follow the charge. So for example, if I had alumina, O3. Well, if I were Lewis, if I were Lewis I would see this as 2Al3 plus. Right? This is how he saw ionic bonds. Ah, I want to make sure this is clear. I'm going to start over. So I would have 2Al3 plus, and then I would have a 3O2 minus. Remember, O likes to be 2 minus, and so if I were to introduce something like Cr2O3, well, this is 2Cr3 plus, so this is 2Cr3 plus, and 3O2 minus. So if I substitute aluminum with chromium, then I'm charge-balanced. Right? I'm OK. But if I substitute titanium, TiO2 is Ti4 plus and 2O2 minus, you know this just from the formula. Because if oxygen is 2 minus, this thing's got to be neutral, and so that's Ti4 plus. But now, if I put a Ti in there, and I get one in for an aluminum, I've knocked the charge by 1. Right? So this gives us an extra charge, and that means you've got to create vacancies, create aluminum vacancies. It's the only way to do it. So for every three titanium atoms that I put in, I've got to take an aluminum one out. Right? Otherwise, I'm not going to have charge balance, not going to have charge balance. Right? So that would be an example of playing with an ionic crystal, but remember, we also had stress-strain curves. Because when we talked about line defects, we talked about how that's a way for metals and other materials to undergo deformation. And then we said, well, how are we going to conceptualize these things happening, and we did it with the stress-strain curve and the mechanical response of materials. So we learned about stress-strain curves and fracture points, fracture and plastic versus elastic regimes. OK, I'll give an example in a sec. OK? And finally, the last thing we did with defects is we talked about their activation. And so we know if something is thermally activated, as are point defects, vacancies are thermally activated, so are other types of point defects. And we said that we need an Arrhenius relationship, and that requires knowledge of an activation energy for the defect to form. So that was another really important concept that we covered. Right? So for example, here is like a defect formation energy question. So let me go over here, stay with this side, where I've got some defects. So one thing I want to make clear, the fraction of vacancies. We talked about this. We talked about an Arrhenius behavior, Arrhenius-like behavior, and the fraction of vacancies was the number of vacancies over the number of possible sites. That makes a lot of sense. Now, if it's Arrhenius, then this is some constant times e to the minus activation energy divided by KBT, where we use the Boltzmann constant if it's per atom. Right? So per vacancy, that activation energy would be per vacancy. But when I first introduced this, I simply let a be 1, but a is a constant. This is a constant, and it can be whatever. It's a constant. It's called the anthropic factor, and it's a constant related to as you would have in any Arrhenius-like behavior. Right? Remember how it's like a frequency factor, this is related to all sorts of complexities that we don't need to know about related to how vacancies form. It's a constant that's in there, and I let it be 1, but I want to make sure we put it in explicitly. This n is the number of possible sites. Right? So maybe it's the number of lattice sites in the crystal, number of possible sites. But you might get this could be related to, for example, or to or from, things like the grams that you have of something or maybe the volume or the moles, et cetera. Right? So these are the kinds of problems we've done. That gives you a sense of how many sites did you have. Right? It could be per centimeter cubed. It could be per moles. You go back and forth. How much stuff do I have? It's that kind of back and forth. If I had a problem like this, then I know I can set it up, because I've got two temperatures. You see the key here, the concentration of 10 to the cubed per centimeter cubed, so don't let that throw you off. It's simply two equations, where a bunch of stuff cancels. Right? So the fraction of vacancies, fv1, would be 10 to the 3rd divided by fv2, and that's going to be 10 to the 3rd over 10 to the 16th, because the number of stuff cancels. Right? I didn't change that. But this pre-factor cancels too. You can put it there, A times e to the minus ea divided by KB times 300 Kelvin-- remember, so it was Kelvin-- divided by A times to the minus ea divided by KB times the other temperature, 900 Kelvin. Right? And so those cancel, the stuff related to n cancels, and I can just take the number of vacancies. Right? So that's what's nice about having something like this. You have two temperatures and a whole bunch of stuff can cancel, and you can figure out then the activation energy which doesn't change. The activation energy is simply the energy that you need, so you can now solve for ea, and I think you get something like 2.3 electron volts. Remember that you could also have plotted it. You'd know that in this equation, in the Arrhenius relationship, if you take the log, then you have the minus 1 over T relationship. So remember that? So if you take the T versus the log, then you're going to get that. Right? We like plots, when we think about Arrhenius, because it makes it easier. We can make it linear, where the slope of this is minus ea, the slope, minus ea over KB, in this case. Right? So we talked about these are all kinds of things that we talked about. And finally, getting to this stress-strain, what if we looked at this? This is the stress-strain, so now we're here. We just did this. We talked about that. We talked about this, so stress-strain. So if you were to be given plots like this, you should know which one is most brittle, which one is the stiffest, the most resistant to changing elastically, and which one is the most ductile? You can see that the stiffest one is going to be the one that's the most resistant to elastic deformation. That's that linear part, that first part. So there, it's going to be material A. It's going to be A, because it's got the highest slope. Right? OK. How about the most brittle? Well, that's the one that's going to break. So it sort of looks like-- without deforming, without being able to undergo defamation, that's brittleness. It's going to crack before doing anything. So that's B, right? That's B. And then the most ductile will be the one that has the most plastic deformation, because that's what ductility is, so that's going to be C. OK? So remember stress-strain curves. You've got this elastic part here, that's linear, and then if it's a material-- not all materials have plastic defamation. Some are going to break before they start deforming plastically, but if it can deform plastically, then it's going to have that yield point. Right? The yield stress, where then it starts becoming non-linear, and then finally, it fractures somewhere else. OK? So those are stress-strain curves. OK, let's move on. Now, oh, with the next topic-- oh boy, I'm going to run out of real estate-- we've got glasses, and here, the most important thing to know about is the curve. That's how it all started. Right? I'm calling it The Curve. Let's go all caps, because it's The Curve, and what I mean by The Curve is the volume per mole versus temperature. We also talked about the effects of cooling rate, and we also talked about the effects of modifiers, and in particular how the modifier changes the viscosity because it cuts the pasta. OK, so but the framing of these concepts was in The Curve, and that's this one. Oh, I have a picture of it. There it is. OK, they did specific-- we do volume per mole is how we're looking at it in this class, so volume per mole versus temperature. Now remember, if it's a crystal, where am I going to go? Let's go here. If it's a crystal, then you've got one melting point that only depends on the material. It only depends on the material, so that melting point is Tm, and it's going to have a big volume change as it goes from liquid to crystal solid. OK? And so for example, I'm just basically replotting it, but you would have something like this and something like this and something like this, and that's your melting point. This would be the liquid, and this would be the solid, and you know that, if it's going to undergo this big volume change. Then, it's the melting point and it's a crystal. But then we say, well, but if it can't find-- by crystal I mean in opposition to glass, which we know doesn't just mean your windows. It means any amorphous material, meaning it's not crystalline. It didn't find the lattice structure. Remember those three reasons, the musical chairs? Whatever you're trying to crystallize, atoms or molecules, they weren't able to find all the lattice sites, not even withstanding some defects here. They simply couldn't do it, and so it's amorphous, and it's a glass. And in that case, what happens is it super cools and then becomes a solid at something called the glass transition temperature. Which unlike the crystal can have different values, and those values can depend on things like the cooling rate. So if I cool it slower, if I cool it slower-- remember this is faster, faster, slower, and please, do not confuse time with this axis. This is just temperature. So we can use the same-- this is volume per mole-- we can use the same The Plot to think about modifiers as well. Because it's not-- this is just one plot, one curve and another. This happened to be cooled slower, so it was able to super cool more, before it locked in the disorder, and the other one's faster. You could also imagine if you cut the pasta, you change the viscosity. And so now you're changing something else about the glass using chemistry, and you're allowing it to find those lattice sites more easily. You're allowing it to try to become crystalline, to try to close pack just a little bit more. So you can also get this, by cutting the pasta you can get these kinds of changes. So those are the things we talked about with glass. Now, next, we talked about reactions. OK. What do we need to know about reactions, reaction rates? So we want to know the order of some general reaction, like aA plus bB-- remember this-- goes to cC plus dD. And we want to know the rate from-- basically, it's the mass conservation, so from the coefficients. OK, so that would be, for example, like minus 1 over a, d(A), dt-- that's a rate-- equals minus 1 over b d(B), dt, et cetera. Now, those come from the coefficients, but the rate law does not, the rate law does not. So these are simply instantaneous rates, but now I say, no, I want a rate law that gives me the dependence on these concentrations at any time. So the rate law would be, for example, like rate equals k times A to the m, B to the n. Now, here's the thing. These come from experiments. Very important distinction, right? Those come from experiments, and so you could get the rate with respect to A which would be-- I'm sorry. The order with respect to A would be m, the order with respect to B would be n, and the overall order of the reaction is m plus n, the overall order. And then, we had one more item here of conceptually what I want you to know about, which is that the rate also depends on temperature. Sorry, the rate constant, rate constant depends on temperature. So if I were to now say, well, OK, I want to know something about the order of a reaction, you might get a table like this, and we did this in class. You might get a table just like this. Here's a reaction. It's W and X and Y, and it goes to Z. Don't let the fact that you have three reactants throw you off. You've got a certain number of reactants, they've got their coefficients, and they go to some product. And now, you're given experimental data, so from this data, you can get the order of the reaction with respect to each one of these. Right? And that would then give you, of course, the overall order. So let's see. If we were to look at that data, then what would we get? So what you want to do in these kinds of situations is, first of all, don't panic. By the way, on an exam, in general, don't panic, and if you have a question, you can ask a question, but also show us what you know. We will give you-- as you I hope have seen already-- we will give you as much credit as we can for sharing what you know. You don't have to get everything right to get a lot of points. Please share your knowledge. So in this case, what would the knowledge be? Well, it would be that I want to compare stuff, like trial 1 and 2. And if I do that, you see, OK, trial 1, trial 2, then two of them stay constant, but W doubles. OK? So comparing 1 versus 2, you got W goes to 2 two times, but the rate is same, so you know the order is 0 with respect to W. But if I compare 1 versus 3, then you know that, let's see, x goes to 3x and the rate goes to 3x, and so the order is 1. And then finally, for Y-- I will just jump to it-- the order is 2, and you can do that just by find one where only Y changes. You don't have to necessarily find one where just one changes. Right? Because if you knew one is first order-- if you knew one, and then you had another one, and they both changed. But maybe the rate didn't change, then you know they must have both the same order, if they change in opposite ways. Right? So you can get into this in different ways, just looking at these changes from one to another and thinking about what must happen in terms of the rate law, in terms of what you wrote out here. But you could also ask questions like with reactions, OK, this is first order. What's the dependence of the concentration over time as a function of the rate constant? Well, that just goes back to our understanding of the integrated rate equation. Right? And so let's see, oh boy, now I need to find-- I'm going to go-- I hate to do this, XRD. I hate to do this, but sometimes being first carries a consequence. OK. So in this case-- first of all, by the way, you look at reaction like that, you're like, well OK, from the coefficients and the mass balance stuff, you know that the change in oxygen, for example, with time, the instantaneous rate would equal minus the change in MO2 with time. Well, that's not what the question asked. The question says it's first order. So that's enough, because this means that if I integrate the rate law then-- first of all, if it's first order-- then d of MO2 with respect to time must equal some constant times the concentration of MO2 to the 1, first order. And so now, if I integrate this, I take this concentration over here, I put the dt over there. You integrate it. You've got all the integrated rate laws. We've gone through this already. You know that you get that the log-- oh, we do logs-- are this MO2 equals the log of some starting concentration minus Kt. OK? Why do we do that? Because we like plotting things, linear things. We love linear. So remember, we did this. We went from these integrated rate laws to plots and back and forth, and so that's also very important. OK. Now finally, so the last couple of things. And I'm purposely giving these last ones a little less time, because they're the ones that we've seen very recently, and these are the ones that we haven't seen for almost a month. Right? Yeah, but you still gotta know about solubility, and so for solubility, remember what we did is we did equilibrium and Le Chatelier, and we did the equilibrium constants, the K's. Like for example, Ksp is the solubility product, solubility product constant. Right? And then we also did ice tables, and then we talked about the common ion. And so as an example, if I were to tell you, well, OK, I'm going to take BaCl2. I might have done this one in class already. If I add BaCl2 to a saturated solution of BaSO4, why does it cause precipitation? Answer-- because of the common ion effect, but you could write these out. So you write out your saturated solution, so you write out your BaSO4, and you see that BaSO4 solid will go to Ba2 plus in solution plus SO4 2 minus in solution. And so the common ion effect tells you that if I now add something else that has something in common with one of these, that gives me an ion in common, then because of Le Chatelier, you're going to drive the reaction the other way. That precipitates, because it forms a solid BaSO4. So I've just done that. Right? I've given you BaCl2 which gives you a source of Ba2 plus, and how do you figure out if you were given amounts what happens, what the new equilibrium is? You use ICE. Right? You use the ICE tables. OK. And then finally, we talked about acids and bases. Oh, I really hate to erase defects, but here we go. Defects are gone. Arrhenius, I'm so sorry. That was painful. And the last topic here is acids and bases, and this is the most recent one, and so what do we have? Well, got the acid equilibrium constant, the base. We did water. We did pH. Oh, we did p anything. Right? p equals negative log of the thing, pKa, pKb. OK, and then we did, like for example, the definitions, so Arrhenius versus Bronsted-Lowry definitions. And so for example, if you're given a solution-- this is one, just a simple example of an acid-- and it's HOCN, and you're given the acid. And you know the pH, so you're given that. Then the pH from that information you can get that the concentration of H plus-- which is also a H3O plus-- is 1.7 times 10 to the minus 3 moles per liter. That's from the pH, and if you have that, then you can go to the Ka definition, which is H plus concentration OCN minus divided by the acid HOCN concentration, and you've got it all. Because if you're right the equilibrium equation, which is what Ka describes, the equilibrium of this acid dissociating into ions, dissociating into H plus and OCN minus, and so you can get Ka. OK. So what I hope is that this got us in the mood, that this got us all kind of on the right page with respect to these topics. You've got two days. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 25_Introduction_to_Glassy_Solids_Intro_to_SolidState_Chemistry.txt | Today, we're going to just start by talking about what glass is and what its properties are. But before we do, this came in-- we've been talking about defects, OK. And vacancies, point defects, and then on Wednesday, we did line defects. Remember those? Right, and they create these planes that come in. They're slip planes that allow atoms to slide across each other that allows you to plasticly deform a material. And so OK, but with regards to vacancies, I got this by email. I'm at the Heathrow Airport in London, and even this aerial sculpture has vacancy point defects in its pattern. Look at that. There you go. And there you go. This is a beautiful thing, right? Thank you, Sophia. And notice that she's looking at it with her periodic table, of course, right. That's a no-brainer. But even such structures can't escape the fact that you always have vacancies. That was cool. OK, now where were we? Right, so we're going to talk about glass. Now, this is glass, and I'll start with just how cool glass is. But this is also not just-- this isn't all that glass is. And I want to make sure we get that by the end of this lecture. This is not the only thing that glass means. But this is what we think of. So here's a really cool video. OK. No and yes. [VIDEO PLAYBACK] There they are. Look at that molten [INAUDIBLE] and it's-- oh, he just added something. We'll talk about that. Oh, and now he's mixing it. - Table man carefully mixes this glass [INAUDIBLE].. Once the glass is blended to his specifications-- OK, there it is. Is it a liquid? Is it a solid? - [INAUDIBLE] And then he just puts it through a little roller. - That will mash that glass flat [INAUDIBLE] Mash it flat, and there is your piece of glass. OK, and we'll see another one. I'll show you another video of a larger scale-- don't keep going. [END PLAYBACK] OK. So what was it that they were just mixing? Why does it have the properties that it has? Why can they just take this thing that's not really liquid and not really solid, it's some viscous thing, and roll it through, and then it comes into this really beautiful sheet that we all call glass? And it all has to do with what I showed you before, and what we've been talking about, which is order or disorder. And so we did this already. We covered, OK, a solid that which is dimensionally stable. That's cool. Classification, oh ordered, regular, long-range order, BCC, FCC. All right. OK, long-range. Remember, we said it just keeps going and going and going. And then we say, well, it may be once in a while, once every 1,000, 10,000, 100,000, there's a vacancy. OK, so we started messing with it. But it still had this long range-- gesundheit-- order. And so they were called crystalline, crystalline with defects, maybe, but still crystalline. That is not what glass is. Glass is disordered. And so that is the topic of today and next Wednesday. What happens when you go from an ordered structure, like this, oh maybe it's got some-- oh these are 2D defects, right, the whole things are kind of-- so it's poly crystalline, but it's still kind of crystalline, and it's very ordered, to totally disordered. All right. And so a disordered solid looks random, certainly over long range. There might be short range order. There might be some short range order, but over a long range, it's going to be disordered. It's going to look random. This is called amorphous as opposed to crystalline. This is also called glass. This is also called glass. So one of the first things we've got, this is really important, glass is not just the window, or the thing on your phone that keeps cracking. No. Glass is equal to an amorphous solid. Well, so it's bigger. It's broader. It's bolder. You can have glass metal, metallic glass. It's a thing. Any solid that doesn't have long range order that's amorphous is a glass. OK. Now in this class we will use SiO2 as-- so what we're going to do is we're going to talk about glass with an example. We will use the kind that winds up being in the windshield of your car or in the window of your house and on your phone. And that is called quartz. So the chemistry that we're going to use is quartz, or SiO2 OK, and what we're going to talk about is how processing effects that chemistry, affects the-- sorry, the solid that forms. So processing. And there is lots of processing parameters and we can talk about. What I want to focus on today is the cooling, the cooling rate in particular. So that's going to be the focus today. We'll talk about the chemistry. We'll talk about the processing. And we'll talk about how those go back and forth to create glass. Now, OK, what am I starting with? I'm starting with a crystal. This is quartz. This is how you see quartz. Now, quartz is this beautiful, beautiful crystal. And it is ordered. That's why I'm saying the word crystal. I'm not saying the word glass. All right. This is an ordered solid. It's not cubic. So it's not one of the ones we've learned. But it's still periodically repeating. All right. And so it's a crystal. It's got long range order. It's called quartz. It's SiO2 So I'm going to hand this out so you can just see it. And that's what I want to start, I want to talk about, is that crystal, OK, because that's what we're going to be messing up and creating short-range. We're going to go from long-range to short-range. OK, so what is it? Well, let's see. In this class, how do we think about chemistry? Well you can think about it now in different ways. Let's pick Lewis. All right, OK. So I say, OK, I got silicon, silicon, silicon, if I think about silicon as it's Lewis, you know, dots, then it's got four. It's got four valence electrons ready to bond. You can see that now if I have oxygen, what does oxygen look like? So oxygen has six. I'm going to put those like that. So oxygen has six. Maybe it's got a couple lone pairs, which has six valence electrons, not four. Now so if I had-- so when you look at this, you think, you know, silicon looks like it wants four bonds when you look at it. And in fact, we know that the crystal in silicon as we've talked about-- gesundheit-- is tetrahedral because of this. Right, it's tetrahedral. Remember, it's a two atom basis FCC lattice which is is diamond. But what if I just look at it with oxygen, come at this with oxygen and say, well, you know, I want to have four bonds. One way to do that would be to add four oxygens. So what if I did this? What if I did silicon and oxygen, and I did plus 4 times the oxygen. Well, then you could see, what could you do? You could do this, OK, and this and this and this. But now, OK, the oxygens, how would they be left? They'd be left like this. And you'll see why I'm focusing on this SiO4 group in a minute. That's what I got. Now, OK, if I go back in and I put my Lewis cap on, silicon is very happy it's obeying octet, but the oxygens aren't quite there. So they need one more electron, one more electron. I'm going to make it a little bolder because I've added it. All right. So I've added an electron to each of those. So if I had SiO4, and I added 4-- if I had SiO4, and I added four electrons to it, I get a really nice stable Lewis structure. Right. That's called a silicate. That's a silicate group. SiO4. And I need those four electrons, as you can see just from looking at the simple dot diagram. I need those four electrons to make it happy. OK. By the way, just speaking of naming, OK, so let's see silicate, right silica, SiO2, we can talk about, so what we do is when we have oxide, silicon oxide, just in terms of naming, you know, like if you have an oxide, often, you kind of add an a, right, so like silicon went from SiO2, we call it silica. So in Al2O3, let's see, Al2O3 is also an oxide, right? And so we called that-- this is something we've already done, alumina. All right, and so on, except for sometimes. And so sometimes, like for example, if you do calcium oxide, well, that's called lime. It's called lime, calcium oxide. Well, what about sodium? Well, sodium, if you do sodium, you're going to need it I'm assuming the oxygen might break off and become to-- so you've got to think about the charge again. So Na20 would be the one that works there. All right, and this is sodium oxide. But see, sodium oxide, but sometimes, it's called soda, because-- and this is an extra step, because soda comes from calcium from-- let's see, Na2O from-- how do you make it? You make it from Na2CO3, which is actually what is soda? Sodium carbonate. But because we make Na2O, that oxide from something that we call soda, sometimes people just call in Na2O soda, it's not. But sometimes you see that. So I just want to kind of make you aware some of these names. Why am I writing it there? We're coming back to that. We're going to use those later. That's partly why I want them to be on the board, and also tell you about the names. So Na2O is sodium oxide. But it's made from sodium carbonate, which is soda. OK. All right. Now, OK, now, here's the thing. That's an isolated molecule. Look at what I'm passing around here. Oh, I wanted to tell you about the window. I almost forgot. In this class, in this class, we don't go around saying that windows are thicker on the bottom, than on the top, because glass flows like a liquid. No. That is not true. That is not true. That may be true down the street. That is not true here. We do the calculation. Glass is not a liquid, not once it's cooled down into a solid. It's a solid. If you do the calculation, and it's a centimeter thick, a window pane, for that piece of glass, under normal diffusion and temperature conditions for that piece of glass at the bottom to become thicker by one nanometer would take 10 billion years. OK. So no, it's not flowing under its own weight in a window. What happened? What happened is people didn't know how to make glass uniformly way back when, and so they came out un-uniform, non uniform. What's easier? To install it with the heavier side up or the heavier side down? It's easier to install it if it's heavier on the bottom. Class doesn't flow. OK. All right. I wanted to get that done early. Yeah. Oh, here we go. OK, so this is quartz. This is crystalline silica. This is groups of SiO2, not SiO4. This was how we made a molecule happy. But watch what happens now when I go from that picture to SiO2. So now, I've got these-- let's suppose I've got these molecules of-- and I'm just going to not draw it 3D, but I'm going to draw it 2D, and OK, there's-- gesundheit-- there's all these charges there. So now these are these molecules, these silicate groups. And they're kind of happy, and they're on their own. But now look what happens here. See, those two oxygen, what if instead of each of these being separate, they came together? All right. Well, you can see that-- you see, if they shared the same oxygen here, if they shared the same oxygen here, then that oxygen is actually now Lewis happy without the extra charge. So this oxygen, this bridge, oxygen, doesn't need that extra electron to be happy. What does to be happy mean? Well, we know what that means. And if we're talking about Lewis and speaking in those terms, it means you've got your octet. All right. So now the bridge oxygen-- oh, but that is what I'm passing around. That is what quartz is. It's all the oxygen is happy without needing the extra charge, because they are all bridged. So that's what we've done. We've made silica by thinking about it in terms of these silicate groups, OK, and just bridging them all together. And I can't draw it. But you have it there, and 3D models are the easiest way to see it. And they all act as bridges between silicate groups. OK. So why am I-- why do I keep this silicate group? So a lot of times if you look up the structure, and this is the 3D structure, you see every single oxygen is a bridge. Every single oxygen is a bridge. And it's nice to think about it in terms of the silicate groups. Why? Why did I start with that? I started with that because these groups are very strong. They stay together. And they kind of act as building blocks for the glass and for the quartz. And you now know those are two different things, quartz is silicate groups ordered in a crystal. Glass is what we're going to get to. It's when they're disordered. OK. And why is that? Well, because these oxygen, you see the silica, the tetrahedron with the silicon atom and those oxygens is pretty robust. But the bond between the oxygens is-- well, there's a whole lot of rotation that can happen here. You see that? That thing can rotate around. So that's a good way to picture this. That's a good way to picture quartz is these silicate groups that are bonded together through these bridge oxygens, but they have a lot of rotational and distortion relative one with respect to one another, that is possible. So I take those silicate groups, OK. Here they are up top, and I bring them together, and I make a perfect crystal. That is what's being passed around. But look, because there's so much possibility for rotation and distortion, maybe that didn't happen. Maybe as they start to form or they find each other, and they start bridging, maybe they didn't quite make it. And you can imagine these things are kind of bulky groups coming together and twisting around. You missed the mark, it might be hard to get back. So how does one happen over the other? All right, that's the crystal. That's the not crystal, the glass. And it depends on a couple of things. And that's what I want to talk about next. It depends on the temperature and the cooling. Depends on the temperature and the cooling. So how does temperature come in? So temperature comes in because temperature makes everything vibrate. Right. It gives these atoms and these groups kinetic energy. OK. So let's look at that. So if we were to plot, for example, temperature versus the volume per mole, so this is, you know, for a given number of these atoms, much volume do they take? Well, you can imagine that that is actually related to how much energy they have. All right. So like if I had like a crystal, you know, maybe I'll start it here. And if I have a crystal, you know, and I increase the temperature a little bit, maybe they're going to move more. And as they move more, they need more room. And this is just a simple-- you can just think of like a simple atom, connected to another on a spring. And more kinetic energy, more temperature, it goes further. And now you're all thinking, well, wait a second, doesn't it go in as far as it goes out? Why? Does it need more volume? Well, that is because of what we have already talked about, which is this energy curve between two atoms, or two silicate groups. This potential energy curve that plots the energy between those two groups and, say, the distance between them, has something very important about it, has something very important. This is the spring. OK, so I'm in there, and I'm at T equals 0 here. That would be like T equals 0. The absolute ground state, the place where everything is just in its minimum energy, but now I start making things move. All right, I put kinetic energy into it, I increase the temperature, what happens? The energy goes up. But look, it doesn't go up. It goes like this. The average-- I'm a little bit higher in energy, right, I'm a little bit higher in energy. So where I am is there, on average. Now I'm a little bit higher in energy, and so where I am is there, and now I'm a little bit higher, and where I am is there. This is what happens to the average. That's why solids expand. It's because of the asymmetry in this potential energy curve, which we have already talked about. It's because of the asymmetry. You see. This is going out differently than this goes up. That's why you have thermal expansion. All right. Things are vibrating. But more often than not, they're going to be farther apart. And so this happens. And so this happens. So we've just understood this from an atomic-- so this would be like if I had a crystal, and I'm going to save three letters and write xtal. That's so efficient. And then I keep going. And by the way, the slope of this is called slope is called the thermal expansion coefficient. Thermal expansion. That is the definition of the thermal expansion coefficient. Sometimes we use alpha. That's the slope of this line. It's the volume per mole divided by-- over the temperature, the change in the volume or the change in temperature. It's the thermal expansion coefficient of the material. OK, so I'm adding temperature, and I'm expanding and I'm expanding and then I get to this point. I get to this point. This point here is where everything changes. Everything changes. That's the melting point of the solid. That's the melting point. Now at the melting point, as we know, the whole thing goes through a transition. It goes through a phase transition. And it becomes a liquid. And the thermal expansion of the liquid is different than the solid, because those, now, you think about those molecules are not rigidly bonded to each other anymore. They're more weakly bonded, and they're moving around with a lot of kinetic energy. And so when they get more kinetic energy, they can expand even more. That's why those slopes are different. But as we know, there's also a big volume change. There's a sudden volume change. In going from a solid to a liquid, there's a sudden volume change. And it would be the same thing if I went back the other way. Right, I'm cooling it down and cooling it down, and all of a sudden, I get to the solidification temperature of the material, and I become a crystal. Except when I don't. Except when I don't. And when I don't, that's when I become a glass. And that's what we have to talk about now is, how do we go from this very simple picture of being a glass-- being a liquid or being a solid, a crystalline solid, how do we go from that to being a glass? And to explain this, to start, I want to show you something really cool that can happen. That actually wasn't an intentional pun. But it has to do with cooling, and it's really cool, because sometimes, if you cool a liquid down, it doesn't solidify right there. Then that's called super cooling. So you can actually cool the liquid down like that, and have it stay a liquid. That's called super cooling. By the way, so this is called super cooling. By the way, you can do that with water. And so I want to show you like one of the coolest things you could do tonight. It's a Friday night. You're going to have guests over, or you'll be out at that same restaurant that knows about you, because you asked them already all about the candle and the oxygen. And you always come with your periodic table, and this time, you sit down and you say I don't want water. Do you want sparkling or regular? I want super cooled. This is what you're going to get. This is what they'll bring you. They should. If it's a good restaurant, this is what they would bring you. You know, so here's who is supercooled water being poured. This is what you should see in your glass. Now, that is a liquid that is stable, for now, below its melting point, below its solidification point. So what happens is as soon as it hits the glass, it's like, whoa, wait a second, I am supposed to be a crystal, and it immediately solidifies. All right, so it immediately went like this. Now, by the way, it's not that hard to make supercooled water. If anybody's interested, I'd be happy to tell you how to do it. You can't just put bottles in a freezer. That won't work. But it's not that hard to do. Now, OK, back and forth, melting point, supercooled-- oh, it's playing again. And it starts freezing. You can also take a bottle of supercooled water and just go like this. It's liquid inside and just tap it, and the whole thing freezes instantly. It's really cool. It's really cool. OK. OK. Now, why am I talking about this? Because you can super cool and go down. You could go below that transition point and go and go down, like I just did there and become a crystal right away, but you could also not. You could also become a solid, so you could become a crystal, or you could also become a solid. And I'm going to need another plot to do this. And you could become a solid right where you were. You might not go down to the crystal curve, but you might start a new curve. And that will be a glass. And so we're going to draw this again and show you a glass, but this is what we need is this framework to understand when a glass forms. So this was our picture, crystalline, crystalline, glass, amorphous equals glass. When does it form? Now I think the best-- I'll show you that one in a second. I think the best way to think about it is let's draw the plot again, and I think the best way to think about it is an analogy that has been used before, and let's see-- and this is temperature. This is volume per mole. By the way, that could be energy, or enthalpy. Yeah, because as the atoms aren't packed in as well, we know this, right, as they're not packed in as well, they're also going to be higher in energy. Right, the bonding won't be as strong. But I think that one of the best analogies to think about is so I'm coming in as a liquid, so I'm now a liquid. And here's my melting point for the crystal, for the crystal. But I went past it. And I went past it, and now, instead of-- I'm just going to draw this out here. There's the crystal. Instead of going down to become a crystal, suddenly or super cooling, and then becoming a crystal, no, I super cool, and then I just become a solid. And then I just become a solid. OK. Now that-- this is a glass. You know that it's a solid. It's got the same slope as the crystal. The thermal expansion of this thing is the same as the crystal or very similar. So that's enough to tell you this is a solid, right? It should be a good indicator. It should have been a nice straight line just for now. OK. Yeah, but it didn't get-- those silicate blocks didn't find the lattice. They didn't find the lattice. As you're cool-- here, they've got all this freedom. I mean, they're a liquid, silicate, happy. Maybe it's an even higher temperature, and the silicons are even dissociated from the oxygen. Let's just assume they're in these silicate groups, and they're floating around in a liquid, and I start cooling it. I think one of the best analogies that I've heard is that of musical chairs. How many of you have played musical chairs? OK, you've got to fix this tonight. For those of you who didn't raise your hand, that's going on the menu tonight, because it's a really fun game. With supercooled water, of course. You line up some chairs, and you get around in a group and you walk around it as the music plays and then the music stops and everybody has to find a chair, but there's a chair missing. So you've got to go fast, or you might not get to a chair. Well, you see the silicate, let's put it below here, the silicates are you. You are the silicates. People equals silicate. And the chair equal lattice sites. And now you can really feel it. You can really feel it, because the speed around the chairs. How fast are you moving? Right. So the speed, well, if it's speed around the chair, OK, if that's fast, then it's, you know, high mobility, so let's see, speed around chair, let's do this. High mobility, maybe you'd find, you know, find lattice sites easier, or faster, at least, but see, if you have a high 1 over mobility-- by the way, that's also viscosity, if you have a high 1 over mobility. So if you have a high viscosity, [INAUDIBLE].. High 1 over mobility, high viscosity, then it's slow. And that's how you have a higher chance of forming glass. So I'm going to talk about what would make a glass form. If you're just walking around slowly, like this, everyone else is running, but maybe you're all walking slowly, and now the music stops, and I'm like, yeah, I'll take my time, it's going to be hard to get to the open lattice site. I might just get stuck. OK, so what else? Well, OK, so how about the arrangement? So the chair arrangement, that has to do with the crystal complexity. And you can imagine that if the chairs are arranged in a straight line, and you're going around the chairs, and it's is kind of an easy loop, right, now I don't have to concentrate too much on where I'm going, and it's kind of-- there's nothing really blocking me, I hope, but what if those chairs were arranged in a really complicated way. So how hard is it to get around these chairs? How hard is it to find them? How complex is the lattice? And you can imagine that it is higher. So let's see. More-- I'll just draw an arrow. Higher would lead to glass. OK, so it fits. Higher complexity of the lattice. It might be harder to find the lattice sites in this musical chairs. OK, one more, and I'll put it here, so that it's close by, because I've got no more room on that board. The third one is how fast you stop the music. So this would be like 1, 2, and then 3. How fast do we stop music? OK. So you can imagine if I, you know, if I slowly were playing musical chairs, and I am walking around, and I slowly fade the music, all right, well, then you got lots of time. You got lots of time, but what if I just stop it. Now, everybody's scrambling. So how fast do you stop the music is the cooling rate. It's how quickly you cool this material. It's how fast you stop the music, and you can imagine that if it's faster, faster, you are more likely to form a glass. You're more likely to form a glass, and so we go back to our picture. Well, maybe I'll draw one over here, because I need more room, and I don't want to block what's there. We go back to our picture and here it is, drawn out for you. You see that is-- OK, there is the liquid line, coming in, volume, OK, enthalpy, visage, and we'll keep thinking about it as volume. The liquid line comes in. You're cooling it down. There's the thermal expansion. It's reducing, reducing, and now I get to the melting temperature, and I become a crystal. That's if it can become a crystal, but all of these factors, now, matter. The melting point of the crystal is always the same number. It's always the same number. The melting point, where the crystal melts, because the crystal is always the same. Right. Now, where that melts is always the same temperature. But notice now, what have I done? I've super cooled the liquid down, and on a, I've got a glass that looks like this. These are glasses. And on b, I've got a glass that looks like that. Those are different. And they have different solidification temperatures. And what do we call it? We have a special name for it, too. We have a special name for it, because it's not crystallizing. Right, so if this is the volume per mole, and this is the temperature, it's not crystallizing. No. What is it doing? So that would be the melting point. It is turning into a glass, and so we call this the glass transition temperature. It's a glass transition. What that means is it is still becoming-- gesundheit-- a solid, but it's an amorphous solid. It's not a crystal. Why did it become an amorphous solid? Well, maybe I cooled it really fast. Or maybe it's a really complex structure, right? Or maybe the viscosity of the material is very high, and it just-- as I'm cooling it down, it just couldn't find the crystal lattice, which is all the way down here. And you can see from this, that you know, imagine I take the extreme limit, and I got my liquid of my silicate groups, and right as I get below here, I'm like you're all frozen. I'd literally-- I'd make a glass right here, right up there, all right. So I can make glasses that are different from each other. And you can see that the volume per mole is increasing as I go up, and you can understand that, because look, I took a glass. I didn't let-- I took a liquid here. This is the liquid. I didn't let any of it find crystals sites. It's a liquid. But I said, ha, you're frozen. And it became frozen and stuck at a high volume per mole. Because it's totally random and amorphous. It's coming right from all those random liquid degrees of freedom are just frozen in. Yeah, but now, I cool it slower. Maybe that's what happened here, right? And I give it more time. And I give it more time to find some of those lattice sites, and it finds some lattice sites, but then it became a glass at this temperature, the glass transition temperature, and everything else is frozen in. So if I label this, so let's label it the same as there, if I label a as this one and b as this one, and this is the crystal, nope, the crystal would have to go all the way up to here. There we go. Right. So if that's a, that's glass a, and that's glass b, then which one did I cool faster? Well, I had to call this one slower-- if all I changed between them is cooling rates, then this one is cooled faster. So let me write it here. a cooled slower than b. Right. Because I get-- and that's this one here. How fast did you stop the music? Right. It's the cooling rate. OK, so this plot, this volume versus temperature really allows us to understand how glass forms, or at least we can plot, you know, what happened, and then try to understand it. And if it did get all the way to the crystal, then maybe it means that it's not very viscous as a liquid, it doesn't have a complex lattice site, and maybe it didn't even matter how quickly-- oh, metals. But that's why it's hard to make glasses out of metals. Because metals have an easy time finding their lattice sites. Metals as liquids typically aren't very viscous. And so even if you cool it faster, it's got time. The musical chairs of metals is easy. It's an easy game. That's why the silicates never played musical chairs with metals. But I can still do it, and the way you make a glass out of a metal, the way you make a metallic glass, which is a really cool material is you freeze it really quickly. You will lower the temperature as quickly as you can, really, really fast, and you freeze in the disorder, and you make a metallic glass. OK, so those are the two curves I just talked about there. That's the glass curve. This would be the crystal curve. OK, two different cooling rates on here. OK, now, here's a picture of float glass. This is the kind of glass-- we're going to move from understanding what glass is and understanding in terms of this temperature versus volume plot and amorphous solids, and we're going to move towards understanding how to modify it, how to control it, because we do a lot of that. Here's an example of a factor right-- I like this. This is flow glass. So this is like when you buy, you know, your windows, or you know, a lot of times, when you need a very clear, beautiful piece of glass, you go up to very high temperatures. All right, so you're taking those silicates. You make them into a liquid. Here he is, measuring the temperature. I don't know what he's doing actually, whatever. Oh and this isn't showing up very well. And there's no volume, so double-- OK, there it is. So there's the liquid glass. Why is it called float glass? [VIDEO PLAYBACK] [INTERPOSING VOICES] - The process is monitored continuously by technicians to ensure quality. But because it floats on top of another liquid. That's how you get the glass because so incredibly smooth. - And ensure uniformity-- And there it is coming out. There's a nice wide piece of glass coming out, and what he says at the end of this, there it is coming out. [END PLAYBACK] They can do kilometers and kilometers per day of this float glass. You float it over molten tin. The tin acts like this really nice smooth surface that you put the glass on top of. So one liquid on top of another makes a very nice smooth interface. And what he talks about, which is what I wanted to capture, is how important that cooling is. So the line has to go in a very particular way after it comes out, as you cool it, and because it's all about the cooling. As you come all the way back down to here, room temperature, right, or as I-- maybe as I went through this transition, what's happening inside? Am I creating a lot of stress? We just learned about stress. Am I creating a lot of stress? Right, or is it going to cool in a nice way? Are there going to be cracks that form? So that is one of the most important processing parameters. What we'll learn next week is that chemistry is also a really important pressing parameter. And that's why you have such good phone screens. Actually, you don't really, because they always break, but they would break even more if it weren't for chemistry. OK, but certainly cooling speed is really important. So let's go to the-- OK, now how-- OK, if you look at glass, remember I said that it's got order or no order, crystalline, amorphous, crystalline, amorphous. How do you tell which one you have? Who can tell me? You just learned about it. What do you do to see if something is a crystal and which crystal it is? You shine x-rays on it. You go to your x-ray source, and you just shine it on, and you measure where it bounces off. But see-- so if you did that, so if you look in the circle, they look the same. They look the same. That's crystalline. That's quartz. This is glass. Quartz, glass, crystalline, amorphous. Right, but if you go to the long range order, there isn't any. So if I shine x-rays, these are two different types of solids. Think about it like DCC FCC, but they're not. They're more complicated of quartz or of silicates. This is what glass would look like. Look at that. There's still like a little bit of a peak there, but not really, right, because this is from the short range order, but here's the important part, which is where-- you can still, even though it's not that everything is where you think it is, right, because it in a crystal, you know everything's BCC or FCC. Every lattice is the same. You know you can count on it for 10 to the 20 second repetitions. Here you can't. You can still control the properties. You can still control the properties. So the properties can be highly engineered. And here's an example. So these are different types of glass. This is a glass cup, and this is a glass bottle. And you might think, it's just glass. No. It's not just glass. It's very complicated mixtures of silica with other things. Notice, it's got soda. This is-- like I said, this is not soda, but it comes from soda, so it's called soda, sodium oxide in Na2O. Lime CaO, magnesia, alumina. All the glass-- you know, unless you pay a whole lot of money for something made of quartz, almost all the glass you buy has this kind of stuff in it. In fact, 90% of all glass is called soda glass, because it's got-- that's this one here. That's this cup, because it's got sodium oxide in it. And the question is why. Why do we put these things in glass? Why did we used to put lead in glass? In fact, the history is pretty-- goes back thousands of years. And I found this. If you go to ancient Rome, they made glass, and you can look at what they made glass out of. They had a little bit less silica. But they mixed all sorts of things in. They did not have 3091 back then, so they didn't really know why this was doing something useful, but it was. Now we mix all these oxides, remember I wrote some of them up here, lime, soda. We mix those in, and we mix all sorts of other things in, but now we know why. And the reason why is all about the chemistry, and it all goes back to this picture right here. It all goes back to exactly this Lewis structure that I started with, which is not on the-- stop it. No. It's stuck here. Maybe it's here. Yes. That made me happy. This all goes back to the chemistry. If I deliver, if I deliver into this system something that gives me charged oxygens, something that provides a little bit of O minus or O2 minus, then I could cut this bond back. I can cut this all the way back to the original silicate, sort of. If I had an oxygen with charge on it, I can go from here back to where I started, right there, at least for that bond, right? And so what every single one of these things has in common is that it can provide into the system a charged oxygen, a charged oxygen atom. If this dissociates, it gives we Ca2 plus and O2 minus. If this dissociates, it gives me two Na pluses and an O2 minus. Oh, a pattern is forming here, O2 minus, O2 minus, O2 minus all the way down. So these modifiers, these things that we use to engineer glass, they give me the O2 minus, and that is the knife that I use to cut this glass apart and modify its properties, and it's where we're going to pick up on Wednesday. But because it's a Friday, I want to throw t-shirts out. We'll do more next week. Have a really good holiday weekend. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | 7_Aufbau_Principle_and_Atomic_Orbitals_Intro_to_SolidState_Chemistry.txt | I want to start right where we left off, which is in filling atoms, filling electrons into atoms. We've done a lot of work to know what it is that we're filling. We had to go all the way to quantum to know what orbitals are, not just orbits. Right. And we talked about the four quantum numbers, right? And so now we're going to use those, and we're going to use the exclusion principle from Pauli and a few other things. And we're going to fill electrons into atoms. So this is where we left off on Monday. There they are, right? Remember, you know, as I said on Monday, if we only had Bohr, what we would have is the left hand side here, n equals 1. We would have just have n equals 1, 2, 3, 4, 5, but we wouldn't have any of this variation of l, right, like the p electrons, the d electrons. That differentiation came in when we solved the equations of quantum mechanics, right, for an electron in feeling the potential of the proton. And then the other thing that happens is these things don't line up necessarily exactly how you might imagine, because there are complicated effects, right? It's not just a naive feeling. Instead there's these effects like shielding. Right, so electrons all the way out here in an atom, they don't necessarily see all the protons. They're shielded. And then to make matters even more complicated, these orbitals can wiggle in with nodes, right? This is what we did Monday, with nodes all the way in so they can see-- some of these orbitals have a little bit of them that see the protons up close. So we showed that with 2s. That's called orbital penetration. And that's why these things split up. They have the same principle quantum number, n is 2, but this has a lower energy than the 2p. But how is the electron moving within this orbital? No. It's not. That's a classical way of thinking, right? And it's so hard to stop thinking that way. I know we want to think that way. But it's not. An electron is being in its orbital, because an orbital is a probability distribution. All right? So the electron, all we know from that orbital is that the electron has a probability of being here sometimes and a probability of being there other times. That's what it tells us. So we know that in that 2s orbital, sometimes it's really, really close to those protons, right? That was that orbital penetration, the peak that you saw on Monday. All right, so with all that knowledge just kind of getting us back in the mood here and our quantum numbers and our Pauli exclusion, we're ready to fill. And the way that we're going to start is the way that most chemistry textbooks would start, which is with a very simple rule, all right, called the aufbau principle. And aufbau means-- so aufbau means filling up in German. And basically the idea is that you fill-- you fill from the lowest levels, the lowest. Those are also the ground state. Right, the ground state would be the lowest, right? That's the lowest energy, lowest levels, ground state level, energy, and up. OK. And you use what is called the n plus l rule. The n plus l rule means that the ordering, the ordering of the orbital energies-- orbital energies-- and let's get a little more space here OK. The ordering of the orbital energies increases with increasing n plus l. All right, you fill from the lowest energy up. So as my n plus l, those are the-- remember, that's the shape. This is sort of related to the distance from the nucleus. This is the principle quantum number. This is related to the shape of the orbital. Right l, and as that number increases, the energy, that's how you fill. Oh, but there's the case where it might be the same. All right, so we have to have another point here, which is when two orbitals-- I did it again. I forgot my r. Two orbitals have the same n plus l, then the lower n has lower e. OK, this is called the n plus l rule, but rules are made to be broken. So as we're going to see, this is actually really only true maybe 80% of the time. But we're going to use it, because it is a good framework, and it captures-- it gives us a way to start thinking about electron occupation and atoms. And so if you take these two points, this filling up and the ordering of how you fill things, then you can draw this as a very simple picture. This is from the textbook Averill, right, where what you do is you follow the arrows. OK, so you go this arrow, I'm filling 1s, this arrow, then I go to 2s. Right, it's not going anywhere. So you go there, and you keep following. OK, so 2s, 2p, so you'd fill 1s, 2s, then 2p, right, because of the n plus l rules is in effect. And then, OK, 3s. n is 3, l is 0. That's n plus 2 is 3. 2p, 2p, n is 2, l is 1, same value. But now I go to this one here, and when they have the same value, the lower n has lower energy. Lower n filled first. Filled first, because the ordering is to go from lowest energy up. OK. All right, so that's our rule, which will be broken. Now, how does this look? Well, if we do this, we're going to get a little notation here and that's important, if we do this, and we just do a few atoms here, so for hydrogen, right, so for hydrogen we'd have-- that's the 1s orbital, and we're putting in one electron in it. Now, the notation here is something that we're going to use a lot. So let me explain this. This is n. This is l. That's the l in, what, spectroscopic notation, right? Remember we talked about that Monday, right, s, p, d. OK. And then this is the number of electrons. And so if we keep going with this notation, then in the case of helium, so for helium you would have 1s2, because I fill a second electron into the 1s orbital. Now, this is where Pauli comes in. Right, Paul said, OK, enough. You're done with that orbital, because I've got two electrons. They do have different quantum numbers. They've got the same n. Right, they've got the same l. Their m sub l's are the same because there is no range here for s, right. Remember m sub l for s can only be, you know, zero. But they have different spins, up and down. And so sometimes, what is also really useful, and we'll go back and forth between these pictures, is you go back to what we drew, you know, with the Bohr model, where we draw this in terms of energy levels. And so if you drew this in terms of energy levels, this is what'd you see. So you'd have like energy. And remember, in the hydrogen atom, this is minus 13.6 electron volts. And so if you wanted to draw this in this way, you would say, OK, there is one electron there, and we'll-- remember, we said you can draw electron spin with arrows. Right, whereas here, if we drew this, then we draw-- I'm not putting the energy there. We don't know it, because Bohr 13.6 good for hydrogen. Right, but here-- oh, and by the way, this would be the 1s orbital. This would be the 1s orbital. But in this case, I've got two electrons, one up, one down. And now, Pauli says you're done. All right. And so if you go to lithium, then you have 1s2 2s1. And so if you drew this in terms of an energy diagram, you'd have the 1s, and we're putting that, and then you have the 2s somewhere up higher where we put one electron. Those are-- that's how you would see these atoms in terms of the filling of the electrons into their orbitals. Right. OK, good. Now, OK, let's keep going. So what we'll do is we'll do this here. So let's see. So beryllium would be 1s2 2s2. And boron would be 1s2 2s2 2p1. OK, this is looking good here, right? I'm really following aufbau now. Oh, I went that way, then I went that way, and then I went up here, and I started on this arrow, which is just a way, graphically, of showing those rules. Good. Now, we get to carbon. And now we have to think about this, right, because for carbon, I've got 1s2 2s2 2p2. That is correct. All right. But if I think about this in terms of the energy, 1s2, 2s, right, well we know from before, right, we know from before, like 2p. Where is 2p? There it is. It's got six electrons that it can take. Because now m sub l can vary from minus 1, 0, plus 1. Remember, that corresponded to the three different orientations of the p orbital. OK. But that means that I've got three, so I can fill this with 2, 2, and then the question is, what do I do? I don't know what to do. Do I do this? Is that right? No, somebody says. And that person knows that the answer is no, because that person knows Hund. Where's Hund? There he is. Would have been more dramatic had he appeared, but there he is, because Hund came up with another rule, which also is broken, sometimes. Not as much as aufbau. You know, so Hund said look, when you have a case like this, electrons in the same p, so the same like orbital, sub shell. OK, it's not the same sub shell, but it's the same n 2p. Let's draw that. It is the same sub shell. I'm saying opposite things. 2p, 2p. Shell, sub shell. But there are different possibilities, and how you fill them according to Hund is you maximize what's called the multiplicity, which means that you want the electrons to come in with the same spin in different orbitals, in different p orbitals. That's how you fill them. Why? Well, that has to do with more quantum mechanics, and it has to do with something called exchange energy that's not part of what you need to know about. And I'm not going to teach it here. But you know, you can think about it simply as the electrons want to spread out, because they repel each other in the same orbital. Right. They want a lot of things, electrons. They want to be close to protons. But they want to be kind of not close to each other. They need their space. That's not a lot of things. It's two things. But that's what they want. And this allows them to maximize that and lower their energy. And remember, lowering your energy means happiness for electrons. So this is the preferred way to fill, right? And that's what Hund's rule tells us. Now, if you go to silicon, and you say, well, OK, what does silicon look like? All right, I'm going to put silicon-- no, don't do that. I'm going to put silicon right underneath carbon, because I think it's an interesting comparison. All right, so silicon is OK-- 1s2 2s2 2p6. They're all filled. Right, and then you go to 3s2 and you go to 3p2. And this looks very different than carbon, but actually, you can abbreviate the notation, and we very often do in these filling notation, and we do it this way. We say, well, this has a neon core. So you go to the nearest noble gas, not with more electrons, because they're not there, with less electrons. You go to the nearest noble gas, and say well, OK, that's basically neon. All right, this is neon. And so you can write the notation is this-- it's the same atom, just a slightly abbreviated way of writing it, where I would write, OK, this is a neon core with 3s2 and 3p2. Well, you can write that with a helium core, 2s2 2p2. And what happens is you wind up having a nice distinction here, because these are core electrons. Remember, we're filling electrons here. That's electrons. And these are called valence electrons, and we'll be talking about valence chemistry for the whole semester and what these valence electrons do, right? And in fact, these valence electrons, these ones on the outside closest to the out-- furthest away, these are the ones that have all of the chemistry that we care about in this class, chemistry. It's so important that I went all caps. I'm not shouting, but I went all caps. Whereas the core electrons are kind of mostly inert. They're mostly inert, chemically inert. OK, and so sometimes it's very useful to write the notation this way to see what do you got in the valence, what's going on? Right, what does this valence-- because then you start to see similarities. The valence for lithium and the valence for sodium look really similar. There's an outer s electron. That's the quantum number, the principle quantum number is different. But the valence looks very similar, one electron in an s orbital. And that allows us to think intuitively about similarities between elements, because if they have the same valence, and the valence is responsible, not same, but similar, similar shapes of orbitals, different principal numbers, but if that's similar, then maybe the chemistry those elements do is also similar. This is a helpful way to think about it. OK. I told you that rules are meant to be broken, especially when we're talking about n plus l, and it happens all the time. And there's two main reasons or important reasons, and then there's a whole bunch of other very complicated ones, that two I want you to know about, which are exceptions to this aufbau filling, OK, which are exceptions to that, come in the following situations. That you get stability-- you get stability. That means lower energy for the whole system. The system lowers its energy and can be more stable when either an orbital is fully filled-- there it is, ns2, np6, nd10. You know those are fully filled, because that's how many elections you can put in each one, or half filled exactly. And so that's why in chromium if you predict with aufbau, you would get 4s2 3d4, but actually you get 4s1 3d5, because you can half fill both of those, and that adds to the stability. In copper, the predicted one would be 4s2 3d9. But actually what you see is a fully filled d shell, which adds to stability and a half filled s shell. So it's willing to make this trade off of a fully filled s show for half filled. That's not a big trade off to get the d, fully filled. OK, now there's more complicated effects, all sorts of effects that also lead to violations of the n plus l rule. I really hesitate to call it a rule, given that, you know, there's a lot of examples where it's not about the half filling or the full filling. It's about complicated interactions that have to do sometimes, all the way with things like relativity, relativistic effects, literally, of these electrons, which will not be on any quiz or exam, with also the complexities that happen with orbital penetration as you add more and more electrons and the levels are kind of closer together. They're a lot harder to see clearly separate energies than in the first two, three rows, as you go down in the periodic table, those levels are closer and closer together. So anomalies happen. But they're not really-- I'm not sure we should call them anomalies, because these are all exceptions to aufbau. So like I said, I want you to know aufbau, because it's a great way to start, but I also want you to know that sometimes things switch around because of the effects that we've been talking about. I don't expect you to memorize all the exceptions to aufbau, OK? But I do want you to know about these two key factors, which are half filling and fully filling leading to enhanced stability. All right, good. OK, so we got valence. We got filling. Now, right away, right away, all right-- so by the way, configurations are in here. It's a beautiful thing. When I look at this, I now can look at this line in here that shows-- that shows using this notation, the filling of electrons for every single atom, every single atom. And it's a beautiful thing because it tells us so much about the periodic table. It tells us so much about why the periodic table is what it is. It was arranged. Right, it was arranged because of things like the combination, remember Mendeleev, mass and properties. But now we see why. We see why. It's incredible, because it goes back to counting. It literally goes back to the quantum mechanical derivations of orbitals and the principles of filling them, and then the filling them themselves. And you see, you know, why does the lanthanides and actinides, why did those things run for 14? Because that's how many electrons go into the f orbitals. Right, that's how many electrons go into 4f and 5f. That's why these are 14. Why are these two columns here? Why do those have similar properties as you go down? Right, I mentioned lithium and sodium, right. Well, because of what we just said. You're increasing the principle quantum number, but the valence chemistry is very similar. They're all s. And we even call them sometimes these blocks by the valence chemistry that's getting filled. Right, so those first two columns are the s block. Here over here, they're the p block. It's not that these don't have p electrons in them. It's that these are the electrons that are getting added in this part, in this region, whereas here, you've got d electrons. And now we know exactly why this is 10. I mean, we know even more than this, because aufbau says, why is this here and not here? Right, because of the ordering of the filling, of the energies of the filling. These three d's come after 4s. That's why they're here. Right, that's why they belong here. We now know so much more. We have so much more insight into the ordering of the periodic table, because of electron filling. And we'll be making those connections. We'll be making those connections a lot. Right, but I just wanted to kind of show you the power of seeing things in terms of filling. The one connection I'll tell you about today has to do with diameter. OK, so we'll be making connections all throughout the class between electron filling and properties. Right, but let's start with a simple one, how big is an atom? Well, now you can understand some things. Right, so for example, if I plot here or if Averill plots it, and I take that from Averill and show it to you, the probability-- remember we plotted this before for 1s, 2s. Now, what they plot here, OK, the 1s orbital for helium, that's got two electrons in it. There it is right there, the 1s, and then the 2s and 2p combined. Right, everything is sort of thrown together here. They're not separated for neon and then argon. So these are the first three noble gases, helium, neon, argon. OK. And a couple of things. One thing you can see is that the s electrons in helium are furthest out. And then the neon ones are closer in, and look at that. The argon ones are even closer. Why? Why are the 1s electrons getting closer in as I add more electrons? Right. Well, it has to do with what we talked about Monday. Those 1s electrons are not screened. So what they see as you go from helium to neon to argon is more and more positive charge. In fact, I've got the numbers here, if you look at the 1s electron energy, remember for hydrogen, 13.6. Now remember, I didn't write a value there. But there's two protons. So that 1s electron for helium. So if we take 1s electron energy, so much abbreviation. For helium, it's 24.5 eV. That's how much it would take. So we can put minus, if you want to compare with this, right, this minus 13.6. That's also how much it would take to ionize that 1s electron. That's where it sits. But if you look at neon, neon is minus 869.5 eV. And argon is minus 3,206 eV. That is a lot of energy. That is a lot more energy. That is much, much happier. If energy-- if lower energy is happiness, those 1s electrons are super happy, and they get closer to all that positive charge. But you can also understand why, like neon is larger than helium, and argon, even though these energies of the 1s are closer and closer, these are still going to be larger, because as you go down in a column, you add a whole shell. Right, so I can't-- right, so remember, here's my plus charge. Here's my minus charge. I can't-- this is 1s. Now I can't add anything in here. I got to add stuff out here and go out. Remember those principle quantum numbers, 1 to 2? Right, we don't need the s here, just 1 to 2 to 3, they're going to push you further and further out. We talked about that Monday. So now the next electrons that come in in neon's case in 2s or 2p or in argon's case, 3s and 3p, those are going to come much further out. That's why when you look at size, which is what's shown here, you see this trend. All right. So you see this trend that as you go down a column, here they are. Here they are. So this is the size in the periodic table, and these are calculated. These are calculated. Why are they calculated? Because you can see, this should be pretty hard to measure. Right, these are hard to measure unless the atoms are in like a solid, and they're bonded together. So measured atomic radii are often taking half of a bond length. But we can also calculate them. And that's why these are calculated atomic radii. So these have no bonding. They're just isolated, and they're calculated and there's helium, neon, and argon. OK, fine. So we just understood that. And we understood this arrow that the radii get larger as you go down the periodic table. But what about going across? Why? I'm adding stuff to the atom. I'm filling lithium here. I'm adding an electron and from lithium there to beryllium. How can it get smaller? Right. How can it get smaller? So that has to do-- what did I do? OK, I'm going over here. All right. So that has to do with the fact-- with the same things that we've been talking about. It has to do with the fact that as I go from lithium. So OK, let's do this. If I have hydrogen, and I go to lithium, you know that it's going to be larger for the reasons I just said. But if I go from lithium over to beryllium, it gets smaller. And the reason is because of exactly this. If I go from hydrogen to lithium, I add protons. But I shield those protons with electrons. OK, but if I am at lithium, let's see, lithium is here. So I've got one, two, three. And I've got an electron here. Right, I've got two electrons in the 1s orbital, and then I've got one electron in the 2s orbital. OK, so that's lithium. But now I go to beryllium. And if I'm beryllium, I've got four. OK, and so now I've got those 2s electrons here. But see-- I'm sorry, 1s. But in the 2s orbital, I can put a second electron. Right, and so what happens is those two electrons in the 2s orbital, they don't really screen each other. They are screened by the 1s orbital. So if I now-- because I went-- even though I added an electron, I also have a lot more charge here. Right, and because these two don't screen each other, but these two do, right, these are going to be closer in, because they're more attracted. They're only screened by these two electrons. So the additional positive charge wins. If I go down in quantum number, then I get a whole shelf filled to screen. But here, I'm just adding another electron to the same orbital to the 2s orbital. But I've got a hold other proton, and the shielding, the screening is the same. That's why beryllium is smaller than lithium, even though you've added an electron. And it's why the trend goes like this, increasing that way, decreasing that way. Now, I like dancing. And so I like this dancing analogy, because I love thinking about atoms as dancing. I told you about the disco and the periodic table. But you can take that all the way to the atom. So you know, you can imagine that you're an electron, and you see a proton. OK, so dance pairs are happy. Dancers are happy if they have more people to dance with. And so if I'm hydrogen, and I'm an electron, and I see a proton, protons can only dance with electrons. And then I say, OK, let's dance. And you kind of start moving, you know, a little bit. And but now, I'm going to go to helium. So there's another dancer in the middle, and someone comes on, and says, oh OK, that's cool, right. I can-- Now, either one of us can dance with either one of those. All right, so I've got two different people. I'm happier closer in. We can get close, because there's choices for who you can dance with. All right. So that's why you get helium much smaller than hydrogen. More people to dance with. Now you're lithium, and you come along, and I'm a lithium electron here, and I'm like I want to dance with somebody. Who's there? And there's only one person, because these two people are totally occupied. I only have one possibility for people to dance with if I'm this lithium electron. Right, but now I come in, and I'm beryllium, and these two electrons, and they're both like, well, OK, you guys are blocking, whatever. That's not cool. But there's four people in the middle. So there's always two or three people, right, that we can dance with. That wasn't like a move. I don't know what that was. But I'm like I can dance with you or I can dance with you. And more dancing choices, happier. Happier, lower energy. So that's how I like to see atoms. OK, so that's one example of filling and screening and dancing, where this kind of picture of the atoms, this electron filling of the orbitals explains what we see. OK, now, we talked about ionization, and we talked about how sometimes those outer electrons can be lost, or sometimes, maybe an atom would gain an outer electron. If an electron is lost from an atom, called an ion. And so a charged atom is an ion, and a cation is a positive charge, positive charge. Abbreviations. Right, and so here's sodium, and it's got all those dancers in there, and it has lost an electron. There it is. It's going to lose-- remember we talked about the first ionization potential. We talked about that last week. It's going to lose that lowest-- sorry, the highest energy electron, the least bound electron, the one that's all the way out there. And so if it does, it's NA plus. OK, gesundheit. Now, you can also-- this is from Averill. You can also lose more than that. All right, magnesium might lose two electrons. So the gray part here is the ion, and the blue part is the original neutral atom, all right. And you can see that if sodium loses one electron, and magnesium loses two, those actually have the same number of electrons. They're different atoms. Those are called isoelectronic. Well, you can also gain electrons. So if you write that down. So you know, if you gain electrons, does anybody know what those are called? Anion, OK, negative charge would be in an anion. OK, and I can ask question. If I lose an electron, it makes sense that I got smaller. This is the radius. And if I gain, it make sense that I got bigger. And I can mix it up. So I could ask you a question, like what's the ordering-- what's the ordering of these atoms. Right, I could say, well, if I had Mg 2 plus and Ca 2 plus, and Ca, what's the ordering of their size? You now know how to answer questions like this, right? If I lose electrons, I'm going to get smaller. If I gain electrons, I'm going to get bigger. But I also know that for example, Ca 2 plus is smaller than Ca, and Ca is bigger than Mg. Right. And so Ca 2 plus is going to be bigger than Mg 2 plus. Right, that makes a lot of sense. Done. You can order them now. All right, so Ca must be bigger than Ca 2 plus, and that must be bigger than Mg 2 plus. So we can think about now adding in ions to the mix. And we're doing this for a reason. All right, we're doing this for a reason. But because, you see, some atoms really don't mind doing this. They really don't. Sodium is one of them. They don't mind losing atoms. I mean, yeah, the electron is there. But it's kind of there as sort of like, I could be here, I can be there. I'm good. Whereas those 1s electrons, you know the 1s electrons are argon, are like you just try to take me out. All right, no way. But we will. We will ionize those next week. But for now, we're taking these outer ones out that may not be strongly balanced. They're happy, losing or gaining. And how do you know? How do you know? If you have ions, how do you know, you know, whether you have them? Well, here's the deal, you now know a way to tell because in this goodie bag, which I'll talk about in a few minutes, you actually have the most accurate measuring device ever made. You have a scale. That's not-- no. It's in here somewhere. You have a conductivity meter, somewhere in here, which I cannot find. And the conductivity meter is exactly that. There it is. Oh, look at this. This is incredible. So you can put this into a solution, and it literally just measures-- it's got two electrodes that go in there, and it measures whether there's any conduction, and you can see, well, if I've got charge species in the solution, ions, then this will tell you, because they'll help conduct electricity. So you now have a way of knowing when you have ions or not. OK, so that's good. But you see, the thing is something very important happens when atoms come together-- when atoms-- don't measure anything yet. But take it to lunch. Take it to lunch. You don't know. People talk about electrolytes. They talk a big game. All right. Now, the thing is when atoms react, when-- this sounds like the beginning of a novel. When atoms react, right, sometimes they will gain or lose. When atoms react, they may want to gain or lose the amount of electrons to get them to the closest noble gas. OK, closest noble gas. Now, that's really important. We just talked about how you use the noble gas notation-- did I just erase it? I did. Because that could be chemically inert. And we can only go down when we do the notation, but now we can go up, because I could add electrons to get to the nearest noble gas, in terms of reacting. And look at this. If sodium is an atom that doesn't mind losing electrons and chlorine is an atom that really wants to gain electrons, and they come together, and they come together, then you can form a bond. Oh, oh boy. Here we are I guaranteed you. I told you on Monday, we would get here. I said, it's our three week anniversary, and I think we're going to bond. First bond, first bond. And it's ionic. It's ionic. Because this is a bond of an electrostatic attraction. This is the bond of electrostatic attraction between two oppositely charged atoms. So between two oppositely charged atoms. And so if I've got a positive and a negative, they're going to be attracted. How are they going to be attracted? Well, actually that's an attraction that we know all about. Oh there it is. Hold up. We just made our first bond. Let's talk about it. What do I want to say? I want to talk about that attraction. Because that's actually attraction that we know very well. It's a Coulomb attraction. Right, and so the way this works is if I'm an atom with positive charge, and I'm an atom with negative charge. All right, so let's take that same example, and this is the energy. And this is say, zero. And this is the distance between the two atoms. If I'm out here, then maybe the distance between sodium and chlorine is very far. But you see there's a Coulomb attraction. And so the attraction is Coulomb, which is going to be minus a constant times the charges on those ions over the distance between them. All right. So that-- if I draw that attraction, this is the charges on ions, charges on ions. And if I draw that, the Coulomb attraction looks like this. That's it. Right, that's 1 over r. But see, the thing is if they get too close, then those core, you know, then things start to repel. It's like, whoa, whoa. Hold on. I'm not going to go any further, because I don't like having the same charges getting too close, right. I don't like that. Your electrons are getting too close to my electrons, and don't make me go nuclear. And so what you get is you get something like this. And this would be the potential energy of those two ions. They can't get super duper close. But when they're far away, they're kind of free, and then as they get a little closer, they start feeling that attraction, that Coulomb attraction. And this energy here, this energy has a name. It's called the lattice energy, because for the solid, right? So if I make a solid out of these types of ions, and now I break it all up, and I go all the way out to just loan ion, so I go from the isolated ions to a solid of these ionic bonds, then I gain this energy. And it takes that energy to break them back up. That's called the lattice energy. And you can see right away, there are some really kind of interesting observations you can make related to this simple relationship of a Coulomb attraction for an ionic bond. All right, so here's two ionic solids. There's sodium chloride. There's magnesium oxide. There's Na plus and F minus. OK, and notice the Coulomb energy here. And notice the differences in the lattice energy. This is like four times that. That is exactly what you get from this, because in this case Q1 and Q2 are both 1, right, 1 and negative 1. And here, Q1 and Q2 are both 2, and negative 2. So it should be four times, and that's, in fact, you can measure how much energy does it take to break this thing up, four times as much. That's the lattice energy. So we know something about this. And you can see trends. Right, so these are now the radii. We just talked about these. Right, radii for ions. And this is the lattice energy. And you can see that if I'm the smallest one, and the smallest one there, I have the highest lattice energy. And that's the 1 over r part. So you got the charge and how close they can get. And you now know from-- just from the trends we've just talked about, the radii trends we just talked about, you know how you might compare how much one ionic solid is bonded, compared to another, because you can talk about how much charge it might lose or gain. It likes to get close to the nearest noble gas. And you can also think about their size or relative sizes. So like the largest one with the largest has a larger r, and so weaker lattice energy. And boy, is this important. It changes all sorts of things about the properties of these solids, and that's why, we developed the goodie bag to help you explore that. So you have different solids. They're not all ionic, because you're talking about another kind of solid next week. Right, but you've got some ionic solids. And look, like, you know, so a lot of them dissolve, but if the lattice energy is so high, maybe it doesn't dissolve. Or maybe it's harder to dissolve. How do you know? Well, you could see it if you stir it. All right. But you also may be able to measure. Are there ions in solution? If there are ions in solution, than it could be a dissolved ionic solid that you've made. And so-- oh hardness, Mohs' scale. You can look that up. Did something scratch something else? Right. That's what that's related to. Melting point, right, the properties depend on this ionic bond. OK so there's the goodie bag, and then I'll do my why this matters in the last couple of minutes of class. All right, so there's the goodie bag. Oh, you've got a conductivity meter. You've got a scale. You've got a scale. I used to give that scale out with white powder right before Thanksgiving for a different goodie bag, and it was a really bad idea. But now it was-- it was citric acid, lime juice. But people traveled like that. And so we've learned. We've learned. And so we give it to you now, but please keep it. It's the world's most accurate scale. Why does this matter? Because look-- you're going to need the scale again. So please keep it. Here you go. Here's another chart of ionic bonds. These are lattice energies. OK, these are lattice energies, oh kilojoules per mole, it's just per mole of stuff. Right, don't get confused by that. Joules, electron volts, energies, per mole is per mole of atoms, right? OK, so we can isolate it down to one bond or a mole of bonds. All right, 15,000 for aluminum. Look at that. That's why aluminum is in your toothpaste. Aluminum oxide is in your toothpaste. It's in your pans. It's in sandpaper. It is a very hard material. It is a very hard material. Why? Because of this. Literally. Those Q's are really high, right? Why does this matter? Well, I'm going to give you an example in hemodialysis. Hemodialysis is something that people have to go into a hospital. People in their kidneys, don't clean their blood efficiently or enough have to have it cleaned by a machine, and 650,000 patients suffer from this and go to a hospital for literally four hours, three times a week. It devastates their weekly schedule. And if you could make this portable, you would change lives in a really big way. One of the big-- there are people out there trying to do this. This is an example of a design for a portable hemodialysis machine. One of the single biggest drawbacks in making it portable is making a filter, because you're filtering blood. You're filtering toxins out of the blood. It's something that the body can't do. So you need to do it for them. But the filters get mucked up and gunked up, and they're not strong enough, and they can't be cleaned. And look at this one. Here is a filter made out of aluminum. All right, and these things are what's needed. We need super strong, super resilient new filters that can filter things like blood and toxins out of the blood. So very, very tiny sizes that we can make over large areas and uniformly. This is the kind of starting material that we need to make new filters, and it all comes back down. Why would that be a good one if we can overcome it's brittleness? Because of the Q's. Because of the atom sizes, because of everything that we talked about today. So have a great long weekend, and see you guys next week. |
MIT_3091_Introduction_to_SolidState_Chemistry_Fall_2018 | Goodie_Bag_7_Defects_Intro_to_SolidState_Chemistry.txt | [SQUEAKING] [RUSTLING] [CLICKING] ISAAC METCALF: In this video, we'll be discussing Goodie Bag 7, or crystalline defects. The objectives are to visualize zero dimensional, one dimensional, and two dimensional defects in crystalline materials. You'll need a double-sided strip of adhesive, two Plexiglas panels, and 500 1/16th-inch metal beads. First, pull off the protective film on the top of your acrylic sheet. Take four pieces of double-sided tape and put one piece on each of the edges of the acrylic sheet. Then carefully put your balls in the middle of the square made by the four pieces of tape. Finally remove the film on the top of the other acrylic sheet and put it on top. When you're finished, you should have a completed defect modeling kit. CAROLYN JONS: Now that you've built your model, let's use it to visualize how material processing impacts the number of defects visible in a crystal. When looking at our original crystal, you can see several defects. These are a vacancy, line defect, and grain boundary defect. Material processing can lead to both increases and decreases in the number of defects in the material. A process that leads to an increased number of defects is extrusion or rolling, because when we extrude or roll a material, we force the atoms to go into certain places and those aren't necessarily the most energetically favorable states. Let's look at our model to understand how material processing leads to an increased number of defects. When we shake the model, we're able to add defects to the system. As you can see now, our model has an increased number of vacancies, line defects, and grain boundary defects. Other material processing methods allow for a decrease in the number of vacancies. One possibility for this is annealing. When we anneal, we heat the material up, and then we cool it slowly and allow the atoms to move back to their more favorable states. To illustrate annealing, let's tap gently on the side of our model. As you can see, the number of defects in our material has been substantially reduced. In this Goodie Bag, we were able to visualize some of the defects seen in many crystals, and we were also able to see how processing conditions impact the number of defects in a crystal. |
MIT_2997_Direct_SolarThermal_To_Electrical_Energy_Conversion_Technologies_Fall_2009 | Lecture_4_Kinetic_formulation_of_thermoelectricity.txt | SPEAKER: The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. GANG CHEN: This course is for pass fail. You should turn in your weekly reading report. And I will give you-- I'll send the email. So I'll give you two buy-outs since I haven't said this before. Two buy-outs means you can skip two reports, but below that, you can't pass this course-- so pass fail, OK? So I'll send that out. I came back this morning from San Diego. And what I want to show you-- OK, this is my towel bag. And that's all I bring because now I feel one day, my son had a homework assignment-- carbon footprint. And he came back and said we will be below the average, except you travel too much. So my footprint is just a tremendous release of family carbon dioxide footprint. My way of trying to redeem a little bit is by traveling light. And I was at a meeting in San Diego. This meeting was on the application of thermoelectrics. So we actually, in the last two days, really-- the key people from thermoelectrics area around the world converged in San Diego discussing the application areas. I'll share some of the meeting findings. Let's see. I want to start by-- some of you read one or both of the papers-- the assignment. One is by crowning one in on the Inconvenient Truth. And the story is he attended one of those conferences organized by Al Gore. And so that was a good title-- Inconvenient Truth's title. And I know he called me at the time and was asking for some slides. So whether you have any thoughts-- the other the other review by Lambell is more optimistic. So one is the pessimistic one. The other is, I think, the more optimistic one. And I don't know which side you're in. Any thoughts on that, or anyone have any opinion? Yes? AUDIENCE: They were saying some of the same things with a different tone because they both were saying that if this is going to be useful, it's for low power, niche applications, maybe some covered applications. And then the difference was in terms of-- I think it was-- Bell was saying this is going to be a great impact if we can do these applications. And Cronin Vining was more like, you can do these applications, but it might not make that much of an impact overall. You're never going to touch the large scale energy sector with thermal objects. GANG CHEN: OK. Yeah. So I would buy into the-- but still, the two-- in terms of the impact-- conclusions seemed to be a little bit different, right? Probably the same no matter what you do. Thermoelectric won't make a difference in the larger scale. So who raised their hand? Who else? AUDIENCE: But there's an argument for doing a lot of little things, right? If you were to, for some reason, take over all of the low power applications, that's a lot of things you add efficiency for. It's not a large scale. So it's like, if you replaced every cell phone with something that saves 5%, that's a lot of cell phones that you [INAUDIBLE]. GANG CHEN: I guess the future is very uncertain. And it's very hard to see what technology will win. AUDIENCE: Exactly. GANG CHEN: So that's a good perspective. Well, I was asked to serve as a panelist in this meeting. I gave a talk, but they also asked me to serve as a panelist. So I thought of what I would say-- anything during the meeting different from what I was going to talk on the scientific side. I pulled out the Cronin Vining's paper. I read it again on the airplane and made a few slides I want to share with you. So let me-- this is not in the handout, but I can put it in the website. So I also try to use-- you can help me think about the last time. I say whether it's incorrect conclusion or inconvenient truth. So I think the erroneous argument is really based on here. This is the efficiency heat source. And they look at, of course, this is the maximum you can get. So there's no mistake in the scientific side. The question is, where do you place the dots? This is fine. This is a cold running cycle. You can even get the 60% if you do the coaching. And although we-- he has now put in there somewhere between around 25% internal combustion. And solar sturdy-- that's a mistake. Nobody has that kind of efficiency even demonstrated. Nuclear ranking, that's the esteemed solar ranking. So solar, these two dots accompanied [INAUDIBLE]. The solar, if you think about what's currently deployed in solar steam generation is between 15% to 18% efficiency. And, of course, in the lower target range for this argument here are competitive technology. This is an organic Rankine cycle, for example, where the organic has a lower evaporation temperature. So you can do the steam engine-based cycles. So that's his argument. Basically, you see the thermoelectrics, even [INAUDIBLE] too [INAUDIBLE] be much dense compared to, of course, if you compare with this talk. Let's say, I thought that the comparison was not on the mark. Because now, if you look at the PV, photovoltaic, and amorphous silicon is about 6% to 8%. The single crystalline silicon deploys about 18% In the real world. Of course, in the lab there has been higher efficiency demonstrated. And the first solar panels generate, that's the fastest growing solar sector is about 10% efficiency. So, of course, even PV you can make the same argument. Will it make a difference? Because they were at least 30 terawatts in the next 30, 40 years. And PV now the deployment annually is about six, seven gigawatt. You can go to see how much you have to have to catch that. Let's say, of course, today nobody would really argue against PV. In the long run, this could be a solution. So you look at the PV cell efficiency progress, this is the best. This is the best efficiency progress against the year. And as I said, amorphous silicon-- this is the amorphous silicon there where this is amorphous silicon. And then deployed [INAUDIBLE] 6% to 8% efficiency in the real world. And single crystalline silicon, 26% here, and deployed is about 8%. And part is [INAUDIBLE]. And so, this is a-- with this low efficiency, you are, of course, competing against the coal generation. And at this stage, it is not competitive. So the government is giving stimulus, trying to promote this technology. And let's see, I think my argument, if you think about why solar, even nobody probably really will think that in the near future you get a 40% solar cell in the deployed. Of course, concentrating is another option, but there are a lot of problems with multi-junction solar cells. So why solar can make a, say, come to the map and people accept it? I think fundamentally is heated-- solar energy is free. So it's not like coal. Coal, say, any petroleum-based, you have to put in more money to get the fuel. But in the case of solar, you have free fuel come in. So now you look at the thermal, how we use the energy now, you see, OK, mostly are fossil fuels. And the way we're using is conversion into heat and then into whatever mechanical electrical energy or just a simple heat. So heat is intrinsically connected in almost over 90% of the way we are using energy. So that's the one part, that's the input part. And now you look at the output part, useful energy is about 35%. That means 65% is thrown away. And of course, thermodynamics will tell you have to throw away some. Let's say, we threw away a lot more than what's need in terms of thermodynamics. Because if you think about combustion process, right, let's take a 2,000 degree Kelvin. In thermodynamics, most of this is a high temperature combustion. The carbon cycle is over 90% efficiency. And we're throwing away 60%-- more than 60%. So, in fact, if you think about your free energy, I said the solar is free, but you're throwing a lot of here also free. And if you can take that-- so, of course, you think of where it could be deployed. Is it here or is it here? I don't have a clear answer, but the gut feeling is probably this is where you will have this is a free energy. So you may have more, let's say, easier acceptance. And of course, there are a lot of you where a lot of heat is free. But let's say, even in the case of heat is not free, I add this into this generated. If you think about the household, and say you burn fuel, but you just take a hot shower, that's where you generate heat. So this is a, say now you're talking about combined heat and power in your household. And fundamentally, of course, the solar is also the heat. So whether we can use it, with thermoelectric, you have to, at the end, calculate the cost. In the real world, they say solar, even solar is still not competitive because it hasn't reached that cost level. So if you look at what's the solar cost. And per dollar watt, of course, it's progressively decreasing. And now the best, I think, the first solar is claiming they're about slightly below $1 per watt. So to reach the grid part, some people say already it depends on where you are. But I think, let's say, this is ideally equal. So ideally, you say this is where you want to reach the grid. When you say grid parity is really compete against the coal generation type of technology. So it's a factor of three. And it doesn't sound to be very far, but it is a lot there. Of course, there are a lot of challenges to get there. Let's see. Thermoelectrics, I give one example. I think it's not the-- actually it could have say, even in the large scale for the application. Because this is actually a measurement found in the lab. So it's just a two legs, one and two. Each leg is about 1.5 millimeter cube. And so, this two leg, you have about 190 degree delta t, you generate about 80 milliwatt. And 80 megawatt, so if I take that, I go to calculate the weight of the two legs and calculate the material price, and calculate the cost for this 80 mega from a material perspective, and they got about $0.10 per watt materials. And that's just to say, this is almost like a bottom line. So you have to pay for the materials. And of course, you have to manufacturer it, you have many other things together. So the question is, do you go from here, say, to build anything that is cheap enough to be adopted? Of course, the efficiency, there are many constraints. In fact, the biggest constraint for thermoelectrics is, I think, of course, efficiency is the one. But the very different from solar is-- you have to have heat incoming and heat rejection. And those systems, of your systems, could be much more expensive than just a thermoelectric one. So again, this is where I think a lot of you engineers really think where it could be used, where there are a lot of room for innovation. And the workshop I went to was organized by the Department of Energy, energy efficiency. And the focus there is the automobile, because the automobile now is the biggest driver for thermoelectrics because there's the waste heat from the engine. And so, if you think about 40% of the heat actually goes through the exhaust. And it's a really government legislation. The government, particularly in Europe, there is legislation, I forgot the year, I have loads, that a few years down the road and the CO2 emission, there's a standard per 100 kilometer. And if you are one gram above that, it's a penalty of 95 euros. So with the 95 euros-- so if you-- roughly you need a 3% fuel saving to get there, to get that three grams. And so that means the system has to be around 300 euros to have a few hundred, watt of 300 watt, let's say, 300 watt of electric power. So roughly, you have a $1 per watt range they are calculating in there. I thought it's a-- automobile is the hardest problem. In fact, I was-- a BMW guy was saying the same thing. I was-- I agree with him, and I thought a lot of in the past. And if you can do automobile, you can do anywhere. So if that area is successful, I think there are a lot more impact. But unfortunately, that's also hardest because it has to be very space-constrained, very efficient, weight-constrained. When you add a weight to the system, you actually create a penalty. The efficiency of the automobile drops. So there are a lot of constraints, but see, the dynamics-- it's there where the government is putting money. So unfortunately, thermoelectrics is very dynamic in terms of dynamic situation where the money came from, automobile, is driving hardest application. And which is not quite sure whether it will be completely successful or not. You are tackling the hardest problem. But I thought that there are many other areas where you could innovate. Maybe it's not as hard, I think. Of course, the challenge is where the money is. So chicken-egg problem. But I think we need more success to stimulate the interest from both government and the public. And clearly, you see, I'm on the-- which side of the fence because I'm biased. Now let's come back to designs. So I know that for many of you, it's a very steep dive. We start from nothing. I assume that the-- I hear from the engineers is a lot of those concepts. I was reading your report, a lot of those concepts are foreign, and we should expect it. Let's see. What I'm going to do-- to hope is to bring you to where I was yesterday, in the least today and the next lecture-- will be part of the next lecture. And where I was yesterday, there was talking about the most current topic in thermoelectric research. So last time we're talking about the energy levels in solids. We talked about the lattice vibration, where the quantum of lattice vibration is a phonon, so basic energy [INAUDIBLE] of vibration. And of course, most of you-- I had a colleague when I was a UCLA faculty member. And he said in my course, three courses down the road, see, I don't still don't understand what is a phonon. It's not a real particle. Phonon is a collective vibration. And you have a lot more easier acceptance to the name of photon. But if you go to think about what is the photon, it's the same-- you must have the same confusion there. What is a photon? Photon is just people talk. It's the electromagnetic wave, the quantized form of electromagnetic wave, the quanta of the electron. And I took a course in quantum mechanics, and midway there was a substitute professor at Berkeley. And he asked the class, anybody understood? No student dared to raise their hand, understand the quantum mechanics. That's correct answer. Nobody else it. But see, what I think-- so take it-- let's go from there. The phonon spectra in solids, and what's important in solids is their crystal structure. So in each crystalline directions, there are different, say, mass springs because the spring constant is different, even though it's the same atoms. Say, electrons in solid in different crystallographic directions, there are different energy levels for the electrons. So we started with the simplest picture. But I said, I want you to get comfortable. If you read any papers, you'll see people show those complicated lines where in each crystallographic direction that's the different crystallographic direction. Very often use the symbols. And those are wave vectors, different wave vectors. Wave vector is 2 pi over lambda. So we think about the wavelengths, the waves. Different wavelengths have different-- each wavelength has a corresponding vibrational frequency. So that's the phonon spectrum. And also shown at the end here, this is another concept that we introduced, density of states. So it's a number of quantum mechanical modes or quantum mechanical states at, in this case, per frequency interval. So it's a differentiable. So you can integrate this density of states over all frequency to get how many quantum states you have. So that's the phonons. And the electron, again, you see here, different crystallographic direction, there is a different energy for the electrons, a different wave vector, which means the electron wave different wavelengths have different energy. And this is the case theta bandgap, not bandgap. In the case of silicon, the minimum of the conduction band and maximum of the valence band, the another safe spot, that's the indirect semiconductor. And it's really this whether you have gap or not that determine whether you are metal, semiconductor, insulator. The insulator is a much larger gap. Semiconductor is a gap that's in the less than 3 electron volts. And gallium arsenide is direct gap, minimum, maximum is at the same wave vector location. And they have very different optical properties. And then we give an example of how to calculate, again, [INAUDIBLE]. So how to calculate how many electrons we have in the-- so here is the most of those electron is in this region, the minimum of the conduction band. And when we do the calculation, we say this sub-states, that's the number of states per unit energy interval. Of course, we went through the summation to introduce the density of states. And then this is also the number of quantum states. And each quantum state, how many electrons we have, that's the Fermi-Dirac distribution. And they integrate. So if you do a parabolic event, that's the-- say parabola, we can approximate-- always approximate, not always, but as approximation, when I have a minimum or maximum, I can say the first order derivative is 0. So the Taylor expansion gave me the second order. That's a parabolic band. So with that parabolic band approximation I have-- I can do a [INAUDIBLE] statistics. I can do this integration to get the relation of how many electrons is determined by ec minus mu. That's this guy. It's this guy is really the [INAUDIBLE] potential. So [INAUDIBLE] potential determine how many the number of particles, electrons. And here is, say, the related to the effective mass, how flat this mass is, the curvature, how flat is this guy. So those are some of the key concepts that we learned in the last lecture. And today I'm going to-- we learn the phenomenon. You read also Goodsmith's chapter on Seebeck effect. And today we're going to have more mathematics, going to have expressions for those coefficients. What determine the electric conductivity? What determine the Seebeck coefficient? What determine the thermal conductivity? So that's the goal here. And I will still use the same picture I presented in the first class where I say, I look at how many particles are going from left to right and how many particles are from right to left, and the difference being the flux. And this flux could be heat flux. It could be charge flux. And so, with that, we will derive will have the mathematical expressions for all the transport coefficients. So let me start with the electrical conduction. And electric conduction, in the simplest picture, would consider an isothermal electrical conductor. So there's isothermal, there is no Seebeck effect, just a simple potential flow. I have an electric field, and the electric field here, this is the most, for mechanical engineering, including me, the most troublesome point is the electron has a negative charge. So when I draw a field and say and charge go that way, that's a talk about electron. And now, of course, under a field, electron experience a force. So F equals ma. Force of the charge experience is the charge times the electric field. Then you can charge for electrons. F equals ma. And so, that's the simple Newton's law of motion. But the electron will not travel very far because they will collide. They collide with each other. They collide with-- by the phonon, the lattice vibration also distort their motion. So once in a while, between the collision, they go straight. Under the field, due to one collision they will change direction, trajectory. So the terminal velocity, I can do a integration of this, So I do a integration. I say, OK, this is the collision part where the free acceleration time, you can think that way. And I say, this is the velocity the electron will get between each collision, the terminal velocity, that's also called the drift velocity. So you integrate that, you get drift velocity. And we can also write the e pi e tau divided by m as a mu. And the mu is called the mobility, it's [INAUDIBLE]. So we have velocity is proportional to electric field, drift velocity proportional to electric field through the proportionality constant the mobility. And now, once I have the each electron drift velocity, I can calculate current easily. Why? The current is just a-- so let me just add one more. The mean free path is the time between the collision times-- now I don't use this drift velocity. This way I put a thermal velocity. So what I mean is electron, they have two part velocity. One is they are doing thermal random motion. And in fact, that's a very high speed, 10 to the 6 meter per second. It is very fast. And so, and of course, so they do a random motion, that's a random thermal velocity. And then under field they have superimposed on that thermal velocity is a drift velocity. And for mechanical engineer, that's the same with convection. Molecules move about 300 meter per second in this room. Let's say, the average velocity is 0. The convection is when the average velocity is not 0. So this is the convection velocity. That's the molecule random velocity. And now, so this is the mean free path times-- so this is the collision, the distance between successive collision. That's the mean free path. AUDIENCE: [INAUDIBLE] GANG CHEN: Yes. AUDIENCE: So this drift velocity is basically [INAUDIBLE]? GANG CHEN: Right. AUDIENCE: And the terminal velocity is already existing between [INAUDIBLE]? GANG CHEN: Right. So even typically drift velocity is much smaller than the thermal velocity. Just the same as when we do convection. You're electric engineer, so that's easier. Let's say when we do convection, it's a meter, 10 meter per second is pretty fast, wind. But the molecule is 300 meter per second is moving around. Of course, the electron thermal velocity could be 10 to the 6. Yes. AUDIENCE: Why is it the thermal velocity 0? GANG CHEN: Thermal velocity? AUDIENCE: Yeah. GANG CHEN: Ah. [INTERPOSING VOICES] GANG CHEN: Anything, unless you go to 0 Kelvin. AUDIENCE: But the average, not the repeat-- is that like the root mean squared? The average density has to be going in each direction, right? GANG CHEN: Yeah. So the average of velocity is 0, but the magnitude is not 0. Right? AUDIENCE: Right. But if any individual electrons use-- which one would that be? GANG CHEN: This is the magnitude. This is not the vector velocity. It's the magnitude of-- AUDIENCE: So of each, it's not an ensemble property? GANG CHEN: It's an average. It's a rough thing. So it's an ensemble property. AUDIENCE: So it's like the root mean squared? GANG CHEN: Right. So, OK, how you estimate it is 1/2 kvt, we said before, roughly 1/2 mv squared. So that's your thermal, because this is the thermal energy. And the kinetic energy, that's the kinetic velocity. AUDIENCE: OK. GANG CHEN: And the mass of electron very small. That's why it's much faster than the molecules. So now it's easy, because the current-- what's a current? Current is the left motion of a charge. So the electron coulomb number density per unit volume, number of electrons per unit volume, times here the drift velocity. This is the convection current. That's the electric current. And you go to check the unit, you should-- I should have a unit here. I didn't put it there. That's the amp per meter squared. If you do a heat transfer, you will be watt per meter squared. This is a flux. You go to look at the unit. So this is a coulomb per second. Seconds come from here-- gives you amp. Here is a meter cubit, one more meter cubit, here is one meter, so give you a meter squared. So m per meter squared, that's a unit of current. What is that? That's Ohm's law. So here is a few steps. You get the picture. You learn from high school, this is the Ohm's law. You have your current. It's proportional electric conductivity times the electric potential. The electric potential, if you do electric field, electric field is the electrostatic potential gradient. So that's right into differential form. And in fact, this is a very similar-- if you take and replace this with a temperature, that's a Fourier law. So even though one is seems like a convection picture, the other seems diffusion picture. But they all collision dominate transport process. It's all collision laws, the diffusion phenomena. So that's the Ohm's law. And now let's go to the isothermal in the previous case, now I'm putting a temperature gradient. I want to consider coupled transport. Thermoelectric is a heat and charge coupling. And of course, when I have a guilty-- say, a potential, I still got that convection drift part. What's different now is that I have a temperature gradient. So the molecule or the electrons-- and I'm used to talk to mechanical molecule. So electrons here go faster than electron here. Now there is also diffusion flux of charge. So to consider that diffusion flux, I have from left to right and right to left. That's what I did for heat conduction before. So the electron flux law-- so here I have previous one from previous slide, the Ohm's law. That's the drift part because I have an electrical potential. But also I have the diffusion part. This is going from left to right. This is from right to left. Again, I carry this negative sign, negative charge. This way should be thermal velocity. I should have been more careful. It should be thermal velocity because they are ranked randomly jiggling from left to right and right to left. And the left for density, and in this region one mean free path you can go zip through. Longer than one mean free path, it will collide, change directions. So I'm only considering those whizzing by mean free path that can zip through from left to right. And then on the right to left, one mean free path can go half of them only. Because it go equally go in all directions randomly. So this is the same way I did for Fourier law, the diffusion part. And now I do the Taylor expansion, the different-- expand over at x. So I do the Taylor expansion. I have e v x square tau even vx. And in this part, I'm just talking. I messed up a negative sign. See? That's what I say, I really don't like a negative sign in the electrons-- historical mistake. Can't have that. And you know that, we just showed before, n is related-- the electron density is related to the temperature and the local chemical potential. ec minus mu is the chemical potential. That's what I reviewed from last lecture. So now temperature depends on location, space. And ec minus mu, I can think of just mu, chemical potential depends on location. Because n really-- or say, mu determine n. So both depend on location. So if I take the dn, dx derivative, I have dx, dt, dx, give you many x. AUDIENCE: The velocity is also [INAUDIBLE]? GANG CHEN: Very good questions. I can't do that simply in here. So I just do 1/3 v square. Otherwise, I'll have to put it inside dm, dx. So that's why my coefficient will not be accurate. AUDIENCE: How much is the thermal velocity different from Fermi velocity, the temperature? GANG CHEN: Ah, thermal velocity is very different from Fermi velocity. This is-- OK. So some of you know all those concepts. But if you think about what's a Fermi level-- I briefly mentioned this before. In a metal-- I can take a simplest case. This is the electron energy level. In the metal, Fermi level is here. So the energy of the electron is close to Fermi level because only those electrons are moving. The electron deep inside don't have any free space to move. So this one is the Fermi velocity. So m vf squared. So that's also-- it turns out this also is the thermal velocity. Those are the electrons, because the electrons do have this much energy. They will move around with this energy. But in the semiconductor, that's the valence band, that's conduction band. And then you dope, and your ef may be here if you don't have it doped. That's what I was drawing before. So this ef, no electrons have this ef. Electrons only in this region-- because the thermally, a small fraction of electrons go there. So those energy is really-- this is just 1/2 [INAUDIBLE]. This is the relatively-- this kinetic potential is-- let's see, relative to the kinetic energy relative to here, 0. So. So the only is about the [INAUDIBLE] about the 1/2 [INAUDIBLE]. AUDIENCE: The key is the difference between the energy of the electron and the chemical potential? GANG CHEN: No, not even chemical potential, only relative to the bottom of conduction band that you see there. AUDIENCE: Oh, OK. AUDIENCE: But can you [INAUDIBLE]? GANG CHEN: Yes. AUDIENCE: Yes, so it could be anything. It could be 0 degrees or like-- GANG CHEN: Yeah. Let's say, the average energy of those electrons near that conduction band bottom is about 1/2 [INAUDIBLE]. Yeah. So I'm going to clean this, take a derivative. My sign might be screwed up. Go to check. I mix it up again here. So here is what I have. I have this derivative. And also I have dt, dx derivative. So now I have the chemical potential derivative. Phi is the electrostatic potential derivative. And it turns out, this is when you do a detailed coefficient analysis, this coefficient and this coefficient is the same. So this is the fundamental. It is also called the Einstein relation, because here is the-- related to mobility, and here is related to the diffusivity. It's chemical potential gradient gives you the diffusion of the charge. And here is the motion. So Einstein didn't do electrons. Einstein did Brownian motion, hypotheses in molecules in liquid. And he derived the relation between the molecular diffusion coefficient and molecular mobility in the liquid. And that relation turns out also to be true between electric conductivity, and here you can think that as an electron diffusivity. So this is a general for the Einstein relations. And if you generalize even more it is the fluctuation dissipation theorem. So Einstein, that's his PhD thesis. If you were born at that year, you may be writing a paper say it's very important. He did the Brownian motion. And really, it's interesting to read that. He didn't even know there was an experiment on Brownian motion. So he studied statistical thermodynamics, the other branch of statistical thermodynamics. Before that it was Boltzmann. Boltzmann doing gas. Einstein tried to do liquid. And go to read. It's not that difficult. So there's-- I said, if you say the conductivity is the same here, there's a relationship. And if we combine these two terms-- and I have rewriting this here into this equation here. And the temperature gradient I write into the next term. So I have an electrostatic potential gradient, a chemical potential gradient, and a temperature reading. So that's-- and so this three term that drives my current flow in the case when I have a temperature gradient in a conductor or semiconductor. And I continue this. I combine some electrostatic potential with the chemical potential. That's called electrochemical potential. Some of you that do battery, you know the electrochemical potential in battery. And in fact, electrical engineers should also know that. It's not the electrostatic potential that drives the current flow, it's the combined electrochemical potential that drives current flow. And most-- of course, most electrical engineers deal with isothermal. So you don't consider this term, isothermal. But you have to consider this term. So there is always electrochemical potential gradient that drives the current flow. And so, it's a very important concept, and it's hard to understand to some extent. Where is this electrochemical potential? So I said, the ec, this separation here is the chemical potential. And electrostatic potential, this is when I'm drawing a conductor or semiconductor. And if I draw relative to an absolute reference here, this ec is parallel. So electrostatic potential depends on where you plot your reference. So I can always shift up and down because this is the gradient that determines really the current. So there's a constant reference point. So it's the gradient of this ec that gives you electrostatic potential. And so I have ec here minus e. The problem is, e energy-- ec is energy. Energy divided by charge is the electrostatic potential. This is always a historical problem, not my problem. So electrostatic is here. Chemical potential is here. So what's left is this electrochemical potential relative to absolute level. It's that gradient that drives current flow. So now I combine my electrical current. In the front, I just say this electroconductivity I generalized with the coefficient L11. And I combine all the two-- say, electrochemical potential gradient. So this is a typical-- the generalized Ohm's law, it should be this one, electric current proportional to electrochemical gradient is the electric conductivity as a proportionality constant. And when I have temperature gradient, I have L12, it's the extra term, the coupling between charge and heat. So that's the L12 here due to the temperature gradient. And it's really here what they created the diffusion. Yes. AUDIENCE: What is the electrostatic potential again? GANG CHEN: Ah? AUDIENCE: You're saying it's the same as the gradient of ec? GANG CHEN: Yeah. AUDIENCE: But then that cancel out [INAUDIBLE]. GANG CHEN: This one is the chemical potential. This is the electrostatic potential. The difference of these two give you the electrochemical. We can discuss more-- AUDIENCE: [INAUDIBLE] GANG CHEN: He's the electrical engineer. OK. Now I'm combining everything together. And Seebeck is when there is no current flow. Seebeck voltage is an open circuit. When there is an open circuit, no current flow, you can say there is a temperature gradient, there is corresponding the electrochemical potential gradient. You put a probe there, you do measure electrostatic potential. You measure electrochemical potential. The metal is connected to a semiconductor. When you connect it, the metal, let's say, is a Fermi level connected to the electrochemical. So you measure-- you can only measure this. But when we are doing one experiment and we really want to measure electrostatic potential, what are we doing? So we are doing, for example, we're trying to do, OK, there is a green boundary, green one, green two. And the green boundary there is an electrostatic potential. We can't measure it with the contact method, but we're putting-- trying to put an atomic force microscope. And there, when they have-- there is electrostatic potential, it will be electrostatic potential determine the repulsion or attraction. And that's the coulomb's law. When you calculate use the coulomb's law, it's the electrostatic potential. So there's a-- still there are ways to-- I said the content method can't be distinguished, but there are ways to distinguish it. So that's where the Seebeck coefficient is. Seebeck coefficient is defined as they-- ah. It really should be negative d phi dx divided by dt, dx, should be. So here is negative d phi dx. Then it should be phi dash minus phi z. And you can say, it's related to the Seebeck coefficient now can be expressed in terms of ec minus mu. That's chemical potential. Three, kbt is the thermal energy. So the Seebeck coefficient has a very clear picture here. And this is the-- so you can set up two part. One is the chemical potential here. Between here and here, that's ec minus mu. And the electrons average is thermal energy in this region. But you ask me the vx is a random velocity. So that two-- 3/2 is not good. Because of vx I have to do integration averaging, and it turns out to 5/2 rather than 3/2. I'll show you more exact expressions. So now any question here? AUDIENCE: Can you go back one slide? GANG CHEN: Yeah. AUDIENCE: Why do we have, in the graph, why do we have the position dependent energy level or the potential energy-- GANG CHEN: Ah, here? AUDIENCE: Yeah. [INAUDIBLE] has a slope. [INTERPOSING VOICES] GANG CHEN: That's the potential of the electrons. Potential of electron is divided by electrical charge gives you electrostatic potential. Under current flow, your electron-- and so, the electrostatic potential is going downslope or upslope. AUDIENCE: OK, I think-- GANG CHEN: When I think about downslope or upslope, I always have to sit there thinking for a long time whether it's positive charge, negative charge so I can [INAUDIBLE]. That's why I don't like the charge, negative sign. By the way. I don't think, let me see, for a mechanical engineer, to understand this part, I have to sit there many, many times. And I think sometimes when you think you understand it, you have to ask the electric engineer, you understand it is problematic. But it's good to sit there and think through yourself. Now, I said in the previous part that was the electric current. Now I'm going to do heat current. Peltier coefficient is a heat current. So I have to think about the heat carried by the electrons. And again, we did this in the first lecture. When I think about the heat flow, Fourier law is from left to right, right to left, diffusion. Here you can see I put an equipotential conductor, low current flow, just a heat diffusion. That's a Fourier law. The Fourier law is valid for electrons and for phonons. I didn't say whether it's electron phonon here in this derivation. So I'm not going to this detail because we're writing down this before. And the thermal conductivity is 1, 0, 3, [INAUDIBLE]. So that's what we did before, diffusion. And now I'm going to do charge focused on heat carried by electron. And what's the heat carried by electron? And here you have to go back to a little bit of thermodynamics. And the first law tells you that du equals dq. Doesn't look typical thermodynamics. For mechanical engineering you do du dq-- du equals dq minus pv-- pd, pv. There we'll not talk about pdv work. There's no volume change. So we'll talk about chemical potential. So if you're a chemical engineer, this part is easier. So du equals dq plus mu pn. That's a chemical potential term. And what this tells you, because the du u is n times individual energy e if you have n electrons. So e minus mu is related to q. The heat-- so the energy of a charge consists of two parts. One is the chemical potential. The other is the heat. So e minus mu is the heat carried per charge. And now, if I take that, I say The heat carried per charge is e minus mu, again, applied to the same picture going from left to right and right to left. So this is a velocity, random velocity, thermal velocity, number density per unit volume. And again, you go to check here, it should be watt per meter squared, the unit. This is a jar. Per second gives you a watt. And then it's a meter and over meter cubed gives you a meter squared. So here I have left to right, right to left diffusion. I add an extra term here. Because when I do-- when I have a potential gradient, I'm looking at this case. Oh, I forgot to draw one. I should have electrical potential going down. I have current flow. I have temperature potential-- gradient. So that's the-- I'm considering coupled electron heat transport. So this is the diffusion term. And this is the drift term. Same as I consider, now I go back a few slides, here. Diffusion term for current flow, drift term for current flow. Diffusion and drift. So let's look at it again. Here is diffusion. And drift is-- this is a drift velocity times the charge density. And each one is carrying e minus mu. So this is the convection term there, convection of the electrons going through the surface we draw. And now, if I combine-- because this term will give me the same thing as I drew before, electrochemical potential. And the second term I will have also the temperature gradient. So here I draw-- I made some jump. I didn't go through each step derivation, and you can go through that yourself. And now the heat flux, what I have from this is the heat flux carried by the electron is temperature gradient terms but also is a potential gradient term. So I have, again a couple-- before I have a charge flux electric current, the potential gradient, temperature gradient. Now the heat carried by the charge is also proportional to the potential gradient and temperature gradient. So under equal, say, isothermal case, this is 0, but this is not. This is the Peltier coefficient. When I have heat flows through-- a charge flow through the current, the charge carries heat. So this is the L21 is the Peltier coefficient. And so, that's the-- now I have the coupled heat and charge flow, and I have the two expression. One, before I wrote the charge-- if you go back to the previous two slides, this jex equals L11 times this plus L12 times this for the current flux. And now, for the heat flux carried by charge, I have Two. 21, L22, and then we'll come back and rewrite this expression-later. So this is a kinetic. I have not considered the average. That's the question you ask. I cannot do that from this kind of picture. When you want to do the detail, you have to say each charge have different energies, slight different energy. So you have to count more carefully, just like we did before, counting all the modes, counting all the quantum modes. And each one is moving at a velocity-- local velocity, its own velocity. And each mode has this number of charge determined by Fermi-Dirac statistics. And of course, this e could be either e, minus e, that's when I do current, or E, capital E minus mu, that's when I do heat. So if I do that, I can get the more rigorous expressions. And this is what-- you still get the same format except the coefficient is a little bit different. So I have this relation I've written before, same format. And those cross-terms, these two terms depends on how you write. If you write generalized thermodynamic forces are equal, but this is not generalized thermodynamic force, so it's related. So that's the [INAUDIBLE] relation. And the course term always related. If you do piezoelectricity, there is also mechanical to electrical coupling coefficient in the case of mechanical motion. Or when you have current flow, quantum mechanical. So the reverse-- this coefficient, cross-coefficient is always related, and that's the also relation-- very fundamental. Also got a Nobel Prize for this. So that's the similar for the charge, flow, and for the charge carry the heat flow. And now I'm going to write this. You tend to think, this is a thermal conductivity. That's not right. Because when I write, I could write my potential, rewrite it into a jex L12. So I'm replacing my potential in terms of electric current because the electric here, potential is related to temperature gradient. And now if I substitute into here, I will say the electrical flux, heat flux carried by the charge is proportional to the electrical charge flux and with a proportionality that's a Peltier coefficient. I said this L21 is Peltier coefficient, that's not right, because this is a potential. But the Peltier is really current flux proportional to the charge flux. And this L21 divided by L11 is the Peltier coefficient. And the next term is the temperature gradient. So only when I write into this form I have the expression for the thermal conductivity. In terms of for electrons, the thermal conductivity involves all four coefficients. L11, L21, L 1.01. Of course, these two coefficients are related. So this is the thermal conductivity. And now if I write more clearly using the parabolic band approximation, again, here I'm not going to give the derivation, just say the same-- I follow those steps and do, in fact, I need to go forward to my equation. We're not going into that. And here is my final expression. This is a random velocity relaxation time, which could depend on energy. And before when we do it and we didn't say it depends on energy. So we have-- so this is a more rigorous expression. And density of states, and this is the other coefficient. So with that, you can actually, using those coefficients, you can actually go to model the solid. And what do you have, the solid is really different solid have different density of states. So the band structure is different. And sometimes it has multiple bands, so you have to consider multiple bands' effects. Electrons have holes is another-- so you can see electrons-- negative charge plus the positive charge, and those could all be taken into consideration using this formulations. Again, I don't expect that you'll fully say, be able to derive it. I didn't derive it. But this is where many people, when they try to interpret the experimental data, they go to those expressions and try to fit the expressions. I say they try to fit because you actually do not know exactly, for example, how to-- what's happening in the solid. I said the electric conductivity, thermal-- electronic thermal conductivity that's in the previous slide, so those are the electronic conductivity here, electronic thermal conductivity. And the L11 is the electrical conductivity sigma. And the ratio of this, if you go to do Fermi-Dirac statistics in metal, you can do this integral. And turns out the ratio for metal is a constant. And this is a so-called Wiedemann-Franz law. And so that means thermal conductivity is very hard to measure. So if you measure electric conductivity in metal, as long as the lattice contributions are large, you can roughly-- I say roughly because this is not always valid-- as we measure electric conductivity, you can go to estimate some of it just based on this interpolation Wiedemann-Franz law. So this L is the Lorentz constant, the Lawrence number. Let's say, if you do semiconductor, it's not a constant. So in thermoelectrics, if you go to actually read thermoelectric literature, many people just take this Lawrence number and say, OK, I measure electric conductivity, and I want to find out what's the electronic thermal conductivity, I calculate. They will take this constant number, and they subtract the total measured thermal conductivity so the rest is due to phonon lattice vibration. And that often leads to very, I say, varied result. You can't interpret those phonon, because they did a wrong calculation. In semiconductor, L actually depends on-- you have to calculate it. You have to go back to calculate the electronic thermal conductivity using this transport coefficient and the electrical thermal conductivity-- electrical conductivity. I'll give some examples later. And again, so let's go back to look at what it means. What's the Seebeck coefficient? So if you look at it, that's from the previous expression there. And it's really, the Seebeck coefficient is the weighted average of electron energy relative to ef. ef really is mu, chemical potential. In electrical engineering, the chemical potential and the Fermi level is typically mixed together. They don't distinguish very carefully. So ef weighed against the electric potential-- electric conductivity at each energy level. And that's really the average energy of the charge relative to the chemical potential. So that's still what I said before. And the electric conductivity is the density of states, velocity, electrical relaxation time, electron relaxation time, and efde. We'll show more later on. So with that, now you can see why metal is not good. Metal is not a good material for Seebeck, because Seebeck is too small. For wide metal, Seebeck is too small. And this is because in the metal, ef is an in [INAUDIBLE]. And the average energy of electron relative to ef-- so the electron in metal is about half is above, half is below ef. The electron below-- because the Fermi-Dirac distribution is over-- roughly over kt, that's the electrons. Only on this region kdt, in this region, there are multiple charge. And I draw too wide. It's really just this very narrow. Above is a positive heat, below is a negative heat. When they move together, the positive heat energy above the chemical potential move forward, energy below the-- electron below the chemical potential carry negative heat forward. At the end, they two cancel. So you've got a very small Seebeck coefficient. So metals have very small Seebeck coefficient. And if anybody has any idea to cut this part off, you can one-- I don't know how to do it. Or cut the upper part off. So in metal, do you have width-- think about width-- how I can place something to scatter potentially half of this. So if you scatter the bottom part, you get a positive Seebeck coefficient, current flow-- heat flow in the same direction as the current flow. You cut the positive part, you get a negative Seebeck coefficient, the heat flow [INAUDIBLE]. So this is where, if you do materials research, you have to think hard on this, how you could potentially manipulate the material to create this asymmetry in the current heat carried by each charge. Semiconductor is better for Seebeck because, of course, here what I'm draw is this [INAUDIBLE]. If you think about this, I said Seebeck is the average energy-- so semiconductor, for example, if I doped it so that Fermi level is here, and the average energy is rapid here. So there is no negative part to pass through. So each of this electron here is-- relative to the chemical potential is a carrier-- a positive heat relative to chemical potential-- no cancellation. Thermoelectrics are typically not here. They dope actually into here. So the bottom part is actually not good, only the top part is better. Why they want to dope it there is because, when you put your chemical potential here, the number of electrons here is very small. And the number of electrons is very small, so the electric conductivity here is, this is proportional to the number of electrons. So what do you want? Because at the end, the zt is s squared sigma. Remember, zt figure of merit is the-- s is the Seebeck coefficient, s squared, times the electric conductivity, because you want the enough electron to carry the heat. You just don't want the one electron to carry heat, it just doesn't carry enough heat. So you see here, this is a-- OK. If I look at this, what is df d? Fermi-Dirac is a very symmetric around the ef. A density of states is here. So it's really, in semiconductor, only this region. This two product is not 0. Only those regions is contributing to electric conduction if you look at this expression. And so, that meaning only this region of charge rather to the chemical potential is the Seebeck coefficient. So you, later on, we will-- when we talk about quantum structure, people want to say, how you can make this region carry more charge. So you create a peak here. The distance of these states is not good. d is a square root of e minus e, ef, we show-- well, you see before. This is a very small density of states. So don't give you many carriers. So that's where people do a lot of things when they try to improve electrons. They say, can I create something sharper in this region so I can put more electrons? So remember that, at the end, you want to optimize s square sigma. Not even optimal, just optimize that. You want to optimize ats so there are still some things were missing. So this is the material side, the basic material physics. And if I go, again, to enforce my statistics, I said that the electrical energy scattering relaxation time depends on electron energy. So there are a lot of different scattering mechanisms. We're not getting into that. You go to do material research, you read literature. But if I do a gamma here, this is a potential power. For example, due to lattice, due to acoustic phonon scattering, this gamma is negative 1/2. So what it means, negative 1/2 is a low energy electron, has a longer relaxation time. They travel further. High energy electron travel less. You don't like that. If you look at that, your Seebeck have a gamma here. And this is a-- not 3/2. If you do the before, that's 3/2. I said, it's five parts. That's due to this vx squared. This gamma is not good if it's negative, because if I think about acoustics phonon scatter, the high-energy electrons scatter. But I want the high-energy electron travel as far as possible. But the ionized impurity, it turns out that gamma can be 3/2. That's very good. Of course, ionized impurity, so that means you doped material very heavy, you can have ionized impurity dominant. The problem is your mean free path become very short. Your electric conductivity becomes small. Thermoelectric is always a struggle. You try to sell to one guy, the other guy is not happy. And that's why the material research is very hard. So and, of course, if I put the ec minus ef [INAUDIBLE] divided by n, and then you will see, because this depends on temperature-- so as the temperature increases, Seebeck coefficient increase. And the reason is that, if you look at this one, you go to check the reason, as you increase temperature, actually the Fermi level shift down. So the average energy now increases relative to the chemical potential-- Fermi level. So that's the reason why the Seebeck coefficient increases with temperature. Well, I gave you a very simple one-band picture, although I know many of you say, this is pretty complicated already. Let's say, if you want-- if you do materials, you go to look at the real material, this is still the among the simplest picture I can give you. But say, in real material, for example, I have electrons, I have holes, positive charge. It cancel each other, because when your electron mode see positive energy form of negative energy rather than chemical potential, they cancel each other. So Seebeck, for electron is negative, for hole is positive, and then you combine when you have two different electric conductivity or electron hole. And what's worse is this thermal conductivity. They just don't not only combine electron hole and lattice vibration, but the combined-- they carry a cluster. This is because when the electron and hole both diffuse from one place to the other, there is an extra energy across the bandgap. That extra energy-- so when electrons go from one diffuse to the next place, they combine with the hole in the next place, they carry a lot more energy. So this is a bipolar contribution of thermal conductivity-- very bad. And it took me several years to understand the bipolar. So even if you don't understand everything, you can still do research here. When I first wrote the thermoelectric proposal, I didn't know whether a metal was good or semiconductor was good, just to make you feel more comfortable. Look at this. This is the real material now. And this is actually out of an experiment. This is a Seebeck-- electric conductivity decreases with temperature. Seebeck increases with temperature. I wonder you start to appreciate-- look at the curve, see what the physics, what's [INAUDIBLE] physics? So I said, Seebeck typically increases with temperature. That's all fine, except that now finally start to drop. See, that's-- you said that was wrong. I'll say why. And if you look at the electrical number-- electron number density, then it will increase with temperature. So you think the electric conductivity should increase with temperature in semiconductor. But this is a completely wrong trend. If you think about n, go to check your n at the fourth slides, second slide I hand out, and that was a review from last lecture. The reason was that we dope this very heavily, and it's not-- almost like a metal. Metal, electric conductivity drop with temperature. So the carrier in the material is somewhere between semiconductor and metal. Microelectronics, you never dope that-- you seldom dope that. And so, in microelectronics and most semiconductor, the electric conductivity increases with temperature. And Seebeck, when I start to draw, so that's because when I go to higher temperature, I got a holes. Electrons go excited from valence band to conduction band and left behind the empty states, left behind this empty state of holes. So temperature increase, more excitation, more energy. You reject the electron from valence to the conduction band. You got holes. Holes is detrimental because they're canceling the electron. So when I go to high temperature, I start to drop there. The holes start to cancel. What do you do? You can dope it heavier. So this will have higher electron number, I shift to a higher temperature-- shift here. You can play with doping for different temperature. But here you see these pics here. You have to optimize it. And you can, if you do mature research, you can just miss the optimization. You may get a goal, throw away. That could happen. And so, there are a lot of details, of course, in doing the material research. OK, I think I will stop here. |
MIT_2997_Direct_SolarThermal_To_Electrical_Energy_Conversion_Technologies_Fall_2009 | Lecture_2_Thermoelectric_effect_and_thermoelectric_devices.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. GANG CHEN: So if I give a very quick summary of what we're talking in the first lecture, we reviewed first and second law. And of course, I assume that you all have seen in various courses. And this is the maximum thermodynamic efficiency of any heat engine operating in addition to the reservoirs. One is cold, the other is hot. So that's the key point. And then for many of you, probably first time is, I introduced if you go to the statistical side, the microscopic picture, the one quantum mechanical state, you count how many particles, either electrons or molecules or photons or lattice vibration photons. And for electrons is given by Fermi-Dirac distribution, and where this mu is the chemical potential. That's the energy you need to take a particle in or out of the system. And e is the energy of that quantum state and t is the temperature. And then say that's for electrons where the Pauli exclusion principle will state the maximum possibility of having an electron in accordance with this one. So this number is between 0 and 1. And then I introduced also the other type of particles, those-- the photon electromagnetic wave or lattice vibration we call [? photon ?] or even molecules. This is the Poisson [INAUDIBLE] distribution where, again, this is the energy and mu is the chemical potential. But here we're not limited by the Pauli exclusion. So in one quantum state, this number can be larger than one. And if you take out this minus 1 plus one, you go to the classical regime where we call the distribution f, doesn't have that one, the exponential distribution, that's the board's mass statistics, horizontal distribution. This is the one you see in [INAUDIBLE] nor in chemical reaction. So that's a review of thermodynamics. And then I went on to heat transfer where we have three modes of heat transfer. The first mode is the heat conduction. The [INAUDIBLE] tells us the heat flux is proportional to the thermal conductivity and the temperature gradient. But I went one step further to show you in a very simple kinetic picture, just count the number of molecules going from left to right and right to left. Then we have a simple expression for thermal conductivity. That's 1/3 c is the specific heat, v is the velocity, and gamma is the mean free path. That's how a lot of the molecules travel. So this is the heat conduction mode. And then we have the Newton's law of cooling for convection or the e-transfer's proportional transfer coefficient area and counter difference. And the difference is this heat transfer coefficient is no longer property of the material. Thermal conductivity is a material property. And h depends on the flow configuration, natural convection, forced convection, what kind of geometry? And then the third mode of heat transfer was the thermal radiation. And I gave the [INAUDIBLE] model, that's the maximum emission of an object in half in the-- So this is called the Stefan-Boltzmann law. It's proportional to [INAUDIBLE] fourth power. A real object, the emission is less than that. So we normalize the real object emission by this number, we get the so-called the emissivity. So it's a number between 0 and 1. So I moved on at the end of the last lecture, I was trying to show you the thermal radiation can be derived. There are continual, and they give you a derivation of the Planck's law again. Now we move from continuum to a microscopic picture. And once we go to microscopic, I use the same strategy, I'm going to do the counting, the number of modes, quantum mechanical state, because I already know from statistics each quantum mechanical state, how many photon. That's the Poisson [INAUDIBLE] distribution, that's the previous slide here. So I know each mode, this is a number at this temperature. That's the number of photons in the mode. And also I commented before for photon, photon equals 0. So we'll use that. And the task is really just count the number of modes. So I will take a cubit, and I assume that the [INAUDIBLE] is perfectly reflected. And this is a [INAUDIBLE] t. And now I want to see how much is radiation energy is contained inside this cube by counting the modes. I want to show you this because we will actually use a lot of those common techniques towards electrons, towards the photons. So it's a general way of doing the counting. And of course, we have the electromagnetic wave. So here are our basic relationships. And, you know, the wavelength times frequency equals the speed of light. So I write it rather than wavelengths, I write it in terms of wave vector is the numerous wavelengths and the angular frequency, so [INAUDIBLE] to near relation between the amplitude and the c of the wave vector. Wave vector is vector, so it's a kx, ky, kz in Cartesian coordinate. And that's what we call the dispersion relation. The relation between wave vector and frequency is dispersion. So now I want to see how many moles in this cavity. And then my simple picture is this is the length in the x direction. and because a perfect wall, so it's reflecting and it's a wave. So the only way that this wave can exist is having 0 amplitude at the end-- on the surface of the wall. So this is-- if you have a string and you want to do the, go to think about that, you play violin and you have the nodes at the two ends and the vibrational modes at the two ends, 0. So we have different modes. The first mode is a half and the second mode is including the [INAUDIBLE]. And I can use this as my constraint in the x direction, that's the length and a half modes or twice or in [INAUDIBLE] and half mode, half wavelengths. So with that I go to, because if we factor in the x direction is just 2 pi over lambda, so I can write this in terms of n and L of x, the length of the box. And if I write that kx, I can same way write ky and kz, so I have a lot more numbers for this kx, ky, kz. And then each set of kx, ky, kz, if you think about the wave, give me a distinct wave. And that's one mode, except that the electromagnetic wave you learned before, it's a transverse wave. The sound wave is a longitudinal wave, right. When I speak here, I'm pressing the air and the direction of propagation of the wave is in the same direction as the air doing expansion and contraction. So that's a longitudinal wave. The transverse wave, electromagnetic wave from the sun, that's direction to the Earth, the k vector. But the electric field, the magnetic field is perpendicular to the direction of propagation. And I have two, so I put a two here. And then I sum up all the modes, so n number, each set of n number, nx, ny, and z is one mole. So I count how many moles there are inside this cavity. So I just sum up all possible combinations of and each combination of one mode. Each mode has a polarization, vibration direction. And then each mode, the photon has an energy, each and every photon has an energy, that's what we got. Yes? AUDIENCE: I have a question. How does the factor two relate to the fact that the electric field is transverse to the wave vector? GANG CHEN: So when I have a set of x and y and z, so in one case, the electric field is this direction. In the next case is this direction, or other is a combination of the two. So it's not a fundamental mode. So you have a factor of 2 is because in this one set of nx, and ny and z, you can have two possible e directions, electric field direction. So I do my summation here. And the only thing, this is the trick I will use many times. I do my summation because I don't know how to do it easily, analytically, so I'm converting that summation into integration. So if I think about n at this [INAUDIBLE] and the wave vector is great, but if this separation between the energy is small enough, so I can say in this distance dkx, and this is the unit of this discrete, so that's the equivalent that a continuous number of n rather than discrete number n. So this is the equivalent of summation into integration as long as this function doesn't change a lot between the [INAUDIBLE]. And I do that for y and for z, so this step is just for my future. What this step means, rather than doing this effect of two hear from 0 to infinite, which is the standing wave, OK, n is an integer, positive integer is a standing wave, I'm going to do from negative infinity to positive infinite because this is a unit function, And this way, I'm really rather than what I miss is this factor of two, I don't have a factor of two anymore. And physically what I'm saying is the standing wave summation is the same as two propagating wave, [INAUDIBLE] propagating wave. When kx is negative, it means going negative direction, positive, going positive direction. So this is mathematically equivalent. And later on, a lot of this minus infinity to positive infinite summation of the standing wave summation. So it's a more mathematics. I can walk you through, but there's no physics here. It's a book doing the latter. Are you ready to hear the dkx, dky, so I wrote it into Cartesian coordinate. Now it's not x, y, z, but kx, ky, kz, it's the Cartesian coordinate. And three dimensional integration, the rest is just the numerator here I had before. Those numerator are lx, ly, lz, so 8 pi cubic. So that's just a copying what I had before. And x when we got f is [INAUDIBLE] copying. So I'm doing all the copying now to simplify again, simplify [INAUDIBLE] the integration, you can say I have a conflict, the conflict is y is integrating over k, but my function is a function of omega, so I have to transfer either y into omega into k or k into omega. And I would like to transfer k into omega in this case, it's your choice actually, when you do it. So I transfer that k into omega, and I want to do spherical coordinate because that's easier. OK, so dk, dx, dk, dx, dk [INAUDIBLE] is 4 pi k squared dk is spherical coordinate. And now you go to look at the kx squared, I hope I read them down in the previous slides. OK, kx squared equals omega squared divided by c squared. So this is how I change k into omega. That's where the dispersion relation, the relation between omega and the k is very important. So k into omega, that's just omega equals k. So dk is d omega, d over c And now I combine everything. I do per unit volume, how much energy we have. And I write all this other factors into the integration and this is what I have here. Everything is a mess. And then this t, I call it density of states. What it means is how many quantum mechanical states or electromagnetic modes we have between-- around omega. And so this is the default count, energy, one photon, one state, how many photons we have. And in this state, in this region, how many states we have. So this is also the energy density. Did I write everything together? It's just an energy density in the frequency. OK, so this is the energy density, as I wrote down. And if I want to write out the density of states, f, and the actual [INAUDIBLE] combined. That's Planck's law, Planck's blackbody radiation. The energy density per unit volume, and these were actually the start of quantum mechanics. Planck was trying to explain the blackbody radiation. And if this is not called-- because the energy state, in fact, most time we don't talk about energy state, we talk about radiation leaving a surface, not just how many energy in per unit volume. And in that case, the definition is the intensity, energy flux per unit, solid angle. What is a solid angle? A solid angle is if you have a two dimension, you know, a polar angle, x, y. From x, y, you want to use a circle polar coordinate, you have polar angle and in three dimension, that angle is a space, angle in space. That's a solid angle, per unit area normalized to the distance r squared. So a sphere, the area is a 4 pi r squared. The solid angle then is 4 pi, because r squared [INAUDIBLE]. In 2D, what's the phi? 2 pi. 2 pi versus phase for 4 pi. That's the solid angle. So that's because that's the terminology we use often, so this intensity is just now, I know the energy density per unit volume, this energy is propagating at the speed of light, So is the energy density times c, and it's going radiation, thermal radiation is going in all directions. So per unit, solid angle is a 4 pi. So this way, I write the speed of light times energy density divided by 4 pi. That's the intensity. And this is just so you go continue substitution, you get the intensity, and sometimes we do per unit wavelength interval rather than per unit frequency interval. So rather than doing d omega integration, you do d lambda integration, and because lambda is times omega equals 2 pi c, so you convert the omega into lambda. Again, this is when you actually take a radiation course, this will be the Planck's law as written [INAUDIBLE]. And what are they? So that's a per unit solid angle. And when I do the total radiation per unit area, so this is the area, [? dense ?] area, so that's the total radiation is a pi times i, that's the mathematical, we're not getting into it. With that, what I want to show you is from here, if you do your integration, you get Stefan-Boltzmann. OK. Stefan-Boltzmann law, if you take a 2 double 5, is where we get-- we start on thermal radiation. And what I gave you is a way of going from basic concept to go to the Stefan-Boltzmann, but say, what's important is for you to look at the curve of blackbody radiation. This is very much relevant to the way we want to do solar cells or solar thermal, because the sun's radiation is about 50-- here I wrote 56 as hundred, but it's really around 5,800 Kelvin, that's the solar radiation. It's like a blackbody, the spectrum is very similar to this curve. OK, outside the atmosphere, to go to measure the solar radiation, it's close to 5,800 kelvins curve. And at 300, I didn't plot, 300 Kelvin, this peak will be around 10. So here you can see the peak start decreasing and that relation is the wave's displacement mode. Wavelength times temperature is about, back of the envelope, three solid. If you don't remember that detail, whether it's 289 [INAUDIBLE] and in fact, every time I go back to check, I remember about 3,000. OK, so 3,000 micron, and that's why our eyes are sensitive to the 0.5 micron, in the solar system. And the solar radiation, you put about 6,000, 3,000 divided by 6,000, our eyes are the evolution choose us to be more sensitive to the peak of solar radiation. And during night, there is no sun. Everybody is emitting, you're emitting, I'm emitting, we're emitting 10 microns. 300 kelvin [INAUDIBLE] 10 microns. Of course, if you take a detector, you can look at it. So you are like that, the infrared radiation measurement. But if you look at just a common-- a solar cell, silicon solar cell, the wavelengths that you can take, the shortest wavelengths is about 1.12 micron. So 1.3 is about 1.112. So that's-- this is the 1 micron. This is the 1.12. So only radiation in this range can be utilized in solar cells. You have a lot that's wasted if you do [INAUDIBLE] or yourself, this rest of the wave length's energy cannot be used to generate the electron holes. We'll talk about that later. So that's a completely what I was talking in the review, basically heat transfer and thermodynamics. What I'm going to move on is giving you a brief introduction of thermoelectric energy conversion. In the last lecture, you have seen a basic demonstration of thermoelectric generation. And in fact, this is the module that used to generate the power. Pass around. Take a look. And basically here we are using direct energy conversion, we're using electrons or holes in semiconductor. We have names for positive electrons and holes. And we're using this as a working fluid to carry it around from one place to another. So let's look at the phenomena, basic thermoelectric phenomena. And then we will see do a simple analysis of that device and pass it around and then look at the current and the potential application. First system I like the effect is not new. And in fact, this were about the same time people started to investigate the electricity. And Seebeck discovered that in 1821, that when they put the two metals actually at the junction, and the one side is hot, the other side is cold, we measure the voltage. Seebeck, now we call it Seebeck effect. And Seebeck himself, actually, you go to read the history, he didn't know what's going on. He thought that was a magnetic effect, because at the time there are a lot of interest in say, magnetic Faraday or the experiment. So he saw thought that was a magnetic effect. And about 10 years later, Peltier discovered another effect, that's the same two, say kind of junction, two metals, was pass the current through it. Actually, one side cools down, the other side heats up. He could make ice at the time. In fact, the Seebeck, if we choose this material to make a device, even at that time, a power generation device, you will get about a 2% to 3% efficiency using this material. Somebody do the power generator, so at the same time, people were starting to develop electric motors. You can see the history was really interesting. This one actually worked out. First, there will be a lot of study, and it will be different, right. Let's say, this effect is reversible. You just pass the current first, one is one direction. It will be cooling, and the other direction you will be heating. And so both those were experimental discoveries, comes in, gives us the theoretical prediction. And Thomson prediction is that based on the experimental discovery and understanding of the effects, he predicted that when you have a material, one side is cold and the other side of the hot. So a material under temperature difference and you pass current through it, there will be heating or cooling along the material. So this is not char heating. Char heating is always heating. This is a reversible. It will be either heating or cooling, depending on the direction of the current. as of Thomson's, Thomson is also Law of Kelvin, and you'll probably see a lot of Kelvin work, Kelvin relation for this, Kelvin relation for that. It pops in more than 500 papers, very productive in this lifetime. So now let's get a very intuitive picture. We have to go to, for those understanding, we have to go to microscopic again and we now have to look at electrons. Let's look at the electrons and I'm going first to just look at the isothermal, there is no temperature difference in the material. Isothermal conductor, a lot of electrons inside, and that's why you have to feel the current will flow. We all know that. So I have a current flow. Now again, I'm doing number crunching of this number. What's the current density? Current density is amp per meter squared is the charge per electron. Hence, the per unit volume or how many electrons I have times the velocity of those electrons moving. It's a very intuitive. So you go to look at the unit, you'll be adding per meter squared. And the velocity is related. I'll give you the relation more later on through a electrical field. And this, the coefficient proportionality, coefficient is called the mobility mu, relating the velocity of the electron to the electric field. Again, this I'll give more detail derivation. And of course that will be the ohm's law, the current is proportional to the electric field and the proportionality constant is the electric conductivity. And sometimes we don't talk about conductivity, we talk about resistivity, just 1 over the conductivity. So this is the ohm's law, and all the definitions, their units are given here. So this is when we talk about the electrical flow, we care about current. But each charge also carry heat with it. I can do the same thing for heat, So rather than the charge, electron charge, I'm going to say the heat carried by each carrier. I don't define how much or how I write that heat yet. I'm just saying per carrier, heel is the heat carried by each charge. Again number, density, velocity, they're moving. And of course I can replace the n by j the electric current. So now the heat current is proportional to electric current and the proportionality constant is the property of efficient. So associated with the current flow, there is a heat flow. So Peltier coefficient can be defined for each individual material. You don't have to have a junction. The junction is only when you do experiment, how you observe it. OK, so this is the-- when I want to observe the Peltier effect, the best way is to put a junction there. I have two different materials, and when I pass a current through, then you can say the current has to be-- electric, current has to be continuous. That's a [INAUDIBLE]. You have [INAUDIBLE] in and of course out. So that's the continuity, current continuity. But the heat carried by the electrons in 1 or 2 of the different from the other, so if I do my first low energy balance, I find out if I have more heat flow out, then come see it, I have the supply with me. So what does this mean, supply the heat. The lattice cools down. The atoms cool down, because I'm talking about electron, The atom can supply the heat to the electron cool down the junction. So this cool down, and on the other side, you have to reject that heat, energy values. So this is the Peltier phenomenon, you observe experimentally. And you can say, if I reverse current, this phenomenon is reverse. So this is the Peltier effect. It's reversible. Now look at the Seebeck effect. There's a seat here. OK. Seebeck effect, Seebeck effect is the voltage generation under a temperature difference. And you think about-- we have charge in the conductor, and then when one side is the hot, the electrons are near the hot side, have more energy they move more violently. So they will diffuse to the cosine. So you have at the end after diffusion, you have more electrons here. When you have more electrons, the electron imbalance will happen, you have electrical field building, The internal electric field will resist the diffusion. So at the end you have a steady state for the diffusion current balance, the electric field will drive the current back. So this balance will give you the Seebeck voltage. That's the process here. OK. The Seebeck voltage is defined, I put the [INAUDIBLE] sign. This is when you do an experiment, you want the one side is hot, the other side is cold. You put a probe, normally you say, OK, the definition, I put a hot probe on the high temperature side, positive voltage on the high temperature side, negative voltage on low temperature side. And if you do that, here is written very clearly, you should have put a negative sign in front, that's the definition. I remember the first time I measured a sample. I just got a wrong sign, I didn't know why. Everybody measured differently, because people normally just say Seebeck voltage is the voltage divided by delta t, but then when you do experiment, you have to see which side it falls. Thomson effect. Thomson effect is when you have a temperature difference and you flow a current through and if you look at every point, there is a Peltier coefficient. The Peltier coefficient can depend on temperature. So that means when current flow in, the energy carried in could be different from the energy carried out. So the heat carried out could be different from current in. And then that difference must be made up, cooling in by either cool or rejected. So that's the Thomson effect and Thomson to efficient is defined the low amount of heat absorbed or rejected versus the gradient divided by temperature gradient normalized to current. [INAUDIBLE] comes into it. OK, so we have this phenomenological explanation. And now I want to do more detail. I want to put this in more-- into more mathematics into it. I forgot to mention here, Kelvin, of course, the concept of Kelvin realized, based on his understanding, those three effects are interrelated, and he actually gives the relation between the Peltier coefficient and Seebeck temperature, and constant coefficient is a Seebeck derivative with respect to temperature. OK. Those are [INAUDIBLE] dependent properties. Like I said, let's go the-- so let's go listen to the device, and we want to say, based on this, what are the material properties I want? So I show this device. And again, you can buy the commercially, particularly some of you may be interested in various competitions, MIT has a different competitions, and if you want to build this device, it's not expensive. This is a $10 to $20 commercially. you Can buy one. And the again, the basic concept, now even cheaper device, you see a lot of legs, they are electric. There are only two electrodes. One is in, the other is out. So electrically, they are connected in series. And in fact, most of this device are made of opposite type, p and n together, a P type semiconductor, N type semiconductor. And in the N type is electron carry the current flow and P type is called the positive charge. So when I pass a current, of course current flow against the electron, along the positive charge direction, so you can say that's in terms of device how it works. They carry heat from one side to the other. And on this side, they reject the heat. On the other side, they cool down. So you can do the refrigeration as well as power production. We show the power, here is a simple demo of the cooling. That's the price, you convert the water into ice. OK, let's look at how the device performance depends on-- we show the summary, let's say three thermoelectric effect, but in reality, the solid state material, so there are other effects goes on, when I have 100 difference. And the first one is, of course heat conduction. On one side is hot, the other is cold, heat will flow. This heat could be heat conduction, electron can conduct heat, lattice also conduct heat. So you have both electron and lattice. This is different than just the Peltier coefficient or Seebeck coefficient. So we have the [INAUDIBLE] law. And if it's a 1b, I integrate it, I get the amount of heat transfer is proportional to the cross section area, thermal conductivity and the temperature difference divided by the length. You see this in [? 2005. ?] And if I group all those parameters together, AKL, I say this is the thermal conductance. We probably define thermal resistance before, but the resistance is just conductance is just the inverse of the resistance. So this is the definition of the thermal conductance in the KA or l. And just as a comment, very often we write it just look like math. But when you actually do it, it matters a lot because here, for example, the heat flow is inversely proportional to thickness, So in the technology development, if I use a film, l is of the order micron. And if I use a bulk material, l could be probably 100 microns or millimeters, and that heat flow is a 0 order of magnitude difference for the same delta t. So whether, in reality you have different problems, sometimes it's clearly space, like the solar, solar radiation clearly is fixed. And you want to build your system to create a delta t, because it's a thermoelectric device is proportional to delta t. So that's one type of problem. There are other cases where delta t is fixed. Give me give me an example. There are people looking to ocean geology generation. In the ocean, there is the top and bottom [INAUDIBLE] difference. You've got the different problems and you can go to back to the most basic picture to think about your system, sometimes that's always the first step. OK. Now let's look at the device. Again, the [INAUDIBLE] device has many repeat units of this t and n. And if I have just an ideal device, how much cooling I can create. I pass a current and the cooling is the difference and the junction of the two materials. So one possible material multiplication times a negative material coefficient times current. So that's how much cooling I can create. But in real situation, what happens is that when this site is cold, this site is hot, the heat will flow back. This is solid material. You can't get rid of that heat conduction. So it will diffuse back by conduction. That's one reality. That's a long idea factor. The other non-idea factor is when you pass the current through, I have [INAUDIBLE]. I didn't talk about [INAUDIBLE] here, right? That's unavoidable part. I don't have a superconductor. In fact, you don't want to use a superconductor here. Superconductor, electrical superconductor is only one quantum mechanical state. They introduce 0, doesn't carry heat. Electrons do not carry heat, in case, superconductor. OK, so there's a [INAUDIBLE] because the electric conductivity. And I have to consider both, reverse heat conduction and [INAUDIBLE]. Now, I think about [INAUDIBLE], the difference between conduction is this side is this side is called [INAUDIBLE] conduction. The [INAUDIBLE] is uniformly generated along the lag as approximation. So in that case, [INAUDIBLE] about half go this way. The other half goes the other way. You can solve the equation and you'll find that the answer. But that's the intuitive argument also. So in that case, the real net cooling I have is, in addition to calculate, I have half of the [INAUDIBLE]. This is electrical resistance. And I have a reverse heat conduction that the thermal conductance of the two legs. So that's the mathematics. The resistance, the electrical resistance is proportional to the resistivity of both legs area, that's and here are thermal conductance, they both p and n combine together. And now you can see the amount of heating or cooling I create depends on the current flow, right. I square, y is a linear, the other is square. If I drive the device too hard, I generate too much heat. I drive too little, doesn't have enough cooling. So there is an optimum. When you drive the device, there's an optimum there. It's easy to find the optimum. Firstly, I want to find out more not only for cooling but also for efficiency. So I have to calculate how much electrical energy I give in. You say. i square r. It's not that simple. There's a little trick here. Because there is a i square r, so voltage drop is i times r, that's the [INAUDIBLE] part. But when there is a temperature difference, there is also a Seebeck voltage. So when this side, one side is cold, the other side is hot, there is an additional voltage you have to overcome, the internal review voltage, Seebeck voltage. That's the Seebeck coefficient difference times delta t. That's the voltage you have to overcome in addition to resistance. So the electrical power is i times v. So that's the electrical I put in. So the efficiency, according to our definition, for refrigeration, we talk about coefficient of performance, We don't talk about efficiency. So this is what I want, the cooling, this is what I pay. The pay is the power, the power is the current times the voltage. So I have i square r, that's the [INAUDIBLE]. I have to overcome the Seebeck voltage. So that's the term there. So that's the bottom part. That's what I pay. And the numerator is what do we derive, the previous slide. Including is accounted for the reverse heat conduction and the [INAUDIBLE]. So now you go to optimize it. And this-- yes? AUDIENCE: Shouldn't Seebeck dominant be reversed? i squared r. GANG CHEN: No, you wish that's the case. You wish that's the case. But it's-- you are-- when you have a delta t, you have to push hard. You can say, OK, we'll have a hard flow for cooling. This is a t, this is n. I know [INAUDIBLE] this way. But this side is a hot, the voltage is against here. Otherwise, you get too easy, your problem. So you go to optimize your current, that's the arithmetic, you can divide ei for 0 and you find the optimal current, and then find the optimum, under optimal part you got to maximize efficiency or CLP. Of course, I dropped all the math. And let me again make a comment. You go to save your [INAUDIBLE], that's easy because there's no denominator here. You got the optimal current for maximum cooling power. Here you get an optimal current for maximum efficiency, but they two do not agree with each other. And in fact, in your operation, you want to adjust your current just between you can't satisfy, you go optimal current and you don't have much cooling power. The other point is, you look at this, you say, it's a linear term. This term is all thermal energy. That's a [INAUDIBLE] cycle for refrigeration. We wrote down [INAUDIBLE] for power generation. But this is a column so it's a column factor, it's a heat engine. And of course, you all make sure this term is less than 1. So you can't get column, and here came to this vector z. So every time people see, what's your zt? This is where it came from, z times temperature. And this t is the average temperature between the hot and cold side. And this is z, again, this is all coming from this mathematical derivative. And it's the Seebeck difference of the material. And the denominator is the conductance, thermal conductance times the electrical resistance. So look at it. It's not a material property at all, it'll equal the material property, Seebeck coefficient, electrical resistivity, thermal conductivity, but it also depends on the cross section area. And they look at this function, you want to maximize your zt to get a higher efficiency. So the first thing is that you go to maximize your z by geography. You can reduce the denominator by changing your l or a [INAUDIBLE], right? That's the least you can do if you don't work in the material. And that's what you want to minimize. And if you minimize it, it turns out this is what you have to do. The k times r is the minimum when they do lens cross-section area of the two legs. So you are engineer. This is where when you do the device, you optimize it. We then-- when we do that testing, own lab, sometimes just hand polishing, all this, trying to make it match to each other. So you want to get a better performance than others. OK, now you that your best case is an idea, that still is not a zt. That's still not the individual material property. It depends on both sides, here and the end materials. So basically, if I want to make a good device, I need a good n, a good p, and the two should match each other. One material you get, you do get the best performance, And you get about half of the performance if you have good one material. But because at the end, using the two means you have to make a lot of choices. In materials research, people just talk about one. You engineer, you know, you have to go to get both materials. Somebody has one. You say, do you have another? You're eager to us. We'll find others who has the other. So this is when material scientists just say, I discovered material with high z, That's s for sigma or k. You want pi, a material with a high electric conductivity. And so you want to minimize [INAUDIBLE]. You want a material with a high Seebeck coefficient. Seebeck coefficient fundamentally is the heat carried by per charge. OK, we'll talk more on that later. So average speaking per charge, how much heat it carrier. And then you want a thermal insulator, a low thermal conductivity because you don't want to heat reverse back. If you know any materials, let me know. People have been searching this for centuries, good material with a good electrical thermal conductivity, poor thermal conductivity. But also you want every electron carry a lot of heat. And this has to happen in one material, [INAUDIBLE] material. AUDIENCE: So we derived the Zmax for a junction. GANG CHEN: This is the only-- Zmax. You need a computer device in these two materials, Well, theoretically, you don't need two material, you can do one material. In fact, in that case, what people choose, the other material, they choose a superconductor. Just don't do anything. So that's how they characterize the one mature if you want to do a good job. All right, let's see, this one is immature research, you don't-- you can't wait to get too mature and positive, You get one but mature taking many years, so you do a simple material to say this is a measure of how good is this material. But as an engineer, you want to build a device, you have to say, where is the other material? I mean, I need both materials. So those are the comments that is made to optimize [INAUDIBLE] optimize index. And there is two types. And then you optimize the device performance, load matching by current. This is a [INAUDIBLE]. So here is some typical [INAUDIBLE] for the business, that's the commercial device mostly you buy. You can only buy this type of material based device. Seebeck [INAUDIBLE] is about 220 microwatts per kelvin. Electric [INAUDIBLE] fence is around 1.51 [INAUDIBLE]. This is all Si units. It's good to keep in mind, example of a micro [INAUDIBLE] including the [INAUDIBLE] zte is about 1. And we'll talk how good is that. Now, what's the problem? If you have all this library, you say, OK, this is-- I think, a [INAUDIBLE] 2 millimeter long electrical interface, cross-section, one millimeter square. So it's a small, if you look at the device that pass around, that's dimensioned roughly, each leg is only 20 million ohms, 20 million ohms, very small. You go to buy resistors, actually very hard to buy this small resistor. What it means? It means if you want to build a single device or two legs, because current is a voltage divided by resistance, It puts your [? heart, ?] huge [? heart, ?] you need a big power supply. So you go to look at you in your lab, have a big power supply, a small driver, small device. And at the end, that means cost, so everything you have to consider. And that's why some people do micro devices. You can actually shrink your area, You do a smaller cross-section area where you put a lot of those together, in that case, you get a higher resistance by putting more of the [INAUDIBLE] together. So in the device, I showed the cost around, this is about 100. So if you multiply, 100 is only two ohms, the total resistance if you own this sort of device. So it's a small resistance, current device. So that's the reason to put them in series is putting getting higher resistance so that they are supplied easier. Let's say it's still pretty large current. And then you put them ceremony in parallel, because the parity from one side to the other, electrically [INAUDIBLE]. AUDIENCE: When you optimize your system, then you really get that simple number, 1, 1, 2. GANG CHEN: Simple number of 1, 1? AUDIENCE: [? 100 ?] millimeter by [? 200 ?] millimeter. GANG CHEN: Oh, you mean-- no, no, no, no, no. This you can charge. You can make a long leg to test it. You will still get the same delta t. AUDIENCE: But when you optimize the system? GANG CHEN: Optimize means also material cost, The? Real world is not just you have to match to your application. You don't have to fix to the geometry I pass around. And that geometry is in commercial production because in fact, each application may be different, but the heart of the market is too small, so people are just selling more standard modules. Well, anything you go along a standard, you have to talk to the manufacturer and see whether they can manufacture. OK, so this is the example, this we built before. This is actually measuring two legs. There's a lot more difficult than measuring device, because any time you put a thermocouple there, you have a feed loss. Feed loss is your performance may diminish. If you want to say, what's your maximum delta t or that, so it's a lot of time brewing and doing the experiment, you have your leg is mediums. So anytime you have missing [INAUDIBLE] the industry, they do much more, much better. I look at a production. I was really amazed at what I'm seeing the worker can do, versus what we can do in the lab. That's a different environment. But this is the 1p1n, and this is a measurement curve. You can see it's a large term. In this case, we go one fatigue along the line through that device. By [? line ?] absent, the use of aggregate 1 million or 2. So these were-- What's amazing with that small thing, on this side is 100 degree, the other side is 0. So you can derive a lot of the other things you have. The efficiency is not very good. OK. The material is dependent-- mentioned this before. An the analysis I gave you is for constant temperature. So if you do real device design, you have to go solve differential equations and match the property, consider all the temperature dependent properties, and that I'm not going to cover in depth. Anybody wants to do that, you can just do the energy balance in differential form. And this is the heat conduction term. This is the constant term. This is [INAUDIBLE]. So that's a differential equation locally. And very often people use differential analysis to incorporate companies. And then the boundary has a plural term and the positive term. So you have a boundary condition and a differential equation. And so, again, this is a [INAUDIBLE]. Y'all know what. I want to talk in details. And now let's look at where those technologies are. I have the expression for CLT and you do the same analysis about power. And you get the efficiency for power generation. And we [INAUDIBLE] power generation and getting ZR inside, and this factor is slightly different from CLT, but you go to check, it's probably your first one. And this is a part of that, Again, it's a thermal engine. The [? power ?] pans to whatever your material. You can try to [? power, ?] you went big on this, you have 5gt, you can go to [INAUDIBLE]. And so this is not-- so I would say, not so accurate. I mean, it's like around 95 to 96. And this is the accurate ratio of hot to cold in Kelvin. And this is the efficiency. And what I put down here is the thermal activities at the bottom. OK, there's another [INAUDIBLE] picture. First, have to do a lot of research, And the power plant is about 40%. The normal coal fired power plant is 40%, Internal combustion engine is about 20%, 25%. OK. What I need now for the [INAUDIBLE] is [INAUDIBLE]. Depends on amount of silicon, about 6%, single crystal silicon about 18% Once you think about that, then your picture becomes a better. You can, in fact see the [INAUDIBLE] use a thermoelectric power generator and that's about 6% to 7% in the deep space missions. And now in the lab, there are people developing similar generator using a zt bottle, one material that is around 40%, copper, 40%, uranium 2 level of that's actually very attractive [INAUDIBLE]. So it really is an application because you don't-- at this stage, we're not thinking you can't it's not at that level to say I'm going to replace major form over in the sense of like, we're not dealing here with an internal combustion engine, but there are many different sources where you can potentially [INAUDIBLE]. And there is refrigeration on this before. and CLT, again, depends on temperature. And a CLT of thermal electric compared to fossil refrigerator is about a factor of 5. So that's actually probably the oddest really want to say the application, say really replacing the screwdriver, I think probably took a long time. But again, there may be other applications where you can think about [INAUDIBLE]. And in fact the current market, and I'll show later is mostly [INAUDIBLE]. So this is where really the engineers, you can really say, I know the technology is where it makes sense. There are different areas that it could make sense, and commercially, mostly standard size, standard size people offer different sides, but the geometry is very similar. The height could be in the case of our generator for about 5, 6 millimeters, and the coolers, the typical millimeter in height in the neck, and the top is a ceramic and copper bottomed ceramic. And [? the head, ?] the next the next are interconnected by copper. So the technology wise, they have to put a diffusion barrier between the copper and material, because copper easily get into the material that composes in the [INAUDIBLE]. In terms of applications, this has one standard small cooler, this is because when you reduce the compressor, the compressor actually, the smaller you make, the hotter-- the hotter you make it. When you make a very small compressor, of course, the northern part is [INAUDIBLE] and this becomes a more attractive, so this becomes more attractive. A lot of the dormitory [INAUDIBLE] refrigerators made for somewhere like this. And still the efficiency, I think, is still not as competitive, but the cost could be very attractive. And then it can be as small as just a little vial. And the people, in fact, that put a lot of multi-stage to create a lot of-- each stage is a maybe-- this to about six feet together, you can get a commercially at least 50, [INAUDIBLE] pounds. So the infrared detector were also needed to be cool with [INAUDIBLE]. But the main laser diode has a somewhat like a cooler. This is because the wavelengths in communication, telecommunication, the wavelengths of this laser are [INAUDIBLE]. So they actually use some energy to stabilize the temperature, not much for the [? human, ?] but for the [INAUDIBLE] macro DNA, we have to compare the initial [INAUDIBLE] naturalization and recycling. And probably a really good commercial example is this company for the American. And they started putting a device, this is where you have to look at where it makes sense, They started to manufacture around 2,000, somewhere like six, and in some high automobiles. Now they have a market, about 3 million models, include the car seats. I still have this in my car. It's a luxury item and but pretty well successful. And now the DOE is trying to even get rid of the main air conditioner. And the idea is that the main air conditioning automobile will require about 6, 7 kilowatt. And the purpose is to turn the engine on, the air conditioning on. You want to cool the whole compartment in about 5 minutes, but normally you do need a high compression, large compressor. So the question is whether they can use this distributed cooler into the automobile. I'm not sure whether it will be successful, but there [INAUDIBLE]. And in fact, this is a-- when you think about where they make sense. They find the seats, not your offices. Your offices, you move around. Once you get the car, it's there. They plug in the power because this means the power there. Right? So any other places it could be using that-- AUDIENCE: Movie theater GANG CHEN: Since that's probably too big. I was thinking about the bed. I say there are actual products there. You go, I was thinking before, I looked at the wrong products already. I guess the key there is whether it's a physiologically comfortable and also how you reject that. If you reject it to the same rule, does it make sense at the end? So there are-- those are all the places where if you're an engineer, you got to think about where it makes sense and where you have a product people accept. And with the more product into the market, of course there's a material is also improving that you have potential, further advancing the technology. Power generation, the main driver in the past is space missions. So in once you go beyond the Mars, the solar radiation diminishes. So in Mars, most of the power still come from the solar panels, although there are some electric cooler or thermoelectric generator. And sometimes those generator proposal was actually to keep the electronics warm, amazingly, in you go to say beyond in the-- it's very cold in the night. There's no power. You need the electrical power to heat up the electronics. So the space mission, most of the deep space mission are thermoelectric power. And that's a good news because Galileo, for example, send back signal for more than 20, close to 30 years. Galileo, eventually the signal was not lost a lot because the power generator, just because of the mission was abandoned. Voyager, all those were based on thermoelectric generator. And in there, they use a nuclear force. So actually, I went to one deep space conference in our pocket. That's interesting, in the conference outside, there are people, space missions. So there are outside people protesting, because it's the most of the space missions are nuclear power. OK. A terrestrial application. This is a standalone generator where, for example, this company, global thermoelectric they become good business because they sell those power generators along to big oil companies. The oil company needs to monitor their pipelines. The pipelines, let's say you go through Alaska. There's no electric power there. So they actually use the oil to generate the heat and the power generation. Body power watch. OK. I don't think it's very commercially successful. The company first company trying to market it was a Japan company, Seiko. I went to visit them at the time, and they said $3,000 is the modern power watch. I for one. I say it's amazing that the watch only 1 microwatt, 1 microwatt, that's what I learned from the [INAUDIBLE]. The temperature difference is 1 degree Kelvin, a degree Celsius to develop for this watch. And there was a lot of companies who see, [? Carlheim ?] I think I didn't go to check her watch, but I heard now they are still on the market. They are selling $300. This one, went to-- I visited and they say, well, what's in the Smithsonian? So today it's free. But I think [? Carlheim ?] it's is still about $300. I don't have much. I don't need a watch. But some people are developing body power for, you think about the low power electronics. Some of these devices don't really require a lot of people. At the other day, I met an MIT professor in the department, I know [INAUDIBLE] student there. He has-- he said that he can generate 300 microwatts, which is pretty good. And I saw Philips trying to also develop it, the body power. And in fact, it might be worth the calculation to say, I think Philips is trying to develop a few, where you put the body the delta t right. And one of the prototype I saw is more like around the head. Maybe it's good if you generate enough power for the poor kids to read like LED, now, how much power LED is. It's interesting to check. This we were actually talked a lot recently, not because this will create a lot of [INAUDIBLE] in the thermoelectric generator, but this will be a technology that will really make the full area of people [? burn. ?] Because look, a lot of places, people are still cooking using wood. I know where I grew up, say not my place now, but when I grew up, I used the wood. And in this wood, if you ever use a wood cooking, you will know that the smoke doesn't burn efficient And what the [? Phillip ?] develop is a fan generated power from the temperature difference of the burning and then drive the burning process clean. So it's a clean, clean wood-burning, wood-burner. And now they-- I heard that they're trying to market, but they still, this is about 10 to 20 bucks and people still couldn't afford it and $20. And so their market is now first person to get to world health organization by giving the people free. So there are a lot of-- this one, if we can gather, all the people use it, it's a tremendous, their analysis actually of CO2 problem right, because about 2 billion people are out of Greek in the world. A lot of people burn this. And if you make those who still efficient, reduce a lot significantly CO2 consumption. So the point is, at this stage, you have to think about where it can make sense technologically. Well, at the same time, of course, none of you are interested in developing more efficient, better materials. So I go to a lot of material conference, several industry conference. And one sense in the past, the last conference was very different. But in the past, you see more than 90% of the talk are materials, and not many talk on systems. So this is where I think the system, the engineering actually can make a good contribution there, because a lot of times people just say, I have the [INAUDIBLE] make a difference. But ZR is only half of the story. Because if you look at particular system, right, where the cost will come from, most likely it's not in the generator. It's an important part, but it will not be the dominant one. So this is an object to be cooled. The difference of similar electrics from solar is a solar cell, you put it in the sun, that's your system. You manufacture out. You just need a frame and put their electrical and the rest is-- it's about half the cost or balance of system, the inverters or those other stuff. But there's a much simpler system. Here for thermal system, you have to establish, you have to be able to maintain the temperature difference, so that means you have to take the heat out or put the heat in. And I once did a calculation for system for. And what I found is the electric, thermal electric material is only about a few cents per watt, electrical. But the cost come from all the aluminum that, I didn't do a very good thermal design, but that cost us a few dollars per watt because how you maintain the DC and do the cooling could be easily dominant in your cost. Not always, you have to do a good design. And the other is if you think about the temperature, so you need, for example, household temperature is about 22. Think about your refrigerator. And inside the refrigeration, refrigerator, let's suppose you want a 0 degree. So you say, I need a 22 degree solar heat. But that means you [INAUDIBLE] 22 is your refrigeration compartment inside, your air outside. Now you have to transfer the heat to the cold side. So this cold side must be lower temperature than 24, So let's suppose the 15 degrees, I give you degree. And then the far side, I rejected him. You don't want [INAUDIBLE] reject has to go through the error. So I have to have another theology from here to here. I give another 5 degrees. OK, that's 12 plus 22. So I need 39 degrees rather than 22 degrees. Now you go to 39, COP versus 22 COP will be very different. So your system is really a very critical part of the consideration. That's the significant difference of those thermal systems, it could be more expensive than the simulated side. Sensor, if you think about [INAUDIBLE], right? [INAUDIBLE], let's suppose you want-- this is a hot gas, and you put a similar [INAUDIBLE] exhaust and then you cool this side. And you see what temperature this system will offer. Then you suppose the device operate at a uniform temperature, not a good design. OK let's say, I suppose, the device offering uniform temperature, then the [INAUDIBLE] gas must have a higher temperature at exist. This temperature of the gas must be higher than this, otherwise heat cannot go from gas to the hot side. OK. Heat only go from high temperature to lower temperature. So that means if the gas is 600 degree, the amount of heat I can take out is from 600 degree to whatever this happened. So if you do an optimization, it's about half, because you need to take the heat out of the gas, put into the device. So that means your [? half-side ?] operator is 300. So you can't do your calculations, 600 degrees, t what's my [INAUDIBLE]? What's my efficiency? And of course, you can be more innovative. You say, OK, I don't operate at the same time, but actually I operate at a higher temperature and my efficiency system [INAUDIBLE]. So there you got a lot of system design that depends on material availability and cost, weight, particularly when you do automobile. And with that cost, and so a lot of really-- so what I say is there's space for both sides. There are a lot of problems. Engineers can make an important contribution and a lot of problems with the material, [INAUDIBLE] improvement of chemical conditions. You look at-- say this is not a serious work on the other side, because you look at where the potential to use it, but maybe car generation most people not working [INAUDIBLE] because it is too small. And the reality is automobile company says 3% fuel saving, you'll be happy to think about it. 3% fuel is less than 1% system. So if we are exhausted, one third of the wasted energy there, one third, let's take one half of the heat, if I can intercept the half of the heat. So that's one sixth. So my system efficiency, if it's 6%, which you usually can be done with a [INAUDIBLE] not system device, 6%. But then the question is, when you put all this together, your [? half-side ?] your co-side is really the temperature, you know, if you internalize naturally. So you have to think all this, put them together and make you have the impact. Yeah. So the [? BMW ?] is a very aggressive things, but if you put in a market when [INAUDIBLE] is before and after [INAUDIBLE]. It's really up to you to find that where it's not competitive against-- I say not competitive against main [INAUDIBLE]. Let's say there are a lot of places where you can be [INAUDIBLE]. Any questions? [INAUDIBLE] |
MIT_2997_Direct_SolarThermal_To_Electrical_Energy_Conversion_Technologies_Fall_2009 | Lecture_6_Thermionic_power_conversion.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. GANG CHEN: OK, well, let me start talking. And I ran out of my slides, I mean, since last week. I was making this a new topic. And you will see, I thought I'm pretty familiar with this, but I still struggle a lot. And there is one place I will show where I'm still confused. Maybe you can help me. And then, that's always related to this electron negative charge that always confuse me. So I struggle a lot. So today, in the first few lectures, we discussed the thermal electric effect. And now we move on to talk about the thermionic engines. And also, I hope we will have time to discuss related topics from the discussion on thermionic engines, and also hoping to actually jump in to introduce you to really how pin junction works. And in terms of Schottky barrier and Schottky diode, that's really just a metal semiconductor interface. Again, I think from the vacuum thermionics, we can probably appreciate the evolution. I hope you will see the evolution. And then, in the mathematical formulation-- formula, the expression for current voltage are very similar. But let me-- I need something that we showed before. So this is the first, the Fermi-Dirac statistics. This was what we did in probably the first lecture, the occupation of electrons in a quantum state. And here you can see, it's a very-- this is the electron volts, 0.051. So the Fermi-Dirac distribution is very narrowly focused near the chemical potential. And if you are away from the chemical potential, the function-- either its state is fully filled or almost empty. And let me just make a remark. Typically, when we talk about thermionic emission, e minus mu, is u weight. So what we-- really, at the tail end, are for the Fermi-Dirac distribution. And the e minus mu is still u weight, kb is a 26 milli electron volts at room temperature. You can see that factor of 1 will drop out because it's insignificant. So use this when we derive the Richardson formula. And the other point that I will use in the discussion is the electron number density. So this is, again, we did before. We did the content of number of electrons counting all the quantum states. Each quantum state have two electron spin-up, spin-down. And this is the average number of electrons, two states, spin-up, spin-down. And from there, if we do the exponential, drop the 1, we get the electron number density is proportional. Here is, I want to point out t 3/2. So we group all this factor into nc. And then it's the difference of the conduction band to the chemical potential exponential. So that's what we had before in terms of electron density. And the other point is to make-- is to, again, review that we say, what's the difference between metals, semiconductors? In metal, we have the chemical potential for into the band. And semiconductor has bandgap. We dope it either n type or we dope the t type. Depends on what kind of atoms, for example, we put theta either boron, which is a t type, or phosphorus, which is n type. And that way we're shifting the Fermi level or the chemical potential. So those were what we covered previously. And now let's think about, if I have a material, meta, for example, I pull the-- I want to pull out the electron. And if I pull out the electron, the ion background, in metal attract the electron. So I have to do some work to pull it out. So the minimal amount of work that I pull electrons from inside immediately to outside, vacuum, is the work function. So from the Fermi level, this is a metal, from Fermi level, I pull electron. I pull it out to vacuum level. But this vacuum is immediately outside the metal. And then why I emphasize that, if you think about the-- I have a vacuum. I have positive charge, a negative charge. Even both positive and negative charge in vacuum, then the electrical potential at these two points, even in vacuum, are different because there is a potential from the [INAUDIBLE]. So I say, this is immediately outside the surface of how we define the work function. AUDIENCE: For metals, right? GANG CHEN: Yes. Semiconductor we have to use a different terminology. And that's how Einstein got the Nobel Prize. So he explained the photoconductivity effect, where the effect was you shine light on a metal, and then you get electricity, you get the electron out. So you can measure current. But what was puzzling at the time was that the number of the current not just depend on intensity, but depend on the wavelengths of the light. And see, so there was no explanation. And Einstein then explained, you have to have the photon energy larger than the work function to leave the electron on the metal surface. So the photon energy, of course, is inversely proportional to wavelength. So long wavelength full time not effective there. So not-- you know Einstein's Nobel Prize is not because of relativity, which was too radical. And this were more or less controversial. And then, in semiconductor, the Fermi level, there's no electron there. When we dope, it such that Fermi level fall inside the bandgap. So there are no electron between the conduction and valence band. The Fermi level is in between. So the way-- you could lift the electron, but then when you lift the electron, usually you lift from the bottom conduction band, which is ec to the vacuum level, and we give a different name. And in that case, we call it affinity. So semiconductor, because the chemical potential level can be changed by doping. But how much energy you need to lift an electron from the bottom conduction band to vacuum level, that's fixed. So this is a number you can look into tables, the affinity of semiconductors, and typically counted from conduction band edge. So those are the basics. And now, of course, if I think about the Fermi-Dirac statistics, there are some electrons with energy higher than w. Even though that f is very small, but it's not 0. So what I want to talk now is how much energy-- how many electrons is flowing out of the surface. And of course, you put a piece of metal there. You flow out. There are also ones that come back in randomly. And then, at the end, there may be some electron flying around the surface. I'm just looking into the flux coming out. So the flux comes out. e is the electron energy-- and say, outside. And I probably should not-- so this is where-- OK. Mu, this is one of the-- if you look at how we derive the chemical potential, you will see the derivation process is really mu is relative to the bottom of conduction band. And when I talk about work function is relative to the vacuum level. So there are some difference in the reference point. So that's the one place that can cause confusion. So I'm looking at, really, for those electrons with an energy above the vacuum level. And those energy, you can think of this kinetic energy inside in the vacuum here, that's the-- when I write this more, hk is the momentum. p squared divided by 2m is the kinetic energy, So those electrons, here, that's the kinetic energy. And I want to calculate the flux of the particle. I'm not putting charge there yet. I don't like that negative e. So just a particle flux coming out. So here, because it's coming out with kx, the wave vector, which is the momentum, it is the positive x direction. I don't do both from negative infinite to positive infinite. So now I do this summation. Because I'm looking at flux, so vx is the velocity in the x direction coming out. And then f is [INAUDIBLE] state. So I'm still doing my counting. The number of electrons, each electron has a velocity, and that gives me the flux there. So if you start from here, you do your normal way of converting this weight, because this is a flux, we should already be divided. That's a typo here. So if you do your normal conversion of the summation into integration, that's what we do here. And then, in reality, because the electron is-- or in terms of weight vector, it's not very large. So you actually can extend this integration really rather than 0 to pi or a 0 to infinite. So what's the vx? vx here is a p divided by momentum. So I'm doing that because, at the end, you can do either energy integration or momentum integration. And in this case, you can say k, I integrate over k is easier. Because if I put this, drop the 1, and e minus mu is work function plus the kinetic energy. And you can see, if I do this integration, I can actually, because of the summation here, I can split each term and do integration kx, dky, dkz. So that's an easier integration than I convert it into energy. I show this because, like I said, the density of states is just a mathematical tool. If you use-- it's sometimes convenient to use it in energy, sometimes convenient to use just as momentum itself. So if I do this, go rather than 0 to pi over a, 0 to infinite, because this exponential function drops out quickly, this is what I get. And the exponential here, w, is the work function. it's the energy. kbt is also energy. This is a very small number because w is a few electron volts. kbt is 0.026 at room temperature. So that's a big negative exponent. But the front side, this term you can see is temperature-dependent. And we could write it in terms of t squared times a constant. So now I put it-- before I said it's particle flux. Now I put the charge, because each particle carries a negative e. So this is my current flux. So the current flux, now I have a t squared. And whatever in the front, I group it into a, so a t squared dot exponential. And this is the Richardson formula. And Richardson got, say, did a lot of experiment, and he was awarded a Nobel Prize for this work in the thermionic emission. But there are also, if you go to check the literature, in fact, the Dushman had a paper before him. So sometimes people call Richardson-Dushman formula rather than just Richardson formula. But the point is, we have a t square, we have exponential function for the flux of electrons coming out of the surface. And this is a constant a is called the Richardson constant. But, say, in more general form, because the quantum reflection, so this is, in fact, not just an a, there is a factor in the front. That's the electron get reflected even though its energy is larger than the work function. But the waves has always a possibility to get reflected. So that's a 1 minus r shows the transmission. And the rest is the-- what's the-- this one sometimes we use w, sometimes you use phi. It's all work function. So any question here? Yes. AUDIENCE: How did he measure that? GANG CHEN: How did he measure that? You put the voltage or, in fact, Edison did a lot of work too. This is this-- the beginning, so you see people develop vacuum tubes. So you can, in this case, you change the temperature, you measure the current that come out of the electrode. And what Richardson did different is before, if you go to read the literature, and people do the charge, this kind of thermionic emission, not in vacuum, just in air, in a, say, gas environment. And sometimes the surface is negative, sometimes is positive. So it made the explanation very difficult. What Richardson did was do it in vacuum. And that made the explanation much more cleaner. AUDIENCE: This is just for metal or everything? GANG CHEN: This is just for metal. And the w is for the work function. Now, next, this is where my confusion starts kicking in. And I mean, let's talk about two metals. They have different work functions. And you connect them together. And once you connect them, let's say, the charge will flow from one to the other. This is a mass diffusion. If the chemical potential are different, they always at the end will pull together equal chemical potential. You don't have any driving force. You just connect and put a wire in between them. So of course, two pieces of metal lie in contact, far away, each has a local vacuum level, no torque. But if I connect a piece of wire between the two, then the chemical potential is flat. So that means initially there is a current flow from one side to the other. And I was trying to think-- you can tell me now-- I was trying to think, now, what's the direction of this current flow? What do you think? What's the direction of the current flow? Sometimes we're forced to think, ask questions, try to explain to you. Even though I can't make it clear, but maybe I can make it clear when we all talk and think. So now I have one piece of metal at larger chemical potential-- larger work function, a piece of metal smaller work function. And I connected them together. If I don't connect them together, I think, when I put them very close to each other, I know the direction of current. When connected, I don't know. If I don't connect-- let me tell you. If I don't connect them, but I put them very close to each other, I think the electron will go from the high work function side to the lower work function side through the vacuum again. Do you agree? Or you have different argument? But when I connect them together, my question was, will the electron flow from the other way? Because that means, the question is, which one is positive, which one is negative at the end? I don't have-- I still don't have a good answer. AUDIENCE: When electron travels from the left to the right, then doesn't it increase the Fermi level of the right side? Then if you-- GANG CHEN: If the electrons go from left to right, it increased the Fermi level-- well, so this is another thing there. What's really increased is furthermore, it depends on how you measure it. I said that the way we calculate from Fermi level before is relative conduction band. Fermi level chemical potential is a measure of number of electrons. Maybe you're right. Maybe this is the way we should think about it. And maybe I should say, far away for the vacuum is the same. And then, so this is the-- because here I have a larger work function. Here I have smaller work function. Far away. So when I connected it together, so electron should go this way, the wire. But that's different from what I was saying. If I put the vacuum, the electron probably will go this way. AUDIENCE: Why? GANG CHEN: Huh? AUDIENCE: [INAUDIBLE] GANG CHEN: Why? Because I was thinking somehow those electrons-- OK, this are they-- here is the energy, and this is the kinetic energy. Of course, you have to-- this kinetic energy must overcome the work function. So what's left over here is still 0. Just those kinetic energy just above the chemical-- the work vacuum is still 0, because the rest is used to overcome this work function. But anyway, so if I think this way, if I connect with the wire, then the electrons should flow this way. And that's what they were saying. In that way, maybe your reason-- so electron now, the chemical potential is raised, but you don't change the work function. So the vacuum level is raised here local. And then you have a, say, so this is the-- if I draw, that might be the local-- this is the vacuum, local vacuum level, this is local vacuum level. And then, my question then, so if the electron is flowing this way, how should I think which one is the positive? Yes. AUDIENCE: Does it depend on the difference between the Fermi level and the work functions, how they interact with each other? Because in this diagram, when you connect them, the Fermi levels become the same. GANG CHEN: Well, after you connect, the Fermi level has to be same. AUDIENCE: Right. So the work functions, depending on their differences, they can-- that you can have no flow or flow in either direction-- GANG CHEN: Yes. But see, I'm drawing a schematic where I have one side is a larger work function. So now with this work function is really relative to Fermi level to vacuum level, right? So my question is, which side is flowing? AUDIENCE: So-- looking at the Richardson formula, the smaller the w, the bigger the current. GANG CHEN: OK, so you're right. So the smaller-- OK, that's a good way to think about it, I think. AUDIENCE: That's kind of-- GANG CHEN: So the larger is the w. OK, good. So maybe this will be consistent. The larger is-- smaller w, more flux going this way and higher, less flux. So at the end, you will have more electrons on this side. So this will be positive, this will be negative. And that's consistent with my connecting wire. OK, now I agree. Thank you. AUDIENCE: The local vacuum level is shifted by the electrical potential. GANG CHEN: Well, that's the-- so, OK. Once you get more electrons here, now you think about, this is the-- electric field, the direction is this way. That's your electric field direction. And you will resist the further going to the other side. And then you draw your potential. Now, if you say, electrostatic field equals the energy, which is the potential divided by charge, which is negative, so when I fill this way, that means this flag here is higher, this flag is lower, that means the energy is higher, and this is lower. That's consistent. It's just this e, you have to think about the voltage opposite to e, to the local energies because of the negative charge. That's the part I really don't like. Every time I draw this diagram, I have to say, ah, electron is a negative. How I'm going to draw? Whether I'm drawing a potential here, here, here, OK, they're really-- OK. Here we have an energy or e. When that's talked about the electron energy or local energy. I say, when I talk about voltage, I talk about electrostatic potential. And electrostatic potential is really divided by charge, u divided by q. This q electron is [INAUDIBLE] to reverse the sign. So when I draw this way, as the potential, I'm drawing this work function. That's an energy unit. So when I draw the electrostatic, I have to reverse this way. See? This has a higher electrostatic than this side because the [INAUDIBLE]. That's where the confusion every time I have. But I do want to mention this. So now, clarify the question. So you get a w, so that's the w minus w prime. So really, this negative e is-- because I did w on this side, if I put the positive electrode, negative electrode, and I should mention negative voltage, that means this positive this negative. AUDIENCE: What does the dotted line mean? GANG CHEN: Huh? AUDIENCE: In the figure, we have a line in the middle. GANG CHEN: Oh, that's not my figure. I should have cited this. I got some-- this one was from [INAUDIBLE]. So what he's saying is that you are checking the electrostatic energy. Because this is a chemical potential, and there is also electrostatics. So here, the voltage is a positive here, but for electron, the electrostatic can move up. AUDIENCE: But in the upper figure, isn't it true that the right side should go up so that the work function level should be the same? I mean-- GANG CHEN: Work function doesn't change. Work function is that you pull electrons from metal-- AUDIENCE: Yeah, so-- GANG CHEN: --immediately out to the local vacuum. AUDIENCE: So when electrons are out from the metal, then the energy of the two electrons should be the same. GANG CHEN: No, that's the point. That's what I say at the beginning. You have a local vacuum. Once you see-- once the electron goes from this side to the other side, right? So here is positive. We got a negative. So the local vacuum locally on the surface here, a metal surface, there is a higher electrostatic potential than here. So the local vacuum level varies. AUDIENCE: But in your figure, in the middle-- GANG CHEN: Yeah? AUDIENCE: --the highest level that is the work function level is the same. It's different from-- GANG CHEN: No, what I put there is thinking at the beginning, far away. No talk to each other. If you put together, they start talking, you have local electrostatic influence, you will create an imbalance in the field, electrostatic field. AUDIENCE: OK, I see. So the upper figure does it mean that the two-- GANG CHEN: There is no talk. This is just a random, because you really can't align it. But this is where contact potential is-- this is actually defined as the contact potential. And I say this because, if it turns out that very often we want to measure the work function. And this is when I realized-- I suddenly realized this is an easy way to-- relatively easy way to measure work function. And one way to measure work function is just to measure the contact potential. You take a standard piece of material where you know the work function. And then you come close to the other piece, and you start to have current flow. Because when you close it, and your potential is the same, but when you change distance, your current-- because they used to think of capacitor, you still have current flow. And that's the so-called Kelvin probe. Kelvin probe, Kelvin method actually measure the contact potential. And using that, if you have a standard piece, you can measure the other contact potential. And I think Kelvin probe is standard [INAUDIBLE] rate if you buy atomic force microscope. You can add a few thousand and it will have a Kelvin probe software with it. So let's move on. Now I'm talking-- I'm thinking about, really, the diode. That's what I drew on the board. And again, I draw a higher and anode and a lower function cathode. So in this case, just as we discussed, you have a local vacuum level higher and then eventually the chemical potential is flat. And at the equilibrium, at the transient, they will have current. But eventually, you establish equilibrium. So this is the-- normally, this is the cathode current going from left to right. And this is the anode current from right to left. And you can say, I do not just do wc, I do a wc plus ec. This is because I'm thinking about the static state. I'm not thinking about the transient initial establishing equilibrium. So you can see, once you got a potential, the electron from here cannot really go here anymore. So only the electron with the energy u on this side, with the energy higher than this point can go to the other side. So at the end you get the-- at the equilibrium, the difference of the two is 0. There is no net current flux. You put the two pieces of metal together, you connect them. Rather than just the initial current, at the end, it should be no current flow. Otherwise you get a perpetual motor. So that's the vacuum at equilibrium. Now I'm going to put a temperature difference across it. One side is hotter cathode, I made it hotter, and one is colder. And of course, I still have the current. And if I write the Richardson expression-- OK. Now, in this case, if you think about it, when you heat it up, there should be more electrons gaining kinetic energy, higher kinetic energy, higher temperature. So the electron will go from cathode to anode. So electron, there are some current go here. And so, eventually, relative to speaking, the chemical potential will no longer be the same. And really, this chemical potential is the electrochemical potential. Because what I added here, the voltage that's generated added to the chemical potential. So this level is the electrochemical potential. And that's what I can measure. When I put the two electrodes, I'll measure a voltage difference. And that voltage difference is these two points, voltage difference. So because of that, my current, cathode current, now-- and this is what I originally had. And this is the additional. And again, this is where I don't like it because I put a-- I should put a negative sign. Because if you think about, the electron here now, the electron that needs to go over this, so from electrochemical potential mu c to the maximum point is-- the distance is shorter. And I put a positive sign here. So ideally, I should get the negative solution v0. That's my open circuit voltage. And I don't think I get that. I still don't know why. My sign doesn't seem to be right. So I give you the challenge. You go back to sort it out and come back and teach me. And this is the cathode. This is the anode. And of course, at steady state, if I don't put any connectors there, I have a voltage difference between these-- real voltage that I can measure. So again, this doesn't relate to the negative charge, but at the equilibrium, I should not say at equilibrium, at steady state, not equilibrium. When you have temperature difference, you have chemical potential difference, you don't have equilibrium. Accurate way to say is at steady state. At steady state, there is no net current flow because I didn't connect it to any load. I have open circuit. So current is 0. So if I go through this steps, put this two expressions, this one minus this one is 0. So I'm going to do that. This are the two terms. And I take the ratio, I take a log, get rid of my exponential. Second step, I was hoping I'd get a negative sign, but I don't get it. So tell me where I should get it. I didn't see, I get a negative sign. I still don't know why. But anyway, the magnitude is correct. The sign somehow, I don't know. But let me also look at this and point out that if you think about before, when we talk about Seebeck effect, is a voltage divided by delta t. So if I do a voltage divided by delta t, not delta t, just an order of magnitude, this term I put it here. And here is kb divided by e. 2kb divided by e, that's of the same order of Seebeck coefficient. So this term actually come from-- is a very similar to the thermoelectric. And this term is the thermionic term here because it depends on the barrier height of the anode-- anode barrier height. So that's the open circuit voltage. And of course, open circuit is useless. You don't generate any power. You have maximal voltage, but you don't generate power. This is the same for photovoltaic cell, So you need to put a load here. Once your current start flow, you don't get that high open circuit voltage anymore. You get your actual voltage. That's the actual voltage. And you can change this voltage by changing your load resistance. That's a load matching. That's for all solid state device, solar cells, thermoelectrics, thermionic, you have to optimize your load or optimize the device itself to match the internal and external. That's load matching. So if I write-- I rewrite the expression because the contact potential is the same as wa. And you can say my current depends on the voltage. And of course, power is just current times the voltage. So that's my unit volume power, unit surface, I think, because j is the surface. And so, that's the power. Now, for thermionic generator, I have to do the efficiency. So I have to look at how much heat transfer happen. I have now electrical power output. I have to look at how much heat transfer. So what are the mode of heat transfer here if you look at this? First, this side is hot. This side is cold. There is a radiation. Any other heat transfer? AUDIENCE: Electron carries heat. GANG CHEN: That's right. Just, we don't forget electrons also carry heat. So first, let's look at-- that's a fundamental site. Radiation is not necessarily due to electrons. It could be due to lattice. But electron, when you have charge flow, always carry the heat. So I have to count that part of heat. So this is the first step. What's the kinetic energy? That's the-- just the 1/2 mv squared. So this is a p squared divided by 2m. That's the kinetic energy above the barrier. So I do my integration in the same way I did before as I do particle flux. I didn't start from beginning here. I just write the expression. And I do my integration. And it turns out the heat flux, kinetic part of the heat flux is proportional to current flux then 2k be, That's the Seebeck coefficient-- times t. So Peltier coefficient, remember, in the thermoelectrics, Peltier coefficient equal Seebeck times t. So you can see, this is very much the Seebeck part. This is the Peltier coefficient, Seebeck times t, times part. So this is the [INAUDIBLE] carried by electrons from cathode to anode. But then heat is more than this. Because, remember, when we define the heat is the energy minus chemical potential, e minus mu, relative to chemical potential. So relative to chemical potential, I have this part is kinetic energy. Then I have really from here to here. This is a cathode site. The energy, the one that really go over the barrier, the rest of it from here to here. So that's my w 4 e minus v here. So this is from here to here. That's my potential part, and this is the 1/2 mv squared kinetic. So that's all the heat it carries. What is this in terms of resemblance? If you're a mechanical engineer, think of this. Give me an analogy. Huh? This is evaporation of water. The water molecule coming out of the surface carry a lot more energy because you have to say the energy is-- say every other-- so if you think about the water, and there are some molecules flying around. If I take all the molecule away, and then there's another molecule we created behind. And the creation of this molecule from the liquid phase to the vapor phase, this is the lock and key. Essentially, those terms, this and the heat of the electrons when you come out of the surface. And this is the kinetic part. But at the end, also I see this. I still have a little bit of trouble thinking this as a evaporation process, because I'm trying to think of an analogy of, OK, here is a potential, the bottom. You come out here, so that's the minimum. So each molecule carry out this. I say, that's an evaporation process. But what about suddenly a bottle-- OK. What about this process? If I have a high energy electron, and the potential here suddenly fall off a cliff? And then that's excess energy, potential energy. Do I consider that as a evaporation process also? Then I'm a little bit confused whether to think of mechanical. I don't know what's the-- whether I can give an analogy of this two different processes in terms of evaporation. And this one is more, to me, somehow, to me, is I have a container of liquid, I open a hole. This is a high temperature. When the liquid come out here to the other side, the vacuum, it suddenly evaporate, change from a liquid phase to vapor phase. I'm making this to this and this to this. You can tell me whether you agree. But at the end, I'll give you one example where I struggle on this question-- why I struggle on this question. So that's the electron heat. And of course, there's also another side from a-- I can reverse this. This is from anode to cathode. This is from cathode to anode. And the difference give me-- this is the electronic part of the heat. That's the first two. And then we have radiation part. And then you have a lead, that lead conduct heat. So that's a conduction heat loss. So if you have a real device, you have to consider all those heat losses. And of course, the efficiency is defined as the power out divided heat in, always what we do when we do efficiency. And if I do most ideal situation, I neglect radiation. Why you can neglect? I say, well, I have my perfect photonic crystal, no radiation. It's impossible. But let's say, I make this small. So if I neglect all other heat, just for the electronic part, I cannot neglect it-- intrinsic. And that's the curve you will get. And this time, I remember to cite the reference. And you can see, in terms of the efficiency you can get, it's very much a e v a. That's wa, e v a. v is the voltage. So it's w work function of the anode side divided by kta. So that's one parameter. And you optimize this parameter. And this is the ratio of the hot to the cold-- the cold to hot side temperature. So that's the ratio. So you can see the heat efficiency, pretty impressive, very high efficiency. Of course, this is when you neglect the radiation. You can't get rid of the radiation part. You also have additional heat loss. But compared to thermal electric, because you're using vacuum, so you don't have full-on heat conduction. That's why this can give you a high efficiency theoretically. And the first experiment was done by [INAUDIBLE]. Some of you are from that lab. He was a-- he did the-- this was in the '50s. So he came from-- he told his story. He came from Greek and trying to get a topic, and he were able, at the time-- it was since, didn't have really research government funding. But somehow he was able to get some funding to do the thesis. And he did a thesis. He went out and open company. He had a very nice book, Thermionic Energy Conversion, two volumes. I was reading some of it. I couldn't read all of this, but very detailed. He actually talk about the detailed electron-- I'm very impressed. So he talk about from basic quantum mechanics to work function. But that's Hatsopoulos and Gyftopoulos. Gyftopoulos is still professor emeritus in nuclear engineering. So they wrote a book together. And you can see, they did an experiment, 2,300 F per high temperature, hot side. And cold side, 1,000 F. That's the challenge with thermionics, it's a very high-temperature device. And the efficiency was, at the time, 1958, it was pretty good, 13%, 13%, 14%, pretty good efficiency. And so, they were very ambitious here. He also put his voltage negative. And you should read this paper. I'll send it as a reading. So his story is, he started-- he was a faculty member in our department. And he went out to start the company-- assistant professor at MIT started a company that's called a Thermo Electron. You go 128, 95, you will see the signs, Thermo Electron. Now it's called Thermo Fisher. And I believe they were trying to commercialize this. The story is, he said, he told me he went to Washington on Washington trade, and then he realized NASA-- the business really depends on whether NASA flies or not. And so he changed direction. Good for him, I think. Otherwise, I can't really say something. And the company, Thermo Electron, really does a lot of instrumentation. So it's no longer doing really thermionic energy conversion. And the recent merger is now with Fisher. Fisher is another big, I'd say, company name. So now it's Thermo Electron-- Fisher Thermo Electron. That's a good story. And the real device was not as simple as I draw, of course, the high temperature. But also there is really a fundamental problem. This is the fundamental problem is charging. If you have an electron in the vacuum, So any electron, any charge, surrounding charge, there is a field, a charge there will free the field. So when you've got a lot of electrons, they will create electrostatic potential. And this potential distribution, you can see now, I draw rather than a linear one, and really depends on the gap, depends on the charge number. But the point is, now your barrier is higher than just work function. You got an additional barrier to overcome due to space charge effect. So this is a big problem. Space charge is a big problem. And in addition to this, there are also many other problems. Of course, the most mature work functions are very large. That means you have to operate at higher temperature, few electron volts. So if you check most material, metals is about four electron volts. There are some materials like [INAUDIBLE], two electron volts. So if you use a cesium, cesium is actually-- you can first, you can generate cesium to neutralize the positive charge system vapor, to neutralize the negative electrons. So they reduce the space charge. But cesium is very corrosive, so there's problems at high temperature. And you can use either vacuum, or there are other modes where you just put a gas in between. When you put the gas, there's a plasma formation. So there's this vacuum mode or plasma mode of operation thermionic devices. But historically, think about this, the transistor, the diode, the solid state were invented in the '50s. And before that, computers were built on vacuum diodes. The vacuum diode is essentially this, except they operate in a near room temperature. And this is just a one site to operate the high temperature outside the room. Problem is, we go to 1,000 degrees. What are the recent trends? The research in this is relatively small, I would say. A Russian actually put this in space station. I tried to dig out a picture, I couldn't find one. But they call it the Topaz. And so Russia had a few space station, space missions that use thermionic engines rather than thermoelectric engines. You can search whether you can find more information if you're interested. And so, some of the idea people explore, one is the negative electron affinity. So if you let-- maybe electron just jump off itself. Negative electron affinity, vacuum level is lower than what's inside. And so that's one, people still look at this. And the other is a small gap. And this small gap, there were actually some programs, so the gap goes to angstroms. It's crazy in the sense-- of course, in that case, what happens is the electron can tunnel quantum mechanically from high side to low side. So you don't have this space charge limitation anymore. The electron don't have to be having an energy higher than the top of the electrostatic. So you can really just zip through, when this one is of the order of angstroms, that's a scanning tunneling microscope principle. Electron can go from one side to the other tunnel. But say, like I said, that's pretty-- technologically, it's too challenging, because think about the maintain hundreds of degrees at angstrom level. And then, the most recent is solid state thermionic where you run into still the heat conduction problem. Once you got a solid, everything solid state, then you have heat conduction. So that's the same as thermal electrics operation. But I just want to give you one sketch on what does the electron affinity means? Because normally you can see, the work function is between chemical potential and vacuum level. That's a work function. And of course, the space charge created a higher. And the negative electron affinity still doesn't mean the electron will all-- for other metal, you don't get the metal anymore. Here is the bottom of conduction band. But this is a, say, in negative electron affinity, which is typically semiconductors, very wide gap semiconductors, the electron actually sits in here, more the Fermi level is here. So it does not come out. So you still need to overcome the space charge to [INAUDIBLE] come out. So that's the liquid [INAUDIBLE] affinity material. So people have interest in this, not only for thermoelectrics and thermionics. So that's a discussion on thermionics, which is in formula, what's important. Yes. AUDIENCE: You said that [INAUDIBLE] is very [INAUDIBLE]. GANG CHEN: Right. AUDIENCE: What happens if you just put some [INAUDIBLE]? GANG CHEN: Dielectric material is-- the problem now you got a heat conduction. AUDIENCE: Oh. GANG CHEN: So run into this classical thermoelectrical problem. And what are we-- now somebody ask, say, semiconductors? And let's think about now we have a metal and a semiconductor. You always have that in any microelectronic device. Because, at the end, you want to make a count. You put the metal on the semiconductor. So that's a typical classical situation. And now let's see what happens. And now, of course, here is the Fermi level. Here is the semiconductor Fermi level. I'm talking about the along degenerate. So Fermi level is in the gap. This is really where electron sits in. And when I put these two together, I still got the same problem. If I collect them electrically or put them in contact with the shutter, the chemical potential difference will drive the charge flow. So eventually, the chemical potential on the two sides will be the same. That is always the driving force for diffusion, mass transfer, charges of mass. Mass transfer is chemical position. Heat transfer is temperature. Mechanical is pressure. So the thermodynamic quantities. And then, when I put them together, so the Fermi level will align to each other, chemical potential align to each other eventually. That's a driving force for charge flow. And here the work function here is the affinity. And then at the point of all the contact, you can think that they have to have the local same vacuum level, same vacuum level. At that point, if I put these two pieces together locally, I pull electrons out to vacuum, that's the same point as the metal side. So there is a mismatch between the work function and the affinity. And this mismatch is delta here, which you can check. For different material you can always define-- not always. Sometimes people have measured it. But in theory, you can find this. So there's delta. That's the-- this delta is the Schottky barrier height. That's the barrier height. There is a potential difference of the electrons in the semiconductor and metal side. Now, of course, if you think about this, electrons here, so this electron will fall into the lower-- the metal side. Electron go from high energy to lower energy. Anything go higher energy to lower energy. So once they fall, there is less-- there are positive charge, less electron here. So chemical potential is measured between ec and here, and less electron means larger distance. So you can see here, I'm going far away from the interface. That's my bulk region. Interfacial region, the semiconductor side, there is a band bending. ec is no longer flat. And that's because intuitively electron fall on this side, less electron, so bigger separation there. Yes. AUDIENCE: Are the Fermi levels-- GANG CHEN: Same. AUDIENCE: Same level? GANG CHEN: Yes. I didn't draw it level. AUDIENCE: So the vacuums are different? GANG CHEN: Vacuum now-- AUDIENCE: It's different. GANG CHEN: Vacuum-- AUDIENCE: It's three levels. GANG CHEN: Yes. Yes, thank you for telling me that. Vacuum level is different. AUDIENCE: So is the drop the same on the other side because the ec will go down [INAUDIBLE]? GANG CHEN: So locally, the vacuum level also changing. So this is where, again, you have to really think, this is defined really, really low quality. This rule is actually sometimes violated, particularly when you put two different semiconductors. You first think about the lack of vacuum level, then look at the affinity, and that's the [INAUDIBLE]. It doesn't always work. AUDIENCE: Because I'm trying to tell-- so if delta, before you bring into contact, you have ec between-- ec and the Fermi level. GANG CHEN: Right. AUDIENCE: The level, right? GANG CHEN: Right. AUDIENCE: So that's the original level. When you bring it together, ef and the vacuum drops down for the semiconductor, then ec drops down as well. Is that correct? So that's why that's why [INAUDIBLE] GANG CHEN: Does the vacuum level change? Let me draw the vacuum level. What I saw with the vacuum level here, this is a vacuum. Let me draw it. This is the metal side. And this seems to be the semiconductor side. And this is the ec. And here is the-- so here is the work function. Here is my electron going into vacuum. So this is my affinity. The difference can be the [INAUDIBLE]. So I didn't have that jump in the vacuum. Locally I put the same. So this is the-- this kind of interface here, is often between metal and semiconductor. And this is a Schottky barrier. And now if I think about the flying flow electron, I put a-- I talk about contact. You apply a voltage, how the electron flows? And of course, you can always say, again, from left to right, right to left, same kind of thinking there. And now here is a p type. So before I draw an electron, and here is a pulse. Pulse is a-- this from here to here is barrier. This is a positive charge-- [INAUDIBLE] holes. And here there are less holes because the electron fall into here neutralize-- make less hole really. Because electron fall into-- wait. Where I'm confusing? Think about-- so positive, so really, so electron should fall into this side, from here to here, because the electron is at a higher energy than here. So electron fall into this side. There's less hole here because the electron fell in the empty state. And the holes are just what's left over, the empty states of electrons. So what's hole? There's really no hole. In the [INAUDIBLE] span, this is the ev, the electron field over here, and left behind the few empty space. Those are the empty area. Now the metal, electron from the metal, electron from the metal fall into this side, fill in some of those empty states. So this is no longer available for full. So this is less holes space. So there are less holes here. This is less hole. So the Fermi level return, the chemical potential, and the span becomes larger. So this is the Schottky barrier for the p type. AUDIENCE: Professor. GANG CHEN: Yes. AUDIENCE: So in the p type, [INAUDIBLE], so you are bigger actually [INAUDIBLE] p type [INAUDIBLE] level will go up. But it seems like in the vacuum-- GANG CHEN: Vacuum level, I didn't draw a vacuum level here. AUDIENCE: The green region. GANG CHEN: Where is it? Where do you mean? AUDIENCE: I mean, the [INAUDIBLE], the ec and the [INAUDIBLE] is the [INAUDIBLE]. GANG CHEN: No, no, no. The p type has to be-- you have to look at the-- I think this is your w on the other side. AUDIENCE: Yeah. GANG CHEN: And you have to think about this will be here. And so, from here, vacuum level-- AUDIENCE: Yeah. GANG CHEN: --vacuum level here. And now electrons start to fill in, because the electrons go from high energy states to lower energy states. AUDIENCE: Yes, you mentioned that [INAUDIBLE]. That's why. GANG CHEN: So the Fermi level will be-- because this is p type. So the p type will be going here. This is the Fermi level. So this is the-- this will be the-- AUDIENCE: Do the ec all the way around? [INAUDIBLE] GANG CHEN: No, ec doesn't go in. This one, ec the background has to be the same. So between here and here, this it hasn't been done. AUDIENCE: Oh, [INAUDIBLE]. GANG CHEN: Yeah. But there are not much electrons in the p type. So you say, why the electron doesn't flow. AUDIENCE: And the vacuum he has right now too. GANG CHEN: Huh? AUDIENCE: The vacuum he has right now too are the semiconductor side. GANG CHEN: Vacuum also has to bend down, yeah. So this is your vacuum level. AUDIENCE: So [INAUDIBLE]. GANG CHEN: Huh? AUDIENCE: That's exactly what I was telling, [INAUDIBLE]. GANG CHEN: Oh, vacuum, I see. Yeah, vacuum is [INAUDIBLE]. I thought you were asking the ec. Very tricky. Now let's check. I'm giving you the formula. I'm not deriving. But the point is, it's a Richardson formula. Negative what we wrote for vacuum. And when I-- but there is some difference also. So the first factor here is the same Richardson factor. And then I have this exponential ev kvt. And then you can see, that depends on whether my voltage is positive or negative. So when I see positive, is I raise the Fermi level on this side, on the semiconductor side. So the voltage is applied, so this will be, in this case, this will be positive electrode, negative electrode, raise the electron on this side. And of course, the electron raised, so there is more chance for this electron going from left to right-- from right to left. So that's where my ev kvt comes from exponentially. So I have-- this is a forward bias. Now, when I reverse bias it, let's say when the electrons go this way, it really tried to go this way. That's what I draw. This is a positive metal, positive semiconductor, and it's [INAUDIBLE], right? And when we reverse pass it this direction, the Fahrenheit is fixed, didn't change. So this current is always the first-- this saturation current here. Now there is less going this way. But that's not much less because you see here, that difference is small. So now, this is the same as a diode. Biased differently, you have, say, different current characteristics. So that's similar to a diode, typical diode. And this is a Schottky diode, positive bias, negative bias, saturation current here, not 0. And this is if we invest too much, it break down. So you can just punch over this barrier and under not reverse bias. And you can see, a metal semiconductor interface, and if you make a contact like this, it's not good. Typically, you don't want to make a Schottky contact. And how you solve that problem? And the way people solve this problem is they dope it very heavily. So let's say, when you dope heavy, this becomes-- say, so basically, when you dope very heavy, few electrons come over, this is very narrow. This is very sharp. So when we draw, this becomes very sharp when you dope heavy. And when it becomes very sharp, so the electron tunnel through rather than have to go over the barrier. So that's one way to overcome the Schottky resistance here. Otherwise you'll see, so this is how people make [INAUDIBLE] contact is you dope the region very heavily. And of course, you want to use the right metal to start with so that the work function, the delta here is not large. So that's why, if you want to make a metal-- a contact, you want to know your work function, you want to know your electron affinity, [INAUDIBLE]. And of course, there are many other reasons where you do make a good contact and other than this. You may have a physical long contact, a vacuum gap here that could also-- vacuum gap is essentially a very high barrier. You have to power through that. You can power through that. Or if it's too wide, there's no chance you can go through. Schottky diode. Since we're talking about diode, I'll continue. This is a metal semiconductor. A regular diode is not a metal semiconductor. It is a semiconductor-semiconductor diode. So let's look at semiconductor-semiconductor. So we make a semiconductor n type, semiconductor p type. And I put the two pieces together, contact, same thing happened. The chemical potential must align each other. So what that align looking like, what it really means is, here, n type got a lot of electrons. So electrons go this side. P type I got a lot of hole. Hole go the other side. And so, when electrons go from this side to this side, what's left behind? Positive atoms there. So what do you have here is positive atom. This atom do not move. Those atoms are just charged there. They don't move. On this side, holes left, so they left behind-- really, just get a holes left is, you get an excess electron to-- for this case, p type or boron. Say you get an excess electron, not called the neutral now in this region. So those, when I draw this, is the ions, positive ions, negative ions on the two sides of the interface. That's a space charge region, not a mobile ion, mobile electrons. And if I draw the bandgap-- band alignment. So if I-- oops, I forgot-- ec should be mu. No, this one, there should be a line here. That's a ef. I disappear. This is a ef. And this ef and this ef will have to align each other. So it's easy to think about the connection. If I have far away from the interface, I have n type. And far away from the interface, I have p type. And now p type Fermi level is close to eb, and type of Fermi level is close to ec. And then connect in between. And at the end it's equilibrium. That's equilibrium. And in between this region, a space charge region, no charge there. No free carrier, no electron, no holes, just a positively ion, negative ions. And that means I got that now I have positive ion, negative ion. I have a potential pointing from positive negative. So that potential is building potential point negative to positive. And this potential will resist further diffusion. So at the end, you've got the equilibrium. So this is the view of the input picture that resists diffusion of charge from one side to other. Change junction. Now let's look at what's the current voltage characteristics. Very similar to Richardson's formula. So I'm going through this. That's what you learned before. So this is a-- I threw at you. Electron number density, hole number density, and the electron in any semiconductor, if you [INAUDIBLE] the electron, you can also derive similar hole. And then you take the product, you found out the product is constant. This p and n, so if you dope the electron, dope for n type, there are less hole because p times n is constant. p times n here is a relative band structure, effective mass conduction, effective mass balance, bandgap, and kvt, right? So p times n is constant. That's the-- now, how I calculate building potential? From on this side, I can say, because I know this is a constant, so I just need to calculate this. I need to calculate this. And the difference is this. That's the building potential. So from-- or look at this side. I have ec minus n. That's ecn minus mu, that's this distance. So I can start from the-- let's suppose each of the donor, each phosphor, for example, I put in give me a electron. And I reverse it, I get this. And I can do the same on this side. Given the number of the acceptor, that's the number of phosphor I-- boron I put here. Each boron create a one hole. So I find the distance between here and here. This is from-- oh, well, I already converted it into ec. That's from here to here. And now this, from here to here, that's here, that's here. The difference of these two give me the building potential. So this is the building potential. And it's the acceptor/donor, acceptor on this side, donor on this side. How much I dope it, because the more I dope it, the higher is the building potential. How wide is the space charge region? Now that you can solve this equation. I'm not solving. I'm just agreeing [INAUDIBLE]. But I want you to look at this. If it's just an electrostatic, that's the-- this is the width of the space charge region. And it depends on the dielectric constant. Larger dielectric constant, wider is the, say, space charge region. So if I look at this one side, either acceptor, donor, that's roughly how wide it is. And that's why I said I want to dope heavily to make a-- to reduce the Schottky effect for contact. If you can see that if I dope heavier, the width of the space charge region is narrower. And this is where you will talk to people with intelligence. They say, oh, what's the bi-length? That's essentially a measure of the weight, the intrinsic weight. And then the bi-length is essentially what we have here, replaced by the building potential by kvt, this spread, thermal spread. So heavier doping, shorter is the space charge region. Some people do nanowire surface effect, depends on you can see whether surface effect [INAUDIBLE] or not, depends on your doping. So bi-length. Finally, this is a-- I want to show you this because it's the same as Richardson formula. That's a pn junction, i times v, current voltage relation, except js-- except js. And that has a lovely price there. The js is no longer a t square. That exponential barrier, no longer the Richardson form. And the one we have here is the diffusivity of hole, diffusivity of electron, recombination time of hole on each side with the electron on each side. So this is a complete different mechanism. And so, this are the definition. Here I'm going there quickly, but I want you to chew on it and come back I will do-- I will explain more. And basically, this is a different-- here you can see, the formula is the same. Saturation current js is different. But it's a complete different mechanism in terms of transport. Because in a pin junction, the electron will go from this side to this other side. There are no electron anymore. p side, there are not much electron there. So how the transfer happened? And it turns out, there's really the hole recombined with the electron. So the holes here recombine with the electron. And then hole has to supply-- keep supplying. So there is a-- so in a pin junction is a recompilation that's important, that drives the current flow. While in a Schottky diode, it's just electron go this side, electron go this side. There's no supply issue there. So one is a majority carrier device. The other is a minority carrier device-- very different. Formula is the same-- almost the same. So just the last slide. And again, I want to show us a-- I'll come back to do a little review. But to go think about this. So this is the electron current, hole current. And they add up to the total current must be continuous. That's a pin junction. And what most people do not know, even electrical engineer, I'm pretty sure, is that inside of the just space charge region, there's no heat. In fact, it's cooling. You look at the Peltier effect, you go to analyze the thermoelectric effect in the semiconductor device, you'll find out that in the space charge region, the cooling effect. And this is really the recombination outside the space charge region that created the cooling-- heating. Of course, heating is a big issue in microelectronics. But in a diode, the center, the interface is actually cooling type. OK, I'll stop here. And I'm thinking about the same topic. Again, a lot of materials. |
MIT_2997_Direct_SolarThermal_To_Electrical_Energy_Conversion_Technologies_Fall_2009 | Lecture_1_Introduction_for_Direct_SolarThermal_to_Electrical_Energy_Conversion_Technologies.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. GANG CHEN: I have a set of handouts. If you don't get a copy, it's all on the web. And if I do this well, every time you should get a handout. And my objective is for you to focus on listening and discussion, not to take notes. So it's a [? similar ?] handout, or you go-- it's online. It's an experimental course, the first time I'm offering it. So it has a special, say, 997 number, 6 units. If you look at the handout, if you print it out yesterday, I changed something and one of the changes is very important. This course will be pass fail basis. And what does it mean is if you are undergraduate student, for example, in course 2, pass fail could count for unrestricted elective. But it does not count for restricted elective. And if you are course 2a, it does not count for your concentration. And if you are a graduate student, this is another entry level course. It's not an entry level course. So those are just for your clarification. And if you look at the handout, the syllabus, what we will have is once a week at this time. And also, I have office hour. And I have volunteer TA. Daniel is here. You can contact him if you have questions. If necessary, we'll ask Daniel also set up an office hour. And what else? I don't have a copy in my hand. Thank you. In terms of the requirement homework, the homework is every week I will assign one reading. And you will turn in your report, one page summary of the reading. You can summarize what you understand, what you do not understand. And that's the-- I do not see will give an extra number of homeworks. So it's a reading based homework. And at the end of the semester, you will write a final report, project report. And the detail of the report will answer in the middle of the semester. Any questions? OK if not, I will start. Like I said already, everything is online. And you can jot down whatever notes you want. But my purpose is not to have you write too much and focus on this. I do not have a textbook on this. So let me start. So this course, if you look at the title, is a direct solar thermal to electric energy conversion technologies. There are few points I want to emphasize. One is a heat-based course. Solar thermal, this is all focused on the thermal aspect. The solar energy is, in this course, is a big, giant heat source. And the other aspect important in this course is the direct energy conversion. So the focus is on how to convert the heat into electricity directly without the moving parts. And when I say the technology, you realize that we move on. Those technologies, some already existing. But clearly is not widely spread used. It's not widely used. So they are still under development. And whether they will work out or not down the road is not completely sure. But that's your opportunity. You're here to do research and future development. And maybe you will push some of this technology to real world. So I will start with this slide to make a case that it's important that we think about the heat. You can start either from the input side, look at how we are using all the energy. Fossil fuel or even nuclear, the energy source from the energy source to convert this energy into electricity, into mechanical energy. You can see more than 90% is [? VIP. ?] Mostly is thermal mechanical means to convert the energy into mechanical energy or electrical energy. And the other aspect is look at the end use, how much energy is lost. And you can only see the numbers here. You'll find out that more than 60, about 60% to 70% of the energy are wasted. So what are we are looking into is whether we can develop alternative ways to either use the input energy or convert some of the waste energy into electricity. Yes. STUDENT: Are there 10% missing or [INAUDIBLE] GANG CHEN: 10% missing, you have to look at some of this in here. We're talking about heat sources. Just a few examples. If you look at the automobile, the driving efficiency as a measurement is less than 20%. And if you want to remember, it's easy to say about one third of the energy goes through the tailpipe exhaust. The other third roughly goes to the radiator. Those are completed with-- so can we recover some of this energy? And in fact, this is challenging because it's a model. You have to feed into the compact systems. And in those cases, it turns out people look at various options by say the mechanical way of converting those energy into electricity is not very feasible. So there are a lot of companies looking into thermoelectric, for example. And now you look at another way that we use energy is buildings. And give me an example. In New England in the winter, home furnace, boilers, most of the time they are off. Very often, they are on. And there's a combustion process. And [INAUDIBLE] what we're using for-- hot shower, heat up the room. That's a low quality source. Once you switch to this temperature, you have high temperature combustion. You quench to low temperature just take a shower. And you are generating a lot of entropy. And in some regions, there are big coal fired power plant or oil fired power plant. In this case, the people take the steam and supply the heat to the community. That's electricity, heat cogeneration, cogen plant. But we don't do this at home. The technology is not there yet. People are still-- people are pushing. And in fact, there is a, say, [? Honda, ?] is fighting, for example, developing home stand-alone internal combustion based system. But it's still not taking advantage. During the combustion, during the heat of the hot water process, we are really generating a lot of entropy. We are wasting useful energy. Can we develop technologies that can do home cogeneration? Industrial process, that's about another third of the energy that's used. And their heat sources are different temperature, where wide temperature range. Once I visited the aluminum smelting plant. So that's just converting aluminum oxide into aluminum. And the process is a high temperature electrochemical process. So you heat up aluminum oxide to molten state. And then you use a electrochemical way to extract the aluminum. And those plants are typically built just beside power plants, the electrical power plants. They travel the highways. There's a 500 megawatt electrical input. That's a huge power generation, power requirement. Just give you an example, how much is a 500 megawatt? A typical power plant is about a one gigawatt. And the world solar installation, solar photovoltaic installation, it's about six gigawatts. So one power plant, one aluminum smelting plant, takes 500 megawatt. And, of course, there is a part of this aluminum oxide to aluminum energy stored as chemical energy. But the half of this is dumped into our environment as heat. And right now there's no technology to really recoup this into anything useful. So now, you think about this is a waste heat, renewables. When we think about the photovoltaic solar, we often think about photovoltaic. But fundamentally, solar is a high temperature heat source. Anyone knows the equivalent temperature of the solar radiation coming to Earth? STUDENT: 20,000. GANG CHEN: Close. STUDENT: 7, 5, 7,000 kelvin. GANG CHEN: 6,000 kelvin, 5,800. And geothermal. Those are renewable. Those are heat sources. So how we can use as a heat? And in fact, if you look at how people, what people are doing now, the solar utilization, the biggest installation is in solar water, not in [INAUDIBLE], just hot water. And I came from China. When I grew up, there was no hot water. [INAUDIBLE] didn't sell them. I take a shower. No water in the winter. And nowadays when I go back home, we have solar hot water. And it was a really fundamental life changing technology, very cheap. But it's actually high-tech. So here I'm showing the solar hot water system that's widely used in China. And give us a order of magnitude, it's about 100,000,000 square meter installation. 100,000,000 square meter. And if you take 100,000,000 square meter, you take a solar energy. One square meter is about one kilowatt. And thermal efficiency, they collect the heat around 60%. So you multiply this together. How much you get? You get about 60 gigawatt solar energy being used. And I said as a high-tech, this is a [INAUDIBLE] tube-- vacuum tube. And so there is an inner layer, outer layer. And in between is a vacuum tube. And in the inner layer tube, there's a coating. This coating will absorb the solar radiation, but the minimize the thermal radiation. And it's done very cheaply. In a typical tube like this high, it's only-- I waste of the ones that are factory, $1.5 a tube, [INAUDIBLE], all manufactured. So because of the really the low cost, it's widely used in developing countries. And of course, you can talk about the electrical generation. This is their installation in US Kramer Junction, where the heat from the sun is a focus through those parabolic troughs. Heat up the fluids. And these fluids are used to drive steam turbines. And there are also the power system that people are developing. Some are used in Spain, and I think also Algeria, where the mirrors focus and heat up the power here. And this goes to a higher temperature. And the conversion is we have mechanical systems. And the question, [INAUDIBLE] here in this course will not-- as I said, we're not looking into mechanical energy conversion. We're looking into alternatives. And I just want-- so some of those alternatives that we're going to discuss, one is a thermoelectric. In fact, Daniel, does that work? If you can turn it on. So you will see that this is a conversion that, as long as there's a temperature difference, you can generate electricity. It's by solid. If you have used thermocouples, those are essentially thermocouples putting together. Thermocouple, of course, we just want to generate an electrical signal. But in this case, you can also, if you make the efficient, you convert heat into electricity. The second-- here, the working fluids is electron. And in this demo, what we have is just a copper. Here is a heat source. Copper coming out heat to this side. The top side is hot. And there is this thermoelectric device, very thin. It's about two millimeter, three millimeter sandwiched between this copper and the heatsink, aluminum block. And hopefully I don't see it. But once the temperature difference, there a difference, you'll start to see the LEDs lighting up. So that's direct, no moving parts, using electronics as working fluids. And the second example is thermionic energy conversion. How many of you know that in our department there is a microfluidic catabolism plant? Fabulous. That's on the second building 3, second floor. [INAUDIBLE] was a professor in the mechanical engineering department, 1950s. And he developed this thermionic engine. So you can see it lighting up. And of course, you have to maintain this temperature difference. So if we burn it for a long time, the other side, the cooling is not there. The temperature difference will be becomes small. Your power, I saw it before, it actually fluctuated before. But this is also another important aspect. It's a thermal system. So you have to put the heat in, take the heat out. And the converter is small, but your thermal [? auxiliary ?] system may be very big. So you have to do a good engineering design. I was mentioning thermionic energy emission and energy conversion. And Professor [INAUDIBLE] started his PhD on thermionic engines. And it's been off thermo electron. If you drive 128, you probably see that [? component. ?] Thermo electron is called thermo feature. And the original purpose was to develop the thermionic engines. In fact, if you go to read it, it's 1956 or 58, I forgot. In the 1950s and 1960s, the efficiency was about 18% from heat [INAUDIBLE]. But it's a very high temperature. So there are a lot of technological difficulties and at the end they didn't pursue in terms of commercialization. Of course, photovoltaic, normally we don't think of it as a heat engine. But it is a heat engine. It's limited by the second law of thermodynamics. And in fact, any time we can get close to the second law, you'd be really happy. But see, this is, of course, using the photon from the sun. And there's another way is using the photon from a [INAUDIBLE] or a resource. You have a home furnace. And you can use the photon from there to put a photovoltaic. That's called a thermophotovoltaic. So those are all ways some of the direct energy conversion, no moving parts here is using the charge as a fluid. And we'll be talking some of those technologies and the basic principles. And of course, when we think about the solar, the advantage compare potential advantage-- doesn't say it's demonstrated fully yet-- is that in the case of photovoltaic, only the energy above the bandgap of semiconductor useful. But when you think about solar thermal, you can think about the full spectrum of the solar energy. So this year we started actually department energy center looking into how we can advance this technology. What is solar thermal photovoltaic? So here, the idea is I take the solar energy, absorb it. This is not directly onto photovoltaic cell. But they absorb it to heat up this and re-emit the photon. Why do you do that? Because when you re-emit the photon, if you can do better control, you can emit photons right at the bandgap of the photovoltaic cell. And theory says you can potentially get a really high efficiency with a single junction cell, but no demonstration. So that's one of the direction we're looking in this center. And the other direction is looking into thermoelectric, just like what we have here. The temperature difference, instead of the burner here, using this anytime you have temperature difference. And again, the k is how we can make this efficient, how we can make that low cost. So when you think about energy, the cost is the main concern. The photovoltaic cost right now is just too high. So most people are reluctant to put on their rooftop. So that's the motivations for this course. And to really understand those technology, we'll have to dive into, for some of us-- [INAUDIBLE] For some of us, this may be unfamiliar, territory, particularly, I think for mechanical engineers, because most of the time, we are familiar with heat engines using steam, using molecules. And here we're talking about heat engine using electrons and solid. So we'll have to cover some of the basic background knowledge in this area and understand the technology, understand the limitations, and I want to bring you to the forefront of research. What are people doing? Maybe you have new ideas in doing new things. Now, just make another comment, this is very different. You can say the dying down process is slow. Thermal has a potential to do storage. With a thermal mass, you can store the energy. A photovoltaic will take if there's no light emitted, don't have any power in it. So that's one potential advantage of thinking thermal. OK, so now I'm going to move into really-- the first what I want to use here is to review some thermodynamics and the heat transfer. And if you have taken thermodynamics, you will know some of the material. If you are from other department like physics, you also know some material. Most groups have some material, maybe enough material. But my advice is just to take it. This is, I would say this is my starting point for future discussions. So when we think about the first law, with thermodynamics we think about two laws-- First law of thermodynamics, second law of thermodynamics. We take a system and we define by defining the boundaries. So the boundary define our system. And across the boundary we have heat flow. We have work flow. And typical convention is that they say heat goes in. We take the positive. And work goes out. We take as a positive. That's just due to convention. If you get a negative sign, it means your heat goes out or work goes in. And then we have closed system means that there is no mass flow across the system boundary, or open system when there is a mass flow across system. In the simplest form of thermodynamics, the first law is just the energy balance. On the left hand side, I have the energy of the system, the energy change. And on the right hand side is the heat transfer into the system minus the work output. Simple energy balance. You can write it in the differential form or you can write into [INAUDIBLE] form. And then you do heat transfer. You actually use the [INAUDIBLE] form most often. See, some of you may still remember what's important is to recognize on the left hand side is a state property that does not depend on the process. On the right hand side is a process dependent. So it's actually very amazing. You take two process dependent quantity. You take a difference. You get a quantity that's independent of the process. Looks very simple. But this is a truly important aspect. And the energy of the system, including kinetic energy, potential energy, and internal energy. Or if you have other forms of energy, like elastic, magnetic, or those. And consider your relation that we're using is specific heat, for example. And here the specific heat is the temperature derivative of the internal energy. Here, the internal energy, I use a small [INAUDIBLE]. It could be either per unit mass based or per unit of volume based. So, that's the first law. Second law. Second law we're written into our equation. But it's not an equation. It's an inequality. And what we have here is the entropy change. Again, this is a state property, entropy, equals the entropy transferred across the system boundary and pass entropy generation in the system. On the right hand side it's process dependent. And on the left hand side is state properties that's independent processes. Now, I think about the heat engines. The heat engine, which when we think about engine, we think about cyclic motion. You don't want to design engine. You only go one direction, doesn't come back. And so when we think about cyclic motion during a cycle, any state property, go back to the original, the independent process. Once it goes back, the charge is 0. So here, we just wrote the entropy and energy, and other state properties. So if the entropy change is 0, now what's the maximum efficiency you can get? And maximum happens when there's no entropy generation. So if there is no entropy generation. [? SG ?] is 0. And I look at this is my system. The heat is positive. Heat output is negative. So I have my second law written into left hand side is 0. Entropy generation is 0. This entropy transfer across system boundary. And this gives me a relation between the heat in and heat out and temperature of the two reservoirs. And if I use this relation, I go to calculate what's in this case, what's the efficiency of this heat engine. You can see the efficiency, of course, we define as the work we get versus the price you pay as the heat in. The efficiency here is the work. Now, the work first all tells you the work equals heat in minus heat out because the internal energy during a cycle change is 0. So you can see you've got a [? power ?] efficiency. That's the maximum you can get for any heat engine operating between two constant temperature [INAUDIBLE]. People sometimes look down into heat engine. It's a heat engine, meaning it's thermal dynamic. It's not-- you can't convert all the heat into electricity. That's correct, unless you have 0 Kelvin. But the important thing is that you put in the lumber and you see where we are. The heat engine is very [? respectful ?] if you have even just a 200 degrees temperature heat source and if you can get [INAUDIBLE] efficiency-- 40%. If you take the sun, 95%. And then more than thermal power plant is about 40%. I said automobile internal combustion engines. The driving efficiency is less than 20% and the AC engine efficiency is probably around 25%. So heat engine, if you can achieve the theoretical efficiency, you're doing really well. The problem is there are a lot of losses that limits us to get to practical [INAUDIBLE], practically get too close to the economy efficiency. So that's what you learn in 2005 thermodynamics. Now I want to go to a little bit microscopic picture. Because what's an entropy, for example? And so let's look at an isolated system. An isolated means there is no heat transfer across system boundary, no work transfer, no mass transfer. It's isolated. And I'm going to look at, imagine I have the magic eye, can look at the individual molecules, the electrons in the system. And there are many different possible configurations. A molecule, so the molecules in the system could be at different states, different position. And each atom could have many different possible energy states. This is from quantum mechanics. So every possible configuration possibility-- you can go back and take a two color, two different color beads and mix them up. That's your experiment on this micro picture. And every possible configuration is a micro state. And let's suppose we know how to count what's the total number of those possible configurations. So the total number, that's omega microscopic state. Those are very big numbers. Now, what are the basic principle of this microscopic picture says every state that is equally possible, equal rights, no discrimination. And so this is the equal probability principle. Every micro state is equally possible. And it's based on this. All the statistical description of the statistical thermodynamics is based on, starting from here. And what Boltzmann did is to relate this number of total microstates to interpol. This is actually in Boltzmann tombstone. It goes to Vienna. And this kB is the Boltzmann constant, 1.3. I don't remember any-- people in physics, they use atomic units. I always remember things in the SI unit. So that's a difficult large exponent, 10 to the minus third [? jar. ?] That's good to keep some numbers in mind. So that's the, like I said, a starting point for the microscopic description. And here, we'll consider an isolated system. And now if you think about a not isolated system, a system that can thermally [INAUDIBLE] exchange energy with the environment. And because of that, each micro state is no longer have equal right in terms of observation. The probability of this system, if you can observe this system in certain specific micro state that has certain energy, e. And in this case, this is no longer equal probability. And we're not going to derivation. But you can go from here, equal probability, to construct an isolated system that made up your system and the environment. And with all that derivation at the end. We get the probability of the system actually inserting one micro state. This is the one micro state that has an energy, e. It's depending on the temperature of the system and the energy here, [? exposure. ?] So you can see higher energy, less probable. And if you do chemical combustion or a lot of Arrhenius law you probably learned in high school, that's the exponent. That's coming from here, [? personal ?] statistics. And so here is a closed system. What if there is a mass flow, particle flow across system boundary? And in that case, this is similar. But there is an extra chemical potential uphill in the exponent. And this is a constant temperature, but open system. And mu here is a [? chemical ?] potential. What is [? chemical ?] potential? Think about temperature. Temperature is a driving force for heat flow. [? Chemical ?] potential is the driving force for mass diffusion, mass flow. Pressure is the driving force for mechanical motion. Those are all thermodynamic, intrinsic thermodynamic quantities-- pressure, chemical potential, temperature. The other way to think about the chemical potential is it's the average energy needed to add a particle into a system or take a particle out of the system. So first chemical potential is the energy look at. This is the equivalent here. And second is driving force for the mass transfer. So if you do, say chemistry, you do mass diffusion or you do batteries, this is a quantity you will be dealing with all the time. So this is a microscopic. And now I'm going to apply what I just wrote down in the previous slide to molecules. I have lots of molecules. And I have an equilibrium, temperature t. I think the one molecule. It has a kinetic energy, a single, simple kinetic energy of the molecule. So it's 1/2 mv squared. [INAUDIBLE], molecule is the velocity is the three components, the vx square, vy square, vz square. So that's what I have here. And this is the energy. Remember, in the previous slide we say this is now a constant temperature molecule in thermal equilibrium at constant temperature. So that's the statistical system I'm using. And this is the energy that's a kinetic energy. And here is a probability, probability I have a normalization factor. So I put in the probability of this molecule having the velocity vx, vy, vz. And the next step I want to determine a, determine that pre vector, a. What I do? The probability-- so the molecule probability is the velocity is from minus infinity to positive infinity. I include every possible chance. So if I equal the error possible chance, that's my normal, the way I find out this factor a. That's all, say, velocity from negative infinity to positive infinite. And this should sum up to 1. They've got to have the velocity in between these. And if you do this integration, you find your factor, a. And this is the famous Maxwell distribution for molecules. STUDENT: It should be a minus. GANG CHEN: Thank you. Yes. STUDENT: [INAUDIBLE] GANG CHEN: There is a minus. There's this minus sign here. Thank you. And see the other one. Yes. Please add [? another ?] [? 7. ?] And here it's the same problem. I just copied. So now if it's a probability, once you have a probability, you can calculate your observation. What's the average once you have probability? So I want to find the average energy, kinetic translational energy motion, kinetic energy of this molecule. So here is the energy weighed by this probability over all possible velocities. And again, it's an integration. You can do yourself if you look at a table. And you find out is a very simple result e equals two thirds, 3/2 kbt [INAUDIBLE]. It's 3/2 kbt, actually. It's a very good language to remember, kbt. But also it's a fundamental. There is a fundamental principle, equal partition principle, which says if your microscopic energy is quadratic-- so if you look at this, kinetic energy is quadratic in velocity and v squared. So if energy is quadratic, each of these quadratic terms contribute to average energy half kbt. So if you look at this molecule there, because it has three quadratic terms, so each contribute one part kbt in total [INAUDIBLE]. So this is the equal partition principle. And now I would like you to get an idea of what is kbt. This is really something that if you want to have microscopic mind, you should remember. How large is it? kbt, 1.3 times [INAUDIBLE] times third, 300 degrees Kelvin. I think I forgot to correct this again. This should be 4, not 5-- 4.14. 4.14. And if you take a 4.14 divided by 4, that's a big number. Too hard to remember. And people-- again, different field people use different ways, different measures. And one often used the unit is electron volt as an energy unit. What is the electron volt? You have a charge. You apply an electric field, one volt, say under one volt. That's a potential how much energy it has. So one electron volt is 1.6, 10 to the minus 19 [? JR. ?] So I translate this into a room temperature one kbt 26 literally electron volts. Now, think about if this is the average energy of the molecule, what's how fast they are moving? So the molecule moving in this room. You can go to estimate, because this energy equals 1/2 mv squared. So you can go to estimate the velocity of the molecule, one half cancel. You do the math. I took the oxygen as an example. And all the number of oxygen atomic number is 8. So it's 16 neutron proton. And each of the proton mass is 1.677 plus 27. So number is about 200 meter per second. Depends on whether it's a light molecule, heavy molecules. So those are the microscopic molecular picture. That's about molecules. Let's think about the electrons. Molecules, what we see, we did not do anything quantum side. The Maxwell distribution is a classical statistical thermodynamics. Now, when I do electrons, the quantum mechanics sometimes is very important. So it's the more common picture. From quantum mechanics, we learn-- you have a physics before, basic idea. I'm sure you all heard about energy levels of quantum mechanics. And some of you may still remember, if you have quantum mechanical state, you can only have one electron-- Pauli exclusion principle. And the quantum mechanics, actually here I have listed the quantum Einstein relation that gives the energy. This actually started from photons. So next slide I have more energy related to frequency of the photon and the momentum relates to wavelengths of the photon. So that's the so-called the Planck Einstein relation. And again, this is another number. If you think about the quantum, you always see h Planck constant at 6.6 10 to the minus 34 [? jar ?] [INAUDIBLE]. Now I'm going to look at one quantum mechanical state. And this quantum mechanical state can have maximum of one electron. It can have 0 electron. You have one electron. And this quantum state has an energy, e, quantum mechanically allowable energy. When the electron is there, this state has energy, e. When it's empty, the electron, the actual energy of the system is 0. But I only have two possibilities. So I have-- there's two possibilities. I sum this up. I want to find this a, the difference between the molecule I did before. And here, for molecule velocity I say, OK, from minus infinity to plus infinity, I include everything. But for the electron here, I only have 0 electron or 1 electron. So I sum this up to what I will find a. I'm not going to detail the math. So from the previous relation, I found a. And now I give you an average. Again, I can give you the average energy. Because if I do observation, it's 0 or e. And if I do a lot of observation, it's a equilibrium temperature t, what's the average time, average energy, or average number of electrons? Each individual observation could be 0 or 1. But there is an average possibility. So if I look at the average, again, say this is an n could be 0, 1. And they sum of two Fermi-Dirac distribution. You look at that. What's the range of this [? F, ?] the average? STUDENT: [INAUDIBLE] 0 to 1. That's so they can't be above 1. Its maximum is the 1. Minimum 0. So it's between 0 and 1. And then you plot this depend on the temperature. And it depends on whether the energy of this level is above the chemical potential or below the chemical potential. The chemical potential is they say, we said before, now you can think of this as a one quantum state that the open state system, the electron can come in and goes out. The chemical potential is what's the average energy you need to take the electron in or the electron out-- take the electron out for put electron in. So depends on whether it's above or below. You can say it's close to 1 or close to 0. Very narrowly distributed near the chemical potential. And in fact, if your [INAUDIBLE] is 0, this will be a sharp position. At the chemical potential above is 1, here below. Let's say above 0 and below 1. And the chemical potential at 0 temperature, there is a special name people give often called that Fermi level. But in a lot of different fields, sometimes people just call Fermi level for energy transfer. It's terminology convention. So that's for electron. Now we're going to lecture two. We talk about molecule. We'll talk about electron. Next I'm going to talk about electromagnetic wave, the photons. I mentioned before, the Planck Einstein relationship is really looking into photon. And later on was generalized for any material wave. And so what we have is if there is electromagnetic wave at a certain frequency, mu, what Planck found is that the energy of that mu, the quantum mechanical state-- at the time there was no quantum mechanics. So electromagnetic mode, of specific mode, the energy has to be quantized. And basically energy quanta is [? hmu. ?] And this other quantum [INAUDIBLE] was called a [? photon. ?] And when Einstein came in, he's saying that the quanta not only just have the wave, but also has a momentum. So this is where the wave particle duality came from. So the momentum of the quanta is related to the wavelength of the wave, and again, the edge. And so each quantum state, the energy can only be multiple of the edge mu. Normally when we think about the continuum mechanics can be any value. But the important difference is that the energy could be only multiple. So n is now integer. And it has a one half. That's also called this one half. Normal is not important. It's called 0 point energy. And it's really-- if you still recall from quantum mechanics, that's the so-called the Heisenberg Uncertainty Principle requirement. So this is 0 point energy. Normally it's not. So I'm not going to give much. So that's a photon. Now I'm going to-- it turns out the photon, you can think of this as the electromagnetic wave. Now I'm going to talk about different waves. That's the lattice wave, the atomic vibration. The [INAUDIBLE] vibration now, you can think about starting with classical mechanics. If you have a mass spring system, the fundamental natural frequency is square root of k over m. And, of course, in classical mechanics, we see the energy is related to the amplitude of the wave, the velocity. And once you go to quantum, it turns out this way, the energy cannot be any value, same as the electromagnetic wave. And the energy can only be multiples-- same, you can say multiples of [? SU. ?] And now there was an analogy made for this compound of vibration. That analogy is [INAUDIBLE]. Electromagnetic wave, the basic energy quanta is the photon. That is atomic vibration or the energy based energy quanta is photon. OK, so what's the statistic they obeyed? At a certain quantum state, if you think about electrons, could be 0 and 1. Maximum is 1. And here, at the mole, at this frequency, it could be 0, 1, to infinite. So that's a difference. Electron, 0, 1. And here, it could be any number of integer numbers. So I need to find the distribution average number of photon or [INAUDIBLE] if I have a equilibrium system. Do the same thing. I have the probability. And now, if I have n photon or [INAUDIBLE] in this microstate, the quantum mechanical state that has this specific frequency, that's the probability. I sum this up from a equals 0 to infinite. That still should give me 1. You have either 0 or 1, or infinite number, or the in between, somewhere in between. So sum up, I have one. I will determine a. And this a, from this a, I go to find the average again. Once I find the probability, I find the average number same way as I did for electron. And this average number now is the Bose-Einstein distribution. And this Bose-Einstein distribution, if you compare it with the previous Fermi-Dirac distribution, the only thing changed is a plus sign now goes to minus sign. And because of that, it's no longer between 0 to 1. It's between 0 to infinite. And in fact, for photon and photon, because the number is not conserved. It's not like the electron is conserved [INAUDIBLE]. So chemical potential concept is not [INAUDIBLE]-- is not a good one. So that's a 0. We take it to 0. And this is the distribution for the Bose-Einstein distribution. STUDENT: What's the reason for setting mu at 0? GANG CHEN: It's not a conserved quantity. It's, say, when we talk of mass transfer particle, that's a fixed number. Total number either going out or going in. But that's a conserved number. Here is another number. But there are actually-- see, later on when we talk about the solar cell, you'll find out there is a quasi Fermi level getting into here. So it's actually a very deep similarity. So this is just a simple argument. That's about-- now, in this time frame, like I said, you're probably familiar with the classical thermodynamic. You're now familiar with statistical side. Like you just take it. This is all the result we'll be using when they get into [INAUDIBLE]. Then microscopically we have [INAUDIBLE] distribution. We have Fermi-Dirac distribution for electron. We have Bose-Einstein distribution for phonon photon. That's what you need to know. And what we need, I'm going to summarize next. This is the heat transfer. Before it was thermodynamics. Now, heat transfer, again, you take the 2005, 2006. This is a very-- should be very familiar. We have three modes of heat transfer-- conduction, convection, and radiation. And describing the conduction is a Fourier law of diffusion. So the heat transfer rate, here for the rate in watts, it's proportional to the area, proportional to temperature gradient. A proportionality constant is the thermal conductivity. And that's a material property. Normally, we think of it as a material property. And the negative sign actually take care of the second law, says temperature, heat will flow from high temperature to low temperature. And you can do that. from watt you can do flux. That's a per unit area based. So just normalize the area. Or you generalize it to 3 dimension rather than writing into one dimension. So that's [INAUDIBLE]. And convection is we have fluid flow in the surrounding of a heated or cold surface. The heat transfer between the surface and the fluid is proportional to the temperature difference of the wall and the fluid. This fluids are actually transition. The temperature goes from the wall to gradually to the center of the fluid or ambient. And the proportionality constant here is h, the heat transfer coefficient. But h is a flow dependent. It's not a mature property. It depends on how fast the fluid flow and depends on the geometry. So this is a big topic in 2005, 2006. And we know there is a natural convection, forced convection, just in terms of the terminology there. So those are conduction and convection. And, of course, the thermal of heat transfer is radiation, thermal radiation. Thermal radiation for blackbody one single surface or black object. The emission is given by the Stefan-Boltzmann law as the proportional to temperature force power. The proportionality constant is Stefan-Boltzmann constant. So this is a for black object. A real object is not black. So we use a emissivity to characterize it. And we'll discuss some more down the road. Now, if I have two objects, exchange heat, in this case, it could be vacuum. That's the difference of radiation compared to conduction and convection, which requires a media. And here, the heat transfer is by electromagnetic waves. So I wrote a simple form of relating the heat transfer is the [INAUDIBLE] to the fourth power. And remember here, you have to use a Kelvin, not degree Celsius. Look at the units and so on in Kelvin. And if the object is not black, we use the emissivity. And here is the factor between two surfaces, because the two surfaces may be different. Like I said, we'll discuss more down the road. So those are three modes of heat [? conduction. ?] And in this course, we'll talk more on conduction and we'll talk more on radiation. You have learned a lot. Some of you learned a lot of convection. So that's probably fear. Now, let's think a little bit more on conduction. If I have a one dimensional multi generation inside, I can say heat flow is constant and secure is constant. You go back to the Fourier law. You integrate it. You find out that the heat transfer between the two points from this to the other side at steady state is proportional to thermal conductivity area inversely proportional to the length. So if you want to do thermal isolation, what do you do? You use a long object. You use a small cross section. That's the thermal isolation. A lot of times when you design a thermal system, not only heat getting in, you also want to isolate it. And we can combine all this right into thermal resistance. That's just a definition. Because this way, it looks very similar to electrical circuit where your driving voltage is now the temperature difference and your current is the heat current and your resistance. So you can use the same way as you do circuit analysis if you know the thermal resistance. This is for different geometries, different. And even for convection, you can express, you go to do thermal resistance. So you can build some resistor network to do some simple heat transfer. And another thing that's important to keep in mind is the order of magnitude. What are the good conductors? What are good insulators? And the best conductor turns out is diamond. Why? That's because the bumping, the spring. So in this case is the wave, atomic wave that carries the heat. So if the bonding between the atoms is strong, the spring constant is large. The velocity is large. So that's why diamond is a good conductor. And metals are not that good. Metals are by electrons. And insulators, you can see the wood, amorphous material, pores, those are poor thermal conductors because the atomic arrangement is random. And that randomness makes the wave harder to propagate in the material. So this is, again, macroscopic. Now I'm going to also give you the other view, microscopic of heat transfer, and the kinetic picture. So what happens, how heat really is conducting? Let's think about the molecules. And the molecules, let's say we have a box. And one side is hot. The other side is cold. Let's forget about convection. How quick it goes from one [INAUDIBLE] side, we add the molecule to the cosine. It's really on the hot side, the air molecule collided with the wall. And on the wall is hot. It means the velocity is faster. So this molecule will get a faster velocity due to that energy exchange between the wall and the air molecules. Now, this hot air molecule will collide with the [INAUDIBLE] molecule. Of course, we have a chance to go randomly. Every molecule is moving a few 100 meters per second, as we talked before. And they collide with the cooler one. And they pump some of that excess energy to the next [? level. ?] The next [? level ?] will go back and forth. Again, they do this, cascade energy from the hot to the cold side. It's a diffusion process. And typically, there is a the distance of this collision. You can go to estimate it. And just turns out that this distance between the average of multiple collision is about 100 nanometer. In this room, the air molecule distance between collision is about 100 nanometer. And so now I give you a few more definition. The energy per particle, let's take energy per particle as e. And the average velocity is v. And time between collision is tau. And distance between collision is gamma. That's the mean free path. So time is the average time between collision is the relaxation time. And this distance and the time is related by the velocity. So what is a heat transfer? When I think about heat going through this imaginary surface here, how I calculate it? I can't. I do [INAUDIBLE]. I will say, OK, only those molecules within one mean free path. vx is a velocity component in the x direction, times the collision time. So only those molecules can go zip through the interface without collision, because tau is the average time between collision. So this part of the molecule is harder to go from this side to the other side. And on this side, the molecule is cooler again, in the vx times tau [? average, ?] they go from this side to the other side. And I take the difference. I count all the molecules. I take the difference. That's my reflux. So let me write down the math. That's what I do. I say on the left hand side, this is a per unit volume of how many. N is the particle number density, the second bullet there. Each particle, how much energy it has? And this is the velocity in the x direction. So I say only in this range, from x to vx minus tau. In this range, you can go through. On the other side is x plus vx tau. You can do more detail. I hear this is a very simple argument. So that gives me a [INAUDIBLE] for heat plotting. Go check a unit is a watt per meter squared, [INAUDIBLE] plus. In the next lecture we'll just do a [INAUDIBLE] expansion of these two terms and write it into a differential form, because this is about 100 nanometers. If I do meters or centimeters or millimeters, that's essentially continuum, a very small delta. So I do the derivative. And I do a Taylor expansion. This one half is because there's only half of this go this way, half goes that way. So that's where the [INAUDIBLE] came from. And after I do the Taylor expansion, I have e and v and vx. vx is random velocity. I pull out this independent of distance. It could be dependent on this. I'm just making an assumption, approximation. In there are different distance. This vx square. vx square is random. So it's one third of the average velocity, one third of v squared. So that's my one third v square replacing this vx square. Tau is here. And the n times e is a per unit volume how much energy, internal energy the molecule has. So I have this n times e is u. And I take a general do, dg, dbdx. OK, what you get? You got [INAUDIBLE]. Because here is one third v squared, tau [INAUDIBLE]. And that's the thermal conductivity. And this is the temperature gradient. You can do that. You can go back and do your Fick's law and Newton's shear stress law. In fact, all the diffusion law you learn in three steps. But yes. STUDENT: You said we could move the velocity out of the derivative because of its constant. But isn't the velocity related to the temperature? GANG CHEN: Yeah, good question. This is later on when discuss the thermoelectric effect. I'll come back to this. Here I'm talking about a molecule. And I just wanted to show you the [? Fourier ?] law. And they're clearly approximation. And what are you talking is more like a coupled mass and heat transfer. And, in fact, if you learn once the heat transfer, you know mass and heat transfer couples. So the thermal conductivity is one third specific heat velocity and mean free path. And the specific heat is per unit volume. If I do capital sales per unit volume based, and typically we do per unit mass based since this is a density price per unit mass basis. So this is the a very useful relation to keep in mind. Simple thermal conductivity depends on how many energy is stored by the molecule, how fast they travel, and how far they travel between quality. So if you want a high thermal conductivity, you want the power of [INAUDIBLE]. Low thermal conductivity, you reduce that. And for most solid, they say is about 10 to the sixth power. It doesn't vary by a factor of 5. So when I do on my back of envelope order of magnitude estimation, I say is 10 to the sixth. You go to check. And velocity in solid is just you can roughly think the speed of sound, few 100 or few 1,000 meters per second. The mean free path is short-- nanometers, 100 nanometers. So that's the range order. So this is I think most of you, probably in 2005, you haven't seen it. It's very simple. But the picture is really the diffusion picture or similar. Let me give you-- I haven't finished radiation. So I come back to-- I said that Stefan-Boltzmann law. And what I want to show you next is how I can derive Stefan-Boltzmann law from the statistics. I just showed. So I'm considering enclosure. Let me imagine the enclosure is perfectly reflecting. There are photons. And the walls is the temperature, t. So it's a equilibrium system. I want to see how many the electromagnetic waves inside this box, how much energy they have, a constant temperature-- equilibrium, thermal equilibrium. So let's go back to a little bit of the basic relations of thermal electromagnetic wave. We know the speed of light is proportional to its speed of light as a wavelength times frequency. That's a high school learn. And definition, I say angular velocity is 2 pi mu. We vector magnitude is 2 pi over lambda. What is the wave vector? The wave propagates in certain direction. And so because the direction is a vector. 2 pi lambda, lambda is a wavelength. It's a scalar. And the rationale is the vector. So it's a kx, ky, kz if you do Cartesian coordinate. And so I can write it C equals mu times lambda. In terms of frequency and vector, it's c times omega equals to ct. So this relation between the vector, magnitude, and the frequency is called dispersion. And in the speed-- in the case of light, it's a linear. C is 310 to the eighth meter per second. And, of course, because the vector, so it's a square root of the components. So that's the dispersion. And now I want to count. My question is, how much energy photon has inside this box? I'm going to count how many modes you have. Remember each mode, the Bose-Einstein distribution already tells me for each mode how many photons. So how many modes? So that's basically I'm counting. Because I'm a perfect [INAUDIBLE], you can see the mode has to be standing wave here, either this way or this way. That boundary is 0. STUDENT: Why is it 0? Why should it be 0? GANG CHEN: Otherwise it doesn't get a steady state. It keeps propagating. So this is my wavelength component in the x direction. kx is a corresponding wavelength in x direction. And the lx is the length here. So if I just say, this must satisfy the rule, then I have to have integer. Those are the possibilities. And that translated in the possibility of my k. kx just is 2 pi over lambda x. So this is n tau pi 2lx. OK, so each combination of x and y and z, each set gives me one mode. And I want to find out how much energy, I just sum up all those modes. So I sum that up. And remember, when I do molecules, I do velocity from minus infinity to plus infinity. And here when I do this, I do discreet summation of nx, and ny, and nz. So I do this summation. This is an x, y, and z. Each mode has this is the number of photons in this area set. And this is the energy of a photon is [INAUDIBLE]. I got the factor of 2 because the electromagnetic wave is a transverse wave. And it has a two component, two polarizations, so that factor of 2 [INAUDIBLE]. OK, if you don't look at this, that's where my starting point. And what I want to do is, rather than do the summation, I'm going to translate it into integration. Summation is over n. So if you look at this interval between n and k, it's 2 pi 2lx, so when I have a dkx, I divide by 2 pi 2lx. That's n if this is large number. If small, you have a lot of problem. So I translate this summation, nx, into this dkx integration. I do same for ky, kz. This is-- I'm just copying. I go the extra step here just for the later purpose. Because my function is a is even function. So I extend to minus infinity to plus infinity. The difference is the factor of 2. So I don't have this 2 here anymore, because the factor of 2 is gone. That's just for my later purpose. The end result, this is just an equivalent. And what it means is I can, rather than standing wave, I can say do two counter-propagating. One goes left, one goes right. That's when you do the maximal velocity is minus infinity, plus infinity. So that's equivalent. OK, I think I will probably have to stop here. There are two more slides you can look at, but I'll discuss. What's coming is the blackbody radiation in each wavelength, how much energy varies with simple [INAUDIBLE]. OK. |
MIT_2997_Direct_SolarThermal_To_Electrical_Energy_Conversion_Technologies_Fall_2009 | Lecture_8_Radiative_heat_transfer.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. GANG CHEN: [INAUDIBLE], we talk about solar cells. So in the rest of this course, we're now shifting our attention to solar power. We're combining solar to this engine and particularly solar thermoelectric and solar thermal [INAUDIBLE] state. And so the-- here, what's important is not, of course, the [INAUDIBLE], but really have a system perspective. You'll find out the spectral control is really crucial for the overall system efficiency if you want to achieve [INAUDIBLE]. So let's review the three modes of heat transfer. This is the first class for mechanical engineering is very simple. But if you come from other field, this is probably the best familiar. But see, we know there are three modes of heat type thermal conduction from [INAUDIBLE] radiation. And conduction is the fact that through [? the ?] thermal conductivity, heat transfer rate. that's the power, and then it's proportional to the [INAUDIBLE] gradient. And convection, we usually characterize with the [INAUDIBLE] coefficient h, so it's proportional to the [INAUDIBLE] difference between the wall and the fluids, which is away from the surface. And there are natural convection and forced convection. And then thermal radiation, we typically talk about the Stefan-Boltzmann law for blackbody radiation. And then we calculate the heat transfer relation-- heat transfer between surfaces in terms of view factor and [INAUDIBLE], which we'll go more into more detail since we're talking more now on the radiation part. So the thermal radiation, Planck's law, is usually considered a maximum. And we define the intensity as the power leaving the surface in certain directions, so that the intensity is different from power. So this is so that power in this direction that we will characterize by solid angle as a power in this direction per unit solid angle and per unit area normal to the direction of propagation per [INAUDIBLE]. So that's the intensity. So this is what we mean by intensity. It's propagating in this direction. And when we think about the area, this is the area, not this area, but the area projected in the direction of propagation. So [? ba ?] perpendicular has a cosine [INAUDIBLE] to the [? direction. ?] And the solid angle is a measure of the angle in space. So if you think about 2D, we use a polar angle. And a [? parallel ?] angle is the arc [INAUDIBLE] divided by radiance. Or solid angle is the area perpendicular to the direction of propagation, [? eaj ?] here, and divide by r squared. So if you take a sphere in space, that's 4 pi r squared, your solid angle is 4 pi. In 2D, polar angle, the maximum is 2 pi. So the solid angle is the space angle. And if we do [? a ?] into a polar spherical coordinates, this is theta, and [INAUDIBLE] with the direction to the phi, then you can show that this is a sine theta d theta d phi. So solid angle really including both theta and phi. So that's the definition in terms of intensity. And when we want to calculate-- so intensity is a user starting point. In fact, it will be really interesting to go back and read the Planck's law. He had a very interesting discussions, very careful, interesting discussion about [INAUDIBLE] radiation intensity. The immediate-- total immediate power of a surface normalized by surface area, that's what they call the emissive power. So now if you look at that, we normalize the emissive power. So power equals intensity times the area ea is perpendicular to the [? emissive ?] direction. So I have cosine theta. And d omega integration, so this is still per unit wavelength. So that's the initial power. And for blackbody radiation, the intensity is isotropic. It's the same in all directions. So it's independent of direction. You can take this out, blackbody. And then you have cosine theta, d omega is sine theta, d theta, d phi. So you go to do the integration. And if we think about the integration, theta is from 0 to 2 pi. That's 90 degree. And phi is azimuthal polar, say, coordinate, the spherical coordinate, that's a 2 pi. So if you do the integration, you find that the intensity for blackbody radiation for isotropic radiation, the intensity [INAUDIBLE] emissive power is a pi relationship. So that's the emissive power. And Planck's law of blackbody radiation could be either written in the intensity form or in the emissive power form. Apologize, I have a-- I missed the c square in my previous first lecture. If you go back to the notes, you can add that. So that's the intensity and the emissive power. And so, this is what we talked-- pretty much what we talked before. And if you plot the blackbody radiation curve as a function of wavelength, usually it is a unit of micro per unit micrometer. And you find out the [? multiplicity ?] peak for a [? shift. ?] As it goes to a higher temperature, the peak shifts to a shorter wavelength. So this is a Wien's displacement law states where this peak happens is the maximum radiation state of [INAUDIBLE] wavelength where [INAUDIBLE] happens. And temperature is about 3,000. So that's the [INAUDIBLE]. Remember, and hence the solar temperature, [? static ?] temperature in a vacuum is about 6,000. So that's why our [? highest ?] is [INAUDIBLE]. That's [INAUDIBLE]. The evolution works really well here. And we have not, say, [? adapted. ?] So it's interesting that the human adapted the sensitivity to the sun. Not a lot at night that you're probably sleeping. So if you think about the emission everybody has is [? 2 ?] micron. Because the 300 degree Kelvin has a peak around [? 2 ?] micron, which is equal to [INAUDIBLE]. So if you do integrate all the energy in this [? under all ?] wavelength, you get the Stefan-Boltzmann constant, 5.67 10 to the minus 8. So this is pretty much what we-- a review. And now, I want to plug this into a more convenient form, blackbody radiation, because it's a function of two parameter, wavelength and temperature. But if I normalize this emissive power by temperature phase power, I change it to a small e meter [INAUDIBLE] change. So if I normalize [INAUDIBLE] phase power, you find out this temperature phase power combined with lambda here is combined with lambda t. So now I only, I think about lambda t as one parameter. So this curve now, the blackbody [? uses ?] power [? localized ?] from pure phase power is [INAUDIBLE] function just lambda t. So I can have a universal curve rather than say each temperature have different blackbody curve. And in fact, if you take your maximum lambda t, you can say, if you think this [INAUDIBLE] at the x, you find out where maximum is. That's Wien's displacement law. Wien got this by experiment. And in fact, you will see that the Wien had an [INAUDIBLE] [? too. ?] But this is from Planck's theory you can derive these things. So now see, using this function, we can also calculate the fraction of energy. This is a very convenient [? particle. ?] In the old days, even now, for me, it's every time I want to see how much energy in certain wavelengths from the sun, I take this function and I look at table much faster. So this is the fraction of energy between 0 to lambda. That's the integrate between 0 to lambda. And of course, using this function, we can say that eventually we could write this into lambda t fifth power of-- this is a force. So there should be another t here. And then say lambda t, t [INAUDIBLE], because here is the fifth power, here is normalized the fifth power. Here I normalized it only to the fourth power, because this is the total. That's the Stefan-Boltzmann law. So after I write this, you can say, I can do this integration. this function is a function-- this integral is a function of lambda t only. Here is lambda t, lambda t, lambda t. So it's lambda t only. Again, so by doing this, I have a universal function that contains this is the fraction of energy with the wavelengths between 0 and lambda for any temperature t. So if you give me a temperature, I look at the wavelengths, I do the multiplication, I go to check the table, and usually quickly you find out. So this is the integrand. That's the small f. That's here. And as a function of lambda t, lambda t is written in micron Kelvin. And here is the integral. So that's the fraction of energy between 0 and lambda t. So you can quickly do an exercise. Again, let me take a solar energy, the sun at 606,000 Kelvin. A silicon bandgap is about 1.1 micron. So 6,000 times 1.1, 66, 6,600. So 6,600. If you look at this curve here-- this is a log scale, so it's harder to read. So the drop is 75%. So the other 25% of the solar energy beyond the-- below the bandgap, which is longer, is not used at all. So that's the case of when you use a photovoltaics, there's a bandgap, and the photons below the bandgap does not electron [INAUDIBLE]-- does not excite the electron [INAUDIBLE] pairs. So that part is completely gone. And in fact, you can also think for that part, it's better even just filter it out if you can do it cheaply. Why? Because if your solar cell absorb it, you will heat up the solar cell. And usually, solar cell heats up, the efficiency drop. So that's the flat body. Now let's look at the sun itself. And then, [INAUDIBLE]. So the distance diameter of the sun is 1.4 2 to the 9, [INAUDIBLE] about 1,000,000,000 meter. Remember, diameter of the Earth, that's a 12 million or 13 million meters. And the distance is close to, what is it, 10 to the 11, 10 to 12. Is there a link for 10 to 12 for distance? [INAUDIBLE] [? Parameter. ?] So 1.5 10 to the 11 power. So that's the distance. And you look at the solar, of course, the orbital changes. The distance between the sun and the earth, slight change. But you see that this might be-- [INAUDIBLE] too large for you. But some of you may not be aware of this. Actually, in June and July, the solar flux is the lowest. From the distance, July is here, and it's the farthest away from the sun. Another thought is, this is the farthest. So we have the lowest flux, but we have the summer. And the reason is not due to the distance, it's because the Earth is wobbling a little bit. When you surround the sun, if you look at it, the Earth's axis is not perpendicular to the plane of rotation. And it has a 24 degree of-- 22 or 24? So this is the direction [INAUDIBLE]. --23.5 degree from the normal direction. So this is the normal direction. And because of that, on each side here, the sun's atmosphere facing the sun. And here, the northern hemisphere is [? facing ?] the sun. So that's the tilting. And this actually created a problem. Particularly, we want the [INAUDIBLE] sun. Say, some people do-- when we use concentration, you have to track your maximum. You want to maximize your intercepting area. And so, that's the tracking. You have to complete the rotation of the angle here of the Earth, of the [? orbit. ?] So here is the blackbody radiation coming from the sun reaching the earth's surface. It's the outside atmosphere. And it's close to 58. Different people have slightly different numbers. Later on, I will show if I use a 5777 Kelvin, I get a [INAUDIBLE] constant very close to [INAUDIBLE] solar constant. So this is the-- it's very close to blackbody overall. And of course, when the sun comes through the atmosphere, you have a lot of things happening. The solar radiation coming in, it will be reflected by the atmosphere. The clouds can scatter light to all directions. Some will go back. And the atmosphere itself will absorb the solar radiation. Depends on what molecular species. Those are the vibrational modes of the molecules. Typically it's of the few micron range. So it depends on what kind of molecule you will have different absorption bands. So if the molecules are happily absorbing, or shooting around all of those molecules, then say those photons cannot reach the surface, it's the heat up the molecules directly. But there are transparency windows. Again, this is a pretty lucky for us, for everything, just absorbed by all the molecules, we need to have direct sunlight and [INAUDIBLE] dark. And of course, there are also ways to reach surface. You will have the reflection from the surface of the Earth. So when you put a solar panel there, you have direct sunlight, you also have scattered light from the atmosphere and from the ground. So this is a [? characterize ?] solar cell. People have to compare orange to orange. So there are standard in terms of what you should compare. And this solar spectrum [INAUDIBLE] by the air mass, which is the ratio of the [? confidence ?] of the air, of the air [? being ?] travelled through to the shortest possible pathway. So if I-- the outside atmosphere is an a m 0. And then a m one, air mass one, occurs when the sun is directly overhead. And a m 1.5 occurs when the sun rays travel through roughly 50% more of the atmosphere than when the sun is directly overhead. So at an angle, there is, say, the confluence of the sun will be longer. And you can imagine, because the longer confluence, the spectrum of solar radiation reaching the surface will be different because the absorption of the atmosphere itself and scattering process. So you look at, let's say, 1.5, the angular relation, the actual distance is 1 over cosine theta. So cosine 48.2 gives you 1.5. So this is, of course, a m 1.5 [INAUDIBLE] solar in this angle, the angle away from the perpendicular is 48.2 degrees. So that's the-- but there are also different [INAUDIBLE]. Depends on whether the solar radiation we measure is directly coming from the sun or there are also scattered, diffuse light. So there are a m 1.5, there are different a m 1.5. I mentioned before, the a m 0 is the spectrum of the radiation outside the atmosphere. And a m one is directly 0 degree. And 1.5 has 1.5 [INAUDIBLE], g, and 1.5 direct [? path ?] circumsolar. So the direct one is really consider a small cone of solar radiation. And this cone is confined to 2.5 degree, let's say here. Let's say 2.5 up and the 2.5 degree is around the sun. And so, this is the a m 1.5 direct. And then a m 1.5g is the radiation from the sun. The entire sky is reflecting off the ground. And so, in this case, the solar [INAUDIBLE] is 1.5 to the 40-- 48.2. But also the panel itself is tilted at an angle of 37. So it's not a perpendicular incidence from the sun to the solar panel. Here is 37. So the angle of the incidence, angle of incidence is 11.2. And so, people are using 1.5g for solar cells. So that's just a property [INAUDIBLE]. 1.5 is the 40-- 48.1, 48.2. And so, the global equal the scattered light. And also, the panel is 11.1 degree away from the normal incidence. So if you look at the spectrum, this is how the different gases in the atmosphere absorbs. And water is actually really the absorption is the worst. So this is the water absorption spectrum different wavelengths. Carbon dioxide here is 10 micron, That's about a 10-micron range. Carbon dioxide in this band is what people worry the most because of the increased carbon dioxide, the 10-micron radiation from the Earth cannot escape. And in fact, if you combine all this, let's say the Earth warms a little bit more. In this, there are more water evaporate from the ocean, and more water will attract more radiation. So there is-- say, that's why it's a very complex problem. And methane [INAUDIBLE] oxide scattering is-- the scattering is the most severe in the blue region. And that's why the sky is blue, because the scattering proportional of omega fourth power, frequency fourth power. So the short wavelength scatter more, and the sky is blue. We look at only scatter light. We don't directly look in the sun. So that's the solar spectrum, say, in terms of the absorption characteristics. And here is the spectrum. So outside the Earth, you can see the blue curve here. And let's say when the radiation goes through the atmosphere, the absorption bands, different absorption bands here, will kick in, and the radiation is diminished when they reach the Earth's surface. And if you integrate all the solar radiation here. And you get that the total radiation energy in the solar spectrum, and the a m 0 that's outside the atmosphere is 1,366 [? Watt ?] meter square. So you go to, say, probably I think Professor Ely Sachs, he has a startup company called 1366. Next time you can [? tell ?] when you saw him, that you know that's the solar [INAUDIBLE]. And this is a [INAUDIBLE] 577 Kelvin, so that's about a [INAUDIBLE], solar. Now let's talk about the surface itself. And I'm giving you the next few slides of definitions. And the key point I want to [INAUDIBLE] not say you need to know all the details. But the key point is that the surface properties, radiation surface properties, is spectral dependent, depends on wavelengths, it's also directional dependent. So those particular data, we'll talk a lot on selective surface. That's the spectral dependence. And in fact, they're also directionally [INAUDIBLE]. So control spectral and directional property of the surfaces. So the appreciation I want you to have is those are-- you can start with the basic property. When I look at the vicinity of a surface, I can look at the different direction on different wavelengths, how it's emitting the emission characteristics. So this is the light coming, the intensity emitted in a certain direction. And I normalize the blackbody intensity. That's the-- usually we consider that as the maximum. So this normalization is the directional, which I put a prime, so notation-wise. And the spectrum, which I put a lambda, so directional spectral emissivity. And it's a function of the temperature of the surface, the material. Temperature depends on temperature, depends on material, and also depends on surface finish. That's how we control the properties. You can have a piece of aluminum, which is a very shiny low emissivity, but you rough it, then your emissivity will change into much higher values. So that's a spectral direction or spectral. And then you also allow you to integrate all the energy. So this is intensity. Here I'm talking about the energy now because it's a emitted [? in all ?] direction. If I integrate together all the energy going [INAUDIBLE] the surface, but still at a certain wavelength interval, only at a certain wavelength, then I get a half spherical. So that's an all directions, that's spectral. And you can calculate. You can calculate from here to here if you know this, you know everything. We just want to go through the concept. And then you can also say, in this direction you integrate the whole spectrum. So we call that total, total is all the wavelengths. Hemispherical means all the angle. So you have direction total. And then formally, when you take an undergraduate course, you define emissivity, we typically talk about hemispherical [INAUDIBLE]. That's the typical [INAUDIBLE]. You look at a table [INAUDIBLE]. So that's the all direction, all wavelengths. So those are the definitions. And surface, when we say [? alpha, ?] it's in terms of the emission characteristics. If it goes all immediately in all directions isotropically, intensity is the same in all directions. Blackbody is the one example. It is a diffuse emitter, so it goes isotropic all directions. And if the emission is independent of wavelength, So the directional emissivity is same as if we integrate all wavelengths in the same direction. So that's the [INAUDIBLE], the independent-- radiance independent of wavelength. Diffuse means isotropic. So diffuse grade means the [INAUDIBLE]. So if it's diffuse grade, then in all direction, all wavelengths is the same, and then all those emissivity are the same. But fundamentally, you can start from here and write the relation between all these properties to calculate all other properties in all this. But this one doesn't always equal to the others. So that's the emissivity. And the inverse of emission is absorption. So if the light comes in, and the thermal radiation of light, and then the power, really-- I think I probably should have written d omega then in a small form how much radiation come in and then power absorbed in this-- the fraction of the power that's being absorbed in this small form. That's the absorptivity. And again, here we have, coming from a certain direction, so that's prime. And at specific wavelength, that's lambda. So we define directional spectral, and then you do the same thing. You integrate [? all ?] angle of incidence. They absorb as hemispherical, spherical spectral. And then all in the same direction, integrate all wavelengths, or you integrate both wavelengths and direction, same analogous to emission, except now the reference is what comes in. But let me just make one comment here. Although what comes in, the overall spectrum of-- so, for example, solar coming into the surface depends on temperature of the sun. But the spectral here doesn't depend on temperature. I can have the surface, this is [? thermal ?] radiating, coming to the surface, or this laser beam coming to the surface. That's my intensity. And this temperature is really the temperature of the material. But it's not a temperature of the source. So this is the absorptivity. And then the next is a very important-- second law. Second law relation for emissivity and absorptivity. That's Kirchhoff's law. And Kirchhoff's law say, this is the same character of the object, the directional spectral absorptivity utility inverse directional spectral absorption. Blackbody radiation is not a second law. The Kirchhoff's law is a [INAUDIBLE]. The [? maxima ?] of the Planck's law, [? I am going to ?] say blackbody radiation, I mean Planck's law. You can break it. Kirchhoff's law, you cannot break it. This is the second law of thermodynamics. Fundamental, but remember, I'm putting down, again, directional, spectral. I don't mean the outer properties are the same. Now if you have a diffuse surface, diffuse emitter, diffuse absorber, then this is fine. Then say all the other property-- the higher level properties are also equal. But in general, they are not equal for higher order. It's only at this fundamental level, directional spectral. So like I said, the message is, it's dependent on wavelength and angle. And if you think about the reflectivity, transmissivity, so you have a piece of glass. So light passing, you have reflection, and the light can go through. Of course, there are also [INAUDIBLE] here inside. And then the light comes in. The reflectivity is usually more complicated because you can go other directions. You can go-- when light comes to a surface-- and this is relatively smooth, but you can go to diffuse it to other directions. So we then have to say, incoming is one direction, outgoing is another. So the fundamental definition is the actual despite directional, what comes in, what goes out is [INAUDIBLE] the intensity. So you have two angle. One is the angle of incident, the other is the angle of reflection. It's a bi-directional reflectivity. Let's say, now, if you integrate the whole reflected intensity, you get, say, the directional, when I say directional spectral. So that's coming in and integrate all angles. So if, here, when I do bi-directional, notation-wise I have 2 prime. And when I integrate all angle of reflection, I get directional spectral. So this direction only means the incoming because I integrate the reflection over all angle ray. And then I have [? transferred. ?] So this now from here I can do the rest similar as I did for emissivity and [INAUDIBLE]. So in fact, there are eight. You can start with more fundamental and define all those other properties. And you can do the same for transmissivity, bi-directional transmissivity, and then integrate over all transmitted light. And that's only one direction that angle of incidence. So all these are [INAUDIBLE] properties. Now, the energy conservation will tell us, if I think about the light comes in, and it's [INAUDIBLE] reflected. That's the reflection of all angles incident at one angle, and absorbed it inside the material or transmit it to the other side. So that's an energy conversion. That's the first law. So in those relations, I've told you the first law and second law. Can't violate them in any of those-- in any cases. So you've used reflector, so similar to the model we talked about, the emitter is when one light comes in, the reflected light go all directions, a really bad, say, surface in the sense of light goes all direction. But in fact, a lot of times you do want this, particularly in some illumination. You actually want the diffuse light that go to make the room feel comfortable. And so a re-emitter is [INAUDIBLE] one wavelengths, and diffuse reflector is both isotropic and frequency-independent. So if you-- some of you, to the undergraduate heat transfer, learn diffuse [? resurface ?] means diffusive re-emitter and diffuse re-reflector. So it means both. We have come back to a solid diffuse re-emitter here with this [INAUDIBLE]. Now the definitions. We're not going to use much. I just want to make you aware that the surface properties are frequency-dependent, wavelength-dependent, and directional. And now, there's another key concept. When I want to calculate how much radiation goes if I have an emitter. Really, how much of my radiation get to you? So that's the concept of view factor. If I have one surface here, another surface here, view factor is defined the power leaving this surface. That's the total power leave this surface, and intercepted by this surface. I didn't say absorbed by the other surface, it's just reaching the other surface. After which you could reflect it, transmit it, that's not including [? the power-- ?] say, the view factor. Just to say diffraction [? reached ?] [INAUDIBLE]. So that gives you a way of, if you think about the previous reflectivity transmissivity, that's near the surface. This is just saying how much goes to that surface. And if I use this definition, the power within the surface here is the i [INAUDIBLE]. So that's the-- I should have multiplied the pi, I think. Yeah, so pi, that's because the intensity affects pi, the diffuse surface give me the total radiation leaving the surface. And then this is the radiation, if I go back to the definition of the intensity. So this is the power in the cone times the solid angle that this area, eaj, suspended with respect to this point here. So that's eaj cosine theta. That's the lower area. And divided by r squared, that's the distance between the two surfaces. And this is what's the power in this cone and reaching that area. So this with this definition, I have-- I still [INAUDIBLE], I still think that needs a pi, but I will check that. So I have a pi r squared cosine theta i, cosine theta j, eaj. So that's a differential, small area, the two surface are small. Does anybody remember if this is pi? And if I integrate both surfaces, I get a total [INAUDIBLE]. The surface is a uniformly [? medium. ?] And so, that's the view factor. Those kind of view factor, again, you can look for tables. One way to look is go to do this calculation, but a lot of standard geometry, people have done the calculation. So it's calculated. And the assumption I made is that, over this surface, the radiation is diffuse. So I said I have a pi times i. And then radiation given the surface, the whole surface is uniform. So that's the assumption [INAUDIBLE]. And I said, you can look at the tables. But there are also relationships. If you look at the definition, it's very symmetric. So from this, you will see the reciprocity relation from view factor from a to j is times the area i. And then [? increase ?] from j to i times the area j. That's a reciprocity relation. And if you do the [INAUDIBLE]. And then the summation. Summation is basically the energy balance. The energy leaving surface i will reach j, combined surface i plus k. Then it's the same as leaving i [? plus ?] j and leaving i plus k. First [INAUDIBLE]. First one. So with those relation, the view factors, most of time you can, for simple geometry, you can find it from table, or you calculate using this relation rather than doing those complicated [INAUDIBLE] calculation. Let's go look at the Earth and the sun. So the Earth and sun, because they are so far apart, essentially I can treat them a very small surface relative to each other, so the d area. And rather than look at the curvature, because if want to look at curvature, each point will be different angle-wise, perpendicular surface. And in fact, you can just treat this as a disk and this as a disk. It's the equivalent. The disk area of which pi d squared over 4. And then I look at the view factor from the sun to the Earth. It's the Earth's area divided by the distance. So very small. And from Earth to the sun, so how much radiation leaving Earth reach the sun? That's the fraction. This is the fraction, radiation leaving the sun reach the Earth. That's here. There is the reverse. And because the sun cross-section area is larger, the view factor from Earth to the sun is larger, fraction-wise more radiation leaving here than here. So that's the [INAUDIBLE] first [INAUDIBLE] check the reciprocity is satisfied. Now I want to calculate the solar constant. I said the 1,366, [INAUDIBLE] check. Radiation reaching the Earth from the sun, so this is the blackbody from the sun definitely by approximately the blackbody, times the surface area that is the area, a s, times the view factor from the sun to the Earth. So that's the total radiation from the sun reaching the Earth, intercepted by Earth. And I can use the reciprocity to get a per unit area. Because the a s times the se, a s, se, [? fs ?] ae versus ae Earth, [? fe, es. ?] So when I do per unit area, I normalize, I use the reciprocity, I will actually write this as the sun emissive power times the Earth to sun-- here it's sun to Earth, but the area gives me the Earth to sun view factor. Oops, I forgot this number. So this is a step, of course, [INAUDIBLE] minus 8. I was distributing an area. Sorry about that. Minus 8, 10 to the fourth power, that's this side, 5.67 10 to the minus 8, 577, that's the sun temperature, times this factor here, [INAUDIBLE]. I did the calculation. What I got is the 1,365, pretty close. And solar is constantly integrated. It's not exactly blackbody. So this is why some people say the equivalent is 5777. Solar constant is 1,366. Now let's look at-- so that's the energy coming from the sun to the Earth. What's the maximum efficiency I potentially can get? What's my best [? luck? ?] If I can convert the solar energy into mechanical or electrical energy, the maximum using thermal process, thermal energy. Thermal energy, of course, is limited by the Planck's law-- or no, the [INAUDIBLE] efficiency. So you can always say, it's the Earth's temperature and the sun temperature. You drive a power efficiency between the 2, and you will get 1 minus 300 divided by 5777, which is about 9%. That's an upper absolute, absolute possible [INAUDIBLE] you can somehow [? tap ?] into the sun directly. Of course, we can't put anything in contact with the sun. And so, let's dream up this process. I have a blackbody of the sun very close to my absorber. Let's say, this is the-- I have a very close, because in reality, it can be close. But I can focus it, focus the solar radiation. And it turns out that if you do the full version, and the maximum you can get is something like the solar temperature. So the blackbody at the sun, and the absorber, if I absorb all the solar radiation, so here the absorber is also a blackbody. And then say, I discovered a t, and then I put a engine in between this absorber and the ambient and the output. And then I can imagine this engine is a perfect car engine. So what will be the efficiency of such a, say, system? And first I want to see how much heat can go from the sun to this absorber. Here, I'm doing to blackbody, so blackbody radiation is from sensing my T fourth power. You can see, now I don't have this very small view factor. I'm just doing two parallel plates. So sigma T fourth area is an [? no ?] area difference anymore. Sigma T fourth power minus this radiation, now the absorber will radiate it back. So [? minus ?] T fourth power. So that's the heat transfer from the sun to this absorber, maximum you can get. And so, that means the thermal efficiency, what comes in, this order, this sigma T fourth power, and the amount of heat transfer that actually should happen is T fourth plus [INAUDIBLE] fourth must T fourth The thermal efficiency to the 1 minus [? come ?] to the fourth power ratio. So the higher the temperature the absorber, the lower is the thermal efficiency. And now, [INAUDIBLE]. So the power efficiency is here. And in fact, as I move on, you should ask questions. Because what are the questions, what if I use selective surface? Why you put a blackbody here? And this will be related to what kind of concentration, how you concentrate. Because here I'm really putting the maximum temperature. Really, this is the sun, this is blackbody, and they are facing each other. And that will require you to put your maximum concentration. And in the case of maximum concentration, the emission of this surface becomes not important. It's really the absorption that's most important. Absorption, the higher the better, the best is 1. So you get a blackbody. That's why. So now I have a thermal efficiency, I have a power efficiency, and the product of the two gives me the maximum-- the combined efficiency. So this is the thermal efficiency. This is the power efficiency. Of course, the power efficiency, you want to operate your engine as high temperature as possible. The thermal efficiency, as you operate the higher temperature, radiation loss increase. So you have a maximum here. So this is the maximum efficiency you can get about 85% of efficiency out of this. And this happens around 2,450 Kelvin when you have [INAUDIBLE]. And what's interesting is that, actually, there is a pretty wide range. The efficiency, this is actually really good, pretty flat, relatively flat. And so, if you operate, if you think about here, it's only about 1,000 degree. Here is the 4,000 degree. And this is a 7%. Pretty good if you can get there. And we know the solar-- the coal plant is only 40%. The combined, that's the coal burning plant. Coal plant is about 40%. And if you combine gas turbines, [? and a ?] combined cycle, combined gas turbine with a steam turbine. So you burn fuel, so you generate hot gas, and gas profile turbine, say, a gas turbine and exhaust generate steam. Steam drive a steam turbine. That combined about 60%. That's the best you can do with the fuels. And an internal combustion engine, the car is about 25% to 30%. So that's the-- so this is a pretty optimistic value here. So if you put your solar-- if you put your thermal engine between the using the heat from the sun. But of course, there are many limitations. Now, the next question I want to ask is, if I put the material here, under the sun, how far they can get? What's the maximum temperature you can get? And I could imagine a case where you could potentially get to a very high temperature. And this is the-- radiation comes in, you will be up [INAUDIBLE]. I don't think about the direction. Let me just say all direction the same. And again, this is, because of that, maybe you have ways to do better, I don't know yet. So radiation comes in. These absorptivity of the surface is a function of lambda wavelengths. And then, also, the emission, you will re-radiate. So absorption, if I put the [? rationale, ?] I know the Kirchhoff's law tells me, absorption equals emission for wavelengths at least. At least if I say the rationale one is the same. So it's all diffuse, you can imagine that. So when I do this, I will have the absorbed energy, this is the-- you can see, this is the spectrum from the sun, and the times absorptivity. [INAUDIBLE] I have integrated the [INAUDIBLE] angle and integrated over d lambda. So that's all the energy comes in. And then, this is the energy that is re-radiated. One is the surface radiation. In [INAUDIBLE] times blackbody emission, this blackbody emission [INAUDIBLE]. But also the ambient is [INAUDIBLE]. So you have a [INAUDIBLE] change between here and the ambient. So this is the radiation to the ambient. t should not be t, a, t0, ta. And here I have concentration. As I said, this may not be-- I think this probably is not the maximum. Well, as I talk, I'm just making comments. Because the radiation from the sun is in a very small angle. It's almost lower-- you say lower incidence. And when I write this emission, I am writing the emissivity isotropic. I could potentially design a surface emission only also in the same angle where the sun has. Then I have to do angular integration. That probably will go even higher temperature than this if I just make that guess. So once the-- in terms of how I-- the spectral characteristics. So if you look at, this is the radiation from the sun. And this might be the radiation emission from [INAUDIBLE] the surface. And at each wavelength, so you can design your selective surface. Ideally, you can design [INAUDIBLE] if you could. Seriously, [? no, ?] seriously, you [? cannot. ?] But it's hard to do. But let's just say, imagine I have that [INAUDIBLE] surface that I have a sharp cutoff. So below certain wavelengths, all the radiation is absorbed. So that could be the solar radiation. And then, beyond that, all radiation is rejected. So that's a spectral idea 1, 0 spectral properties, selective surfaces. And so, if this selective surface is a 1, 0, and I can look at, if I design the selective wavelengths here, if you look at it, this blackbody from emission reduces rapidly. So if I make my wavelengths here, this radiation, Kirchhoff's law-- and now I have to remind you of Kirchhoff's law. When I see this spectrum, this absorption is 1, I also mean the emission is also 1. I can't violate that. So this radiation will be going out. This blue in this region will be emitted if I absorb everything from the sun. That's the second law tells me. So if I have-- I check this wavelength. You can say. Now I make the wavelength shorter, there's less emission from the blue curve. So here is almost 0. But there's still a fraction of the solar radiation that comes in absorbed. So I want to do my energy balance and say, what's the maximum temperature I can get just with a flat surface, one [? sun? ?] So c equal to 1. And I do, rather than do a blackbody spectrum here, I was trying to do this by hand. And then later on I found that I still couldn't solve the equation. So I get [INAUDIBLE] that came into my office. He has a code that you possibly [INAUDIBLE] you'll see later on. So here is the radiation from the sun. And I say absorption is 1 becomes 0 lambda. So this is 1 times whatever comes from the sun. So view factor from the sun to Earth's surface, the surface area of the sun, blackbody emission of the sun, t lambda, that's what comes to the Earth's surface. And this is the emission radiation out from the surface. So assuming this surface is facing the sun directly, there's no angle [INAUDIBLE], and emission out and the emission to the ambient, [INAUDIBLE] ambient. And of course, let's say, I've used this trick before. This area times view factor is the same area on Earth times the view factor from Earth to the sun so that I can cancel the area. So it's the independent of area here. I can cancel the area. And that's why I was doing the universal blackbody curve. So if I do my universal blackbody curve, this is the radiation between the sun temperature-- I mean, wavelengths normalized to sun 10 to the fourth power. This is the function I want to find the time temperature. I thought I could do this by hand. Again, this function is not [INAUDIBLE]. I can't-- you have to do a [INAUDIBLE] to solve for the [INAUDIBLE]. So but this is definitely solvable. You can go to solve the equation. You can even take the real spectrum to solve it rather than doing blackbody spectrum as I'm doing here. I've written down, in fact, that's what [? King ?] has done. This is the curve a m 0 condition. And if you choose your wavelengths, it's interesting that the shorter the wavelengths you choose, the higher the temperature you get. That's because the loss is smaller. You look at it, you go to shift to small wavelengths, you absorb the radiation, but you lose very small. Because this curve, once you go to very short, this curve is always below the blackbody curve of the re-emission. And so you-- but see, isn't this amazing if you have that surface here? AUDIENCE: My old house, I think I bought my [INAUDIBLE] something. GANG CHEN: [INAUDIBLE] AUDIENCE: No, it's black. [INAUDIBLE] Yeah, no conservation, and no, let's say, no optical treatment, no filters. GANG CHEN: Right. AUDIENCE: It's definitely not 600 Kelvin. GANG CHEN: No, no. AUDIENCE: It's 81 Celsius. GANG CHEN: Yes, what is that, is it a flat panel? AUDIENCE: Yeah. GANG CHEN: Flat panel? Yeah, flat panel, you got a lot of losses. Firstly, the surface is not bad. The [INAUDIBLE] industry has done a pretty good job. And you can buy-- I visit-- this is the part of the things I'm going to talk later. You can buy commercial this hot water surfaces, have a 95% absorption, 5% emission, not a [? shock ?] [? product. ?] We'll have an example later on. And another flat panel, it's not a vacuum. You take a vacuum tube that the channel use, you put it inside, you go to 200 degrees Celsius. That's when you don't have water inside it, just dry in [INAUDIBLE]. Yeah, so this is the-- I guess the point is, if you can design, in here is-- if you recall, for thermal engine, 85% efficiency range, or 70 to 85, you need 1,000 to 4,000 temperature range, theoretically. And here, it doesn't seem to be that hard if you've got that surface. Of course, that's a really ideal surface. If you can get it to the 1,000 degree, even with one sun, [? flat-- ?] sorry. And here is another example. [? In ?] 1.5, that's the [INAUDIBLE] Earth. And again, [INAUDIBLE]. Now, for those ideal situation, you can do almost-- this doesn't give you any power. What we did is just to say, it's something equal to the loss, no power. So now let's look at the more realistic case where this is still-- the surface is ideal. But I don't optimize the temperature, trying to reach maximum temperature. I try to say, given a temperature of [? operation, ?] what's the-- I choose the wavelength. So the way I choose the wavelength, if I say, this is the curve here, then what radiation come in from the sun? And at this wavelength, the blackbody, at this temperature, the blackbody intersect the solar radiation at this temperature, at this wavelength. So I cut out this point. And then, below this wavelength I have 1 above, I have 0. So the difference from the previous discussion is, previously I say that each wavelength what's the maximum temperature? And here is, I gave a temperature, I choose my wavelength. And then I say, if I operate this engine at this temperature, what's the efficiency I can get? And so, this is the, again, the same picture here other than any of this [INAUDIBLE] and the blackbody, two blackbody working together, I'd say I'm looking at the one sun [INAUDIBLE]. Again, like what I said. Now, in this case, the thermal efficiency is what comes in at the total, and then this part is lost. The blue curve part is lost. So if I calculate the thermal efficiency, and this is the [INAUDIBLE] mechanics calculation. And as you-- here is the thermal efficiency. So the loss, this is a larger-- the blue-- the purple area is the-- or dark area, this is a 0.9 [? high ?] thermal efficiency. So naturally, if you want to have a good thermal efficiency, if you want to [INAUDIBLE] out [INAUDIBLE]. So you want to operate, again, at a low temperature. You go the low temperature, [INAUDIBLE]. So that's the-- but of course, you get a 0 power efficiency. So you still got the more optimum. So you multiply this to the power efficiency, and this is the transition wavelengths are here. And so, this is the-- in this case, what you have is 0.5. So still pretty ideal, pretty good, very good if you can get even half of that. Because 50%, let's assume you have a thermal engine, a power engine operating between, in this case, let's say, here is a 50%, it's a power engine operating between 800 Kelvin to room temperature. And of course, if you have a real efficiency, [INAUDIBLE] that's the power of-- so most likely you're shifted to lower temperature. This is actually a real selective surface. With selective surfaces from [? sun ?] [? select, ?] this is the emissivity, the absorber in the solar region, and in the IR region. Problem is, nobody could make this really sharp I think. And it's a transition. There are always a transition, the sharper, the better. And so, if you use this surface to do the calculation, real surface to do calculation, and then, again, assuming a [? power ?] efficiency. So here is the system efficiency now becomes about 30%. And the operator, you don't want to operate at high temperature. You can see here, the [? upper ?] one is actually pretty low temperature, except that you need a device converter that work by the power efficiency here. If you recall, the solar cell, [INAUDIBLE] sharpening [INAUDIBLE] paper, [INAUDIBLE] limit for single junction solar cell, silicon solar cell, the maximum efficiency is about 28%. Now look at this. Yeah, this is very good. And of course, solar cell is also all thermal energy. It is limited by this efficiency, power efficiency of whatever limit that we're discussing here. Except that the emission [INAUDIBLE]. So not this one, I'm sorry. The emission, solar cell do not have a re-emission problem. So it's a matter of the what's above the bandgap times the-- the current efficiency limit of solar cell is the solar cell is itself operating at low temperature. So the current efficiency limit is really from the sun temperature as an [INAUDIBLE]. And if you do-- it turns out that if you do infinite multi-junction solar cell, the maximum efficiency is still that 85%, same upper limit. So those were-- the previous discussion were done for the concentration in first one, one [? sun ?] condition. And you can add other concentrations and the-- use various kind of absorbers. And this is the-- we have a blackbody absorber, these different [? conservation ?] [INAUDIBLE] [? concentration. ?] And that's the temperature of operation here is the efficiency. And now you use a more realistic absorber. The blackbody is [INAUDIBLE] overall spectrum. Let's say here is a selective absorber. Let's say you get the higher efficiency. And if you compare these two, you also operate at higher temperature. So here you got to operate at a higher temperature. If you selectivity do it, you can push to a higher temperature. If your selectivity is not good, spectrum is not good, then the blackbody, you don't want to operate at high temperature. You want to operate the low temperature because you're losing too much heat. So the solar thermal energy is always the fight between the losing heat by radiation and the [INAUDIBLE], the thermal energy you want to operate at higher temperature, your surface you want to operate at higher temperature, so optimum there. And now, if you go back to the ideal absorber, ideal absorber means 1 and 0 selectivity spectrum, you push to higher and higher temperature depends on your concentration ratio. So those were generally [INAUDIBLE] I think the messages. If you can do good spectral control, you can get the high [INAUDIBLE]. So that's to say, spectral control is really a critical problem. And we're [? finding ?] out, doing Department of Energy Research Center, we have two parts. One is we look at the, for example, [INAUDIBLE]. The other is really look at the spectral control. And in fact, later on, we talk more to thermal [INAUDIBLE]. And you realize that you're transforming one difficulty into another difficulty. In solar cell, the difficulty is the single-junction cell is limited to sharp [INAUDIBLE]. So you try to do multi-junction cell, which people have reached about 41% three-junction. But say, infinite junction you can get to 85%. So but the growing material to more than three junction is very difficult. And now we transform one problem into one-- this difficulty into another difficulty. If I can have a selective surface, really ideal, that work at a high temperature, then I can [INAUDIBLE] get high efficiency, even with single junctions. So but of course, this has now a lot of engineering aspects. Because let's suppose, when I have this ideal surface, selective surface of absorptivity, emissivity, on the other side, that you can probably design a surface of really 0 emissions. So this side I will lose radiation. And also, you have to hold it somehow, 1,000 degree, 2,000 degree, you have to hold it. And of course, you can use, let's say, a suspension. You can use this thing called-- this is actually, if you do the solar in that, this is the one, and the one that I consider solar. If you do with a combustion chamber, it has to [? gas ?] contact [INAUDIBLE] or a very sturdy structure. At least in the case of the sun, what comes in, let's suppose this is a vacuum chamber. What comes in, then it goes-- photon can go through the glass. Today you have much less in terms of the complexity in dealing with the [? top ?] gas, the physical contact. I think you can use a really-- say, very small [? stream ?] to suspend your surface. So that's one advantage, that [? view, ?] which is probably better, they do the solar compared to thermal source. So here is the suspension. And of course, if you have suspension, then you have heat loss. You have to account for other heat loss, suspension, and other emission losses. And so it's a, say, then becomes a careful engineering problem. That's why at the end, now even you work on the [INAUDIBLE] thermophotovoltaic, you have to go back to basics on how the conduction, [? key ?] leakage through-- any [? key ?] leakage [INAUDIBLE]. --degree making factor for your system applications. Any questions? |
MIT_2997_Direct_SolarThermal_To_Electrical_Energy_Conversion_Technologies_Fall_2009 | Lecture_9_Solar_concentration_and_solar_thermal_technology.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. GANG CHEN: Today, there are three more lectures. And we get two guest lectures on one of the reliability of the materials. I prepared this original and the last lecture. I also have to juggle a lot the schedule for everybody. So instead of what I will do is next Friday, I will be talking about the reliability of the materials. This is Professor Christopher Schuh in the Materials Department with-- has been doing research in this area for long time. [INAUDIBLE] I forget it. And then the week after last week, we will have Dr. Ivan Simonovich talk about selective surface. He has been working on several photovoltaics. And so he spent a lot of work in the spectral control area. So help you, again, better talk to me. And then see the final lecture, we will be-- the content, come back a little bit to talk about how we combine selective surface with thermoelectrics and with also photovoltaics. So we have learned thermoelectrics. We have learned photovoltaics also. So that's a solid state solar thermal energy conversion. And today, what I want to cover is the mechanical energy conversion, solar to electrical, solar, thermal but there will be mechanical means. So let's recall what I was talking in the last previous lecture. We talked about blackbody radiation. We mentioned a little bit on the Earth's motion and how we usually test solar spectrum AM0 or AM1, AM1.5, et cetera. And then we define relative property, this definition, many different definitions. So as I said, I don't expect you remember all those details. The point is for spectral control, it's very important to understand the wavelength dependence of the properties. And then with that, we discuss the maximum efficiency of solar thermal engines and maximum achievable temperature. And again, see we emphasize the silicon surface. But we do know or you do not how to do it yet, but it's important. So in today's lecture, like I said, I want to cover, really, the solar thermal mechanical part and what we'll start with is the hot water system, solar hot water. And then we'll talk about-- to bring it up to a higher temperature, we'll talk about the solar concentration. And we'll talk about the maximum, how much you can concentrate the solar radiation. And then we look at how you could in practical to do different ways of concentration. And to do concentration, you need to track the Sun. That's one of the problem challenge. So I'll talk a little bit tracking and long tracking and then go to mechanical. If I have enough time, I'll start a little bit, give you a preview of what controls the surface properties, the action of microwave. So solar hot water, right? It's very mundane, but I really appreciate this. As I probably mentioned before, when I was a kid, I grew up in China. I didn't have a chance to take a hot shower. And now when I go back to my parents, they still live in China and they use solar hot water. Concept, of course, is very simple-- solar collector. They absorb the solar radiation, and you heat up the water. Very simple concept, right? And also, I want to give you a very simple backup [INAUDIBLE]. So if you're a mechanical engineer, this will be pretty easy. But I did this for the preparation of the class, also out of my own curiosity. Of course, you are from Materials, so this is probably a little bit more, say, not very familiar with you. But it's a simple energy balance. So what I want to ask is if I want to put in my own house, what's the area I live? Right? Solar hot water. And typical household water tank is about 80 gallon. It's a larger house. If you have smaller, so ranging from 40- to 100-gallon range, I guess. And if I take a start temperature of water at 15 degrees, and I want to keep the water tank at 60 degrees Celsius-- so during the day, right? Let's suppose I deplete all the hot water during the night, and I want to store 80-- during the day, I want to heat up the water from 15 degrees to 60 degrees. And so you want to see how much was this solar surface area, hot water surface area you want. And it's simple energy balance, right? During the day, how much solar radiation is it. And, of course, you consider the efficiency of the absorber and then that balance to heat up this much water, how much energy you need. So you do that energy balance, the area, what I want, the solar radiation insulation, and time of the day efficiency. And of course, the heat capacity that's stored in the hot water. So I put in those numbers. Typically, when people calculate, you can say it's a little bit optimistic. If you live in Boston, you don't get the 1,000 watts, OK? And I took a standard calculation, 1,000 watts 5.5 hours. So that's an average. That's the whole day average. Typically, that's what people take when they test the solar cell. 1,000 watts, 5.1, we calculate solar cell output. And efficiency of solar thermal hot water system ranging from-- I'll tell you how to estimate efficiency, but I take a typical number, 60%, OK? So I need a 5 meters here. And in China, I know it's very cheap. And typically, people install somewhere 1.5, 2 meters. So it's not a very big tank. Most people live in apartment, and they use less water. OK. So I have ordered a different format. One of the format is a flat panel, right? So here, what we have is the absorber. Typically, it's a frequency selective. Meaning, they absorb solar radiation but do not or you want to re-emit a very small amount of thermal emission when it heats up. And so you can see the back side is the insulation. And this is the absorber tube, could be up on top or on the bottom. And typically, they do not-- so if we change it here. OK? And the tank could be on the top of the absorber itself. So in that case, you don't put a tank in the basement. And you can use the gravity of the water to drive the water flow, to drive the hot water down. Let's say the problem is your water flow rate depends on how high you put the tank. So if you live in low buildings, your pressure is low. And this is an example. Clearly, the water tank is not on the roof. It's in the basement. And in that case, you use a pump to pump the water. So it's a pressurized system. So they are pressurized. And this one is not a fossil system, just natural. And the one that-- what I thought was really interesting is this vacuum system. And this is predominantly used in China. And it's glass-- so it's a double-layer glass. Inside is a black coat. What looks black is actually inside. Well, outside is another glass, clear glass. So it's a circular. And the inside glass is first coated with selective surface. And then you evacuate it. You see on the other side. So it's a double jacket. And on this side, because of the vacuum so it has a getter. The getter, what does getter do is, say, the evaporation is actually collected by this metal reacted with whatever evaporates. And so you imagine so putting vacuum under the Sun, how long it lasts. And I actually visited some of this as a-- they really is the father [INAUDIBLE] the father of this technology in China. And he said that he put this stuff early in the '90s in Tsinghua. He's from Tsinghua University and still working. So after 20 years. And that's conceivable because if you think about in my house, I bought a TV. I remember my first TV is still working, about more than 20 years. And that's a CRT. It's a vacuum technology, right? Say the screen of the TV, so the cathode ray tube inside is actually ultrahigh vacuum. So the vacuum is pretty high. And you can do a calculation of 5 to 1. If the vacuum is too low, the air conduction is too much heat. And so this is an evacuated tube. That's glass. This glass is really cheap. I visited the manufacturing site and the whole tube is about this high. The diameter is 50 millimeters. And the manufacture of the double glass tube is about $1.5 per tube. So it's really cheap. This one is a little bit more expensive. It's not a glass glass. And what it has is a metal absorber. So it could be a copper or aluminum. Again, this is a cost, right? This is really the-- see, the cost is crucial here. And this is selective absorptive coatings again. And so what it does this is a heat pipe. So the heat absorbed conducts to the heat pipe. So the heat pipe's diameter is not this big. That's too much copper, right? It's a small diameter heat pipe. And so the heat is evaporated, the liquid inside the heat pipe. And on this side, you have forced water, heat exchanger, forced water take the heat away. So this is one of the cheap forms unpressurized. So what it has is the double glass layer. You fill the inside of the glass with water. I go back here inside, this is empty. So you fill with water. And when it heats up, the natural convection, so the hot water goes up here, and the cold water flows down. So it's a passive system-- no pump at all. And you put on the roof. And then you can take a shower. So a typical family in China, probably, is about 15 to 20 cubes. And this one is a forced pressure, right? You put a separate tank, you put the pump through and forced cooling at the end. OK, so that's what the system looks like. Again, out of my own curiosity, I say, OK, I'm going to estimate what's the efficiency of the system, flat panel and vacuum systems. So this is a model for a vacuum system, where I have a cylinder inside the cylinder, inner cylinder. So this cylinder, there is one-- so actually, this is a self-tracking, right? No matter what Sun direction is, there is always side wall is facing the perpendicular to the Sun. So does have some advantage here. In fact, there is a startup company called Solyndra. And they are putting solar cells like this. So circle along the circle of the cylinder. And it's a vacuum in between. So incoming solar radiation is just the cross section area. That's like capital-- the outer diameter. And say the insulation, solar insulation. So absorbed, how much is absorbed? In terms of absorption, the first thing you have to go through is the outer glass layer. So that's a transmissivity [INAUDIBLE]. And then inside the surface is absorbed. Alpha is the absorptivity. So that's the absorptivity to solar radiation. Now this is a disadvantage here-- the radiation loss, right? Radiation loss, because the whole inner tube is heated up, so the surface area is actually is pi times the inner diameter. It's not just the cross-section flat. Compared to flat, it's only the flat area. Now it's the whole surface area. And then that's the emissivity And I do a simple blackbody calculation of the surface. So what I did not consider is the multiple refraction, the surface emission, I just say, OK, everything goes to the environment. And in reality, the sum of those cannot go. In fact, the glass may trap some of the radiation back. The reflection and also the glass is non-transparent in certain wavelength range. So with that, I can say this is the absorbed radiation minus loss. The difference is what I actually get into the hot pipe, and then divided by what comes in. So I have my efficiency. So this is a simple, again, back of envelope for fun. Yes? AUDIENCE: I have a question. So if the tube is treated [INAUDIBLE] I think only the 1/2 the total surface get the solar radiation? GANG CHEN: That's not-- AUDIENCE: [INAUDIBLE] reflector. GANG CHEN: That's the 1/2, right? It's cross-section area. But the emission is pi D. So that's a disadvantage. OK? And so that's an idea. I'll show you the curve later on. Now I'm going to look at the hot water flat panel type. So a flat panel type in between the air. I was actually curious whether people use double glazed windows here or not. And what I found is since they do not use it, so it's just one layer of glass. So this is air and selective surface. And of course, if we think about heat transfer, this absorbs solar radiation and then will conduct through the air, conduct through the glass and outside the lateral convection. So you have the resistance in series, conduction air, conduction resistance, glass resistance, convection resistance. So that's one part of the heat through conduction and convection. And backside, the thermal insulation can still leak heat. So backside, I have conduction. And then I have radiation goes through directly. So I have three parts for the heat transfer. And this is my simple circuit. The solar radiation come to the selective surface. The energy now, one part is going directly through radiation. And the other part goes through the air-glass convection outside, and then the backside, the insulation. OK? And I put the resistance because this is assumption if you're not the mechanical engineer or didn't take heat transfer before, this is probably less familiar. But this is similar like a circuit, OK? Thermal resistance. You do the resistance calculation for air, the thickness divided by thermal conductivity. So the gap thickness divided by conductivity divided by area. And they say if you do that, why don't you use thicker? And the problem is once you [INAUDIBLE] start the circulation, convection starts to happen. So you look at the double panel window in the typical window, that thickness is not-- it's a few millimeters thickness. So you don't use too thick here. So here, I took a 10-millimeter. And then, say, glass, glass is a-- few weeks ago, there was a talk, somebody is going to install a flat panel system in one of the MIT building. MIT, I don't know. Boston? OK. I was surprised. The cost per was just a surprise for me because that's like a 13-, 15-year return. I don't understand why, but let's say it looks like they use a very tough glass. This is a steel. What's the glass like? You harden the glass. You can walk on it. But anyway, but glass resistance is very small. It's really air resistance, convection resistance. Those are the resistance that cost the total. That's this side, the resistance, and then the insulation resistance on the back side. So I put all this lumber together and did the energy loss, again, considering all the release. So I have radiation loss. I have the conduction loss in conduction/convection loss. And with this loss, I can, again, go calculate how energy comes in, absorb, mass loss divided by what comes in. And that's what I have. OK. Those numbers, I assume, for example, in the next graph, I assume 95% absorption, 5% emission. This is what the industry can do. So we have a research group on selective surface so we can see how we can do better. Industry can do that at the meters per second, selective surface-- 95% absorption, 5% emission. And this is how water is, of course-- the typical operational temperature is about 70, 80 degrees. So they do not operate at very high temperatures. And then when you go to higher temperatures, if you can go 500 degrees, still keep this, you will be the best in the world. It's very hard to do that. OK that's the curve I have. That's the vacuum system. And this is the big flat panel system. And efficiency depends, of course, the temperature of the water. Essentially, it's the temperature of the selective surface there. And I went to the web, and I found this. Depends on which website they're looking at, people have different curves. But I found the one that I thought was closer to what I have. So this is one company that a vacuum system that compared to a flat panel. A flat panel goes to almost 0, I think probably, yeah, that's probably too small. But solar thermal is efficiency compared, of course, to the photovoltaic is many times better. And so in terms of if you look at what has been used, this is a chart I got from this publication. And here is this side that's solar thermal. That's here. It's compare all the renewables. Solar thermal here is one. Here is geothermal photovoltaics, solar thermal power plant, ocean tidal power. And you find out that the total capacity, and the red is actually produced energy. So you can see the solar thermal is actually a pretty large part of how people are currently using renewable energy, OK? Just heating up the hot water mainly. And this is in terms of technology. The vacuum tube is dominant because China-- and really China use it. And flat panel, 32%, unglazed [INAUDIBLE] collector and air collector. So this is where it's being used. You can see that China is off the chart here. And it has about 100 million square meters installed. I don't understand why we didn't use them here. But I think I see Hawaii now has legislation, if you have a new house built, you have to put in hot water system. Tim Young from Hawaii, right? You can go back, check this. [LAUGHTER] So I was once talking to-- every time I have a chance, I always ask the people what-- and also, price gets very high. That's probably because in China, your typical household is $300 to put a thermal system rooftop. OK, so that's a thermal. Now we want to go to higher temperature, right? Solar thermal is 80 degrees, 100 degrees. And we want now go to higher temperature. And to get a higher temperature, you need concentration. And now we need to look at the concentration. Let me remind you, I showed this before, the Earth itself, see the orbital is not just a perpendicular of this plane. There is a 23.5 tilt of the Earth, right? So on this side, this is tilt. On this side, the same tilt. That title is fixed. That's why we have summer and-- oh, the wrong direction. We have summer and winter. OK? And the degree suspended of the Sun by, say, if you view from Earth, the degree is 32 second. So that's about 0.5-- so that's about 0.4, 0.5 degrees. And now the question I want to ask first is, if you do concentration, how much you can get. What's the maximum concentration, right? So we can look at theoretically, what's the limit? The maximum theoretical limit is I cannot heat up an object hotter than the Sun. Right? AUDIENCE: [INAUDIBLE] GANG CHEN: Now, so that, you all agree. But if I ask you, can I concentrate radiation even with the intensity hotter than the Sun? And not hotter-- the intensity higher than the Sun? Is that crazy or not? Huh? OK, you'll see later on. OK, so first, we have to look at the first or just look at energy balance. If I think about the spherical radius from the Sun and the area of the sphere times the energy flux and area of the-- see, it's a constant. Once the energy leaves the surface, the larger you draw a circle, the less is your intensity. So the intensity of this Earth, you think of a circle here, that is the capital R 4 pi R squared times whatever insulation outside the Earth's atmosphere. So this is the insulation outside the Earth's atmosphere. This is the intensity on the surface of the Sun, radius of the Sun. So that's the energy balance, conserved always. Now, of course, you say there are something in between scattered a little bit like-- assuming the space is pretty much all empty, right? So now let me concentrate. If I concentrate this concentration ratio and this intensity on the Earth's insulation, and this is the energy I put in onto an object that receives this concentrated light, right? So energy balance will tell me in this case, again, the object should be blackbody, actually because you want to absorb every drop of solar radiation. And because you absorb the solar radiation, you also reradiate. So you want to absorb maximum. You also reradiate maximum. So it's a blackbody. So actually, I put the-- so the energy absorbed, the balance the energy reradiated, I do perfectly insulation. So this is per unit area based. This is per unit area based. So that's the energy balance for the object that receives the concentrated light. And of course, as I said, my condition is that this object cannot be hotter than the Sun. Right? So this is the concentration. And from here, I get my maximum concentration happens when this is equal. And that's the solar flux intensity divided by whatever [INAUDIBLE] in the Earth. So that's R over small r based on my energy balance. And if you look at R small r, and this theta here, it's half angle sustained to Earth. So that's a sine theta squared. So that's my maximum concentration. And if you plug in the number, I said the 2 theta is a 32 seconds. So you plug in a number, that's about 46,000 times. It's the maximum you can get. It's not the maximum. People actually have achieved higher than this one. OK? This is because the emission inside the object is proportional-- I have one mistake here. Should be proportional to n square, not proportional to n. OK? So the emission inside object is proportional to n squared. So if I have a medium like a water or glass, the intensity inside could be higher than the vacuum. Blackbody intensity, n squared. n in glass is 1.5. So square is 2.25. And if you use a silicon, the n is about 3.5. The square is about 10, right? So if you do that again, your limit is the temperature limit is not the intensity limit. Your temperature of the concentrated surface cannot be higher than the temperature of the Sun. You do that, you get your maximum is n squared sine theta squared. So that's higher than the previous one. And indeed, this was achieved in the lab. This is Roland Winston. I'll talk about this in more detail. This is a compound with the focus and non-focus, and they have achieved 56. So there we go. OK? So that's from the second law perspective, your absolute upper limit. And now let's see how we can get there. OK? And I think I'm going to use a lens. And this lens, if I think about a lens, so all parabolic trough, we know that you have [INAUDIBLE] comes in, you focus on one point. So the question is, What's the concentration limit if you-- those are the image formation, right? Using a lens or using a parabolic trough, you have a focal point. That's a focal point. This is the image. So the image forming optics. And remember, the Sun sustained angle. That's the k, right? Even though we say solar radiation is a part of light. But when you want to do focusing, that angle becomes important because previously, you've already seen. So what I want to think about is if I have either lens or parabolic trough, say the opening here is D, and the focus here is a small d-- why is it big D, the other small d? And I want to find the ratio. OK, let's look at it. So if I do 2D, this one is a flat panel. So if you do two, you have a different result. But here, I'm a flat panel, and this is the half of it. And now I look at-- this is a vertical-- I say perpendicular line here. So R times sine-- see that here is 2 theta. So this is really-- if you think about parallel line comes in this way, and that's my reflection here. And this one is my this line reflection. So that's the edge of the solar cone there. And so R times sine theta gives me this one, right? Perpendicular line. And D over 2-- so that's here-- times, this is phi. And here is phi. So here is also phi. So D over 2 times cosine 5 is all giving me this segment. So I have an equation geometry here. Geometrical relation. And then I have this R, and this R is about the same. So R times sine theta gives me D over 2, the opening there. And then from here, I can get my relation, how much focus I get, capital D over small d. I have sine 2 phi divided by 2 sine theta. So this is the maximum you will get. This is the concentration you get. And you will see the maximum happens is assigned to 2 phi equals to 1. Right? So 45 degrees. So when phi equals 45 degrees here, I have the maximum here. And that maximum is 2 sine theta. That's 107, my calculation. Maybe you can go back to check. And if you 3D axisymmetric, you square that. And that's 4 sine theta squared. The problem is now you can see this is a factor of 4 smaller than the limit, right? Thermodynamically, it's 1 over sine theta. Here, if you form an image, your limit is a factor of 4 smaller. And how you can reach the maximum number. OK. So imaging optics is a little bit too constrained. Ideally, you want everything goes to one point, right? And you want to form an image. So this is a-- here, there's another slide that I just copied from Winston's book. And if you want to focus on the cylinder, that's 1 over pi smaller than the previous slide. Now there's another way to do it. That's a lung image method. This is Roland Winston's baby. And if you read, his rationale is starting from this 1 over 4, how you can get to the maximum thermodynamic limit. And here is one where there is no simple mathematic in terms of the shape of the concentrator. But this is how he would explain it. So this is where the area you want to-- this is a flat plate example. You want to focus on a flat plate and B prime to D, OK? And this is the angle of incidence, right? So that's the maximum angle. Let's suppose this is the maximum angle that you accept the radiation coming in. If you have another angle higher than this, you will not get in. You will be blocked. So this is the maximum angle. So you take the string here. At this point, you take a string. This is the opening limit A. So you draw-- you keep this string fixed. Let's fix, and you draw the curve down. So this one will slide. So the angle of incidence equals reflection. That's what it's following. This surface gives a surface such that the angle of incidence equals angle of reflection. Of course, it's a specular surface that's being assumed. And at the limit, this one when it's here, the reflection angle as [INAUDIBLE] goes to D prime. OK, so that's the image surface. This is the image method he developed. And now we can check whether this one gets to maximum or not. So this is, again, simple geometry. I have AC, this section, plus A B prime-- that's the [INAUDIBLE] length, right? And the outer limit-- so this is the edge. You look at the edge, you'll need A prime to B plus B B prime, right? So this is the [INAUDIBLE] length. This is the [INAUDIBLE] length, and the same [INAUDIBLE]. So it's actually a very convenient and simple way to do the geometry, whether the surface is harder to express analytically. And you know, say, the symmetry A B prime, A prime B from symmetry, they're equal. So you get rid of two variables in the above equation, and your AC is this 1 times sine theta. So A A prime times sine theta, and your concentration then is 1 over sine theta. Remember, before is I'll say sine 2 phi over 2 sine theta. Now you got rid of that factor of 2 in the two-dimensional geometry. And then you go to 3D. You actually reach 1 over sine theta square. So if you put your theta angle of acceptance, right? If you just say I only accept the solar cone, theta S, when that's your maximum, you have to accept those solar within that 32 arcsecond. So that's the maximum there. But this isn't convenient because let's say if you think about your design, you don't need that high concentration, but you want a large angle acceptance. Particularly when you have also streetlights, right? If you think about you don't always have parallel light, you also have the scattered light. And typically, people take 80/20. It depends on where you live. And so scattered light is actually approximate isotropic in all directions. So tracking, if you track only the solar part, maximum concentration, anything beyond that angle, you can't capture it. So that's an AD-- that 20% is gone. And so with this, these are the other examples. And if you read the paper, it's interesting. This is when you concentrate to cylinder using the same string rule. You can even do this, and the effect, those curves show different acceptance angle theta and the concentration ratio, and also the height of the-- height versus the aperture. So this height, aperture, right? And if you look at it, if you want 3 times concentration, that's about the 21 degrees, and the height, the aperture is about 2 to 1. And also, what's important is you can never make a perfect surface reflector. So every time a light bounce once, you lost a fraction of a light, right? So this surface is never perfectly reflecting, 100% reflecting. So you want to minimize that level of bounce because every time it bounced, it's absorbed, a fraction of it. And let's say aluminum is-- you can get about 97% reflection, but that's 3%. If you have 10 bounces, 97 10 to the power, you'll get some more numbers, right? So you want to minimize it. So this is the average number of bounces we calculated for these geometries. OK, so this is the conservation, right? Now, you concentrate it, particularly when you go to higher concentration, you have to track it because only certain angles are accepted, right? So you have to go track it. So let's think about how we would-- typically, how you should track it, right? Again, think about the Earth has this 23.5-degree tilt. And in fact, I haven't figured out all this a little bit detail in the Earth's motion yet. That tilt of 23.5 is an average number, which it turns out the Earth is actually wobble a little bit. So that wobble happens between 22 to 24.5. OK? And this will be important when you do the high concentration ratios and the acceptance angle. So if you think about the-- so that's what I listed. In the summer, Northern hemisphere, and in the winter they come from-- so in the winter, Southern hemisphere get their share of the sunlight. And now let's think about the why it's easy to think of why you need the packet. So [INAUDIBLE] let's say your radiation coming perpendicularly. And when you come at an angle, the [INAUDIBLE] energy you get is times sine theta. So you don't get all the energy you could get if you make this plane perpendicular to the ray direction, right? But see this installation not only just a sine theta change. Also, if you think about the departments through the atmosphere, different angle of incidences, the departments going through the atmosphere changes called 1 over cosine theta thickness derived cosine theta direction relation. So your insulation also changes. And if you combine all this together, those are some of the places people measure per square meter how much energy comes in. And so this is the Northern latitude, 45 degrees. Anyone remember Boston latitude? I thought I was going to check, but I forgot. OK. AUDIENCE: I think 42. GANG CHEN: So we're close around this number. You can say this, not very good solution. So my calculation was too optimistic. I need about 7 or 8 square meters for my household. OK, so which direction is your track? If you have a solar-powered or a solar panel, should you render the source loss East-West? So first, this is the East-West. This is the rotation, the angle of rotation in the East-West direction. So of course, if you do only one x tracking, you want the maximum amount of energy received be your solar cell area or your aperture. Then you want to do the axis in the south-north direction. And then your rotation is the-- so that it's the East-West direction. And then say in the other direction, you remember the Earth is the tilt during the summer and winter, that degree will go from-- go back here, right? So this angle will fix your solar cell here this on the summer. And then when you go to the winter side, this side your angle changes. So ideally, you want to track both directions, OK? There is one case people have done before that I think it's interesting where you do not need tracking, or you do seasonal adjustment. And this is when you have a low concentration and people use a weight ratio. So these are the-- you put a solar cell here. Those surfaces are reflecting surfaces. It's a wave group. And now if you read weight in the East-West direction, if you think about East-West direction, those groups in the East-West, so you will use this one means you can do light here. You don't need to put the solar cell. So you can actually increase the flux here by a factor of 2. You use less-- 2 times less of area. So typically for this geometry, this group geometry, you can get, let's say, concentration ratio is 2.5 to 3 times. So you can operate in that range, low concentration range. And if you do a little bit, what they needed here is a seasonal change because the acceptance angle in the south-north direction will change. But there are also people now developing this local. I will check in there, people actually do tracking, a little bit tracking. So there are even people doing south-north direction tracking. But in theory, so for this low concentration looks like you put it in south-west direction. You still don't capture that sine theta, the solar installation. That's still will change because your angle coming at an angle, you project area. You're not tracking perpendicular there. But you use less. In this case, if you do this concentration, you use two or three times less solar cell. So you cut the cost. OK, so this what we talked about is the concentration and tracking. And in terms of PV, People do [INAUDIBLE] concentrated PV. There's effort, there are some companies, start-up companies-- I'm not sure how successful they have been in deploying. This is particularly when you think about the high efficient solar cells, like gallium arsenide, the maximum here you can get about 40%. That's very expensive. So what you want is to use a high concentration ratio to reduce the cost per unit area, so income and cost. But see, I'm not sure how successful they are commercially. And I know one of the problems is actually the heat transfer is a problem. And the other is it's very-- I think it's very important to keep the light uniform. So if the higher the concentration you have, if you're tracking is not accurate, you light goes off your cell, then so you're screwed. So that's the tracking accuracy becomes very important. Now I'm going to shift next to the energy conversion system and the mechanical energy conversion. And the most popular in use is the trough. So the absorber is a tube, and you concentrate the light. And so this concentration could be either by parabolic mirrors or the Fresnel arrays. In this Fresnel, so you can see the mirror is individually adjusted. So this is not a one, say, parabolic, but the angle of each mirror is adjusted so that the light focuses to the pipe there. It's called the Fresnel, but I think the Fresnel-- my understanding of Fresnel [INAUDIBLE] real object that's a diffraction side. But anyway, and this is a dish where you concentrate-- you track 2D. Here, you could just do 1D daily tracking, or I'm not sure if people do a seasonal adjustment or not. And here, you need a 2D tracking is a dish structure. Or you tau. You have, say, station the mirror reflecting and to the power. And trough is the most common use to look at the more cost structure where you have a parabolic mirror. And you have the pipe inside in the focal plant. And this is a typical pipe, the solar collector pipe. Again, it's a vacuum. So the vacuum is a pretty high vacuum here, 10 to the minus 3 millibar, right? And you can imagine the challenge there because the pipe itself is a steel pipe. Steel pipe. And then outside is coated with selective absorber. So typically, you can see the absorptance is larger than 95. Emittance is less than 14% at 400 degrees C. And the outer clear glass is also coated so that it's under reflection coating. So the typical glass transmission is about 93%. I'll get to 96%. And the watts-- if you look at this, what's important, how you make the seal, right? High-temperature seal is really-- so you have a tube, 100-meter long, and thermal expansion and seal. Those are the really critical technologies. So tubular trough plant, you've got collectors. You have the steam turbine, and you also have-- very often, you have to have the gas fired. So you have to do cogeneration. What do you do in the light? Or what happens if you have clouds come in? That's actually a big headache there, particularly for PV. Solar thermal actually slightly better because the thermal inertia, thermal heat capacity damping them. That's one of the advantage that solar thermal has. And you could also do some storage, I'm not sure here as storage, but you could, you could do storage. And the solar thermal plant here, the fluid itself, you can see there's a closed loop. The fluid through the pipe is closed. Or steam generation, steam turbine, similar if the water-based steam turbine. So this is the outer loop, the steam loop here. And in fact, in the pipe, they use oil. And the oil is one problem. You can't go to too high temperature. You go to too high temperature, you decompose. And so the oil has a stability. This is a steel area where people are doing research, stability problem, and how you can develop a higher temperature oils. You can get to higher temperature, but your oil will not unknown. OK, and this is another just a picture here. If you look at it, they typically installed and typical size of the plant is around 100 megawatts. 100 megawatts is small in terms of if you compare it with the coal-fired. Coal-fired plant, typically one steam turbine is one gigawatt. OK. And this is too small. You can read your handout. So those are examples, and what I want to point out here are the concentration here. It's typically, those concentration is between 60% to 80% Stirling. That's the concentration ratio. And they look at the cost of the solar thermal, that's the one case where different generations, 30 megawatts. This is a US dollar megawatt hour. So if you do kilowatt hour, you divide it by 1,000. So that's about $0.19 per kilowatt hour. Coal fired is about $0.05. power plant. So here is the $0.16, and there's some projection to scale up that will go down to $0.10. The efficiency, again, I'm not expecting to read it. Just let me just quoted the fuel efficiency numbers. The collector efficiency here is 50%, receiver efficiency, 82%, and the cycle, that's a steam turbine cycle efficiency here, 45%, right? So right now, it's a 38%. So at the end, if you combine all this efficiency here, this is about between 13 to 17%. The collection efficiency is not good. Collector efficiency, here it's only about 50%. You lost a lot of light in the collection. So there, you want to minimize the number of reflection. The other problem is your street light. If you design a high concentration, if it's not parallel, you don't fall into the focal point. Interesting to look at the cost. The cost here, if you look at it here, we have 58. That's the collection system, 58. And structure, this one does not include the engine. So you have the storage. Thermal storage is here. And we have 14% is the power block. I do not know what exactly is that. OK. And now here is the-- so this is the parabolic power plant. And again, this is a breakdown different materials, the cost of structure. And so that was kind of fun. I thought I saw it before. Where is the cycle? AUDIENCE: [INAUDIBLE] GANG CHEN: Huh? AUDIENCE: [INAUDIBLE] GANG CHEN: Yeah. OK. So that's where it is. Yeah. So the cycle itself is not the-- AUDIENCE: Bottom one is just the soil so it's just so. GANG CHEN: Yeah, this is-- yeah, this is-- so the first one is the overall. The bottom one is the field component breakdown. So the cycle itself is not a big part. The collector-- look at-- I think it was interesting. Look at this here. Metal support structure, 29%. All right. This is a big stuff here. The steel used a lot of steel. And receiver, 20%. Mineral, [INAUDIBLE] 19%. Mass support is the big one. So if you can cut that cost, that's good. AUDIENCE: Where's the engine? GANG CHEN: The engine is here. 14, that's nothing compared to the solar side. OK. So if you were in Jacob Carney's talk a few weeks ago, he gave a mature-- he does a lot of the [INAUDIBLE]. So you have mirrors tracking inside each one and reflect to the power, the center there. So it's like a big array of mirrors reflecting the light. So there you have the solar tower. And what's interesting is that this is what I learned from Jacob Carney is that the finger size-- and you put a question mark, but I think the [INAUDIBLE] size is actually smaller than 12. The tower, you don't want to build too big. OK? And a lot of-- but the temperature is much higher. So the [INAUDIBLE] side there, efficiency should be higher. And the problem is the focus. And he showed some very interesting pictures. You don't always hit your target. You have a lot of missing light. So if you look at around the focal point, you see a lot of white light that missed the scatter. And so this is the absorber, two different absorber ways to absorb the radiation. And what in this cases-- so I showed I think the last lecture, once you get a higher concentration, your absorptivity is more important. The emissivity is less important. So I think those are pretty-- it's not a really spectral control surfaces. There are really not-- I think there are only one or two [INAUDIBLE] probably has. But in the US, there are the [INAUDIBLE] junction mostly is 12, but it does have a tower system. And so here, it gives you in terms of the cost, they feel they sell 40%. Power is 14%, p2p storage, 11%, and the engine is 32. And so here is a structure, reflector. Structure, 10%, reflector, 36%. Drive for the tracking is 30%, and the foundation. So that's the tracking system itself. OK. The car efficiency-- this is from the number here-- is not much better than the-- this one is not power, I think. Sorry for that. This one is not power. This is a steel car. So that was going to delete this one. OK, the other form of the generator is a dish. So dish, you make smaller. This one is, say, tracking. And the problem is the generator here. So you have how you can make a small power generator. And here, it is a Stirling mechanical system, so Stirling engine. And if you're a mechanical engineering student here, the second year, they always do the Stirling engine. It's a burner Bunsen based Stirling engine. So the Stirling engine is very different from an internal combustion engine that's used in the car. You can heat up one side of the Stirling engine from external-- so external heating. And then you generate the mechanical motion to generate power. The cost is much higher, I learned. So I do not know anybody who is using this. It's not mature. The most mature is the 12th. And so we have an efficiency number here. Right here is the efficiency. This efficiency is the-- I thought I would try this. I don't know. It's a pretty high efficiency. I see. But I can't believe this number is 29%, 30%, 39%. And so this is the Stirling. And that's all I have in terms of the thermal mechanical systems. Any questions? Now I will go back. So this is a big stuff, and I'll go back to a simple-- question? AUDIENCE: Are these technologies too immature to talk about in terms of [INAUDIBLE] GANG CHEN: I think the last two-- well, at least the Stirling, I don't think it's mature enough to talk about dollar per watt. And I don't have exact number. I asked the people around. This is 10 times more expensive than 12. But the heliostat, I have not looked at how many deployment-- I see there was one running together with I think the Kramer junction, the 12th system. So what I want to do in the rest of the time now is coming back to-- we have too many simple [INAUDIBLE] stuff now. Now I want to come back to a little bit more math. And the reason we want to do next is, What are you going to say is a selective surface? And it's important to have the-- in the case of concentrating parabolic mirror, it's important to have less reflection. Each reflection, you lose very small amount of energy. So you want a high reflectivity. You want a high transmissivity going through glass. And once it gets onto the surface, you absorb solar radiation. You want to spectrally absorb the solar radiation, but do not reradiate. So now this is in terms of technology, some of the key technologies. And for those technology now you have to come back to the basics, how you can understand what people have and develop further. So the basics is maximally efficient. And I'm not going to spend much time on the Maxwell equation. If you want to really learn, you should take a course. But you all heard about Maxwell equations. It's a set of four equations for electric field, for magnetic field edge, and displacement and magnetic induction. Right? So I gave you the definitions of those symbols-- electric field, magnetic field, displacement. And j is the current. And they are considered relation related displacement, electric field, magnetic induction, magnetic field. And those constants, the material properties, electric permittivity, magnetic permeability. Yeah, Maxwell equations. And it's a wave equation. It's describing electromagnetic waves. So if you go to solve those waves, and because that's important, what's important-- say, when you do the particularly selective surface, very often, it turns out they are based on controlling the phase of those waves. So if you look at a simple solution, a wave has a direction of propagation with vector. And it's an electromagnetic field. So electric field, magnetic field, turns out the solution tells you they are perpendicular to each other. And in fact, they form a right-hand rule. So you go from the electric field to magnetic field to some point to the propagation direction. And to describe a wave, we have wavelengths, and here in time. Here, the inverse is the frequency. Angular frequency is 2 pi times period, say, 2 pi times frequency. Wave vector has a direction, but it also has the-- it's numerous wavelengths. So wave number is 1 over lambda. 2 pi over wave number gives you the magnitude of the wave vector. And the direction of the wave vector is the unitary vector here. And then if you solve the equation, turns out at the end that epsilon and mu combined give you the refractive index of the material. This is a property. You can look at different materials many times people have measured. Sometimes it's called the optical constant, but it's not a constant. It's a frequency-dependent quantity. So you have the real part and your measured. And since it's a wave, so the planned wave, simple solution of this wave is the electric field is a function of location time. This is the direction. It's a vector, direction of the field, right? And the wave vector frequency, complex refractive index. So that's your solution for electric field, similarly for magnetic field. And these two, e and h, perpendicular to each other. And so that's your plan with solution. This one, you can substitute back into Maxwell equation. You find this satisfies Maxwell equation. So this is the field. What's the energy? Energy is the product of e and h. And particularly, this time area average. This is time dependent, but to do time average, you can simply calculate the e cross h complex conjugate. If you do complex conjugate, you see this i omega t cancel. There's no time there. So that's your Poynting vector. And if I do that, I will get-- for propagation in a homogeneous medium, the Poynting vector is the real part of the refractive index [INAUDIBLE] exponential decay function. So this exponential decay, that's the problem silicon we said, right? Silicon photovoltaics takes the alpha is a problem. It's too small. We have to use a very thick one to absorb. And this alpha is the absorption coefficient we mentioned before, and that's related to the major part of the complex refractive index [INAUDIBLE]. OK. So this is the, see, [INAUDIBLE] inside the one medium. Now, what I can control is really with the interface to control. So you've got the interfaces. When you have light comes in, reflection, refraction, right? High school, you will learn that. And because the light can actually have two-- see, you decompose it into two directions, it's a transverse wave which become the two directions. One is your electric field is independent of incidence. So direction of coming in norm form the plane of incidence, right? So if your electric field is within the plane or your electric field is in the other direction. So we have the electric field. If it's in the plane of incidence, the magnetic field is perpendicular, right? It's always perpendicular to electric field. So we'll be in the perpendicular plane. So in this case, we have transverse magnetic wave or parallel. That's for electric field, different lengths. I can remember this. I always just say TM, then I say, ah, transverse magnetic. That's more intuitive. OK. Edgefield. OK, that's if the magnetic field is a plane of incidence, the electric field is perpendicular. That's TE. And so perpendicular and [INAUDIBLE]. So you have expressions when you look at or calculate the interface. You have to calculate for both solar radiation. It's a typical combination of 50/50. We say we calculate, and we say, oh incident is 50 TE, 50 TF. OK. Now, what happens? First thing is reflection and if you have a smooth surface, right? Angle of incidence equal angle of reflection. So that's the reflection rule. And then refraction is the angle change based on the refractive index. And then you can actually calculate the magnitude of the field reflected and refracted. And again, I'm giving you the solution. The point is, I want to tell you, those could be calculated. OK? And if you want to learn how to calculate, you need to take a course or pick up the book, read yourself. Let's say it depends on-- what it depends on is the refractive index of the material on both sides. And so you need to know the property very often, although, you think people should list it. You go-- when you need it, you don't find it. That's one. And even people list it, you do solar thermal, you want a high temperature, you don't find it. So there are a lot of things you can do. Find them. And so when we want to design, the first step, we say, oh let's design-- let's deposit a film and try to measure properties. So those are just a few of the reflection. And the energy, that's reflectivity, transmissivity. So you do reflection coefficient squared. You get reflectivity. Transmissivity, you have to plug in some factors in the front. But again, those all depends on the complex refractive index. So the final one is just one example, where it's a calculation given the refractive index is between vacuum or air, which is one. And let's say dielectric to the constant, gold also to the constant. But remember, it's not a constant. It's wavelength-dependent. So each wavelength, this can change. But here, it's calculating only the angle dependence. So if it's a dielectric, so copper is, say, 0. And here is for the TE wave, transverse electric. And here is for the TM, transverse magnetic. And then what's interesting is this crystal angle, and there are certain. So that's how people make the polarized glass coatings. And you can choose the polarization, the certain angle. You can get, let's say, no reflection at a closer angle. Gold is a very good reflector. You can look at that. Reflectivity is almost close to 1. So when people do selective surface, very often, they start with aluminum metal. It's a good reflection in the infrared. The problem is reflected up to about 0.2 micron or 0.5 micron or 0.4 micron. But you don't want that. You do not want to reflect the solar radiation. You want to absorb it. So you try to put materials on the surface to make them absorb between the solar radiation up to 2 microns. And then you still keep the old-- say, infrared is very reflecting, which is good because that means emissivity is small. So people have developed, like I said, let's say, the different coating even for the solar thermal. Solar hot water, I said they're very cheap, right? $1.5 per two. You go through the factory, you will see their vacuum deposition. I saw one vacuum deposition is about 100 meters long. Continuously deposited in different layers and do it at that low cost. It's amazing. OK. OK, so I'll stop here. |
MIT_2997_Direct_SolarThermal_To_Electrical_Energy_Conversion_Technologies_Fall_2009 | Lecture_7_Photovoltaic_cells.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. AUDIENCE: Yeah. So you have basically two semiconductors to start with-- one n and one t-- one p-type and-- p-type. So for example, on the left-hand side, you have a p-type. Right-hand side, you have an n-type. And since you have a high concentration of electrons on this side, the electrons will basically jump to this side. And you have a higher concentration of holes on the p-type. Holes will basically jump to the other side. So you will end up with this space charge that [INAUDIBLE] talked about last week. And this is basically the characteristic of [INAUDIBLE]. And I just throw them there. And I see these [INAUDIBLE]. So you have a neutral [INAUDIBLE]. So [INAUDIBLE] you have your [INAUDIBLE]. And then you have the contact [INAUDIBLE] which basically [INAUDIBLE]. OK? So you impose [INAUDIBLE] or something to this junction to get rid of all the drivers or close the circuit to generate [INAUDIBLE] power. I just talked about the open circuit voltage. So this is the first thing that basically happens when you [INAUDIBLE]. This space just [INAUDIBLE]. For [INAUDIBLE], and this basically also includes [INAUDIBLE], both in voltage [INAUDIBLE]. So you have the space charge region that's right here. And then basically, for example, the [INAUDIBLE] concentration. And so for the electron, this is p-type and n-type electron [INAUDIBLE] concentration. And so you're basically saying you have neutral [INAUDIBLE]. You have a field generator to do [INAUDIBLE]. So in case you have-- for example, a photon comes in. And we'll talk about this later at some point. When a photon comes in, in general, it's [INAUDIBLE] exciton. So they're still very close to each other. These excitons are just generally-- they have to use [INAUDIBLE] space [INAUDIBLE], but it's separate. So for this case, for example, we have [INAUDIBLE]. So basically, the electron will basically track through an electron's force in this direction. So we're having to [INAUDIBLE] to basically get separate from the [INAUDIBLE]. And it jumps on the other side. And you end up with just a positive charge here. And due to the neutral region from here, first, you have the diffusion of the electron [INAUDIBLE] two-way contacts in here. So further, so this is the carrier concentration. And then you can also draw the charge in there. So [INAUDIBLE] charge here, positive charge here. And the new field looks like [INAUDIBLE]. So if you have-- this generator decided to [INAUDIBLE] about this, or more accurately, it's called a [INAUDIBLE]. And I think this was also-- when he talked about the [INAUDIBLE] last week, he said something that-- this is important [INAUDIBLE] talk about the buyback. The buyback was just the spatial expansion of distribution [INAUDIBLE]. So he basically said this [INAUDIBLE]. So just say up to a certain point it basically drops down to a certain level [INAUDIBLE], especially [INAUDIBLE]. Up to this point, the exciton-- when it comes, it will start feeding the space charge region, and will start to separate the electron [INAUDIBLE]. And then I wanted to talk about-- so in the [INAUDIBLE] case, when we actually have a photon coming up to [INAUDIBLE] electron, [INAUDIBLE]. So when we joined [INAUDIBLE] environment since the [INAUDIBLE] have to be the same. So [INAUDIBLE] somehow. So this would be the [INAUDIBLE]. So that's [INAUDIBLE]. And see, what happens is most of the times, an electron is basically excited [INAUDIBLE]. But at the same time, first, you have the [INAUDIBLE] state. And then when the particle goes through the state [INAUDIBLE], et cetera, that's [INAUDIBLE]. And this basically changes the [INAUDIBLE]. So what you end up with is the [INAUDIBLE] will actually increase this to [INAUDIBLE]. And when we remove the electron here to end up with a hole here-- so we're basically keeping this model here. So we end up with some type of region just [INAUDIBLE]. This is basically the whole circuit board. So the difference between these two [INAUDIBLE] is due to the exchange of [INAUDIBLE]. This is basically [INAUDIBLE]. So the idea is, obviously, you could, basically, go all the way. So it's maximum open circuit. And just [INAUDIBLE]. AUDIENCE: So the electron goes through the conduction band [INAUDIBLE], right? Is it because that's easier? AUDIENCE: Yeah. What was the question? AUDIENCE: So the [INAUDIBLE] band [INAUDIBLE]? Yeah. AUDIENCE: That's a good question. I was thinking about that, too-- whether it's actually a step like this that goes-- and then it goes down. AUDIENCE: Yeah. GANG CHEN: Very good question. I think I can provide an answer. AUDIENCE: You want to answer? GANG CHEN: Well, it's a distributor. So at this point, you should not look at the spatial. So it's a locally-- it could be anywhere along that curve-- the induction curve. So I think that's a good place. I think back to the same point that I [INAUDIBLE]. AUDIENCE: [INAUDIBLE] GANG CHEN: So yeah, that's where we stopped. The last lecture is about the, apparently, [INAUDIBLE] junction diode. It's a Schottky barrier. You can think about-- on the left there's a Schottky barrier. On the right is the induction. And when you look at the current voltage characteristics, it looks very similar except that the coefficient-- because remember, the [INAUDIBLE] for [INAUDIBLE]. And saturation [INAUDIBLE] quality-- this is a metal semiconductor junction-- a Schottky junction. And in the case of a metal semiconductor Schottky barrier, the saturation current [INAUDIBLE] the barrier module. And in the case of the [INAUDIBLE] junction, the saturation current include the band gap and include the [INAUDIBLE] of hole. From the t side is the [INAUDIBLE] hole that carries the current. On the inside is the electron. So if you see an electron on the inside, you can see the problem inside. And then we have this one-- recombination time. This is, say, when you create an electron hole pair, you excite the electron from valence to conduction in the conduction band. How long they can be separated-- and after a certain amount of time, they will recombine and give off either photon, as it were, laser or light emitting-- that, you want-- or give off the general forms. So this is the recombination time for hole electron. You can see that the difference, really, in this saturation [INAUDIBLE] current can sometimes [INAUDIBLE] mathematics. But I want to emphasize another fundamental difference. Yes? AUDIENCE: Sorry. I would have thought intuitively that the longer the recombination time, the more current you would have. GANG CHEN: The longer the recombination time-- yes, let's see. Recombination is also related to longer recombination. Well, longer recombination-- smaller [INAUDIBLE] current. But I see your point, where if the longer recombination-- essentially, recombination happens in this region. OK? So I think in that case, what happens is the diffusion supply is also limited-- your current flow. So this is where I want to emphasize. The important difference between these two formulas is the Schottky barrier. When electrons go from here to here, it's electron. It's still electron. The way I draw it-- here is the electron. And the metal side is the electron. Whole problem. Now, if you look at the t junction, on the inside is electrons. And then p-type's the hole. So if electrons go [INAUDIBLE] here, electron goes from here to here, go through this space charge region, there's no more electron. You have two p-type [INAUDIBLE]. So what happens? And you have to have current continuity. The charge has to be supplied, right? So what happens is that the hole will recombine with this electron. OK? So the hole recombines with the electron. That means you have to supply both from this side. So you draw a hole from this side, recombine with whatever electron comes to this side. And this is where the recombination really happens. It's not only the space charge. It's slightly just offset the space charge. So it's a minority carrier device. That's a really fundamental difference between a majority carrier device and a minority carrier device. Minority carrier device-- recombination is crucial. And later on, I'm going to talk about solar cells. All people are trying to refine solar silicon. It's trying to increase this recombination. OK? So that's your problem. This is the point. That's really important to appreciate the minority carrier [INAUDIBLE]. OK? And of course, I mentioned-- so this is what I drew on the card. You can see electron diffusion is on this side. So the hole diffuses with the-- once it comes to this side, there's no more electrons, right? So the electrons will recombine with holes. So there's less electron flux forward. So there is more. So if I look at electron carbon, the decrease in this recombination regime, and then I look at the hole current-- on this side, the hole difference in this region. That's where the supply recombines with electrons. So the electron goes through this [INAUDIBLE] region recombination. This is as approximation. So the concept-- and then after space charge, they recombine with the hole. So the electrons diminish. Now at this side, it's just overflow. So that's how we go from an electron to a hole. So the PN junction theory-- Shockley-- let's see. The PN junction with the-- let's see. It's three people who got work with [INAUDIBLE]. One is Shockley, and the other is Bardeen. And the third is Brattain. And I think it was Shockley and Brattain who did the invention. And Bardeen was the manager at [INAUDIBLE] building. And then if you look at the history, Bardeen was one of the guys that [INAUDIBLE]. Superconductivity-- also [INAUDIBLE]. Right? So he was very-- the history of the [INAUDIBLE] invention-- Bardeen had the [INAUDIBLE] field. This was so important, he had to do something [INAUDIBLE] theory, OK? But this is really a very ingenious theory. Think about how electrons on this side and those on the other side-- by the end, the your current has to be continued. The summation of the electron current, the hole current-- that's constant, right? And in the last lecture, I also said if you look at the energy flow and when electrons right here-- yes, I would like to know what happens [INAUDIBLE] because [INAUDIBLE]. It's a matter of-- there's no hope. Again, there's no hope. So typically, contact, people want [INAUDIBLE] contact. So what happens is that when-- so the whole motion is really majority of electron motion in [INAUDIBLE] region. So when [INAUDIBLE] move this way-- so to move from here to here, that means this electron goes out to the metal. So that's what happens. And they have [INAUDIBLE] potential-- electrochemical potential. That's what the [INAUDIBLE] junction is-- sort of like [INAUDIBLE] barrier, but [INAUDIBLE] at those sites. And if you think about the counter effect, counter [INAUDIBLE] is the kinetic energy part, which is a 2 KG of the charge average [INAUDIBLE] minus [INAUDIBLE] minus ef-- the chemical energy. And you look at this-- [INAUDIBLE] minus ef is growing. Here, on the other side, [INAUDIBLE] minus ef is here. And the space charge region is growing larger and larger. So electrons carry more heat. Per charge, it's carrying more heat. It's carrying heat out of this space charge region. That's why the space charge region is actually [INAUDIBLE]. And then immediately outside space charge region, recombination happens. Recombination dumps the heat just immediately outside the space charge region. It will flow back when the space charge region heats up. The physics is it's not the junction-- the space charge that really heats up. The heat is actually generated outside the space charge here in this recombination. This is where heat is generated. That's a fact that most people probably don't know. So to finish what I was talking last lecture, the [INAUDIBLE] more research. This is the basic physics. And so we're talking about the Richardson formula, thermionic emission. So there are people doing, for example, in the-- this is the conduction band. If you don't have the Fermi level around this place, the energy electron below the Fermi level is actually carrying negative energy, and the electron above carries positive energy. So my idea is if I do a Schottky barrier or potential barrier, and I just scatter this part of the electron, that's called the energy field term, or it depends on if you have an interface. You also call it thermionic emission. So some people call this thermionic emission [INAUDIBLE] thermoelectric [INAUDIBLE] thermionic effect. The so-called solid state thermionic. But say essentially, it's cutting out the low energy electron, or sometimes it carries latent heat because the electron below the [INAUDIBLE]. And if I [INAUDIBLE] states, most [INAUDIBLE] states need sharp feature. The sharp feature helps increase the thermoelectric effect. So the way I view is, in this region, I cut it off by transport. I cut out the transport electrons. I still have the electron, but they are scattered. They don't move. So then the middle ones up here, and I can [INAUDIBLE]. So that's the way I say, OK, it's all really coming back to the idea, if you can create the sharp features in these states, [INAUDIBLE] a lot of electrons there, and you can potentially improve the electron power factor [INAUDIBLE] sigma. But see, the key is you don't want that. You only want to scatter this part. You won't affect this part [INAUDIBLE]. That's [INAUDIBLE] lot of states. I think that this idea [INAUDIBLE] fully convincing yet to me. And I made a few slides on this. That's all my own [INAUDIBLE]. I published a paper before. There's no experiment, and maybe some of you will be interested. But basically, an issue I'm wrestling-- what is the evaporation? I said that the molecule coming out of the liquid surface, that's like an evaporation. That's equivalent to thermionic emission. Let's say, how about I have a vacuum here. I got a liquid here, and the [INAUDIBLE] of liquid drop into the vacuum side area and suddenly expand. Is that corresponding to evaporation? And I have a solid state case, where I'm comparing these two examples. And these two example are-- why is the thermionic barrier, as everybody have described before, with electron [INAUDIBLE] this way. The low energy electron in this region, [INAUDIBLE] surrounding [INAUDIBLE]. The other case-- I consider [INAUDIBLE] physics-- is what about I flow electron this way and suddenly drop a [INAUDIBLE]? And that means to me it's more like I take a droplet and put it in vacuum and suddenly evaporate. And this one is more like I have a water surface and the water molecule coming out of the surface like [INAUDIBLE] water. So that's a solid state analogy that we're trying to make. So what I did is I did some energy balance analysis, which is essentially a recursive formula [INAUDIBLE]. And you can do-- you can assume the two sides has different temperature on both sides and different type of potential. The real way to do this is [INAUDIBLE]. Very complicated how to do it. And so you solve this, actually, problem-- even when I try to [INAUDIBLE] a very, very detailed, very hard computation. [INAUDIBLE] you can do your derivation. No [INAUDIBLE] statistics, energy balance, current balance at the interface. And what I found is that this is the electron temperature on this side. The electron temperatures one and two just right at the interface. It turns out that when I go down a step, there is an amplification factor. This is the potential high [INAUDIBLE], and this is the mean free path. So normally, this is very small because the mean free path time [INAUDIBLE] just a temperature drop within [INAUDIBLE]. That's a very beautiful inversion. But what I thought interesting is if I have this structure, suddenly, an electron expand-- actually reach this temperature-- or create a large temperature drop. There's an amplification factor. I can amplify how much electron temperature drop at the interface. So that's what I call potential amplified. On the other hand, if I put this structure, same analysis. Didn't have that amplification. So I follow up. I said this [INAUDIBLE] statistic. I could have Fermi [INAUDIBLE] statistics and solve some equations. And this is the equation. This is the electron temperature at the interface. And you can see you can actually drive thermally-- with a temperature gradient, you can drive electron and proton out of equilibrium. And this is the electron temperature jump at the interface. But if you do the reverse structure, there's no discontinuity at the interface. So this way, since it can help the efficiency of cooling. But so far, I have not-- like I said, this is just a theory. No experiments. Now I'll move on. That's just an example. Sometimes it's interesting to think whether you can have new ideas in this field. Now, coming back-- now we talk about the theory-- the photovoltaic cells. And this is the PN junction I just showed, and the saturation current. Sometimes, people [INAUDIBLE] current [INAUDIBLE] detector current. And now, we have a photon comes in-- generates electron hole pairs. So if the photon energy is larger than the bandgap, the bandgap is always the same. Everywhere, bandgap is the same. So if the photon energy is larger than bandgap, you generate-- lift the one electron from the valence band to the conduction band. And this could happen anywhere in the device. That depends on how vertical [INAUDIBLE]. We'll talk about that later. So what I'm doing-- OK. My photon happened to be absorbed within the space charge region, then the electron-hole pair is generated within the space charge region [INAUDIBLE]. Now, what happens after this generation? Of course, they were-- they have a tendency to recombine. And if you do recombine, your photocell is [INAUDIBLE]. You don't want them to be combined. We want to separate them. And now, if this PN junction [INAUDIBLE] the space charge region, there's one [INAUDIBLE] because electron-- now there is an electric field inside. So the electron wants to go lower energy this way. The higher energy as a whole, positive is a higher energy. Electron [INAUDIBLE] higher energy. So hole go this way [INAUDIBLE]. OK? It just go down the potential field. Now, this is the potential [INAUDIBLE] hole. This is potential field for the electrons, going down here. That's natural. And this natural tendency. So now, you've got more electron on this side than, say-- normally at equilibrium, there's no voltage. Now you accumulate more electron on this side, accumulate more hole so you generate a voltage. And once you generate voltage, this voltage will drive your diode. Just like on this side, if you got a voltage difference, you have a diode current. So now, if I have light comes in, this is the current-- the electron-hole current generated due to absorption of photon. And now, once I do the voltage, this is the current-- moves forward and balance it. Now I'm going to have voltage. I drive the diode, and this is where I have an electron-hole pair generation-- the counter going this way. And of course, in a solar cell, I want to go this way. I don't want-- I want this less than this. So that's-- if you understand that, that's a solar cell. And so this one, I want a [INAUDIBLE] short circuit current. For the short circuit, the voltage difference is 0. You shorted this. There is no resistance, then this has to equal this. No voltage drop. That's the short current-- short circuit. Short circuit voltage [INAUDIBLE] for 0. That's your current. jl is your short circuit current. Now, I'm going to, with that-- starting from there, the rest is simple. It's math. I can see what's my open circuit voltage. And my open circuit voltage is just the same. There's 0 current. So if I set this side equal 0, I find out what's the open circuit voltage. That's the open circuit voltage. And you can further simplify because we know [INAUDIBLE] to dark current. Look at-- this is called dark current or saturation current. And then [INAUDIBLE] high open circuit voltage. What do you do? Smaller is better. [INAUDIBLE] smaller is better. You look at it-- smaller js, higher open circuit voltage. So it's js-- how recombination time is longer, js smaller. So I don't want the recombination. I want the long lifetime for the electron. Basically, I want long lifetime so I can stretch it out. And you put again the js [INAUDIBLE] related to [INAUDIBLE] diffusivity recombination time, and I'll write it down out here. Basically, you substitute in [INAUDIBLE] approximate [INAUDIBLE]. That's what you get. That's just electronics [INAUDIBLE] on this. Now, of course, open circuit is useless, because there's no power output. In real operation, we always have a load. We have current flowing. So when we have a current flowing, [INAUDIBLE] power. And so before I do that, I want to show you, if you draw this current voltage-- so this is your short circuit voltage for 0-- short circuit current negative. When current is 0, open circuit voltage. So it's more likely you're shifting [INAUDIBLE] down. If you don't have a light illumination, it's a diode equation, right? That's a diode equation. [INAUDIBLE] to the negative [INAUDIBLE]. And of course, you can look at this. The power output is the voltage times current. So somewhere, your voltage [INAUDIBLE]. That's [INAUDIBLE]. It depends on which one you operate in. You'll have maximum, and that's how you optimize. You change your load resistance so that you can control the current and the voltage by changing external load resistance. So that's a little magic. We do that for thermoelectrics, you do that for solar cells. All the same. You need to optimize. So some people just design their circuit to optimize. And the [INAUDIBLE] this way just reverse it just for convenience. So that would be another way to [INAUDIBLE]. That's the [INAUDIBLE] actually [INAUDIBLE]. Now, we can [INAUDIBLE] maximum power. When the power output is maximized, this is the current times voltage. That's the power. So you just take a derivative of your voltage and say when it's maximized. When you take derivative, physically, what I do is I'm changing my external load to find out where is the optimal voltage so that power is maximized. So that if you take this derivative, you'll find out where the optimal current, where is optimal voltage. And again, see, this is just a mathematic detail-- not important. But very often in the terminology, people say, what's your fill factor? What is that? So this is the rectangular-- this is where you optimize-- you run your optimum current voltage point. So that's the actual maximum power you have. And if you draw another rectangle, where you draw the-- this set up, the [INAUDIBLE] short circuit current and open circuit voltage, you draw another rectangle. This ratio of this extra power divided by the outer rectangle, that's your fill factor. And you want to fill as much as you can-- you can see-- to get the maximum power. Now let's do the source [INAUDIBLE]. Source, this is the electron-hole generation. So I have the deletion coming in. Only electrons-- the photons with energy larger than bandgap can generate electron-hole pairs. And also let's suppose this is blackbody. Let's treat the sun as a blackbody. So the solar radiation comes from the sun, and this is the fraction of solar radiation actually-- so you have to look at the distance, the fraction of solar radiation. Because the blackbody of the sun is 5,800 Kelvin. It's not a-- it's close object-- 5,800 Kelvin [INAUDIBLE]. It's very far. It's diluted. So that's this fraction in relation for the effect. But it's not the radiation intensity, it's the number of photons, because each photon generated one pair of electron-hole. So I'm looking at the energy-- because this is the energy. That's the energy per photon. So divide that. That's-- lots of photons coming. It's not a flux of energy. It's a flux of photon coming from a solar cell. And this is the reflectivity. I assume that the [INAUDIBLE] cell is absorbed and generate electron-hole pair. That's, of course, an idealization. You [INAUDIBLE] different losses. This is just best case. And of course, the maximum best is no reflection for that one. So this is source term. And the good thing is that the source term depends on only the temperature if you idealize. It depends only on the temperature of the source. And when I calculate efficiency, this is the energy of the photon coming towards the cell and this is the power. So of course, I want to maximize the efficiency. Everybody wants maximum efficiency. And then the question is, what's the maximum? How much is in there? Second law. First, the solar cell is limited by the second law column. You can always say [INAUDIBLE]. The maximum efficiency you can get [INAUDIBLE] between the sun and the Earth. So T [INAUDIBLE] is 300 Kelvin. T hat is the 5,800. So that's [INAUDIBLE] efficient. That's an upper limit [INAUDIBLE] upper limit. You can't get anything beyond that. Yes? AUDIENCE: So I'm just curious about-- so the way that the picture you have on that slide-- so it's actually instead on the p-type [INAUDIBLE]. And then don't we want it to hit the space charge region ideally? GANG CHEN: So the photon can be absorbed along the [INAUDIBLE] absorption. So some of them penetrate. Before the [INAUDIBLE] here in the space charge region. They have-- less probability goes on the other side. Electrons [INAUDIBLE] there are some diffusion happening here. Just to take a more [INAUDIBLE] chance of getting there is less. So that's why we want it that way and not [INAUDIBLE]. AUDIENCE: You see how that is [INAUDIBLE]? GANG CHEN: Uh-huh. AUDIENCE: So-- GANG CHEN: So you want to have a longer potential field to drive the-- actually [INAUDIBLE] field [INAUDIBLE]. AUDIENCE: No, I'm talking about this is a picture right here [INAUDIBLE] the really thin and flat [INAUDIBLE]. GANG CHEN: Really thin PN-- of course, the good thing is your electrode is very close. You can get to the electron, but your photon may not get through. So this is another [INAUDIBLE]. That's your backup. Your solar cells [INAUDIBLE] better. [INAUDIBLE] absorbed. Two, you want to get the charge to the electron before they recombine. One is thinner, the other one's thicker. This is essentially the thing people are trying to solve. But the question we want to ask is, what's the maximum possible? What's the maximum possible efficiency? And this one [INAUDIBLE] is two solar cells. The first [INAUDIBLE] by Shockley-- the Shockley-Queisser limit. And then that's the Shockley-Queisser limit I will discuss next. And of course, the real diode is hard to get what's maximum, because the saturation current depends on the field material recombination time, resistivity and electron-hole. It's hard to get your maximum. And the ingenuity of Shockley [INAUDIBLE] very, very smart guy. And [INAUDIBLE]. So this is what Shockley [INAUDIBLE]. So if you think about a PN junction, put it on the other side-- open circuit. Of course, the solar radiation will create electron-holes continuously. Every photon will generate electron-hole pairs. If there is no recombination, no defects will combine. What happens? You can't have an [INAUDIBLE] recombination process-- spontaneous recombination. [INAUDIBLE] So that's your maximum limit. No other recombination, only spontaneous recombination. But what's [INAUDIBLE] spontaneous recombination and coupling that up. And I'm really impressed. I read a few papers. This is where every time, I do this, I have a place where I don't feel comfortable to tell you. And I want to tell you because I want [INAUDIBLE]. And this is not just I read it yesterday. For many years, I tried to understand this. I still don't feel fully confident. This expression where it came up, and this is actually another Shockley paper. This is Henry's paper, and Henry had another paper [INAUDIBLE] and we all discussed different-- three different-- Shockley's paper, Henry's paper, Ross paper. Three different ways to write this expression. And so when a PN junction has a voltage [INAUDIBLE]-- and what's the-- so we see the quantum state. The Fermi-Dirac-- the Bose-Einstein statistics. But what [INAUDIBLE] the photons which obey Bose-Einstein statistics? In the PN junction, there is, it looks like, a Bose-Einstein, except that we add [INAUDIBLE]. [INAUDIBLE] is if you look at it, [INAUDIBLE] is really the Fermi level here in the [INAUDIBLE] in this region. That's the voltage [INAUDIBLE]. Expanding this term that I just feel I can comfortably derive it. You can go to expand so each paper you start from different place. And I encourage you to read. And again, if you have any idea to make my self comfortable [INAUDIBLE], because I, for many years, we were actually looking into this, seeing how we can really make [INAUDIBLE] derive some near-field-- [INAUDIBLE] near-field where it is surrounded [INAUDIBLE]. But see, so I can only just say take it from here [INAUDIBLE]. And the [INAUDIBLE] recombination in PN junction and [INAUDIBLE] you do later or [INAUDIBLE]. This is very important. This is very often where we will start. But I haven't [INAUDIBLE] thermodynamics very easy. So that's the [INAUDIBLE] how mechanically [INAUDIBLE]. Now, starting from here, you can derive what [INAUDIBLE] recombination. You have a voltage-- voltage. What's the recombination part? So you just integrate it again from the bandgap to any energy. And if you do [INAUDIBLE] statistics, you just elected one factor, one. And this is the-- I believe I didn't write the equation right-- copied from [INAUDIBLE] paper. And this is solid angle integration, so that's essentially the photon density of states inside the [INAUDIBLE]. So the Fermi-Dirac Bose-Einstein statistics integrated [INAUDIBLE]. So this is the spontaneous emission due to the current [INAUDIBLE]. And everything is very similar to what we had before, except that [INAUDIBLE] is no longer due to-- if you look at the diode right here, this is the a for the diode. That's determined by actual material recombination. And here, the ingenuity is now this is the [INAUDIBLE] dielectric constant refractive index bandgap of the material. So this is the maximum you can have-- the best you can have. [INAUDIBLE] really do recombination. no other defects [INAUDIBLE]. So because of that, it's free of any material. So now, you can derive the [INAUDIBLE]. But it's still one material that's [INAUDIBLE]. This is the only other material you can have. But while I talk, you can also say, ah, how about the photonic crystal-- inside the photonic crystal? In this chapter, we checked [INAUDIBLE] spontaneous form. And maybe you've got a different [INAUDIBLE]. Maybe [INAUDIBLE]. So now, I have translated that if you look at the term, this is the current term, the diode equation. One is the diode current, the other is the photon generated current. This is the photon generated current. In this case, this is 1. So I put in the best scenario. And the [INAUDIBLE] is the independent material except the bandgap-- only the bandgap. So that's where Shockley could derive what's the maximum efficiency. And this is the curve not from Shockley, but from [INAUDIBLE] paper. And using this [INAUDIBLE] recombination current, photon generated electron-hole pair, so this is [INAUDIBLE] photon generated current. And you do the rest [INAUDIBLE] we did before. We can find out what's the maximum power and what's the maximum efficiency. And this is what [INAUDIBLE] calculated. And here, c is a concentration. What concentration? [INAUDIBLE] concentration. So what does is change jl. If you concentrate more, you generate more electron-hole pairs per [INAUDIBLE]. So if you concentrate, you get this larger-- you get a larger open circuit voltage. So concentration actually can help you increase efficiency. So not too much increase, but reasonable increase. And what's interesting when you look at it, under one [INAUDIBLE] condition, where is the optimum bandgap? About 1.5. If you look at the material, where is [INAUDIBLE] material? 1.5. A lot of MOSFET is actually very good [INAUDIBLE]. 1.1. Not too far from the optimum. So this is about 31%. So that tells you, if you build a [INAUDIBLE] cell, this is about the best you can do in terms of efficiency. [INAUDIBLE] AUDIENCE: [INAUDIBLE] energetically focused compared to the [INAUDIBLE]. GANG CHEN: You mean the-- how many photon [INAUDIBLE] compared to bandgap or? AUDIENCE: I mean, you have [INAUDIBLE]. What happens, for example, [INAUDIBLE]. They are absorbed [INAUDIBLE] energy as the [INAUDIBLE] 1,100. So the [INAUDIBLE] is [INAUDIBLE]. GANG CHEN: Exactly. So this is a blackbody curve. This is your bandgap somewhere here. And really, what's usable [INAUDIBLE] energy is because this is above the bandgap energy [INAUDIBLE]. So if I look at-- this is the fine structure. If I have photon generated electron-hole pair this way, this energy electron relaxes [INAUDIBLE] very fast. Hole relaxes here very fast. So this is a key. And in fact, the people have been theorizing how the electron [INAUDIBLE]. If you can extract this electron-hole with this kind of energy, that's [INAUDIBLE]. Question of how you get it out before you relax. This is about picosecond. So picosecond is very fast. If you times 10 to the minus 12 times diffusivity, [INAUDIBLE] travel nanometers of-- [INAUDIBLE] nanometers [INAUDIBLE] have no way to get it out. And let's say if you have a good idea, that's something people have been thinking since the '70s. So that's a single junction maximum you can get. And exactly on this issue, how about we do multi-junction? One cell-- so all the junction shown here-- the first one absorbs blue, so larger bandgap-- the topmost layer. And the lower-- longer wavelengths, lower energy [INAUDIBLE], so gets absorbed in second layer and get absorbed third. [INAUDIBLE] idea. And of course, you can see with this stacking, each one generates a current. If you look at this, the key is that your current has to be continuous because you put [INAUDIBLE] on top, [INAUDIBLE] bottom. So this still has to generate the same amount of current as a design [INAUDIBLE]. But people have done that. It's a material nightmare and the design nightmare. Design in the sense you have to get a current match. So the electron goes-- come from the top layer, has to go to the hole of the next layer, or hole of the top layer has to merge with the electron-hole of the next layer. The current is continuous. And you have to group different materials, because the recombination is the killer. So that's why people do molecular beam epitaxy and lattice-match. And you generate defects, you generate dislocations. These were just the centers for the recombination. And then you kill the electrons before you can take it out. So to group those material, people have done molecular beam epitaxy like spectral diode-- spectral diode lab. And they were doing laser before. Now, they're doing probably 40%, 41% photocells-- [INAUDIBLE] cells, three junctions, and that rule match this. And so this is-- I got one graph from the internet. I think it gives some idea [INAUDIBLE] in real world, about 15% to 18% [INAUDIBLE]. And then [INAUDIBLE] 30%. And now, this is a triple junction indium gallium phosphide, gallium arsenide, germanium bandgap. 1.8, 1.4, 1.7 [INAUDIBLE]. You have to look at what are the materials that match your current and that you can grow. So you have to look at both material and bandgap device. And if you [INAUDIBLE] much [INAUDIBLE] infinitely. So this is a paper that shows the 72% [INAUDIBLE] multi-spectral [INAUDIBLE] 36-state. And if you have an infinite states, the theory is about 86%. So that's the best you can do [INAUDIBLE]. That's still lower than [INAUDIBLE] 300 and 1,500. That's about the line. If you do 1 minus 300 divide by 1,500. So that's the-- what people can do. Because in theory, it says now you look at [INAUDIBLE] in the lab. This is the best in the lab. It's not the best deployed. Silicon [INAUDIBLE] crystalline silicon. Crystalline silicon down here. This is a function [INAUDIBLE]. The best is about 95%, 96%. Remember, [INAUDIBLE] probably the biggest [INAUDIBLE] lab. They told me about [INAUDIBLE]. Remember, Shockley-Queisser gives you 30%, 31%. So [INAUDIBLE] really well compared to [INAUDIBLE]. Amazing. And also, that's why your potential is also limited. That's the best you can do. But the fact that you can really do that at low cost, I think your problem's solved. [INAUDIBLE] you can always work in [INAUDIBLE] is only 10% [INAUDIBLE]. Of course, [INAUDIBLE]. And so the best multijunction cell [INAUDIBLE] lab. Here, it's less than [INAUDIBLE]. I think that now, the best that I heard is about 41%. And this is a multicrystalline. This is a single crystalline, multicrystalline. And let's look at amorphous silicon [INAUDIBLE]. It's-- well, the real world, it's about 6% to 8% [INAUDIBLE]. If I say the lab, it's about 12%, but that's [INAUDIBLE]. And this is a, say, copper indium. [INAUDIBLE] copper indium [INAUDIBLE]. And so the best is about 19%. And [INAUDIBLE] telluride is [INAUDIBLE] here and here. And because, as I said, it's backwards because most people talk about 64, and now [INAUDIBLE] is probably the best most profitable company in the [INAUDIBLE]. And their CEO I think is about-- [INAUDIBLE] from lab to real production [INAUDIBLE]. And then this one is the polymer, I think, with the organic cells. And then there's a [INAUDIBLE] cell [INAUDIBLE]. So this one gives you [INAUDIBLE]. Not the most up-to-date in years, but pretty close. What are the problems? I mentioned the one for solar cell-- the number one is [INAUDIBLE] device. That's [INAUDIBLE] cell. And our device, very sensitive to recombination. So what are the reasons for recombination? Those [INAUDIBLE] bandgap. Deep level. Those are recombination [INAUDIBLE]. [INAUDIBLE] And [INAUDIBLE] where you've got a lot of [INAUDIBLE]. So those are regions where recombination happens. And that's why people purify silicon. Crystalline silicon, for example, [INAUDIBLE] large [INAUDIBLE]. It's about $2 per gram-- per kilogram [INAUDIBLE]. If you want to make a solar cell, you can recrystallize it, purify it and [INAUDIBLE] $40 per kilogram. And then, of course, during the process, you introduce a lot [INAUDIBLE] heat up, melting [INAUDIBLE]. And in fact, last year, before the market crashed, everybody was on silicon. So the silicon was short, and that got $240 per kilogram. So that's one. That's the same [INAUDIBLE] device Achilles heel. And in that sense, I can argue thermoelectrics is good because it's a majority carrying device. It doesn't have severe recombination. And there are other problems with this [INAUDIBLE]. In fact, this, to me-- I say, OK, [INAUDIBLE] absorb. The second step is to charge it where it goes. So this is really the first step. And look at where the whole photon goes. Of course, the relay interface and reflection [INAUDIBLE] reflection [INAUDIBLE]. [INAUDIBLE] coating and surface treatment. And once you get an input material, you don't have almost no other ways to control this profile anymore. It's material-intrinsic absorption. So this is the absorption coefficient and-- because the intensity decays exponentially. So now, you think about it. In your [INAUDIBLE] photon generated electron-hole pair, let's say, in this region. And you want a space charge in this region to overlap it so that you can separate them. You can use your field to drive the electron from one side of the hole to the other side of the hole. So the depth of this is 1 over r-- gives you exponential minus 1. So that's called the penetration depth. So this is where you look at the material, what I should choose. The absorption coefficient. Absorption happens above the bandgap. So bandgap is the first step. And then you look at the material. You say, oh, silicon, we love so much because that's the [INAUDIBLE] of silicon. Why silicon is a problem? Because it's really just-- firstly, because this absorption. You see here, absorption-- this is the photon energy. Bandgap is about 1.1 electronvolts. It's a very slow curve, and it's 1 over [INAUDIBLE]. So this is centimeter. So here, it's about 10 microns. So if I use a 10 micron-- so the energy is below 1.6 eV-- then those part of the energy is absorbed for very little. It goes through [INAUDIBLE]. They don't get absorbed. So it means [INAUDIBLE] this part can go out. So that's the absorption. And that's why [INAUDIBLE] say like a [INAUDIBLE] sigma six [INAUDIBLE]. They have very sharp [INAUDIBLE] 5th power. So if you have 1 micron-- this is the inverse-- is 1 micron. 10 to the 4th inverse [INAUDIBLE] 1 microns. It's a centimeter. So if you want to use [INAUDIBLE] the bandgap [INAUDIBLE]. And that's why when people make amorphous silicon [INAUDIBLE]. It's pretty good here. This is useless here. You can generate that from [INAUDIBLE] but then you don't have [INAUDIBLE]. So solar cell is [INAUDIBLE]. First, you want to absorb it. Second, you want to get a charge to the electrode. So why is it so different from the [INAUDIBLE]? This is the one I had mentioned before, whether it's a direct semiconductor gap or indirect gap. [INAUDIBLE] all the others that have sharp increased direct gap. So the photon goes from here to here. If the electron goes from valence to conduction band, you need to satisfy the energy conservation. So the photon energy [INAUDIBLE] the difference of electron-hole energy. So that's an energy conservation. And second is the momentum conservation. So momentum is h over lambda. Lambda photon is a micron. One level of electron is lattice constant [INAUDIBLE]. So in this, gives us 1 over [INAUDIBLE]. But so one more lambda is essentially vertical, so the photon doesn't give you much momentum. That's the problem. In indirect gap semiconductor, the bandgap is here. When you want, let's say, the momentum doesn't match between electron-hole. So if you want to leave this electron here, the photon don't have enough momentum. It has an energy. It doesn't have a momentum. It has to have something else. That's photon. You made a photon absorb a photon. Photon doesn't have much energy, but it has momentum. So this is a widely-- So you've got to have either emission or absorption of photons. So you have to have [INAUDIBLE], like I said. Very much scattered [INAUDIBLE] together. So the absorption is weaker because when the photon wants to do it, then there is no photon there. So that's the dilemma-- why they [INAUDIBLE] absorption there and near the bandgap. And of course, once you move to this region, it goes up. It goes from here to here. So that's when I said solar cell, essentially, you solve one problem, maybe the other problem. Thick or thin and how you can address those issues. It's really, the electron thing is better. You just take it out. Before they recombine, you send the electron [INAUDIBLE]. Photons get absorbed. Of course, that's related to efficiency. And the single cell is directly related to how much material you use. That's a cost issue. To have a low cost, you need a thinner. Less material, so always the material cost will be better. And we look at the cost. Now the lowest [INAUDIBLE] from [INAUDIBLE] and they have a manufacturing cost of about $1 per watt. And this is where the [INAUDIBLE] assuming certain amount of years you're using [INAUDIBLE]. [INAUDIBLE] you have $1 per watt. And it's a good curve here in terms of the cost. But saying, look at a [INAUDIBLE]. It takes a long time. Right now, the worldwide installation per year is about [? 67 ?] gigawatt. [? 67 ?] gigawatt. China put 100 gigawatts steam turbine every year. [INAUDIBLE] really just [INAUDIBLE]. It will take a lot of effort [INAUDIBLE]. Different type of solar cells. And a silicon cell is the most predominant. [INAUDIBLE] cost. Single crystal [INAUDIBLE] crystal [INAUDIBLE] is considered [INAUDIBLE]. And what's interesting is all those crystal surface [INAUDIBLE]. And that will trap the light so that we use refraction. We use refraction, not only reduce refraction. Once you get into the backside with metal, you reflect back. You don't want to go out. Some of the photons [INAUDIBLE] more [INAUDIBLE]. And the difference between single crystal or multicrystal is single crystal, you can get a regular [INAUDIBLE]. Just the manufacturing process [INAUDIBLE] crystal [INAUDIBLE]. And then you got, let's say, electron and silicon here. You can see this question here. If you generate [INAUDIBLE] it has to diffuse. It can go both ways. You don't have a field. You have to design so that the [INAUDIBLE] field [INAUDIBLE] go one direction. Yes? AUDIENCE: [INAUDIBLE]? GANG CHEN: It must be related to the material [INAUDIBLE]. I would imagine probably [INAUDIBLE] recombination, but I didn't check. But most of the time, people use [INAUDIBLE]. AUDIENCE: [INAUDIBLE]. GANG CHEN: No, the material is not silicon, then, no. That I don't know. I didn't check. [INAUDIBLE] And you can see here, [INAUDIBLE] contact. We use that Schottky barrier. And you have [INAUDIBLE]. That's a shielding effect. [INAUDIBLE] the photon couldn't get through. That's not good. So sometimes, people even bury the grid down or make it on the back side [INAUDIBLE]. So there are a lot of [INAUDIBLE] improvement. A little bit of improvement people done over the years. [INAUDIBLE] here say [INAUDIBLE] amorphous silicon [INAUDIBLE]. Amorphous silicon [INAUDIBLE] in particular, so you can use it. But the electron amorphous material-- cell is very hard to move. A lot of [INAUDIBLE] recombination. So you've got [INAUDIBLE] the amorphous silicon cell is about 200 nanometers. And if you can reduce that even thinner, it's probably better. Too thin, your manufacturing is a problem [INAUDIBLE] always [INAUDIBLE]. But this is actually showing you just a multijunction. So if you look at it, this is [INAUDIBLE]. So unlike silicon, silicon, you look at the electrode on the top. It's spaced apart. So electrons go here-- actually diffuse to the electrode, naturally going to the electrode, right? Silicon is reasonably good in terms of recombination. Amorphous silicon doesn't work. So you got to have electrode right on top of it. So it's a [INAUDIBLE] conductor. And this is [INAUDIBLE]. The active region is about 6 microns. And this is [INAUDIBLE] space after [INAUDIBLE]. It's a trade-off between recombination and absorption. If you increase absorption, [INAUDIBLE] always better [INAUDIBLE] theoretically work better. If you can use [INAUDIBLE] the first [INAUDIBLE]. You can capitalize on those methods [INAUDIBLE]. AUDIENCE: [INAUDIBLE] the number? GANG CHEN: [INAUDIBLE] here, it is, about 0.5 microns. I'm not sure whether that's the best way to optimize it. I'm just copy from where I was trying to show the scale rather than [INAUDIBLE]. So if you are [INAUDIBLE] you're also trying to improve the conductivity. Basically, you want a very highly conductive. So if you're highly conductive, then thinner is better because the free electrons in this material also absorb photons, which generally, you don't want that. Polymer cell-- polymer, ideally, [INAUDIBLE] polymer reduces cost [INAUDIBLE]. The problem with polymer is that it's even worse than amorphous silicon. You generate-- it immediately recombines. Recombination is very fast. In fact, when you first generate the bonded together, they are not even a separate electron or hole. This [INAUDIBLE] state for the exciton. So it maybe has an exciton center, [INAUDIBLE] center, [INAUDIBLE]. I have, let's say, maybe a solar thermal center [INAUDIBLE]. And there is-- the whole issue is to find how I can separate this exciton and make the electron go one way and the hole go the other way. [INAUDIBLE] So in that case, you look at this really thin 220 nanometer [INAUDIBLE] separate. But that's still difficult, so people are now doing the so-called bulk junction cells. So you put a [INAUDIBLE] of multiple. One conducts electron, one conducts [INAUDIBLE]. So locally, you generate electron-hole pair. Like here, you want [INAUDIBLE] separate so the electron goes one model, the hole goes another model. [INAUDIBLE] cell. So this is not a solid cell. It's an actual liquid cell. And what it has is titanium dioxide nanoparticles, [INAUDIBLE] titanium dioxide nanoparticles [INAUDIBLE] all three electron-holes. Too large. Not much hole [INAUDIBLE]. And so what [INAUDIBLE] cell does is put a dye on top of that titanium dioxide, and this dye is a photosynthesizer. So this is where photons are absorbed. And photons are absorbed [INAUDIBLE] monolayer generated electron-hole pair exciton. Before they dissociate, they are exciton. And you want them to dissociate-- separate into electron-hole. And electron will eventually flow into nanoparticle, so the nanoparticle diffuses. Now there is no field. Really, diffusion process. So the [INAUDIBLE] run out, so that's why people try to do nanowires so that gives a better path to the electron rather than nanoparticles. But the problem is that if we do nanowire, your surface area for absorption also changes. Here, you can have this surface area [INAUDIBLE]. Your nanowire is only this surface-- vertical. So the surface area [INAUDIBLE]. The particle has advantages. It's almost like a 3D wrap-around particle. [INAUDIBLE] There's always a [INAUDIBLE] trade-off. And the [INAUDIBLE] liquid. And this is a chemical reaction now. It's not just a hole diffused [INAUDIBLE]. So the hole oxidize the molecules in the case of [INAUDIBLE] iodide. Iodide in the liquid [INAUDIBLE] iodine(III) and this charged molecule diffuse to [INAUDIBLE]. So again, diffusion. So electron diffused [INAUDIBLE] electrode, hole diffuse to the other electron [INAUDIBLE] reduce, and the reduced diffuse back. Sounds like a very cool process, right? And the idea is, again, you can [INAUDIBLE] pretty good in [INAUDIBLE]. And see, so this is a flat surface versus the [INAUDIBLE]. So that's pretty much what I want to say. And then we're right on time. And this is the dream. The dream, in a sense-- here is the cost, here is the efficiency. You want a low cost, high efficiency, Silicon is cheaper in the first generation for gen 1. Thin film is here-- lower efficiency, but lower cost. And in fact, they make the business, at least for first order [INAUDIBLE]. And people are [INAUDIBLE] gen 3. No winner in gen 3 yet. This is where you guys come in later. OK? I think that's it. |
MIT_2997_Direct_SolarThermal_To_Electrical_Energy_Conversion_Technologies_Fall_2009 | Lecture_3_Energy_states_in_matter.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. GANG CHEN: How about we start by reviewing what we talked in the last lecture. We introduced the thermoelectric effects, including the Seebeck effect. When we have a temperature difference, a voltage is generated. That's the phenomenon. And that will give a microscopic picture of what happens with the diffusion of charge on one side of the path, the charge to the other side. And the total effect is when we have a junction that made up two different materials, we pass current through it. You will have either cooling or heating effect, depends on the direction of the current. And this is because the charge itself also carry heat. So when the charge flow from one material to another material, if there are more heat flows out, then the junction cools down. And if there are more heat flows in the junction itself and you reverse the current direction, you can see that the effect is reversed. And Thomson effect is when we have a uniform conductor, or semi-conductor, we have a different imposed time and the flow of current through, then along the material there is uniform cooling, there is a distributed cooling, or heating, that happens along the material. And this is because, really, at each section of the material, because the temperature difference, the charge, the electrons, carry different amount of heat. So that's the Thomson effect. And then we did the analysis, simple analysis of a device. We have a real device. We pass current through it. There is a thermoelectric effect. There also undesirable side effects. The side effect is one, there's [INAUDIBLE] heating. And two, there is a reverse heat flow conduction, and this diminished degree the device performance. And if we do the device analysis and find out that the efficiency, or the coefficient of performance, we found that those depends on, eventually, the material property which you will call the figure of merit ZT. And, of course, those are heat engines. So the pre-factor and the caudal factor, hence, something that's less than 1. If it's less than 1, it's dependent on the material property, and that 1 to be larger, as large as possible. And then we discuss applications. I mentioned that at this stage, the materials is not competitive to replace the main refrigerators, or main power generator, like an internal combustion engine. So there are two directions to go. One, if you're an engineer, think about really using those there. You have to look at the system and the innovation where it makes sense from both the efficiency and, particularly, cost commercial product on the view. And the other direction is go to improve the materials. So what we'll be talking in the next today's lecture and next lecture is on understanding what really determines the microscopic aspects of the materials, and what are the directions which you go to improve the materials. What are the ideas? So let me, again, start with this figure of merit ZT. I said that when you do thermoelectric, so people say I work on thermoelectric. So when people ask you what's your ZT, right? That shows you know something about it. And I have three parameters-- the electrical conductivity, the Seebeck coefficient, and the thermal conductivity. And the thermal conductivity, the heat conduction in the material consists of two parts. One is the topical vibration. The lattice in the material, the atoms in the material, vibrate and conduct heat. And the other is when charge itself, it carries states also conducted. So here, we decompose this k into [INAUDIBLE] and the electron. So those are when you think about ZT, you look at those parameters-- electric conductivity, Seebeck coefficient, and the [INAUDIBLE] contribution to thermal conductivity, electron contribution, thermal conductivity. And what I want to do next is to give you-- go back into more details. So what are the microscopic pictures determine those properties? And what are the physical more together with some mathematical formulation for those properties? So today, we hope-- I hope you all power those topics. We will first review some basic concepts in solid state. And then using the simplest picture, that's the same picture I talked in the first lecture how we use a three step. I derive [INAUDIBLE], right? It's a really popular going from left and minus particle, from left to right, from right to left. So with that imbalance, you will see we can come up with expressions for the electric conductivity, Seebeck, thermal conductivity, all those. And then I will do a leap of faith. I'm not going to the detailed derivation from this. You know we go-- we can go to more formal transfer theory and have the more rigorous expressions for the transport coefficients. So let me start. Like I said, this is a review, but I know for most of you it's new. Because if you are, for example, from mechanical engineering, you have not seen this. So don't worry about it. You just follow-- let's follow through. Some of you will take this other result from wavelengths. Others, I hope the picture is simple enough for you to appreciate those pictures. So let's start with the basic concept. First, in terms of the atomic vibration, because atomic vibration in the solid carries [INAUDIBLE], right? So now, I think about the bonding between atoms. You can think of this either solid or even just a hydrogen molecule. The bonding between the two atoms, two hydrogen atoms in the molecule, when they are far apart, they are attractive. And when they are too close, they repel each other because the electrons start to overlap, and they-- all the mechanics will say the basic principle is the exclusion principle. It doesn't allow the electron to take this [INAUDIBLE]. So we have a typical potential curve in the far separation between the atoms. It's attractive. And when they close, it's repulsive and the atoms sits in the equilibrium position. So when we have many atoms, they will have constraints on both sides. So they have the most stable position. That's where the energy is the minimum, right? Now, because this curve, the energy is a minimum. If you look at the mathematics, when I have a curve, minimum or maximum, the first derivative is zero. So when I look at the expansion, I can expand this potential into the minimum point, plus, the first derivative is zero. Then I go to the second derivative. So you can go to higher order. But for the understanding here, we just go to second order, and then the force between the two atoms is just the derivative of the potential. So F equals the potential derivative. So I get a force equals the spring constant between the-- So now, we can take this two atoms vibration as a spring. So it's a mass spring system. And for mass spring system, we all know the simplest picture is mass print. And there is a natural frequency for this vibration. The vibration frequency is the spring constant and divided by the mass of the atom. This is very relevant when you think about what material you want to choose for thermoelectric. And typically, the larger is the spring constant between the atoms. You look at this, the higher is the frequency, the more vibrational energy, the more-- the bonding of the atoms is stronger. There is a larger spring constant. And this will lead to higher thermal conductivity. And diamond, for example, has the highest thermal conductivity among the materials, and that's because the bonding between carbon in diamond is very strong, and the carbon atom is very light. OK. So that's just a-- it has implications for this, even though it's very simple. And this is what we learn in classical mechanics dynamics, the natural frequency of vibration. And if you go to quantum mechanics, what it tells you is just a small addition, you know what I'm saying? The frequency is still the same. New is still the same. But what it tells you is that energy cannot be any value. We all know quantum mechanics, the basics, the energy is complex. So if your oscillation is either this frequency in classical mechanics, the oscillation amplitude, the velocity, can be any value. And in quantum mechanics, we say the energy must be integer of this [INAUDIBLE]. OK. So that's the new thing coming from quantum mechanics. So it must be multiples of the Ishmael. And so this Ishmael, you can think of this as a basic energy quantum, and that's-- a solid is called a phonon. And this one-half is also-- is related to the so-called uncertainty principle in quantum mechanics. It's also called the zero point energy. So if you recall before I talk about the photon, right, when we do electromagnetic waves, and this atomic vibration is analogous to the electromagnetic wave. So those are all waves. We'll extend this more to waves in the next step. So that's where the name coming from, photon for electromagnetic wave, and the phonon for lattice vibration. And that's the basic energy quanta of vibration. OK. So this is a-- we're talking about the individual oscillator. And now, I look at the atomic chain. So when I think about solid, the most simple picture is a crystal. Crystal, a periodic arrangement of atoms. So now, I started with one dimension of atomic chain, and the distance between the atoms is the lattice constant, A. So we have N atoms. So the length of this chain is [INAUDIBLE] A. Now, again, you recall in the electromagnetic wave, the last lecture, I derived the Planck blackbody radiation [INAUDIBLE]. I say that when we consider a cavity in the length direction, I have standing width. I'm going to make the same argument. If I clamp the two ends of this atomic chain, do not move. Again, take a rubber band as an example. You fix this end, you oscillate it. You see the waves, different waves. And the violin is another example. So the length gave me-- those are the wave of the constraint. The wave has to be multiples of lambda, which is the wavelength over 2. So I have-- the length must be more than 2. It could be any integer. And then the weight vector. Again, I'm talking about because of the weight, so the inverse of the wavelength is the weight vector is 2 pi over lambda. That's the definition. And the fundamentally is like the Fourier transform. If you're an electrical engineer, the time signal, or sine omega t signal. And here is key and frequency is omega. The relation is omega equals 2 pi over t, the same. This is a spatially periodic vibration, so the weight vector and wavelength relation is 2 pi over lambda. So that's a Fourier transform. And so this is the same picture. I clamped the two ends of the [INAUDIBLE]. I have those other [INAUDIBLE] wave vectors. And I went this pretty fast in the last time I talk about electromagnetic wave. But I did say the standing wave. I did write the mathematics. You go to check. When I wrote the mathematics, I say I can do the summation of two traveling waves. One goes from left to right, the others go from right to left. So here k is 2 pi over lambda. Those are the values. So now, I do the traveling wave. So rather than clamp the two ends, I say, OK, I don't have that constraint. The waves can go both positive direction and negative direction. OK. So in this case, the weight vector is pi over-- or minus pi over 8, 2 plus pi over 8. And two traveling wave will make a standing wave. So if you have the same wave lengths there. But also, if you compare what I wrote before, OK, I didn't put the [INAUDIBLE] limit, right? | this wave every vector can have any value. On the other hand, when I think about very logically vector, what it means in a crystal, because the two atoms, the smallest distance is here. Now, if you have wavelengths shorter than that, it doesn't mean much. So that's why I say my wavelength is limited to here. If you look at this, this is a two wave. The two-- one atomic distance is A, so this is the minimum you can have in terms of wavelengths. So those are the region of my weight vector going from negative pi over A 2 plus pi over A. Now, you think about the frequency. When I talk about the one atom vibrating, the frequency is K or M, square root of K over M. Now, I have the whole lattice, different wave vector. So each wavelength corresponding to one mode of vibration. And the thing about, if my whole atomic chain is one way, one way with pure, one pure, the frequency is higher, or lower, or same as my individual K over M square root. Anybody is making a guess? AUDIENCE: If an atom was moving fast, yes, I'd say it's the same as the [INAUDIBLE]. GANG CHEN: Well, let's suppose-- again, let me take this as right. Now, all the atoms moving just one pure, all the moves. And then let's say is a one pure, and more pure since the frequency. Do they change, or they don't? You probably can go back and do an experiment and put some beads or a rubber band, and then you'll find out the relation between the frequency and the wavelength. In this case, it's changing, right? When all the atoms move in unison, when, say, only one pure, it looks like the whole atoms-- all the atoms in the chain is moving same time. So it's slower. The frequency is smaller. So that's-- in the long wavelengths, this is the weight vector. Long wavelength frequency is smaller. And when you go to larger wavelengths at the end, it's closed. Coming back to here, you can see, here is a K over M square root. So come back to individual atom and high frequency. OK. So this is the relation between the wave vector. And the frequency is, again, the dispersion. Remember, for photon, for the electromagnetic wave, omega equals [INAUDIBLE] k. That's a linear dispersion. And here, the real one we have is approximately half of a sine. This is like a sine function. And often, we do a linear approximation, take the slope. So the speed of sound is that that's-- the slope is for the longitudinal wave. That's the speed of sound. So in solid, you lock one end, propagate to the other end. That's the speed of sound. And that's this slope here. And at a very high frequency, when you go to look at the mass of the atoms, you can see this frequency can go as high as terahertz or tens of terahertz. So very high frequency. Of course, the acoustic wave that we can hear is only a few 100 hertz. So this is a simple one-- monatomic atomic chain. And now if I extend this a little bit-- suppose my chain has two different atoms. Gallium arsenide has a crystal. One atom is gallium. The other atom is arsenide. So in this case, this is a white one here. So from the 22 atoms-- the same atom-- that's one here. Of course, this one simple picture is-- the outer atom is moving in unison with this atom. So they form the same face into two sets of subatom lattice. And in this case, the dispersion-- the relation between the wavelength-- wave vector and frequency-- is similar to what we drew before. And in the other case, we could have this set of atoms moving up and the other set of atoms moving down. So it's a much harder stretch. We are very close. You are forcing them in terms of a different phase. It's much harder [INAUDIBLE] than the higher frequency. So in this case, we have two type-- I'm going to have two atoms. One is the acoustic branch. That's the bottom part. That's this picture here, roughly. The other is the other phase. That's the [INAUDIBLE] approach. So that's the atomic vibration in one dimension. And of course, a crystal is not the just one dimension. A solid is just one dimension. It's three dimension. So any direction you look, it look like you got a one-dimensional periodical arrangement. It's just that space in between the atoms will be different. So how we express it? So here, we'll go back to what I said before-- Fourier transform. And the periodic signal-- if the period is t, We see the frequency is the 2 pi over t. And in terms of wave, we want to mention wavelengths of the lambda. And the Fourier transform is 2 pi lambda. But in three dimension, each direction has a lambda. If I look at the crystal, a cube-- take a simple cube. Put them periodically arranged. You look at the other direction. The period is cubic 3-- no square root of 3 times the lattice constant a. So that's your periodicity. You look at, say, a different [INAUDIBLE] you have different periodicity. So what I do is I transform this periodicity in space. Let's say I take a crystal-- and say, here is a, say, face on the structure-- face on the cube. And we add a face on the surface. There is also an atom. And I do a Fourier transform of this space in three dimensions. That's what it looks like. Now it looks a little bit ugly. But I can say each crystallographic direction here is a p direction, And the shortest distance, L-- this point-- is the longest real space because the k is 2 pi over lambda. So if I think about this from center to this point, the shortest, I know for a cube, that's a 111 direction. It's the universe of the atomic space. So this is a way that the physicists like to use the reciprocal space. I joke with them-- I say, you guys don't live in the real world. They like to express things in the reciprocal space. So now let's look at a simple-- in terms of mathematics-- expression of this, You go to electrical engineers. They like to do signal analysis and frequency. They don't do time signal. Time signal-- you have to plot the time. You do a hard spectral analysis, you can use frequency. It's a much simpler mathematical language. So now you get to the real world because that-- before, it looked very simple. Now you can start to look at a string. For most mechanical engineers, you look at this, and you'll get the-- what is that, they'll say. It's really looking at the different crystallographic direction, the vibration frequency. And if you look at this all silicon gallium arsenide or phase-centered cubic, as I do before, different crystal has different-- those different directions-- different reciprocal-- the shape of this basic unit cell in reciprocal space. So I look at the gamma direction. So this is diagonal, from this to this-- diagonal direction. So if I look at that direction, what's the vibrational situation spectrum? So if I have the lattice wave propagating in the diagonal direction-- so that's from gamma to L. And those are the optical phonons. Those are the acoustic phonons. Acoustic phonons start from 0 here. Optical is from the top of the acoustic. And why I have different branches-- I have different branches because the vibration-- the sound wave you receive is longitudinal. Here, is longitudinal wave only. But in solid, you can have longitudinal as well as transverse. You can sustain the shear. So I have a longitudinal wave, transverse wave. And really, transverse-- if you look at the electromagnetic wave, I said there are two polarizations. See, acoustic wave transfers also two. But in highly symmetric direction, like from gamma to L, in this case, this is [INAUDIBLE] symmetric. So the two directions-- this direction and this direction-- the same wave vector-- they have same frequency. So they become degenerate just for [INAUDIBLE] one. But in some other direction. It's not highly symmetric. You can say there are three polarizations-- longitudinal, one longitudinal, two latitudinal. So acoustic branches, optical the branches. That's the station from 1D to 3D. AUDIENCE: What is the spacing between the different wavelength vectors? GANG CHEN: Different wave vector-- the spacing-- those are experiments. The data is comparing experiment. AUDIENCE: The frequency should be in hertz or [INAUDIBLE] frequency is in the hertz? GANG CHEN: Frequency is-- if it's omega-- so this is another thing that, very often, people get really confused because it depends on who likes to plot out what. And sometimes-- here is k-- omega. And sometimes you'll see this one is not [INAUDIBLE] terms of the frequency. This is centimeter inverse. Well, that's not the frequency unit. So what they mean? And in this case, if you want to look at-- the first frequency is 2 pi mil angular frequency. You have to say what exactly is a better plot, the angular frequency or frequency? And mil is-- if we practice wave vector inverse, we plotted 1 over lambda. So you have to have to convert the frequency equals to c. What about lambda? So people have different habits. Yes. AUDIENCE: This is speed of sound or speed of light? GANG CHEN: Huh? AUDIENCE: Speed of sound or speed of light? GANG CHEN: This is the speed of sound. Oh, I'm sorry. I'm sorry. Speed of light. AUDIENCE: OK. GANG CHEN: Yeah. Not speed of sound. When they do this kind of expression, they always go the speed of light. Speed of sound is different for different materials. So you're asking, what are the wave vector numbers? So this is where the number of waves-- when I go from here to here-- well, this is called from pi or a minus pi over a to plus pi over a. That's called the first [INAUDIBLE]. And the number of wave vectors, k, equals the number of atoms you have. So these points-- from here to here-- the number is n. That's actually important later. This number is more important when you do counting. Once you go to microscopic picture, you do the counting. AUDIENCE: In silicon dispersion relation, we have a mode that seemingly correspond to the optical mode in gallium arsenide. Then what will be-- the mode will look like for the silicon at the top of it? GANG CHEN: Oh, they'll be optical modes. Well, it turns out silicon is a one lattice point. How many atoms? And the silicon is-- in one lattice point, you have two atoms. So there is another atom-- see that the one quarter, one quarter, one quarter. If you think this is one, there is another-- this is a lattice-- this is the point-- a lattice point. So there is another atom at the one quarter, one quarter, one quarter. And it's not-- it's an FCC, but there's two atoms. If there are two atoms, that-- say, in the lattice point, if there are two atoms, then there's three acoustic branch-- three acoustic branch. And then there are another-- so two atoms, n minus 1-- so times 3. So there are three optical branch. If you have three atoms, you have three acoustic [INAUDIBLE] branch, then you have six optical branch. So the total is 3 times n. That's n. At every lattice point on an atom [INAUDIBLE]. AUDIENCE: What's the direction of the two transverse wave? GANG CHEN: Roughly, it's perpendicular. But when you actually do it, you have to solve the dynamic matrix. You have to solve the matrix to find out the exact direction of the vibration, so the eigenvectors. You can always decompose. But there is an eigenvector that you should go to solve the equation of motion. You can solve the Newtonian equation of motion to get those eigenvectors. OK. That's all about crystal and vibration. And now let's see the electrons. I started with crystal vibration by hydrogen molecule. I have the two atoms and the separation spreads, [INAUDIBLE]. And when talking about electron, let's also start with hydrogen. This time, I'm starting the hydrogen atom. And the hydrogen atom-- again, you know quantum mechanics-- this is the energy levels of a hydrogen atom-- electrons, in this case, I'm talking. And the electron-- the energy level-- I didn't even write it down. You can see I wrote-- so E of the electron is dependent on the n-- is minus 13.6 electron volts n square. So what is the one-- so let me say-- if my electron volt is one electron mode in-- one charge of electron in one mode and one voltage-- for electric field, one voltage. And that's the energy of the electron. And this is the inverse, 1.6 to the minus 19 charge. So that's the energy of the electrons in the hydrogen atom. And its electrons are moving in three dimensions. And so when I solve the Schrödinger equation, I'll find the wave function. Again, this is going in quantum mechanics. We're not going to do any quantum-- I'm just introducing the basic concept. And it turns out because we have a three dimension, r theta phi is in three-dimensional space. So there is three quantum numbers, n, l, m. Those are all results of quantum mechanics, except that in s, you have to go through even relativistic quantum mechanics. And those n-- you can see now-- all integers. The quantum number-- n is [INAUDIBLE] 1 to 3. All those integers-- L must be less than n. m must be less than L. This is all mathematics-- coming from mathematics. But what it really means is I wrote down that-- I forgot to write the energy. Energy is minus 13.6 electron volts divided by n squared, so energy doesn't depend on L and n. It depends only on n. So when n equals 1, I have-- L has-- you look at it. L has to be 0 L has to be 0. And the S could be plus or minus 1/2. So there are two quantum states. So here, I have two quantum states. So if you look at the theoretical table, that's why you have hydrogen, helium, failed. Helium is stabled. Two failed. And the next quantum level, you go to a plus 2. It's pretty far away from the first level. So you look at the periodic table again. You fail. This is the one quantum state. One quantum state can only have one electron, [INAUDIBLE]. That's the one I've seen as a different state. So hydrogen, helium, lithium. Lithium is [INAUDIBLE] 2s now. So when you go to continue to look at the periodic table, you fill in. Here, each energy level is 2n. Once you go to n equals 2, the number of-- total number of electrons you can have, up to filled all the orbital-- the quantum state is n plus 2 is what? 8, right? 11. So that's the hydrogen atom. And this is a basic model for the periodic table. Look at the periodic table. You can use this and pretty much interpret a lot of those first 30 atoms of [INAUDIBLE]. Now, I'm going to, again, say-- starting from here, I'm going into a [INAUDIBLE] chain because I want to understand electrons in solid. So a [INAUDIBLE] chain-- I'm thinking about hydrogen as a simple example, but the hydrogen [INAUDIBLE] gas, and then thinking about the solid. And so electrons-- they are waves in quantum mechanics. So both waves and particle characteristics. So waves-- each electron has those wave function as part of [INAUDIBLE]. This is the wave function. Quantum mechanics will give you this function. And now they are waves. So let's plot a plus 1, approximately. And they spread out-- not just confined to one electron. And now, this is the next atom, a plus 1. They start to overlap with each other. What happens? And these two electrons from the two atoms both will have the same states because once they start to overlap-- once they start overlapping, the state of the two atoms are the identical states. And the basic quantum mechanics doesn't allow it. So the state will start to adjust a little bit and form back. So now n equals 1-- this is originally n equals 2, 3, 4. Think about-- they started forming bands. So there are-- if you have n atoms, again, in the chain, between here and here, I have an n point. And each point now-- they start to-- you can see here [INAUDIBLE] different and those same allowable weight vector-- now electron is a wave, extended through the whole lattice. This is one quantum mechanical state. So the waves can have different vectors, different period, same as we argued for the lattice. So that's-- a basic quantum mechanics dictate. The energy has to split a little bit, forming, now, a band. It's quasi continuous with [INAUDIBLE] continuous because that k-- there are many k points. But they still return here. So here, the endpoint depends on the n for your number of atoms in a chain. This is very interesting. because even this simple picture-- I said the hydrogen picture tells you the periodic table. What this picture tells you-- whether you have an insulator or conductor. Let's think about-- again. If I think of hydrogen as a simple extreme, each hydrogen has one electron, So when the electron field go to the energy, the lowest energy level fails first. They're lazy. Now, thermodynamics tells you-- everybody wants to go to minimum energy space. So if I have n atoms, I have-- between here and here-- n point. Each point has a spin [INAUDIBLE]. The spin is not reflected in this curve. So each point has two states-- to spins, s equals plus minus half. So now I start to fill the bottom electrons-- if I have hydrogen, There are really two end quantum states. I have only an electron. Where it goes? It fills about half, right? So the electron maximum [INAUDIBLE] energy will be somewhere in between the band. So this is where-- it depends on [INAUDIBLE] field here depends on how many electrons you can fill to within the band. And it's either 0 temperature-- 0 Kelvin, this point-- where [INAUDIBLE] just think about the water-filled bucket. The [INAUDIBLE] energy level is [INAUDIBLE]. And so if it's a non-zero temperature, those electrons do not move. They need a lot of energy to move because there is no quantum state in this, here, quantum mechanical-- no state allows energy to move to-- when they move, they change state. The state has to be adjacent, allowable in the solvent. So only electrons in this region are movable. And in metal-- because the next state-- just close to this point-- they're very close. The energy difference is very close. So the thermal energy is large enough to allow them to go to the next state. So only when your electrons fill to level their empty state in the band it's a conductor. It's [INAUDIBLE]. That's [INAUDIBLE]. And what happens if my number of electron filled here? Then it needs a lot of energy. Depends on the difference of the energy here and here. If this difference is very large, you've got the insulator. Electron cannot move. So this-- from the [INAUDIBLE] of electrons to the empty state-- that's the band gap. And if this band gap is very large, that's an insulator. How large is large? A few electron volts. And this is because at room temperature, we said that, in the first class, the energy is about kdt. k is the Boltzmann constant times temperature, thermal energy. And that's about 26 milli-electron volts. And so if this one is a few electron volts, much larger than kdt. So very, very few chance electron can come over here due to thermal. And that's insulator. But then for this gap, it's not large. Plus, for example, this is an insulator. And the gap is a view of about 4 to 5 electron volts. Now, silicon-- that gap is about 1.12 electron volts. [INAUDIBLE] So silicon becomes a semiconductor because this gap is not very large. And if you cool silicon to 0 Kelvin, it should be an insulator because there's not enough thermal energy to kick the electrons from here to here. But at room temperature, the thermal energy, kdt-- because of the thermal energy some electrons are kicked from here to there. And if they are kicked from here to here, what's left behind are some empty states. Now, those electrons-- those other electrons are happy now. There are some states they can move. And these outer electrons that kicked up is also happy because there are also empty states they can move. So the motion of those electrons-- those electrons-- it's much easier. So you can think of this-- the thing that's really moving is the electron moving because there are empty states. But it's much easier to think them as positive charge. So don't think those are electrons. Just think of those empty states as holes. There's no hole. No positive charge. It's electron that's mobile. But we just think this as a whole positive charge, and those are electrons. And you can even put-- so that's an intrinsic silicon [INAUDIBLE], right? But you could even put the impurities into silicon, into semiconductors to add more electrons or add more holes. So when I add-- for example, if I put a phosphor in silicon, phosphor has 5 charge. Silicon is 4 in the outer shell. So there's one extra pair of silicon. So that extra-- because it's very close. The energy level of phosphor very close to here. So that electron can easily be excited to this region. So it's an n-type semiconductor. And on the other hand, if I put boron into silicon, the outer has three. So the born energy level is somewhere here, close to the bottom of this band. This is the valence band. This is the conduction band. So the energy levels here-- those electrons are much happier to go here, so left behind holes. So boron becomes an acceptor, entrap the electron. And they create the holes in those bands, anticipating in these bands-- that the electrons [INAUDIBLE]. So that's how you do doping in semiconductor. You create the energy states. Sometimes even impurity-- even frequency could create those additional states. In silicon, we do artificial doping, when we do thermoelectric material's weakened state, a missing atom could create those kind of situation. So the doping in material-- in thermoelectrics, doping is very important. Of course, this is true for all semiconductor device. Any questions? Let's move on. Look at the real material now. That's a one-dimensional picture. And I look at, again, three dimensions. Go to the virtual reciprocal space. And 111 is the gamma 2 L point direction. 111-- I only do one half of it because it's symmetric. Positive wave vector, lengthy wave vector symmetric-- I draw half to save space. So this is a gamma to x that went 00 direction. And it turns out the silicon-- the maximum point where this electron fill to-- at the valence band. And the minimum point-- that's the conduction band. They do not agree in terms of the wave vector. So this is an indirect semiconductor. And gallium arsenide, on the other hand, is a direct [INAUDIBLE]. There's a big difference in these two materials, direct and indirect. Microelectronics industry is established on silicon. But the lasers-- semiconductor lasers-- nobody can use silicone to make semiconductors. People are making progress. It's much more difficult to make a laser out of silicon. It's a direct gap. Semiconductor is much easier. The reason is that in the gallium arsenide device, microelectronic device, you reject the electron here. You eject hole. That's by design. You put a current, and you actually inject it into the material. And then the electrons flow off to the bottom of-- recombine from here and here-- give off a photon. It's much easier to do that in direct gap because there's no wave vector mismatch. And here it's very difficult for electron recombine with the hole. Well, you see that's good for solar cells, Solar cells-- you don't want recombination, really, if you want to get a charge. But there are various points. This one-- solar cell is not good. It doesn't absorb well. You can't create-- when photon comes in, a photon-- The wave vector-- the wavelength is about 1 micron. Wavelengths of photon-- a half micron. You think about the visible light-- half micron. And the wave vector of photon is 2 pi over lambda, 2 pi over a half micron. And here is pi over aa is lattice constant, a few angstrom. So this is a few angstrom. If it's a 2 pi over a half micron, that's exactly-- almost right at the center. The wave vector of photon, very small. Wave vector hk-- h over lambda is momentum. Momentum of photon is very small. So in gallium arsenide, the photon comes in and go-- you go from here to-- if the energy is large enough, you can lift the electron from here to here. That was all. But in silicon, because they interact-- the semiconductor, the photon momentum is here. You cannot leave the electron from here to here. Doesn't have enough momentum. It has enough energy. What can solve that problem? Some method. You solve a huge problem. See, this is a really-- even in the microscopic world, it's the energy conservation, the momentum conservation. You learn either in continuum or in discrete. That always [INAUDIBLE]. So silicon don't absorb well. Gallium arsenide absorb well. If you look at a gallium arsenide solar cell, people use the same field. They absorb well. Silicon-- it's very hard to use the same field, because they don't absorb. So your compromise is make thicker for that chance to happen. AUDIENCE: [INAUDIBLE]. GANG CHEN: Ah, yes. But that's exactly how absorb happens. You need a phonon. Phonon has that wave vector. Phonon doesn't have enough energy. The omega is too small. So phonon plus the photon-- phonon gave you the vector. Photon gave you the energy. Can lift it from here to here. But you've got three particles to work together, Think about your coordination at MIT. You get three professor together. Much less chance. That's the probability. We'll have that experience. Now we're going to do a little bit more. That's the basic idea of the energy levels, vibrational energy level, and the electronic energy level in solids. And I'm going to do a little more. And I approximate-- first, I think about the free electron. The kinetic energy-- 1/2 mv square. And mv square-- quantum mechanics say-- first, m comes away as the momentum. So p squared 2m-- that's the relation from p as the m wave. And in quantum mechanics, p is hk. Oh, I got h squared. Sorry. It should be h squared k squared over 2m. So that's a free electron. For the electron in the solid, I'm going to do a similar thing. Because this-- here is a maximum. Here is a minimum. Again, the first-order derivative is 0, unless this is a very sharp. There are some cases it's not-- the derivative is not continuous. So the band is not a part of-- what I say is it's not a continuous curvature here. But see, most of-- this maximum-minimum-- the first derivative is 0. So I have second-order [INAUDIBLE]. Mathematics I extension. So here, if I, for example, take a [INAUDIBLE], of course I put a k. I didn't put a k. This is not 0, so I should put kx minus k0 square. Here, I put the center at the center. So this is the sigma general, not the most general form of doing the mathematics, And here, it looks very similar to free electron. This is just a second-order derivative of the curvature. Now, if I treat that just as a mass, equivalent of mass-- if I call it effective mass, then it's a free electron. I can do the rest of my work as a free electron. So mathematical simplification. Here are the effective mass of the electron in the crystal. If I take the effective mass, I write this pretty much like a free electron. So I'm going to do free electron, most of the time, except replacing the mass. So this is, again, my dispersion relation. Before, I said dispersion relation between frequency and wave vector. And now, if you think about energy relate to frequency, that's new. So frequency, wave vector. Except for electron, people don't talk to a frequency. People like to talk about energy. Close relation between the vector frequency or the vector-- and energy dispersion relations. AUDIENCE: So the effective mass is different for every direction? GANG CHEN: It could be different for every direction. In some case, it could be isotropic. Next one-- let me show you. If we take the energy as constant, I do this. So if you look at this, this is-- OK, what's the constant energy surface looking like? In 3D, If these are the same, it's a sphere. They are different. They are ellipsoid. So this is what's in gallium arsenide. This is a silicon electron. End of story. So in this case, we say, OK, there, you got six pair of pockets, different directions. And this is just a review. I showed this before. Now we know what is phonon, what is electron. And Fermi-Dirac distribution is for electron. Bose-Einstein distribution for phonon and photon. Most of the time, this [INAUDIBLE] is 0 because phonon and photon are not real particles. That's another way they are not real. They can disappear. Electron also disappear. But the most of the time-- and not disappear. Electron cannot disappear. Electron is the real part. As we go from conduction band to valence band, we combine-- just [INAUDIBLE] the electron still is there. OK, now I'm going to do math. The constant. I'm going into more detail constant. I say, OK, I know about semiconductor. Yes. AUDIENCE: Sorry. Is there any [INAUDIBLE] mass conservation or effective mass conservation [INAUDIBLE]? GANG CHEN: Mass conservation for-- good question. I don't think we can talk about mass conservation for-- under the effective mass picture. I'm sure this is the only from-- I think mostly from [INAUDIBLE]. Electron-- of course, the true math is still the [INAUDIBLE] 0.1 to 0.31. But in the solid, when you look at the transport, I see the free electron. Then usually fill the [INAUDIBLE] of the background, lattice. So now they feel either heavier or lighter than the free electron mass, which is 0.1 to 0.31 [INAUDIBLE]. Yes. AUDIENCE: When people do the [INAUDIBLE] structure calculation for electrons, do they assume independent national approximation? If they include [INAUDIBLE] GANG CHEN: I think they should include the electron-- for example, you do a density functional theory calculation. They will include the outer electron effect. And you don't do-- just an independent electron. You have electron influence the other electron in the same atom. So now I'm going to do a spherical-- I'm not going to do parabolic or elliptical, but I'm going to do a spherical [INAUDIBLE]. And it's the same in all directions. So that's a really like a free electron, except that this mass could be different. Like I said, it's good for gallium arsenide, Let me suppose-- I want to find out-- if I dope the material, how many electrons I have in this conduction band? And I'm going to do counting. This is a strategy that I use. And I use before same for photon. I did that photon in cavity. I count how many photons in the cavity. So now, if you remember, each quantum mechanical state can have one electron. And in the state, what's the average number of electrons is given by Fermi-Dirac? Maximum one electron, right? We said in the-- this is the average given a certain temperature, quantum potential, average number of electrons in one quantum state. Now I just need to see how many quantum states I have, right? One quantum state, this is the average number of electrons. So I just need to-- really, all those counting looking very complicated, it's just counting quantum states. And before, when I do electromagnetic wave, I was counting electromagnetic modes. Same way. So all that counting-- counting how many modes, the modes is essentially a state. So I'm counting. So in the x-- so it's 3D-- kx, ky, kz. And kx, I have nx number, n atoms in the x direction; ky, ny number; kz, nz number. And the factor of 2 is because I have spin, electron spin 1/2, positive/minus-- negative. Electromagnetic wave is a factor of 2. If I want to do full arm, maybe it's a factor of 3 because my [INAUDIBLE] if they are all the same. And so this is the number of quantum states. And this is, at each quantum state, what's the average number of electrons. You can think that for a phonon or for a photon, it's all the same way, all right? AUDIENCE: Gang, so it's counting the wave vector, it's counting the number? GANG CHEN: Ah, well-- AUDIENCE: Why is it number minus the nz over 2? GANG CHEN: This is a-- yes. It's counting the number. But see, then I do the integration conversion. AUDIENCE: Yeah. I see you're counting the wave vector. GANG CHEN: OK. Here is-- I'm counting n. These are the discrete states, right? And then this is a standing wave picture. So everything's positive. And when I do traveling with two directions, I said that these are equivalent, OK? So that's why I go from negative n 1/2 to positive n 1/2. It's just equivalent. OK? Why I count the numbers? Because my next step is convert the number into integration. Because we're counting states, right? It's better to do integrated states rather than fractional number. This is the number space. That's where I convert this-- if I think about wave vector, k is a continuous function. And between two states, that delta k is 2 pi over l. So this is converting this integer summation into-- no, this summation into integration, right? I did that same thing when I do electromagnetic modes. So in the x-direction, if this is the length, but the next two adjacent k-- n is 2 pi over [INAUDIBLE] x, so this is the number converted into here, [INAUDIBLE] down here. So here, I have the integration kx, ky, kz. Those are all mathematics converting-- so the conversion and the counting. And then the reason I want to do this now is I have an integration dkx dky dkz. And 2 pi now, I have is an 8 pi cubic. lx ly lz is the volume, factor of 2. And now the f-- I'm going to just ignore that 1. Plus 1, that's Boltzmann approximation. So the Fermi-Dirac distribution. OK, so I'm plugging this f already here. So that's my number of electrons in this region, in the conduction band. OK. I encountered this classical problem. My integration is over a wave vector. My function is over energy, right? Same thing in the electromagnetic wave. My function, my integration is over kx, ky, kz. My function is over energy. And I have to choose either energy or wave vector as my variable to do this integration. I already have the relation between E and k. That's my problem here, E and k, right? So I'm going to do that conversion. In this case, it's much easier for me to convert dkx, dky, dkz into E because you can say that's a Cartesian coordinate. I can convert it into spherical coordinate. And so I'm doing this conversion. Now I do per unit volume number density. And I do that dkx, dky, dkz. In spherical coordinate, k would be magnitude. So it's 4 pi k squared dk. So that's from Cartesian to spherical function and variant. Now I replace because I know the k, k square is 2 me minus Ecx squared, right? That's the expression. This is a ac squared times k squared. Now, k is my radius, like r, right? So I replace this k square. k square equals 2 pi [INAUDIBLE] minus c over x squared. dk becomes this d. And everything is in [INAUDIBLE]. This will be same thing you do all the time in counting the numbers in crystal. You convert the states-- number of states summation into integration. You choose your integration variable, either in energy or wave vector. And if I group everything all together, this is my f. And this one, I call it the density of states. OK. It uses-- it's mathematics. [INAUDIBLE] nothing-- physics wise, you don't have much physics. It's just a mathematical way of converting my summation into integration. So this is the number of quantum states per unit volume because I normalize the volume. And per energy interval, because here is d, that's a physical interpretation of my derivation. And in fact, if you go to look at this one, you can do the integration [INAUDIBLE]. That's why I use the Boltzmann distribution. If I put a Fermi-Dirac-- if I put a, say, 1 over this, not this, plus 1, I can't do this integration, OK? So the reason I use the Boltzmann distribution is that I can do this integration, and this integration gives me-- here is a kpt, and here is E minus c maybe exponent. So how I determine the chemical potential mu, if you give me the number of dopant-- so I put a phosphor in the semiconductor. Per unit volume, I put how much. I know that. And if I assume each phosphor donates one electron-- assumption, right? And there are more rigorous statistical ways to say how many electrons are activated. But because phosphor energy is so close to silicon, so you can roughly approximate this. So every atom that you put in there donates one electron. So with that, I know the left-hand side. I can do the inversion. This is where my chemical potential is. So chemical potential is determined by the number of carriers you put in. And then you will see thermoelectrically, you have to optimize this carrier density to get the maximum performance. OK. So let me summarize this, right? This way of doing the-- converting it to the number counting, you can always start with that summation, I said. But at the end, the most times you will end up with this density of states. It's just that you derive [INAUDIBLE] state. Every time. Once you become comfortable, you can say, OK, this, I can write it directly the number of electrons per quantum state. This is the number of quantum state per unit volume per energy interval times [INAUDIBLE]. That's the number of [INAUDIBLE] quantum states there are. And that gives me the number of electrons near energy E. And I integrate over all possible energy states. That should give me the number of charges. So once you gone through this a few times, you can just start from the [INAUDIBLE]. And I wrote down for Boltzmann-- so then this is just under Boltzmann distribution. I can do this integration, and that's what we have. And what it means is if I gave a number of dopants to the phosphor, I can see where my potential is, [INAUDIBLE] potential, from here given n [INAUDIBLE] inverse, I know E matches c. And the Boltzmann-- so this is only valid when this mu is far below Ec. Once the mu gets into here, you have to use the Fermi-Dirac statistics. That's a degenerate semiconductor. This is the only approximation. But here, it's general. You can always do that math to inverse the given number. You can get the [INAUDIBLE]. And the density of states is a mathematical concept. But see, it's very convenient. And really, it's the number of quantum states per energy interval that we-- and in the parabolic band, it's a problem. It's a square root relation. It's obviously increasing as square root. And of course, below Ec, this is where Ec is. At the bottom here, there's no states in this region, in the forbidden band gap. So that's a typical bulk material [INAUDIBLE] substrates. And then when you fill, it turns out that most of the time, there's only a small region because here is kt. So the electrons are mostly in this region when you do the measurement. Once nE becomes very large, that is [INAUDIBLE] factor very small. Much larger than kt, this is a very small. OK? So that's what we went through. And I saw that I was going much faster. I will do that next time. This is-- actually, just write down my [INAUDIBLE] in the original syllabus. Any questions? Now, if you're a mechanical engineer, again, I'm sure this is a [INAUDIBLE] normal mechanic [INAUDIBLE] talk about this. And I don't expect you to completely master it, but I think what you'll get is say, oh, I know the periodic table. I know the metal and dielectric difference. And I know that [INAUDIBLE] vibration in the crystal quantize and the basic unit of that quantization of the phonon. Now you know electron, you know phonon. And then the thermal conductivity, remember, comes from electron phonon. And in thermal electrics, the s squared comes from electron. So you have to really get this basic picture [INAUDIBLE]. Any other questions? OK. |
MIT_2997_Direct_SolarThermal_To_Electrical_Energy_Conversion_Technologies_Fall_2009 | Lecture_5_Current_research_on_thermoelectric_materials.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. GANG CHEN: --not a question, I will continue. So, again, I think what you're saying is exactly-- that we start-- so it's a very steep dive. We start from very basic thinking. And then we're trying to understand the materials-- what determines the material properties. And this lecture-- after I told some of you this is topic of thermal conductivity, what I'm hoping is to real [INAUDIBLE] from here, what people are thinking in making materials better. But let me start with a very brief review of last lecture. We showed what is a simple-- remember, my picture is always simple. It goes from left to right and right to left. And the difference gives you the flux. So from there, we show the general relation. The current flux is related to electrostatic potential. So that's the charged driving force for charge flow. And then we will have a temperature gradient that is comparable to the [INAUDIBLE] gradient. So now you see the electricity to heat properly. And then this one shows the peak current. So really, now, electron is carrying the heat, and the heat current is the electric current. Heat current is, again, coupled, in this case, not just to parameter gradient, but also to the actual potential gradient. And this is a very general [INAUDIBLE] relation. If I have written-- really, this gradient in terms of generalized force, and this coefficient-- across coefficients are equal. And this is what the [INAUDIBLE] before. And then I gave some-- we use some of the concept we learned before-- for example, density of states and the-- we have expression for this coefficient. So one is the electric conductivity. And this other coefficient from this other coefficient, called the Seebeck coefficient, and the [INAUDIBLE] equations. Let me just make one remark that those expressions are not the most general. You can have different transport regimes. This is the regime that most people do-- the most common transport regime. So those expression-- you can-- rather than [INAUDIBLE] derivation, you can access from Boltzmann equation to derive those coefficients. And I show examples. One thing that I want to emphasize is that the properties depends on temperature. So different temperature-- you can see the property [? attribute. ?] For electric conductivity, if you follow these simple Boltzmann statistics, you actually get the electric conductivity. In semiconductor, we know typical microelectronics type of doping. The electric conductivity increases with temperature. But for thermoelectric material-- so this is not quite right for thermoelectric materials. You can see the thermal conductivity, electric conductivity increases with temperature. And that's because of the doping, very heavily-- metals like this. Metal-- the electric conductivity decreases because when you get hot, they are not more photons. The atoms are vibrating more violently, and that's scattered electron. And that's why the electric conductivity will decrease. And they see that this typically will have a peak. And if you just talk simply, one-band transport. So one-band means if I have a conduction band and each electron only go one to one conduction band, then the series say it should increase with temperature because as the temperature increases, the separation of the chemical potential from the bottom-- so the [INAUDIBLE] chemical potential [? from ?] the center of the band. But in almost all materials, the behavior is you will pick some temperature and start to decrease. And the reason is, as the temperature increase, you excite-- intrinsically, you leave the thermal energy and the electron from the valence band through the hole behind the electron conduction band. So in this case, now, you have both electron and hole. So when you flow current-- electron current energy in one direction, flow current energy in the opposite direction-- so they cancel each other. In this case, you actually could get a higher electric conductivity, but the Seebecks cancel each other. So this is-- when you look at experimental data, it turns out that [INAUDIBLE] we do that and say, we cannot calculate exactly. So we look at experiments, and this is what happened. This is why it died. And this is where-- OK, how you can move this to higher temperature? If you have more electron, you dope it more heavily-- you will go to higher temperature. And if you have larger bandgap, it's more difficult to generate the thermal electron-hole pairs, and then you can move this to higher. So that's how you look at the different configurations with different materials. So that's where we were in the last lecture, and those were mainly discussion on electrons. And we also talked about the whole concept of lattice vibration. And we gave a very simple expression, before, on the phonon thermal conductivity. And again, my picture is left to right, right to left. So when I want to find out the heat flux going through any band, I say, how much heat is going in from one side to the other without scattering? So this is one relaxation time. And how much from left to right? [INAUDIBLE] one relaxation time. So difference gave me the heat flux. And what I want to do is a little bit more-- better [INAUDIBLE] meaning before-- I will just say this is the number density n per unit volume. Now, I'm going to do modes, so I'm going to sum up-- so this-- it's the same as what I drew here. But here is more rigorous. So this is the energy of one Coulomb lattice vibration-- [INAUDIBLE] of lattice vibration, so one phonon. And then this is-- per Coulomb mechanical modes, the Boltzmann-Einstein distribution Per state, quantum state, how many phonon you have? And this is the velocity. So this is really telling me, for quantum mechanical modes, how many phonons is moving to the forward, positive x direction? The one path gives me path-- random motion. So the idea is the same, but now I'm trying to do a better job counting-- better counting job. And here, what do we have-- this is-- remember, in the second lecture, I was doing blackbody counting-- blackbody radiation. So this is counting all the quantum mechanical allowed modes, kx wave vector. It's a [? phonon ?] wave. kx, ky, kz. So I'm summing up all those modes. And that's from left to right, minus from right to left, giving me the left flux. So that's what I showed before in simpler form. But if I do this-- and I will get the integral-- no longer just a simple one third of cv gamma. That's what we had before. And now it's the integration of all the frequency, each frequency. Remember, [? phonon-- ?] so the acoustic wave-- for example, low-frequency acoustic wave-- that's what you hear. And they have the different mean-free path, different group velocity, the higher frequency ones. So this can all change as a function of frequency. So what we're writing now is a more rigorous form for the thermal conductivity carrying heat conduction by the last waves. That's really important. So this is a typical [INAUDIBLE] bulk material. And people have measured the thermal conductivity. And this is a typical curve of thermal conductivity, of material versus temperature. And this is specific example of gallium arsenide. And you can say there is a peak. So there's a-- in bulk material, its peak is typically between 10 to 20 kelvin. And then after the peak [INAUDIBLE] it drops below, it increases. And this actually reflects different physics. So at a very high temperature, each mode is carrying energy kt. So the number of phonons is also proportional to t, temperature. And in this case, what happens-- because as the temperature increases, you excite more phonons, and they scatter each other. They scatter each other, and they drop the thermal conductivity. So that's why the thermal conductivity is dropping. And in fact, if you follow the [INAUDIBLE] same, the number is proportional to t, the thermal conductivity is typically inversely proportional to [INAUDIBLE] temperature. But see, that's a simple picture. And then in reality, this exponent is not 1. It's typically, depending on the material, between 1 and 1.5. People still are trying to figure out why it's not exactly 1. So this is a higher temperature range. And as you go to lower temperature, the phonon-phonon scattering-- the lattice wave interaction with itself-- becomes weaker. So in this case-- so the thermal conductivity will increase, even though-- if you go to check specific heat-- specific heat of-- which, in this dielectric is due to phonons. Specifically, this is typically this curve. I forgot to show one. And it's not a constant. It increases from low temperature to the high temperature. But thermal conductivity-- so even though there are less energy-- specifically the measure of phonon energy-- less energy in this region compared to this region-- but because less scattering, the phonon goes longer. So the thermal conductivity actually increases. So in this region it's typically phonon-phonon scattering-- no longer dominant [INAUDIBLE] you've got an [INAUDIBLE] atoms. So for example, in silicon, if you have a phosphor or boron, you dope it, they are lighter. So you think about that-- you have a string. Again, phonon is the lattice vibration. You have the same mass [INAUDIBLE] And suddenly, you put a different mass that created some scattering of the waves. And so this is the region where the limit-- the peak here is very much limited by these impurities in the material. And then when you go to very low temperature-- so the mean-free path becomes very long. In fact, even you can measure-- make a fault material, the mean-free path is longer than your sample size. So in this case, the boundary scattering becomes dominant. And the mean-free path is no longer dependent on the frequency, but dependent on-- is limited by the sample size. So in this region, the thermal conductivity is exactly due to the specific heat. So the behavior-- just specific heat and sample size. So-- yes. AUDIENCE: No question because I just think-- because of [INAUDIBLE] the power. That's why t-- GANG CHEN: That's exactly the reason. So, by the way, this is why blackbody radiation to the fourth power. Specifically, the third power-- the energy is dU dt. So if you integrate, the energy is the temperature fourth power. And the [? phonon-phonon-- ?] this is a regime. Unfortunately, it's a very low-temperature regime. But this regime, phonon behave almost identically to photon. Now, scatter-- very weak scatter. So you can see the temperature-- the power is the same. So that's a typical behavior of lattice thermal conductivity in crystals. And just give one example. The highest thermal conductivity material is diamond. And if you purify the diamond, you can change this peak by-- from this peak. Diamond-- [INAUDIBLE] temperature is about 2,000. And that peak go very high. And if you purify it-- just to say even isotopes-- carbon 12 versus carbon 13-- you can significantly limit this peak. [INAUDIBLE] room temperature, the thermal conductivity will be influenced by the [INAUDIBLE] of isotopes. So now if I combine the thermal conductivity, it turns out lattice has a contribution. The electron also carry heat. So if I have an electron and I have four-- four [INAUDIBLE] So I have electron hole. And then I mentioned-- when they move-- when the electron hole or diffuse, the higher temperature region-- you may get more electron hole. When you go to lower temperature region, they have less electron hole. What happens? The electron hole will combine each with each other and give off heat. So in that case, they carry the potential energy from one place to other. And this is the bipolar contribution, which is a really bad for thermoelectric. So this is an example of-- most people who do thermoelectrics don't pay attention to this, and that's wrong. For example, the electronic contribution-- you can calculate it from Wiedemann-Franz law, The electric conductivity divided by thermal conductivity is proportional to temperature [INAUDIBLE] constant mentioned in Wiedemann-Franz law. So many people will do this. They will calculate this term and this term, and they measure this term, and they subtract the difference, gives you the lattice. But they don't calculate this term. And this is actually difficult to calculate, and that's the reason. But you can see that the thermal conductivity at a higher temperature goes up. The lattice thermal conductivity always goes down. And the reason for this goes up is due to this term-- a very strong signature of bipolar thermal conductivity from electrons and the holes. And correspondingly, the Seebeck goes down-- the Seebeck-- because the electron hole cancel that. But the thermal conductivity is still getting both terms, [INAUDIBLE]. So now let's go back. I've talked a long time on this [INAUDIBLE] which parameters-- electrical, Seebeck, thermal conductivity. And we see that the thermal conductivity in the electronic part and phonon part. And now if I divide it by the electric conductivity-- so here, I have Lorenz number. And here, I have phonon divided by [INAUDIBLE], so sigma t. That's just the simple manipulation. And the Lorenz number in metal-- that's the exact number. But when you go to semiconductor, this depends on doping. So that's why I read it out as a function of n. But see, if you look at it, in metal Seebeck cancel each other. The electron above the Fermi level, electron below the Fermi level cancel each other. You will get a Seebeck [INAUDIBLE]. So you plug in those letters, and the best you get-- the most metal is about 0.01. When I first got into the thermoelectric proposal, I didn't know whether metal was good or bad, or semiconductor. Well, now you take a good thermal [INAUDIBLE]-- Bismuth telluride-- that's a typical example. And Seebeck is about 200 microwatt per kelvin. And [? phonon ?] thermal conductivity is about 1. Electrical conductivity is essentially 57 per meter. So Seebeck is just the inverse of [INAUDIBLE]. So that's the unit of electric conductivity. And you plug all those numbers, and then you find that's of the order of 1. So from 1950s to 1990s, the best thermoelectric materials have a figure of about 1. People have-- because experimentally, you can't get better. So people are reading a lot of papers on why we can't break-- this is the absolute maximum. So they're questioning whether their theory-- trying to say, why this is the maximum. So this is the-- really the theoretical foundation. We've talked-- and the last slide on this is a mature-- I said sometimes, you can get a really good one, or you throw away because the doping isn't right. You can say that-- say, because the doping determine where is the chemical potential, So the properties are so intertwined with each other-- Seebeck coefficient as a function of doping. Here is a bipolar effect. In fact, this one-- why it goes down? I have to check. The electric conductivity-- this looks better. The electric conductivity increases, and the thermal conductivity-- and you pick your zt at a certain value, So this is clearly-- so you have to really increase your doping. That's adding other materials or creating-- there are different ways, creating doping carriers. So this is the difficulty with thermoelectric optimization. A lot of work one has to do. Now we're going to really just say, what are the materials? Yes. AUDIENCE: That graph was actually measured experimentally? GANG CHEN: These curves? AUDIENCE: Yeah. GANG CHEN: These curves were calculated. AUDIENCE: So they're calculated, GANG CHEN: Yeah. Typically, when you see, like, two dots, that's mostly experimental. OK. Those are experimental-- what are you seeing here in this frame. So this is the summary. Not always updated, and people always are reporting progresses most days. So here is a summary of zt of different materials. And what people have traditionally been using-- this one's tenorite. That's in all the modules-- commercial module-- [INAUDIBLE]. And this is the function of temperature, So this commercial zt takes around room temperature-- 1. And [INAUDIBLE] tenorite is in the intermediate temperature-- 400 degree, 500 degree. And the peak is less than 1-- 0.8, 0.9. And the silicon-germanium-- this is what NASA has been using. NASA used it for space missions. They use nuclear as a heat source. But they build their system so they operate at a very high temperature. So silicon-germanium is what NASA used in the past. And remember, if you want to build a device, you have to have both n-type and p-type materials. So it's not all shown whether it's p-type or n-type. But for each type, you build a device-- you need both n and p. And also, ideally, if we go back to look at the notes in the device section, when we talk about device, you want these two materials closed-- properties closed. Otherwise, it's not the material that determine your efficiency. So you look at-- let's say the same temperature range-- what materials are available? And like I said, zt has been 1 most of the time, and there are some recent progress. We'll discuss these progresses. This is a more traditional material. Like I said, bismuth telluride, then tenorite, silicon-germanium, n-type and p-type materials. And skutterudite is a newer material. So the commercialized bismuth telluride-- bismuth telluride-- go to periodic table. I can't remember all those positions, every time I'm in my office I look at the periodic table. But those are heavy. So bismuth, [INAUDIBLE] are heavy elements. And why we want heavy elements in this case? Why is it good? So you look at it-- as it becomes heavier, the bandgap decreases here. This is the bandgap. The bandgap decrease is typically about-- you need-- for the bandgap versus temperature range-- around 10 kg. 10 times kdt-- kt. So at room temperature, the kt is 26 million electron volts. So 10 is the 0.26. Turns out the bismuth telluride-- the bandgap is about 0.15 range. So you go to, say, heavier material-- your band gap decreases. And now you think about thermal conductivity. Recall that when I say the vibrational frequency, vibrational frequency is the spring constant divided mass square root. So we have our mass-- you got the lower vibrational frequency. Lower vibrational frequency-- you look at the group velocity-- it travels slower. So the phonon [INAUDIBLE] to go more slowly than lighter atoms. So that's why the thermal conductivity is lower when you go to a heavier material. So bismuth telluride has a very complicated unit cell. This is really-- in terms of full periodicity, when you consider a hexagonal unit cell, it's really 30.7 angstrom-- many atoms here. And it's interesting. This is where you really want to understand the material. So it's a polaron bismuth terms of linear structure, And this is the polaron. This is the bismuth polaron. And in between the layers-- you got you see here-- this [INAUDIBLE] was bonded. And here is more a mixture of covalent and ionic bonding. And there are a lot of study-- people look at which layer is conducting electron, which layer is conducting hole. And it's really-- we're still working on bismuth telluride. And it's important to understand those details because people have been working this for many years. You want to do something, you need to look at detail. Where are they? Which are the layers of electrons? Think about something that you can-- example-- still got the electron transport well in this layer, but screw up the phonon, kill the phonon propagation in the materials. So people calculate the band structure. I'm not going to detail. This is really-- [? obviously, ?] you do this. This band structure calculation is also relative standard. You do a density functional theory calculation. And this is a pure-- bismuth telluride-- If you just do a pure bismuth telluride, [INAUDIBLE] the best is about 0.6. And how you get the 1 is really by alloying. Alloying-- so you can further reduce thermal conductivity by alloying. So bismuth telluride is a compound semiconductor. And you look at the same column of the periodic table. When we are thinking alloys, same column, periodic table. And then you substitute, for example, bismuth by antimony. And that way, you could reduce thermal conductivity. And the best example of alloying reduced thermal connectivity is silicon-germanium. And silicon-germanium-- this is not a thermal conductivity unit. The inverse thermal conductivity unit. So here is the silicon [INAUDIBLE] conducting. The thermal conductivity of silicon is about 150 watt per meter per kelvin. And the germanium is also very conductive-- about 60 watt per meter per kelvin. Now, you mix a little bit. You can see this is not a linear curve, You mix a little bit of silicon with germanium. Your thermal conductivity drops sharply. And the lowest thermal conductivity here-- your [INAUDIBLE] is about 67 watt per meter kelvin. So you drop from 100 to about 67 by alloying. And this is because-- again, if you think about it, you have an [INAUDIBLE]. First thing you do-- you can try with pure atoms there. Secondly, you mess up the atom periodicity. The wave becomes frustrated. And so the thermal conductivity will drop. So this is one way of scattering. And that's-- all thermoelectric materials that are made of alloys. And the typical [INAUDIBLE] of the alloy is in terms of scattering-- the additional scattering. And you can see here, this is a Rayleigh scattering relation. What is Rayleigh scattering? You go outside on a sunny day. You see blue skies. And the blue sky is-- the blue light gets scattered. And high-frequency blue light compared to red-- solar radiation. The high-frequency scatter more. So here is the Rayleigh scatter regime. That's the Rayleigh scattering, high-frequency. And the sound wave is the typical-- that's a typical model. High-frequency is the inverse time to scatter time. Relaxation time is shorter. So that's a Rayleigh scatter. And then this depends on the mass contrast, and also it depends on lattice parameter, lattice constant contrast. Those series were done in the '50s. And in fact, I have a lot of suspicion whether they are truly valid or not. And I say that's what people do. This is just the current status. The problem is you go to think about thermal conductivity-- I showed k equals one third cv gamma integration. And integration is very forgiving in the sense-- if you make the error in one place, you make up the other place. And when you don't have exact theory there and you got a lot of [INAUDIBLE] parameter, and that integration can help you publish your papers. But the details, very often, are missing. Like I said, this is-- typically, that's how people make the material-- bismuth in [INAUDIBLE] bismuth, antimony. And this x is about 0.3-- 0.3 to 0.5. And then the impact material, bismuth, antimony, and selenium. And if any of you, actually, are in front of physics department, you probably went to-- I remember a few weeks ago there was a seminar on topological conductor. Turns out that this material is very hot in the physics-- people are looking at antimony, telluride and its topological conductor. So very hot topic there. But see, we don't know whether that has any implication of thermodynamics or not. And doping-- you can always add other materials as doping. But say, here, typically, is by defects-- the so-called anti-site. So if you look at it, it's possible that bismuth go to tellurium site. And this is a particularly-- because bismuth and tellurium, their so-called char-- electronegativity is very similar. So when they exchange positions, the disturbance will increase very small. But the number of charge is different, so that create either holes or electrons. So that's the traditional material. We're now moving to more current materials people are talking. This is a oxide-- sodium cobalt oxide. And these were originally started in Japan. Terasaki made a lot of contribution to this. And it turns out people are interested in oxide since 1980s. Why? That's because the high-temperature superconductor was discovered out of oxide. It's a surprising amount [INAUDIBLE] BCO, boron, C, copper, oxygen. And that got lower [INAUDIBLE] So the transitional temperature, now in this high-temperature superconductor, has reached probably around 130 kelvin or 140 kelvin [INAUDIBLE]. But the superconductor itself is not interesting for thermoelectrics. Why? This is because there's only one quantum state. So the entropy is 0. Log of quantum state. That's the Boltzmann relation. Entropy carried by as a-- is a log state. So superconductor is not interesting. But this one is not a superconductor, and people have achieved about [INAUDIBLE] one. I think this is around 800 degrees Celsius, so it's a high-temperature application. [INAUDIBLE] I mentioned this material because if you go to Japan-- I talk a lot to Japan thermoelectric guys, and also, from company. Whenever you work on-- you look at the bismuth telluride-- Japan is very sensitive to-- because they have limited amount of materials. It's a small country, and the resources are limited. So they don't want-- they always say, can you make, out of this abundant material-- what are the abundant materials? Silicon and aluminum. So those type of materials. And so this is the one example where the material is called abundant-- nickel, tin, and titanium. And it has reached a reasonable figure of merit. And this is where-- for example, you look at [INAUDIBLE] this figure [INAUDIBLE] thermal conductivity is still very high. Why it's still very high? Why is it very high? Because you always think of bismuth telluride. That's a good number to remember. Thermal conductivity is about 1 or 1.5. Seebeck is about 200 microwatts per kelvin and electrical [INAUDIBLE]. So if you look at this, the thermal conductivity is in the denominator. And you get this good zt-- that means the electron is really good. x square sigma-- that's the power vector-- it's pretty good. Do you [? kill ?] this guy? Do you [? kill ?] it? Thermal conductivity, reducing. Particularly, Can you reduce the phonon pump. You don't just want to reduce it-- because the electronics are also part of the thermal conductivity-- this is part of thermal conductivity. So people are now looking into-- if you look at the pump materials, it's typically a bismuth telluride-- then telluride silicon-germanium. Those are simple structures-- simple compounds-- two atoms, binary. And now, people are looking-- going more into more complex structures. And why complex structure? From thermal conductivity point of view, there is a good reason there. This is a few lecture back. I was talking about the modes, in terms of optical phonon, acoustic phonons. Optical phonon is very flat, and acoustic phonon is really the one that carries heat. So when you got a lot of atoms in the unit cell, you have a lot of upper branches. So those upper branches don't carry much heat. So that's-- the more complex the crystal, typically, the lower is their thermal conductivity. And that's the only very simple hand-waving way of saying it. And of course, you can see some of this material has really low thermal conductivity. Bismuth telluride is here. And it's a particularly [INAUDIBLE] now. This [INAUDIBLE], zinc, pyrithione. Zinc is abundant. If you can make zinc-- zinc is a good one if you can make it work. And it has a very low thermal conductivity. And in fact, I think the antimony thermoelectric figure of merit, now has reached-- I think it's about 1.4. Pretty good, except that there are also issues-- material stability. And when you heat up to high temperature, zinc speed up. I have some pictures showing that under temperature gradient, the material grew here. Here are zinc coming out. So you've got a lot of material reliability issues. But generally, there are a lot of people looking into a lot more complex structures, crystal structures, and the broader-- a much broader range of thermoelectric materials. I mentioned this already, but now let's look at the ideas. We said that complex structure-- and this is another idea that's really interesting because the alloy-- if you think about silicon-germanium alloy-- and it's really-- germanium go to silicon atomic site. So you can think of that as a substitution, So silicon-germanium substitute silicon atom. The lattice-- if you think of a cube. Like, let's take a crystal [INAUDIBLE]. Think about cube. And the silicon sits in the corner of the cube. And now you take out the one off the corner, and you put in germanium. And so this is the substitutional type of-- I pulled that-- you can think of the impurities. And skutterudite is a simple, interesting material. This is a skutterudite that is cobalt antimonide, or rubidium antimonide, iron antimonide, that type of structure. And what is interesting is that it has a very large unit cell. A lot of atoms there. And also, in those unit cells-- so here, we got two [INAUDIBLE]. And so you can see this-- each is a square, But in this eight-- How many? We have eight cubes. Six of the square has atoms inside. Two of them don't have anything inside. So there is an open space there. The open space gives you an opportunity-- let's put another atom there. [INAUDIBLE] Another atom-- this atom has a very different [INAUDIBLE] from the host atom, cobalt antimonide. So the idea is like a rattlesnake. You put the atom-- the rattler on. And this rattler will disturb the normally periodical lattice strips in the surrounding material surrounding the lattice itself. So that was called phonon rattlers. And the next one-- is an example of-- you can see cobalt antimonide. And you fill this with other regular lanthanum. And people tried many different ways. This is-- I was in a meeting last week. And people are still putting-- first with one rattler, one atom, one type of atom. Now, people have gone to three rattlers. Well, the hope is each rattler have its own rattling frequency, and you kill the whole acoustic frequency of the whole lattice waves. So this is clearly a very dramatic decrease in the lattice thermal conductivity. And people have done neutron scattering to actually look at the atomic displacement in this material. So you can see here, this is a lanthanum cobalt antimonide. So here, you have iron, cobalt-- that's their atomic displacement. So inside, it's doing Brownian motion. And this is measurement that can actually measure what's the amplitude of the-- so the [INAUDIBLE] square is here. And what's the amplitude of that motion? And the lanthanum-- they put it into the cage there since we have a very large amplitude here, you can see. And the theory is that this scatter deforms, Again, the theory is still evolving. And the explanation-- people came up with different explanations. But you try to do the modeling experiment to really make this material [INAUDIBLE]. Here, I forgot whether this is a-- what type, n-type, but I do want to tell you a story. Many years ago, people reported the p-type skutterudites, cobalt antimonide, figure of merit for 1.4. And now nobody can get 1.4. Everybody-- the best they can get is 0.95. And the reason-- this will bring up a very important point. Thermoelectric research-- measuring this property is tricky. And you have three properties to measure-- electric conductivity, thermal conductivity, Seebeck. And even if you are very good, each property-- you say, OK, I have about 5% uncertainty, which I think is pretty good. And you go to [INAUDIBLE] and that's about 15%. And it turns out that many times, your system can take the same sample. So you do thermal conductivity. You take a sample. You go to pick another sample for electric conductivity. And you measure the wrong things. And sometimes, if you look at the-- telluride-- it's a layered structure. The property is very isotropic. So this will be really dangerous. You measure-- even the same sample. You measure properties-- thermal conductivity in one direction. You measure electrical in the other direction. And you can just get a good zt. That's not true. So this is-- every time you go to a meeting, people will be debating. And most of the time, people are skeptical of any report. It's just difficult because measurements are difficult. Now I turn to my favorite-- nanostructure. And I said that there are bulk material. People look at the different material, different crystal structures, phonon rattler, all working on crystalline materials. And there was-- say-- this nanostructure approach was really Millie Dresselhaus in the EE department at the time. And the story-- this is a really good story. Around the 1990-- US navy start to rethink thermoelectrics. So they went around-- asked around and they came to Millie, whether she had any idea on thermoelectrics. Millie didn't work on thermoelectrics before. And she thought about it. And there was a physics student who walked into her office, wanted to ask her whether she had a topic. So she said, oh, why don't you look at quantum well-- whether it can give better thermoelectric properties? So the student looked, and-- it's really a homework problem. I'll show you later. But the calculation turns out to be really interesting, and a lot of people started working on that. So sometimes a simple idea really stimulate the research. And here, this is a slide that I tried to make and seeing the potential of using metal. So you think about the electron and phonon. Electron is what carries the heat around. And of course, the quantum mechanics, they always-- we mentioned this [INAUDIBLE] before, but they have different wavelengths. And it turns out it's a spread. In real material, it's always a spread. And we need to look into the details. But in general, actually, electron movements can be longer, average speaking, than the phonon-- heat-carrying phonons. And so that's an order of magnitude that I often use. But again, each material is different, and each country is different. And the other is the mean-free path. That's the average distance in electron-phonon [INAUDIBLE]. So we look at-- one possibility is to look at the wavelength difference because the wavelength difference means-- an intuitive picture is if you have a surface, whether it's a smooth or rough-- really depends on the feature, on the surface versus the feature of the wave. If you have a visible light coming towards a surface like this, it look rough. You scatter light all direction. But if you have, really, a microwave coming in, very long wavelengths, the surface will be smooth. Much-- wavelengths larger than the roughness features. So while this idea was-- if you recall, the thermal conductivity-- the curve-- at a very low temperature, the thermal conductivity increases with the specific heat because the mean-free path equals the sample size. It's limited by sample size. So low temperature already have the size effect. And if I make the structure very small, that size effect-- I can shift to the high temperature, as long as it's small enough so that the-- let's take a wire or film-- if the diameter or thickness is comparable to the mean-free path, now, the phonon will no longer-- is a phonon-phonon collision. They will collide with the surface. And another simple intuitive picture is-- OK, what happens at the interface of the material? You have the lattice wave coming in and at the interface, you hear echoes. Well, we hear echoes for the visible-- audible range. But say-- and the peak current is [INAUDIBLE]. So those are high frequency. You still got the same phenomena happening. And the echo means some [INAUDIBLE] don't go forward anymore. So you could use those phenomena to [INAUDIBLE] the thermal conductivity. And Millie's idea at the time was squeeze electrons. So quantum effect-- which I'll talk more. But-- now, think about the quantum effect. Electron is a wave. Wave function is a function. The s orbital-- wave function is a ball. And let's say this is the size of the ball. Let's say [INAUDIBLE] angstrom. Now, you see this-- this is electron. You make a film only 10 angstrom or 50 angstrom. What happens? The ball is no longer a ball. I once-- when I was a student with [INAUDIBLE] there was an interesting analogy made by Professor [INAUDIBLE]. But think of this. You put a basketball into a piece of box. You squeeze the ball. You have energy quantization. So you have this wave, and then when your size is small, you quantize the wave. And ideally, what you want is this wave is-- under ideal quantization, you want this smooth surface, You don't want the interface-scattered electrons. And this is possible, again, using the wavelength range because whether the surface is smooth or rough depends on wavelength. For the case of electrons, the scatter is electrical potential. Your potential-- whether it's rough or not. And for the case of phonon lattice wave, fundamental or electrostatic-- but it's really-- you can think of mass spring, whether those-- the force is [INAUDIBLE] now. And so you actually have the potential-- if you think about the phonon and the electron-- your electron-- if you quantize it-- is the smooth. And because the wavelength difference-- you can potentially actually have the same surface will be rough for phonon, the acoustic wave. So this will be ideal. [INAUDIBLE] You want to screw up the atomic vibration, but electron can go as a waveguide. That will be one scenario. And the people started a lot-- and in fact, I should say at the same time, maybe go to other places, and people have a similar idea, although Millie did really a good calculation. This superlattice idea was from Rama, [INAUDIBLE]. And what he did is-- the perpendicular direction, the superlattice. And this is-- you can see-- the experiment-- thermal conductivity and the drop. So the bismuth skutterudite [INAUDIBLE] about 1.2, and they drop to 0.5. Very difficult measurement. And this is where I start getting into thermoelectrics. So my thesis-- my PhD, I was looking at the superlattice heat conduction. I was worried about lasers, not thermoelectric. I didn't know anything about thermoelectric. And the electron here decreased a little bit. This is s square sigma. The power factor, s square sigma, decreasing a little bit. And in the case of-- this is a [INAUDIBLE] map. He grew the telluride-- the [INAUDIBLE] on that telluride. And because the lattice constant is very different, the [? strain-- ?] so you grow certain thickness, the strain will break up the film and form dots. So he grew these multilayer structures. And this picture-- I don't think it is from him. I don't remember. But what he showed was, again, no thermal conductivity. And the power factor [INAUDIBLE]. So at the end, it turns out that, historically, the thermal conductivity alloy-- institutional. And now with this nanostructure, it turns out that the thermal conductivity seems to be a very effective strategy. But let me, again, caution-- almost now everybody questioned those results. It's a very interesting result-- very intriguing, but it's a very hard measurement. And [INAUDIBLE] retired. He's a very respected. And I talked to him a lot. [INAUDIBLE] He has a book. If you want to understand thermoelectric, he has a very good book. Not many people read because it's too difficult. [LAUGHTER] But very good if you read. And the first-- I remember the first two pages was talking about electrostatic computer. As a mechanical engineer, [INAUDIBLE] always been hard to understand, what's the electrical potential, what's the chemical potential? And Raman [INAUDIBLE] is from RTI. They have spin-off company trying to do the superlattices for microelectronics [INAUDIBLE]. And if you're interested, you can check the [INAUDIBLE]. They have [INAUDIBLE] start-ups [INAUDIBLE]. And the superlattice-- I guess the challenge is grew very thick, and also reduced the cost. So this is where my story is. I said I would be working on this for a long time. Since my [INAUDIBLE] was really useful leaders. And so those are experiments. In the along-the-field direction, you can see it's isotropic. This is the [INAUDIBLE]. But see-- the superlattice structure [INAUDIBLE]. So this is perpendicular to the [INAUDIBLE] plane direction-- parallel to the [INAUDIBLE] plane direction. And the problem is, if you think about this superlattice, basically, superlattice was invented by [INAUDIBLE] and Esaki. Esaki got the Nobel Prize for his work in resonant diode. That's before, even, the superlattice concept. And the name of superlattice means you have a super crystal. And so you see-- I say this is one layer. There's another layer. Now, you look at-- a unit cell is no longer cube, but the tetrahedral that cover from here to here. One period of a crystal. So that's what the name implies. According to the name, this is a new crystal. You can calculate a new energy phonon spectrum. You calculate group velocity. You have [INAUDIBLE] states. You do all this calculation. And then you do one third of cv gamma. I don't know gamma. But I know I can-- from that calculation. I know c, and I know v, from this state. [INAUDIBLE] That's the curve. You get as a function of thickness. Just-- it's not correct. So basically, the picture is not about forming a coherent structure. People are still debating, so I don't want to prejudice you. You should read literature. And that's my own prejudice because I've been doing research on that. But when you get into it and say, oh, [INAUDIBLE] look into the literature. Look at the different sides of the argument. And so when I go to look into the other picture, I just say, OK, now I got a surface. I only just consider not this supercell-- just two layers. And of course, periodic structure. We consider all that. And then we got an interface roughness. And that roughness scatter phonon diffusely. And that can cause the thermal conductivity reduction. And here, I got a fudging parameter. Yes. AUDIENCE: When you say, roughness, are you talking about just like obstacle spacing, like an electron diffraction or [INAUDIBLE] diffraction experiment, or you expect something to be scattered by something with spacing [INAUDIBLE] wavelength? GANG CHEN: Right. And so that's a very good question. And what's the mechanism of that roughness? And it turns out that when you grew superlattice, molecular beam epitaxy, and then you find out that they're in the 80s. There are a lot of-- gallium arsenide is the best studied system. And there are diffusion of atoms between the layers, about the one to three monoatomic layer. And they said the phonon weakness is very short. So my picture is really the atom-- if it's a perfect periodic, you see the atomic spacing is all periodic, the same atom. Then you should not have that. But now if you just have one atom [INAUDIBLE] and the random distance, later, another atom [INAUDIBLE], and that's the surface-- whether is rough or smooth-- is about 1/10 of the wavelength. So if you go to optics-- so here, again, we don't know exactly. So you go to optics. When people say whether surface is smooth or not-- it's about a 1/10 weakness. 1/10-- you look at the phase of the wave. So here-- because the wavelengths are falling, is about 1 nanometer to 10 nanometer range. So a little bit of atomic diffusion can cause that. But there are a lot of puzzles at this stage. You should read [INAUDIBLE] paper from UIUC, and they have done constant diselenide and some newer material system. And they found that they can have thermal conductivity really, really [INAUDIBLE]. The noise they have is twice [INAUDIBLE], solid, fully solid. Anyway-- so I say this is the picture I have, but I think there are probably more things that need to be done. So what I see from this picture-- what I conclude is, OK, we do need a periodic structure because if it's not a unit cell type of picture, then I don't need a periodicity. So that's why I need us to do this random composite structures, because that way, I can do [INAUDIBLE] lower cost. So we try this, and it works in some materials. This is [INAUDIBLE] telluride. This is about [INAUDIBLE] telluride. And this is-- this is skutterudite. This is skutterudite [INAUDIBLE] and nano. And [INAUDIBLE] I say what we do is just a [INAUDIBLE] material, and we contact them in nano form. Bismuth skutterudite works on [INAUDIBLE] geometric also. Didn't quite work on that telluride. That telluride grid growth. And skutterudite didn't work out either because of grid growth. So we're still trying to figure out how we can prevent-- stop that growth of the grid, and we drop the thermal connectivity. Yes. AUDIENCE: What about the quasicrystals? Is that [INAUDIBLE] GANG CHEN: That's a very good suggestion. And I didn't look at quasicrystals. But there are people looking for [INAUDIBLE]. Clustering is a very complicated-- like, you can have silicon [INAUDIBLE] germanium [INAUDIBLE]. So it's a-- it has a-- say, it has holes inside, and they have a little similar [INAUDIBLE]. But further crystal decay-- [INAUDIBLE] decay is whether you can make the electrons in your [INAUDIBLE]. Remember, you can kill a phonon if you just want to kill a phonon [INAUDIBLE]. But you also want to keep the electron moving. So the idea-- very good terminology came from Glen Slack. And he said, what you want is an electron crystal phonon glass in one material because the phonon in glass-- this glass here-- amorphous random atomic arrangement. So the phonon thermal conductivity is very poor. But this superlattice is different. It's still crystal. So you are-- where we can-- for example, we're looking into whether a randomized wave-- localized [INAUDIBLE] and localized wave. Very difficult, I think, but it's worth trying because you need this crystal structure to get the electron propagating. Very difficult because the short wavelengths. You can do photonic crystal, all that, for electromagnetic wave because the wavelength is pretty long. But it's very hard to do this for phonon because wavelengths are short, and they don't propagate very long. You need a long coherence to have the waves [? disrupt ?] each other. So those are the difficulties. So here are more detailed example. You can see the thermal conductivity dropped. So when I started this, everybody was saying, electron-- what are you going to do with the electron? I said, I don't know. I wish it would work out. But when I think back, you actually have good reason-- the general idea is, OK, let's don't get too much electrical potential offset at the interface, That's what the scatter electron [INAUDIBLE]. But see, it turns out in bismuth skutterudite, you can see the electron is better. When I first look at this, I couldn't believe it. And the reason is, of course, it's a semiconductor. The electric conductivity depends on the carrier concentration, It's the mobility times n, the carrier concentration, times charge. So turns out the bismuth telluride-- when you warm [INAUDIBLE], you generate defects that generate more electron. Or in this case, it's more holes because it's a positive Seebeck in the positive poles. Again, you can see that's the typical behavior. So at the end, if I plot the s square here-- so higher electron density, lower Seebeck. Higher electron density. Fermi level is closer to the [INAUDIBLE] edge, or you can get into the band. The Seebeck is the average distance thermal energy of electron [INAUDIBLE] So higher electric conductivity, lower sigma-- that's [INAUDIBLE]. But because you have higher carrier concentration, the bipolar decreases [INAUDIBLE] later, you can see here. It goes to higher temperature. And then-- so if you plot s square sigma, this is actually better than this. We try that very hard in superlattice. We didn't get a s square sigma. Both equally thermal conductivity increase [INAUDIBLE] factor, and this actually work. But we still don't know how exactly the model [INAUDIBLE] very difficult to model in random structure. We're doing that. And we're not the only one. In fact, [INAUDIBLE] has been working in this material system, which they call [INAUDIBLE]. And so [INAUDIBLE] telluride-- and the single crystal-- he grew single crystal. He spent several days growing single crystals, pulling heat in, and got the very good properties. And again, let me caution you-- you can't believe at this point because they extrapolate it. No, he doesn't show this anymore. And measurement is always a hard problem. One has to realize-- sometimes I scare people away by saying that, but that's reality of this. And what they found later on is that this material actually-- even though it's a single crystal, but a lot of precipitates out of nanodots. And with that, the thermal conductivity is also quite low. So we got a good figure of merit. This is a really interesting system. I've been talking to [INAUDIBLE] And I still can't really give you a very simple picture. Basically, I was asked give a very simple picture so I can explain to my students. And I don't think that I got a simple picture yet. But the idea it's looking at is-- here, is a silver [INAUDIBLE] It's a cubic structure. Silver antimony telluride [INAUDIBLE] similar. And what it's saying is the electron clouds between the atoms have overlap. And when one set of atoms moving that squeeze this flow of clouds, it's very significant. So the electron clouds are very significant. Squeezing cause the force anharmonicity. And this anharmonicity gives a very low-- so what you want is-- you want to make the atoms strongly scatter each other. High thermal conductivity-- if you just have a pure spring harmonic oscillator, you have infinite thermal conductivity. And the finite thermal conductivity really comes from the third term of the potential. It's not a f equals kx. For f equals kx plus a perturbation, kx squared. That square makes the two different frequency wave interact with each other. That's so hard for optical wave. It's much easier for phonon. But when you want to push to very low, very small limit, you got to [INAUDIBLE] get this anharmonicity very large. And one way to gauge that is this recent parameter, which is related to thermal expansion. And thermal expansion-- so higher [INAUDIBLE] thermal expansion material typically is more anharmonic. And what is showing is this minimum thermal conductivity theory. That's also another thing I never believe-- minimum thermal connectivity. The idea is one third-- k equals one third cv gamma. That's the kinetic theory. And the minimum thermal conductivity is replacing the mean-free path by the wavelength. So that's the minimum mean-free path you could have. And this was a Glen Slack original theory. And then David Cahill and Bobby Powell from Cornell said, oh, it should not be just a mean-free path wavelength. It should be mean free path equals plus wavelength. Einstein-- based on Einstein's argument. So that reduced thermal conductivity minimum by another half. And so very often, people will draw this thermal conductivity against the minimum. So this is the one example we say, OK, you've reached the minimum. You can't reduce anymore. But in fact, I said David Cahill because he measured layer structure. His thermal conductivity is twice the error. It's an order of magnitude smaller than the minimum. So his own minimum theory. There is a minimum-- his experiment, is an order of magnitude smaller. So that's why I say I don't believe in it because you should go look at their better formulation. When you have an [INAUDIBLE] you don't have one [INAUDIBLE] mean-free path. And if you're interested in this, I'll point you more to literature. This is something that I really want. [INAUDIBLE] OK, I bring you from the basics to, really, the forefront. Just a few months ago, there was a paper, indium [INAUDIBLE]. And what it's saying-- this is a charged density wave. And the [INAUDIBLE] instability. And this got really-- So what happens is this direction [INAUDIBLE] this is more like a layer structure. And then into the z-axis, that's the covalent bonding. But because the charge distortion-- so in the z-axis, actually, charge a strongly distorted and create a very large anharmonicity for the last wave scatter. So the thermal conductivity is really strange because when the [INAUDIBLE] direction, force weak force typically means low thermal conductivity. But in this case, what you have-- this is the z-axis for one direction. You have lower thermal conductivity in the [INAUDIBLE] direction. And very interesting, and definitely worth more study here. Now let me go to [INAUDIBLE]. This is the quantization I mentioned and the pizza box problem-- the possible [INAUDIBLE] in the box. So you have a simple-- this is a simple homework. I said before this is the z direction. Energy is quantized. And you can do the same standing wave argument I did before and see what's the wavelength and what's the corresponding energy. And so xy direction, is called the continuous wave vector. So now, for different n, we have the sub bands. So this energy quantized different energy levels in the-- due to the thickness in the z direction. So that's the basic [INAUDIBLE] idea. And now you go to do this. You calculate the density of states. I don't want to go through this math. But you can say this is the same thing we did before, except in the z direction. Now. I don't do-- I do not convert this summation into integration in the z direction. I convert the xy direction because the z direction-- that separation is too large, and the summation into integration-- that conversion is not a rigorous-- not mathematically accurate. So I still keep this summation here. And at the end, what you show is the density of states now is a staircase, or in typical parabolic band approximation, we got this 3D. This is 2D. And now, with that, there is advantage. Where is the advantage? Let's go back to the picture I showed before. Seebeck is a measure of the average energy of the electron relative to chemical potential. Electrical activity is really proportional to the number of electrons. So when I have this-- if you look at it, this [INAUDIBLE] says this is df Fermi-Dirac derivative. Now this is very small. This type of state-- that's the region where it's contributing to electrons. Because of the parabolic, this is a small number times small number. So the electric conduction-- the number of charges that carry the Seebeck-- most of them-- it's very low energy. The average energy is here. Those are low-energy electrons. They carry more electrical charge. So that's a problem with the [INAUDIBLE] band, with the density-- the [INAUDIBLE] density of states. OK, now you do the [INAUDIBLE] You have a sharp staircase. And you can put it-- and you see here-- you can put it in more charge or still have the same Seebeck. Or if you have similarity [INAUDIBLE], then you can increase your Seebeck At the end of what you want is for sigma to be larger, So that's why those quantum structures-- like, in the quantum wells, it can give you larger s square sigma. And then you look at the other quantum structures. The density of states-- this is the staircase for the two-dimensional. And then wires is a sharp peak here. And dots are just discrete lines. So progressively, going from 3D, 2D, to 1D, 0D, progressively better. So that's what the theory says. And at the time, Millie and [INAUDIBLE] in Lincoln Lab did the experiment. And they showed that-- I want, first, to show the trend is all right, except there is one caveat. You see there? This is not s square sigma. This is s square n. Electric conductivity is n times mobility times the carrier charge-- electron charge. And the reason is that the superlattice was not-- the quantum well was not very good-- interface scattered electrons. The electric conductivity-- if you plot-- I don't know the detail, but let's say there are still questions here because this is s square sigma and s squared n, not s square sigma. But the trend is there, and that generate a lot of interest. And this is just a more calculation I got from a book that [INAUDIBLE] and myself are writing and never have finished. I did not finish this. If anyone wants to read their draft. 3D, 2D, 1D, progressively, is better. That's a theory. And if you do an experiment, like I said, you can't scatter electrons. You have to be very careful. And there's another problem. So of course, superlattice-- the quantum well is just the [? film ?] that's conducting electron. And the problem is you have to sandwich this. Or you can use the air or vacuum or quantum wire in vacuum. Then there's no leakage. But normally, when you use another material to create a barrier to confine electron in the field, then you've got the phonons in the barrier. The phonons in this barrier will kill your conduction. So they'll kill your-- create a more [INAUDIBLE] leakage here. And there are solutions on this, theoretically. I've worked with Millie on this before. Just as a caveat, one has to pay attention. For example, there are also study by [INAUDIBLE] and others say if you reduce the thickness-- you want to reduce barrier [INAUDIBLE], you reduce thickness. And then you have another problem where these two [INAUDIBLE] will start talking to each other-- colony. And that colony, actually, is not good. The [INAUDIBLE] becomes less sharper than the pure quantum well. There are potential solutions. I encourage you to read the literature and work on this. But it turns out this general idea of [INAUDIBLE] featuring density of states is a universal, not only in thin films but in bulk materials. And there was one paper by [INAUDIBLE]. And he wrote a paper around '97. And he's a very good theoretician. He just took a simple math problem. He said, I'm going to optimize the density of states. Just write the expression I wrote before. Treat it as a function that d [INAUDIBLE] states as a function of mathematics. What's the function that will give you maximum s square sigma or give you maximum zt? And at the end, he concluded it's the delta function. So delta function is the best. And if you go back to what I showed here, quantum dot is best. Except the electron model-- how are you going to move quantum dots-- the electron going to [INAUDIBLE] delta t or quantum dots. So wire is also good. So this is a general principle. Another interesting comment I want to make is you go to check the Lorentz number. And that's what I suspected-- the Lorentz number in quantum dot can be 0. Very strange. Nobody demonstrated that. Very interesting experiment. Electrical activity is not 0, but the electronic thermal conductivity is 0. Very easy to check. Let me go here. So this is really the reason [INAUDIBLE] and he said-- we talk a lot, but very smart guy, and would say-- He worked on nanowires-- bismuth nanowires. Worked with Millie and-- And he say, I'm going to turn to bulk material. So one day-- he read Russian literature, and this is all, actually, in Russian literature. It turns out there is some doping. Particularly when we think about doping-- we think about doping-- the energy level is inside the band, in the bandgap. But there are some other atoms. The doping level is within the conduction of valence band. And in this case, is in the valence band. And this valence band-- the doping form additional energy states that create a sharp feature in the density of states. And this [INAUDIBLE] feature led to very good Seebeck coefficient and very high [INAUDIBLE]. Very nice idea. The question is whether you can replicate it in different materials because [INAUDIBLE] nobody wants to touch it because it's very poisonous because-- it's not that terrible. I have a friend. And he said, I focused on it for more than 50 years, superconductor, and nothing happened. But the person-- be careful. Now, this is another idea I think, again, come back to-- you can interpret the same intensity of state. This is Seebeck coefficient. If [INAUDIBLE] position is here in the band, this [INAUDIBLE] is a positive. So they cancel each other. Of course, this is larger. So you still have-- the cancellation is small. But if you put a potential-- so you have a particle, or you have an interface, that just happen to have this potential height, then those electrons are scattered. So that's for the energy field or thermionic emission. I'll talk about thermionic emission in the next lecture. We'll talk about thermionic engines. That's a different way. But this is a solid. So you scatter this low-energy electron, and then you only have high-energy electrons [INAUDIBLE]. And if I draw the [INAUDIBLE] state picture is-- although there [? are high ?] electrons in this region, but I scatter them-- they do not move. So now, in terms of this state, what's really moving is-- you still have a sharp picture. So that's my interpretation of this picture-- this different idea. So you can always think about the different states, features for electron. But really, the one that has win so far is-- the only successful example I know is this one. Truly work in materials. So most other approach that's successful is important. So with that, I [INAUDIBLE] will from the very simple things I don't know. And you can figure out. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | Tutorial_Photoconductivity.txt | [MUSIC PLAYING] PROFESSOR: Hello, everyone. In our last demo, we demonstrated how the electrical conductivity of silicon can be changed by over six orders of magnitude by adding dopants that can increase the number of free or mobile charges in the material. Today, we'll show how we can use light to break electronic bonds and silicon, and create free mobile charges. The principles we'll be using today can be applied to everything from sun screen, to of course, solar cells. We'll use the undoped, or intrinsic silicon sample from our last demo, and measure how the conductivity changes when we shine light on it. Our set up is identical to that of last time. We'll take a piece of silicon with metal contacts. We'll use an ohmmeter that we connect to our sample via metal wires to measure its conductivity. The measured resistance will be determined by the conductivity and the size and shape of our sample. Finally, we represent the connectivity in terms of our measured values and relate it to the number of free, mobile charges, and the material properties of silicon. We'll first measure the conductivity of our sample in the dark, and then shine light on our sample and see how the conductivity changes. Our ohmmeter is hooked up to our sample, and we measure a resistance of around 120,000 ohms, which is equivalent to a conductivity of around 0.0002 inverse ohm centimeters. Now, let's flip on the light. We can see that we measure a slightly lower resistance of around 40,000 ohms, but what is light doing to affect the conductivity so much. Let's zoom in to the atomic level and explore why. We see here a 2D representation of a pure silicon crystal where all the valence electrons form rigid covalent bonds, are immobile, and don't allow the flow of electricity. This material structure is identical to our intrinsic sample when in the dark, which has a very low conductivity. When light hits our sample, photons of sufficient energy can break these covalent bonds, injecting the formally immobile electron, giving enough energy to move around. The mobile electron leaves behind a mobile hole, which can move through the crystal by swapping positions with neighboring covalently bonded electrons. This explains why the light increases the conductivity of our sample. Again, our conductivity is determined primarily by the number of mobile charges. Light creates additional pairs of mobile electrons, and holes, thus increasing n and our conductivity. We've demonstrated that light is able to generate fee carriers in our ultra-pure sample. The same effect still happens in dope silicon, but the light induced change in conductivity only creates a small relative change that we can't measure using our ohm meter. Generating these extra mobile charges by breaking covalent bonds with light is the source of the electricity that we eventually collect in our solar cell. In the next video, we'll explain how these light generated mobile charges will be collected and converted into electricity. I'm Joe Sullivan, thanks for watching. [MUSIC PLAYING] |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 20_RD_Investment_Innovation_in_PV.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from 100s of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: I promised everybody that we would have the capability of testing a multi-junction solar cell. We've been working so far with a single-junction crystalline silicon. And just to refresh everybody about what a multi-junction device is, it is a high-efficiency concept. The efficiencies that are obtained under concentrated sunlight are above 40% now with some of these high-efficiency concentrated devices. And again, the way this concentrated device works is if you have, for example, germanium, gallium arsenide indium, gallium phosphide stack-- three different materials, three different band gaps ranging from somewhere in the range of 0.67 eV all the way up to a few eV. The largest band gap is placed at the top, because the short wavelength light is absorbed by that one, and the longer wavelength lights go through and are absorbed by the layers underneath. So you can think about this as blue, orange, and red, if you like to see things in colors, the short wavelength light being the blue absorbed at the top, the sunlight obviously coming in through that side through the top, the middle cell absorbing the light somewhere in the middle of the solar spectrum, and the bottom cell into the reds. So the notion is to have a device that is capable of minimizing thermalization losses. So the short wavelength light, instead of being absorbed in a low bandgap material and generating a ton of heat in the process, is able to be absorbed very efficiently via that top cell. And only the longest wavelength light, the lowest energy light, makes it down to the small bandgap material underneath. So this is representative of the stack of materials that we have. I believe what we're going to do is do a direct comparison of silicon versus multi-junction. Is that right? Yeah, so right now we have installed the silicon devices. Those are the very nicely encapsulated ones. We had to change the light source. As you see now, we have a little bit of a flashlight there. Why is that? Why did we have to go from an LED, a monochromatic light source, to a broadband flashlight? AUDIENCE: Well, if the point is that you have these different absorbing layers that absorb different wavelengths of light, if you only put one [? group ?] like that, the other layers wouldn't have any effect. PROFESSOR: And it kind of defeats the purpose. Plus the current outputs of these three different cells have to be more or less matched. So because they're connected in series, if you have a poor performing cell-- or one component, one sub-cell, if you will-- that is generating a small amount of current, that will limit the combined current output of the entire stack. Good. So why don't we go ahead and connect them to our computers. We have our tech support staff available on call-- who just walked out, conveniently, but will be back momentarily. We'll connect them to our computers, fire it up, and team up with somebody if you don't happen to have a computer with you. We'll get this demo started. So the notion here is to first test the silicon-based device. Once we have a good working IV curve out of the silicon-based device, we'll take a pause. We'll talk about what we would expect to see from the multi-junction device when we hook that one up. So why don't we go to it and give it a shot. So let me dive into what is effectively our last in-class lecture before we are graced with some really nice presentations. I'm looking forward to those. So global trends-- what I decided to talk the last day about-- really we have a couple of topics left, which we don't have time to cover. We won't have time to cover both. We'll have time to cover one but not the other. And the two topics are the future of R&D in solar. And the other topic is solar in developing countries. And I think both are equally important. I decided to pick the former rather than the latter. Global investments, trends in solar and other renewables-- what I wanted to do was to briefly walk through some of the recent trends in R&D. So energy companies traditionally are not R&D spenders. They reinvest a very small fraction of their profits in R&D, in research and development. And solar, because it is far from grid parity right now-- factor of two or factor of three in terms of cost-- not price, cost-- we have to invest R&D to get the cost down. And this is both manufacturing innovation and, of course, engineering scientific innovation. So this right here is financial investment in clean energy, global trends by quarter. I decided to compile as much data as I possibly could into the slides so that you can go on afterward, if you're really interested in the topic, pursue it further. And what we see is as a rise overall of investment in so-called clean energy. And by and large, by the G20-- these are countries that have access to resources, to capital-- larger GDPs on average. You also see a trend and in non G20 countries, more recently, an uptick or recognition that this is an important area. And perhaps there's room to play, niches if you will, that certain countries can adopt that would provide a competitive advantage. This is an interesting chart as well. This shows the investment in-- I believe this is government R&D. Oh, this is financial sector investment only, excludes corporate and government R&D-- small distributed capacity in both-- so financial sector investment in the US and China. What we see in the US is relatively stable investment-- picked up in the mid 2000s, but relatively stable throughout. And in China, just a really steady increase here of a R&D funding. Note the role of the market in the United States. Right around '05 and '06, this was when the price of PV began to plateau. The costs continue coming down, but the price of PV modules began to plateau because of the silicon feedstock shortage. So people saw opportunity here, especially the financial sector, private capital, and said, hey, if prices are remaining high and the costs are coming down, that means our profit margin is growing. This is a good industry for us to get into-- a high profit margin industry. So there are many folks getting in because of that market condition. Some of them saw the future in the market and said, this is trending towards commodities. We have to adopt that mentality, and really squeeze every penny out of our cost structure that we can do. And others went into it thinking that this would be a bumper crop-- a really high yield investment. And then as the margins began to get squeezed, they got scared, and some pulled out. So it's an interesting trend, following the market perturbations in the United States and having this fluctuation. And in China, from what I can tell discussing with business leaders and politicians, a much more premeditated, long-term strategy saying this is a strategic industry for our country. We are going to invest in it. And this is of fundamental national importance. So a little bit of difference in the investment strategies of the two. The EU is a little bit in between-- a mixed bag. Again, this uptick in the middle of the 2000s, but a continued investment in PV and renewables. This shows the investment type by sector, broken down on the right-hand side between the renewable energy types. So you have wind, solar, other renewables, biofuels, and so-called negawatts, so-called energy efficiency. And on the left-hand side, we have the different types of investment into clean tech. And interestingly, here in the United States-- this is in 2009-- venture capital is comprising a surprising total of the investment in renewable energy. And in terms of the sector itself, we can see solar here in the United States comprising a large percentage, again, of the total investment. I want to bring caution to one data point up there-- Spain. That was a little bit of a short-term fluke. The Spanish government instituted a feed-in tariff similar to what Germany has implemented, but a little less successfully. Let me dive into the details. Germany did a careful market analysis, was the first to move into the space, and began dominating the market for PV-- 50% of PV installed. So if Germany set the price a little differently, the PV market would adjust accordingly. Spain, a new entrant, seeing Germany's success, decided to replicate it. The first time they attempted a feed-in tariff, their feed-in tariff came in too low, and the market looked at Spain, shrugged their shoulders, and continued installing in Germany. The second time they decided to come in, they put their feed-in tariff a little bit too high. And as a result, the market said, really, you kidding? OK, we go there and install. And there was a slush of modules over to Spain, flooding the market. And over a period of about a year to two years, they got much more than they bargained for. In other words, they had many, many more modules installed than they had expected. And now the government has to pay out to these installed systems. They have to pay out a certain rate based on the feed-in tariff. And it was more than they expected to have to pay out. Then the financial crisis hits. So it was a little bit of a disaster, because they ended up killing the program, killing in the process about 10 years of work to design the program, 10 years of institution building to think about how to create a feed-in tariff for Spain, and without the desired result, which was a slow, steady increase of the photovoltaic installations in Spain and hence, the local industry. So it was a little bit of a flash in the pan-- a huge market that burst very suddenly, and then quickly extinguished itself. So it's a good example of how not to perform a feed-in tariff, how not to design a feed-in tariff. So in 2009, that was the end of the bumper year investment. Spain also did something that was pretty nasty from a government policy point of view. They decided to retroactively change their feed-in tariff. AUDIENCE: Ooh. PROFESSOR: Ooh, so a feed-in tariff is a contract between the government-- or the utility-- and the installer, saying that we will pay a certain amount per year for a number of years. And to go back on that, renege on the feed-in tariff-- that's a no-no. It undermines market confidence. So the PV module prices-- I just wanted to highlight this slide once again, to highlight the market conditions that are being experienced right now. We had, in the mid 2000s, a rise in, or fairly steady prices. And meanwhile, costs were coming down. This, by the way is price, but in thin films for solar cad-tel, these blue dots here being crystalline silicon. And the prices remained fairly steady. And then in, say, 2008 to 2011, prices have dropped precipitously. The 2011 numbers here are at or slightly below $1. So you can see, for somebody who is being driven by market conditions, this is extremely unsettling. And it causes capital to fluctuate back and forth. Now the same thing-- what is happening here in the solar modules, the inverse is happening in the installations. So the installers are seeing the price of their modules go down and the price of their installations declining steadily, but more or less staying fixed because of the investment tax credit staying fixed and so forth in the US, and utility prices rising. And so the installers are saying, hey, this is great. So their profit margins are really big. Warning to anybody trying to get into the sector right now-- think carefully about these market dynamics in the United States and in Europe, and how that will affect your business. If you're trying to get into the solar absorber, as in module, manufacturing business, or if you're trying to get into the equipment manufacturing business, or in the installation side in the grid, think about how these market dynamics are going to affect your business as it grows and tries to gain a foothold in this environment. It's a great time to be a new installer company, but in three or four years, when we're in an under-supplied condition again, and prices might even go up, what then? So think about these topics. We're talking about renewable energy R&D and the technology pipeline. So it's important to recognize the path that many of these new technologies take to go from concept or idea into full-scale manufacturing. So in general, this is the path followed in the United States. And we have technology research happening in places like this, at MIT. The roll-out, in other words, the installations and large-scale manufacturing on the other side. And there are many funding sources available for that-- not so much this last one here in the United States, but maybe in Australia or Europe, in select regions of the US. This technology development right here in the middle has been what is referred to as the Valley of Death. Does anybody know why-- what that means, Valley of Death? What is it? AUDIENCE: A period when it's difficult to get funding from any source. PROFESSOR: It's a period when it's difficult to get funding from any source. Now imagine you're a group of postdocs and students in the lab. You come up with a new, fancy technology, and you can't quite get the venture capital or private equity necessary to kick off your company. That would be one mini-valley of death. It's fairly avoided here in US. Good ideas have a tendency to get funded at that stage. But if you don't get funded, then the postdoc gets another position over here, becomes a professor at university across the country. The student goes off and does a postdoc in another place. And all of a sudden, three, five, six months later, the venture capital swirls around to the professor, who's still here at MIT and kind of has a hollowed-out group at this point, and says, wow that's a great idea. I'd like to invest in it. Where's your team? Yeah, poof. So that's one possible mechanism wherein technologies don't make it forward. And so keeping the team together is extremely important. The second Valley of Death can happen right between technology development and manufacturing and scale-up. We've seen some of these. We've seen pictures of some of the factories. They're around here in the countryside. These are oftentimes $10s, more likely $100s of millions investments. And venture capital funds tend to be on the order of $100s of millions to $1 billion. They don't like investing in large asset goods. They don't like investing in factories. They'd rather invest in a small group of people with the computers set in their garage, maybe put in a couple $10s of thousands and turn around, profit of a few million. That's the type of investment that makes a lot of sense for venture capital. When you start investing in fixed goods-- in brick and mortar-- you need other forms of financing. And some countries around the world have been very adept in providing this financing. China, in particular, has done a great job at making that type of financing available for companies that are small and looking to expand. But in the United States, it's very difficult to access these essentially money from banks, especially, for new technology. The bank will say, well, why am I investing in you? I could be investing in something that's much more sure of a bet-- today's technology, instead of investing in something risky. So the government has stepped in with a variety of programs to try to ease that inefficiency in the market. And so one of the means are loans-- loan guarantee program, for example, is one mechanism. There is a [? sunPATH ?] program that is coming down the pipeline as well. So we have venture capital and private equity, but that can only take you so far, typically. They invest a few $10s of millions, in a few rare cases, a few $100s of millions. But then they reach the end of their credit line, if you will. They exhaust their ability to invest. And what's needed to expand manufacturing is typically on the order of $100s of millions to billions of dollars to reach 100s of megawatts-- gigawatt scale. So that's the second big Valley of Death in the United States that exists. And the way some companies are transitioning across that valley-- they find a variety of means. Some go overseas. They say, hey, the Central Bank of China is willing to invest money in me. I go there. I set up manufacturing. Other companies, small companies, might say, well, GE, you have a big finance group within your big umbrella company. Why don't you buy us? And then we can gain access to capital and expand our manufacturing plant. Others form partnerships with banks. It's a mixed set of business strategies. But if you look around at some of the start-up companies that are now entering small-scale production, this is where the business side of creativity comes in play. And that's where you really need a good business developer at hand to arrange those deals for you. In terms of where the money is, in terms of clean tech in general, these are some figures in terms of 2009 data. I'm sure we can get some more up-to-date figures as well. And in terms of growth and investment, we see some countries that might be rather surprising in terms of the five-year growth of investment in renewables. Obviously, if you start from a very small number, you can grow pretty quick in terms of percents. But it's still interesting to see the development of some countries here. This is sobering. So this is the US government R&D by budget function '55 to '97, the most comprehensive data set I could find. I'm sure that there are graphs that extend this into the future. If you happen to have one, I'd be happy to see it. But the basic story of this graph is the following-- the lion's share of R&D is going to the Defense Department, so the Army Research Office, Office of Naval Research, DARPA-- that's the advanced research program-- a variety of night vision labs, and so forth. These are a variety of defense R&D. And there is trickle down. There's some trickle down of technologies being developed in defence into civilian uses. And so I don't want to point to this and say it's a bogey man-- by no means. But it does indicate national priorities in terms of R&D research. The other big one is health. So NIH, National Institutes of Health-- that's growing and expanding. It's very easy to go to congresspeople and senators who might be advancing in their years, and say, hey, we need money for Alzheimer's research, or we need money for cancer research. It resonates. It's easy to convince people of that. Energy is small, traditionally. Let me dive forward into this going a little bit further. This is non-defense R&D funding pushed out to 2004. Again, you can see health really driving things. Energy, traditionally-- in the Jimmy Carter years right around here, it expanded a bit. This was a renewables burst. That's where the National Renewable Energy Laboratory was founded, originally called the Solar Energy Research Institute or SERI. And then got crimped down again and really experienced a bit of a pinch in the US right when solar was really beginning to take off in Japan. So this is when solar cell production by Sharp was really beginning to climb in the late '90s and early 2000s. And then, of course, Germany followed and the rest of the world. So we've been a little bit behind, step by step. And the interesting thing that many people have looked into is, what sort of correlation exists between US government R&D spending and output of new ideas. And the output of new ideas, the metric that they're using for this are patents. So you could dispute that. You could say, well, patents aren't the best indicator of new ideas. Sometimes new ideas are diffuse benefits, and they help all industries, but you can't really patent the idea. Fine, but this is, I think, the most quantitative comparison that folks have performed. This is an interesting study where folks looked at the number of patents granted and energy R&D funding and plotted it as a function of year, and saw a strong correlation between uptick of national priority and government funding and innovation. So this was the first wave of innovation in energy, and in PV specifically, in the late 1970s when the OPEC oil crisis hit in the United States and there was a big push for renewable energy development. Yes, question. AUDIENCE: Just patents in energy or-- PROFESSOR: Yeah, exactly, so these are pushed down. And we can dive into patents just for PV, which is this graph right here. And this is an updated version of that earlier study by Dan Kammen. This earlier stuff study by Robert Margolis. Robert Margolis is now at NREL, National Renewable Energy Laboratory, and he works on market assessment there. Gregory Nemet was a student at the time with Dan Kammen, produced this wonderful continuation of the study published a few years later. And again, broke it down into specific technologies, including photovoltaics, and saw, again, some correlation between the number of patents and public R&D. In the mid 2000s, this really-- the number of patents and started to grow as startup companies got into the fray. Remember when prices stayed flat and costs continued coming down, and those margins increased, a bunch of players got into that space and said, we could make money. So a number of patents were filed. A number of startup companies got off the ground. Again, funding-patent correlation for energy in general-- I provide you the data. You can dive into it in more detail if you, not only PV but other technologies-- interesting. Global trends in venture investing-- since venture investing is important for you specifically, it's one of the pathways to get ideas out of the university and into a startup company. We talked about this so far. And OK, so we're diving a little bit deeper here. This is for the venture capital private equity financing by sector in 2009, looking specifically at solar in the United States, comprised a large fraction of it. Solar continues to comprise a large fraction of venture investment, surprisingly, despite the market conditions right now. Because folks, especially in the VC community, are looking at today's market as an opportunity. They think that if enough people are scared out of the market, that they'll be able to remain there and pick up the good ideas. And if they're only a little bit smarter than their competition, they can pick up the right companies and the right sections of the value chain-- maybe an equipment manufacturer, maybe an installer with a new idea-- and avoid some of the pain that's going on right now in wafer fabrication and cell fabrication upstream. This was the VC investment in solar right here in those boom years that I mentioned. So during the years when prices started to plateau because of the silicon feedstock shortage in the mid 2000s, you saw this massive uptick of investment in venture capital funding. That scale is in millions of dollars. You're exceeding $1 billion of VC funding in solar in 2007. And that trend continued in 2008. So still to this day, we're seeing $100s of millions plowed into solar by venture capital funds. It's interesting, really interesting. The number of startup companies in the United States has proliferated. There are, I think, somewhere on the order of over 200 solar startup companies worldwide. There are a few that have failed. So Wakonda, Solasta, SV Solar, Synergen, Optisolar, Solyndra, SpectraWatt, Evergreen Solar-- these are all I'd say failed companies, by the definition of failure-- bankruptcy. I mean, by that metric, so is United Airlines failed, and American Airlines failed. They're still around. They restructured under bankruptcy protection. Some have closed their doors entirely. Other ones have restructured, or are in the process of restructuring. Why each of these companies have failed? Different reasons. You can't claim that each one had the exact same trajectory. But you can definitely point to certain market conditions as influencing or precipitating the failure. Let me be more specific. These companies right here are all wafer or thin-film device and module manufacturers. They're the upstream components. These aren't installation companies. These aren't installers failing. These are upstream manufacturers failing-- and a few high-profile ones at that. So that was precipitated by the recent market conditions-- the capital crunch. It begins with some of the companies, say OptiSolar in 2009, I believe. They were a company that was producing amorphous silicon modules. And as we studied amorphous silicon, the efficiencies are low, right? The amorphous silicon module efficiency's on the order of 6%. And they said, well, never mind that our cost structure's a little high. We're going to scale up like nobody's business. We're going to ramp up manufacturing capacity to over a gigawatt, and do it really, really quick. And just by sheer scale alone, we'll be able to drive down costs and get us to the point where we're competitive on a cents per kilowatt hour basis with crystalline silicon. There's a great business plan in theory. But what happened to them was, when they went out to try to raise money, they couldn't find any right around 2008. It was the beginning of the financial crisis. So they had the business plan in mind. And on paper, it looked great. But when it came time to raise the funding to grow, they didn't have it, even though they had a guaranteed customer-- PG&E. That was the Pacific Gas and Electric, the California utility. They just could not get the financing to expand their factory. So what ended up happening was they folded, sold the supply stream, if you will, to First Solar, who picked it up for pennies on the dollar. And First Solar modules ended up going in the PG&E field installations, instead of the OptiSolar amorphous silicon. Each company has its own story, and have failed for different reasons. What's clear is also, there are more failed start-ups coming. This is a time of a financial pain for them. The prices are very, very low. And there are a few people who are tracking these startup companies. I would say Eric Wesoff from Greentech Media is probably one of the most active in publishing his insights. That said, there are many promising companies among here, and some of these, hopefully, will become household names for good reasons in the future, as they have an innovation that significantly drives down cost over their competitors. We're going to get to that in a few slides. We're going to talk about how to evaluate a company. Because it's going to be important for you-- near term because you might want a job, or you might want to form your own company, long term because you might become an investor in PV. And you have to figure out what types of companies make sense to invest in, and which don't. In terms of startup companies in the New England area, we often think of ourselves as kind of maybe second fiddle to Silicon Valley. But there's a lot going on in the region, and a lot of good work. So if you go to cleanenergycouncil.org cluster map, you'll see a map of the local clean tech companies in the region. I'll leave this slide up there, since I see a few of you jotting notes. It's a useful map, and you can select by sector as well, if you look at solar, look at biofuels, and so forth. Trends in renewable energy manufacturing-- this slide is a little bit outdated, but it's the Greentech Media research map of manufacturing in the United States of the different solar technologies-- again, a bit outdated. There are a few companies that have changed. But it gives you a sense of what the distribution is. What are the latest trends of manufacturing in the United States? The latest trends of manufacturing, if I were to point to a few of them-- Mississippi has emerged as a big manufacturing state. Why? First off, who's from the Southeast here? Show of hands-- one, two. All right, what do you have in this region right around here that the rest of the country doesn't have? AUDIENCE: [INAUDIBLE]. PROFESSOR: Close, coal-- you have a lot of coal. You have nuclear as well. The TVA-- does anybody know what the TVA is? Tennessee Valley Authority-- that's a big public works project, in fact, that got started. It provides low-cost electricity to this entire region right up here, including northern Mississippi, including some of the northern portions of the far Southeast states. Bottom line-- you have depressed wages, by and large, in these rural communities, and low-cost electricity-- which if labor and utilities matter-- which it does in solar manufacturing-- it's ripe for manufacturing. And you have many startup companies-- Stion, Calisolar, Twin Creeks, and others that have moved into the Mississippi region in very recent months for that reason. You have Suniva was based in Georgia, Atlanta. Ohio still continues to be. The Northwest as well-- cheap hydro, not exactly cheap wages, but cheap hydro. And the technologies do tend to stay closer to the places where they are born. You can see in the San Francisco Bay Area, there's a propensity to form startup companies in new technologies as well as in the Massachusetts area. So these are focusing on some of the medium-scale manufacturers. In terms of manufacturing support, this is what local state governments are doing to help form new companies in the United States. You have, as well, a variety of mechanisms-- grants, loans, tax credits, and so forth for new factories that are trying to start up. So keep that in mind if you're going for it. And market incentives-- in terms of the three-- Germany, US, and China-- you also have to consider Korea, Japan, India, Brazil, other countries as well in this mix, if you really want a global perspective. But this dumbs it down to three. In terms of what incentives are available for different countries, there is a plethora of different incentives which can help the market pull. OK, so we go back and have a clearer picture of what's happening, at least in the US with its investment-- the VC investments as well as manufacturing, the next step on the state level. So we're collecting some data points there. This is interesting. If your money is from the bank, let's say, and you're paying an interest rate on it and you want to start a new factory, but then your local inspector comes back to you and says, well, we're going to have to delay by three months because of reason x, y and z. Now you're paying interest on the money, potentially. But you're not generating profits off of it. So delays cost money. Project delays cost money. It's also an opportunity cost. And so there was a study done a few years ago looking into why it is that renewable energy projects are delayed. And they came up with some interesting region-dependent conclusions as a result of this study-- whether it's transmission limitations-- Texas, there have even been cases in New York of transmission lines having limited capacity for renewables-- financing constraints, power purchase agreement weaknesses, permitting-- that could be a big slow down if that's not streamlined-- financing and permitting, and negligible local market. And again, we see the TVA popping up here as a negligible market for the renewables, because, well, you have cheap electricity. It's hard to compete against that. But in other states with more sun and more expensive electricity, there's the potential to install it. But you may be limited, in fact, by the grid. There's a big new study released, I believe it was this week, by the MIT Energy Initiative on the future of the grid. Yeah. AUDIENCE: Could you say again what PPA stands for? PROFESSOR: Power Purchase Agreement-- so that's the incentive mechanism whereby you can begin putting the solar panels on your roof, and the installer pays for the panels, gets the money from the bank, installs them on your roof. You sign the contract to pay a certain price for the electricity over the next 12, 15, 20 years. So global trends in R&D-- this is, again, a data dump of several sources, one from the NSF showing industry R&D expenditures, government, federal government, and other. You can see in the United States-- this is across all sectors here-- but really an inversion of the role of industry and federal government. So when you hear MIT, for instance, going after large companies and saying hey, you should invest in R&D here at MIT, this is one of the fundamental driving forces-- this inversion here, the decline of federal government spending. And as well, has resulted in MIT looking elsewhere for funding, not only industry. The greatest gains in R&D intensity in terms of the R&D expenditure have been in Asia. You can see on the right-hand side of this chart, China, Japan, and Korea increasing the expenditure from 1997 to 2007. This was right before the financial crisis hit. In the US, holding relatively constant as a percentage, as a share of GDP. Science and engineering interest in Asia-- this is science and engineering degrees as the percentage of new degrees. You can see over here we have Germany, Korea, and China. Germany makes a lot of sense. The Germans have about 44,000 engineering jobs in renewables right now that are unfilled. So they need people. They need people to go to Germany to get high-tech jobs. China is a bit more precarious, in the sense that the supply and demand is much more evenly matched. And the growth of both are increasing at steady rates. So if the growth of manufacturing and R&D does not continue to rise in China, there's going to be an oversupply of people. And that can lead to a number of problems. And so it's very important for China over the next several years to keep a strong, steady handle on this growth, to make growth manageable. It's a good problem to have. It's managing success. But it could also lead to some catastrophic consequences if the system gets out of balance. In the United States, well, I believe this data came from 2005. That was at the height of financial engineering. I would hope that this number here has increased a bit since the collapse of the financial markets, and a recognition that there are productive ways of investing one's talents and mental gifts. Global research output shifts towards Asia. This is global research R&D, share of science and engineering articles. So if you're starting from a small amount and growing, you're going to be, if you just look in a percentage basis, by necessity taking away a share of the pie from another entity. So as China grows, India grows, the rest of the world grows. The share of science and engineering articles in the US and in the EU begin to drop-- essentially the dominant players-- and in Japan as well. Now, that's not necessarily a bad thing if the quality of the articles is maintained, and the total number of articles continues to expand, and we have the capacity to keep up with the information. So there's a threshold, or a limit, to how much we can absorb-- how much new information we can absorb per unit time. So people are working on more sophisticated ways of gathering and assembling this information, especially using computers nowadays, that can help expedite R&D. So a number of trends happening on the scientific side-- you're involved with that. High-tech trade balances continue to widen. This is the trade balance in high technology goods, US and China. And so obviously, Chinese economy is trending away from-- still heavily invested in raw materials-- but trending away from that toward high-tech manufactured goods. And the US entering a region of trade deficit as a result of purchasing those products. So these are all trends that have policymakers, in particular, concerned, and looking at the future of global competitiveness. Technology evaluation-- let me spend a couple of words bringing all of what I've just said, and everything over the entire course, home to you. So far we've talked about these things in very ethereal terms. It's useful information. It will prove useful once you begin applying it. But we want to run through a quick, little scenario right here, where you're asked to evaluate a new PV technology. Why? Well, you might be applying for a job at this company, and they say they have the greatest thing since sliced bread. And you want to put on your thinking cap, and evaluate whether that's true or not. You might have a new idea or new innovation. And you're trying to make the tough call-- do I go forward and establish a company off of this idea? Or do I have to go back and turn the crank a few more times to come up with the next better idea that's more worthy of investment? You could be an investor. You could have money at your disposal, and you could decide what company to invest in or what not to invest in. So how do you go about this? I'd say three fundamental components. And venture capitalists might disagree with me, or other people who have different skill sets might disagree with me. But me, from this perspective as an engineer, I say first analyze the physics. Figure out how this technology works. Because if you understand how the technology works, you can understand some of the fundamental limitations. You can understand efficiency limits. And you can predict, based on pattern recognition, how hard is it going to be to obtain or to approach those limitations? Analyze the cost scale potential, meaning the potential to scale up, and manufacturing. This is really more in the engineering science side. And we've been getting more into this during the second and third parts of the course. And analyze markets-- this is definitely the third part of the course. So this begins to pull it all together. Let's start with analyzing the physics. We talked about conversion efficiency being a strong lever for cost. That's way we're analyzing conversion efficiency. The way I would recommend analyzing any PV technology that they throw at you would be thinking about conversion efficiency in terms of output energy versus input energy. And think about losses along each step of the way. If you have time with the R&D department, sit down with them. Sit down with some of the chief engineers and walk through each of the steps, going from a light photon entering the device to charge being collected on the other side with a certain current voltage characteristic. And keep in mind that the total efficiency is going to be limited by whatever the worst performer is. And keep that picture in mind, too-- [INAUDIBLE] big advice. Customer needs-- the next is analyzing what makes your product special. So where is it going to fit into the big picture? Is it going to run with the big dogs? Is it going to be an on-grid application, in which case cents per kilowatt hour really matter-- the price for the electricity for a power purchase agreement? If you're just selling the module, maybe dollars per watt would be an important metric as well. Or are you going to one of these other niche markets right over here? And are you going to be a player there? How big is that market? These are some questions to ask. Cost-- all right, so now we know what our intended market is. We know what our product is going to look like. We know the physics behind it. In a bit, we'll know more about the manufacturing. We think about cost. And we talked briefly about this during class. We had more of an in-depth discussion with Doug. That Excel spreadsheet, by the way, will be available to you so you can look through it, and see how a cost analysis was done for crystalline silicon, and how you might adapt it to your technologies. But it's really important to perform a cost performance model for your technology so that you can understand what levers to pull, what levers need to be pulled, to increase your parameter of merit, whatever that parameter of merit happens to be for your potential customer need. Manufacturing technologies and scale really do play into this quite a bit. And for many technologies, or for many companies at least in the Silicon Valley area, there might be 50 companies working on the same material-- copper indium gallium diselenide, let's say, CIGS. But they each have their own deposition process, and they're each trying to develop their own pieces of equipment to manufacture this material. So understanding the basic differences between the different deposition systems is important. And we walked through that during our thin-films lecture. So again, you have access to this information, and you can parse through it in greater detail should you need to. Scaling of manufacturing-- we talked about resource availability. We even had an in-class debate about it, about cadmium tellurium. There are reports out there which you have access to, like this APS report on critical energy materials. And we understand that the manufacturing and reserves of these elements are not equitably distributed. In fact, they're concentrated in certain regions of the world. And so that might influence the ultimate potential of a certain technology to scale. And again, this is another intelligent set of questions that you can ask. We understand a bit about the market dynamics now. We understand this is price. We understand that there was a bit of a plateau in price in the mid 2000s, a precipitous drop over the last three years, and that's really changed the way that investments and equity look at the solar market. In the mid 2000s, it was all gangbusters. Everybody was really happy to throw any money they had at solar. Now people are a lot more selective in terms of what they invest in. And this trend of oversupply undersupply is probably going to continue for some time if the integrated circuit industry is any example and model. So we're likely to see this continue. What is your market timing? It might be a really good time to found an installation company right now, if you can scale, and grow quickly, and you have the right niche. But if you have a new idea for a thin-film absorber, it's important to think critically about where the market is going to go over the next few years, and how that impacts your strategy as a company. You might decide, well, the market's kind of cold right now for modules. So why don't we hold back on building that large-scale manufacturing plant until, say, 2014, and invest really heavily in R&D right now, and stay small, without the financial obligations of a big manufacturing line, until we really nail the technology, and have something good to go. And plus, think about our exit strategy. Maybe we won't ramp up to be a gigawatt or two gigawatt company. Maybe instead it's more important to form partnerships with companies that are manufacturing cad-tel or CIGS right now. So our business developer's going to spend more time chatting with the business developers of First Solar and other companies. We also know a bit about the financial incentives from Germany and some of the largest PV markets in the world. We know that China is going to ramp up significantly in terms of installations over the next few years. We know that the United States will, as well, as the prices continue to come down. These are the PV feed-in tariff rates shown in blue and red for different types of installations, blue being the large, freestanding systems, the red being mostly rooftop mounted systems. So we saw that the feed-in tariff rate in Germany has come down versus time. And the average electricity price has gone up. So we can begin predicting what the role of market subsidies will be when we try to roll out our technology onto the grid. And we can plot this, and see how that will impact our business model as well. And lastly, we know that about 99% of the solar panels have yet to be made. 99% of the solar panels have yet to be made. So there's a lot of potential here. This, again, going back to the very first lecture, where we had new energy installations, new PV installations growing significantly. Convergence is coming. Who's going to comprise-- who's going to the bridge that gap? Who is going to be the maker of the technology that will ultimately bring PV to a massive scale in the grid? That's for you to decide. So other intangibles in terms of evaluating companies would include the team, especially the leadership team. What is their track record? What is their philosophy of running the company? The financing that's available-- how much cash is at hand-- the patent portfolio, how protected are they, and so forth. Patents, by the way vary in importance from the US to certain other regions of the world. It's very important in the US and Europe, less important in China. But it depends. If a certain idea is patented, you will have difficulty accessing that market. You will have difficulty selling product into that market, even though you can continue to sell a product into markets that value patents less in IP. So let's go through a few examples. I'm going to throw a couple of examples at you real quick, and just spout off the first ideas that come to you. How would you analyze this company? Solar paint. All right, so I have my big spray paint system here, and I go to the side of the house, and I go shh. And that saves me a bundle on installation costs. It's easy. You can do it yourself, and you just connect a few wires and voila, you have solar electricity. What would be your first instinct? AUDIENCE: How does it work? PROFESSOR: How does it work? [CLAPPING] Bravo, bravo. What is the physics behind it? How do you separate charge? If you go back to this right over here-- light absorption, charge excitation is important. What's the [? bang-up ?] of the material? How does it excite charge inside of the material? Where's charge separated? How do carriers reach those separation points? You're just spraying on one homogeneous layer? Did I get that right? There's not two layers? Does it phase separate? How does that work? And in charge collection, how are you collecting the charge over that massive area? What are your resistive losses? And so forth. You're equipped now to ask those questions as a result of this course. That's pretty awesome. You think about it, you're pretty empowered. So it could work. Solar paint could very well work. So it's important not to ride into this discussion on a high horse and say ooh, won't work because of x, y and z. It's important to keep an open mind, because new ideas are really quite startling, and they can be game changing. But it's important to have a critical yet respectful approach to this. A critical mind is always a good thing to have on your shoulders. Wundermaterial-- so I'm arriving to you, and I say, this is the wonder material. It's all earth abundant, totally scalable, but I'm not going to tell you what it is. So you're going to have to invest in my company because I have this great team right here-- a great team coming from Intel. I used to be a head manager at a national laboratory. I know my stuff. I really know my PV. I can wow you with a few presentations about PV device physics, and talk all this fancy stuff. But you're going to have to invest in my company based in a few SCM images, plain-view SCM images of the material structure, just to prove that I can actually deposit it. But I'm not going to tell you what the material is, because US VC's could run off with my idea and go sell it to somebody else. So I'm going to hold that very close to my chest. Do you invest in me or not? AUDIENCE: Depends on what you're asking for. PROFESSOR: Depends on what you're asking. Depends how much you're asking for. That was a real situation. I was in that room. I was evaluating that company. And as a scientist in that room, I was, like, are you serious? You can't be serious. The venture capitalists, however, thought differently, and said, look, the team is really good. And for reasons that were described-- I can't get into all of them without giving the details away-- there were reasons for investing in that particular case, in that company. And even though the physics was not understood, and even though the ultimate efficiency potential was not understood, the VC firm made an investment. So far it's been going OK with that company. So just to point out that this is a base, a foundation of which to make decisions. It's not a prescription upon which to make decisions. A lot of things factor in. Use your best judgment. You're the one best equipped to make those decisions. Path forward-- this is kind of my last few words on the soapbox before I hand over the microphones to you. The path forward, from my perspective-- the markets are going to drive a lot of the public story. But you know that cost matters more than price. Price is the short-term market pull. During the mid 2000s-- actually, down here when the price was still very, very high, solar was kind of this hippie, tree-hugging group of nerds that would get together at the Muddy Charles and come up with the facts-based analysis, which formed the MIT Energy Club-- really small thing. And then right when the prices began to stabilize and there was this view that energy was a huge gold mine waiting to be explored, a massive amount of interest came into the field-- and growth, accordingly. And it was good. But in the 2008 and so forth, we had this precipitous drop in prices, as well as the collapse of finance to allow these technologies to scale up. The Valley of Death widened and deepened a bit, if you will. And suddenly, energy looked a lot more precarious. I started having students come into my office doing interviews, saying what sort of career can I have here? Tell me, seriously, are their jobs waiting for me when I exit? The reality is, there are jobs waiting for bright, smart people regardless of the market condition. There were jobs back here. There will be jobs in the future as well, if you're good at what you do and you ask the right questions, conduct good experiments, and know how to disseminate your work. But this is kind of a return to the roots now, where we have people who are in solar and interested in solar who are really there for the long run. And so knowing the market conditions really helps you put everything into perspective, and see how the situation might evolve going forward. In terms of tipping point, it's largely agreed upon that $1 a watt system installed is really where we want to head to. So we know the cost right now of manufacturing a PV module of crystalline silicon is on the order of $1 per watt. And if you're inventing a new technology, you have to do half of that. And crystalline silicon has a roadmap to get to about $0.50 per watt peak by 2020, 2025. And so if you're developing a new technology, you have to undercut that and some way, shape or form. And you can do that in a variety of ways-- improving the efficiency. There's a lot of headroom in efficiency-- lot of headroom there-- maybe at manufacturing scale and so forth. But if a technology, some technology by 2020, can get to $1 a watt installed, we're looking at, by 2030, a pretty massive, good penetration of PV across the United States. And that's just the start of it. The US is just the drop in the bucket. This is the world. There's a lot of people-- 1.6 billion of them, approximately-- without electricity right now. They're coming online. And they're driving the majority of the growth in CO2 emissions. Majority of the growth comes from two points-- one, we're outsourcing our CO2. If you look at the amount of CO2 the United States contributes to the world, it might be 1/5, but if you look at all the manufactured goods, all the clothes we're wearing right now, all of the apparatus that we're using, these were manufactured probably not in the US, but they're here for our consumption and use. We own that carbon. That's our carbon. We're responsible for it. So our footprint increases further. So we're outsourcing the carbon. It's growing in certain developing regions that are exporting to us. And secondly, they're also growing. They're also consuming. They're also getting cars, and computers, and so forth. So there's an overall growth of consumption around the world. We agree that climate change is an issue. And we look at the solar insolation map, and we say, wow, compared to certain minerals, or rare earths, or petroleum, this is fairly equitably distributed. Even regions, say in Patagonia or up in northern Europe, compared to the Equator-- we're looking about a 3x, maybe 4x delta in solar resource, not a million to 1. We're looking at a 4x delta in solar resource, which is not that big of a spread. And furthermore, if we go back and look at the countries that are really coming online, just using this loose parameter of human development index that we referred to on the first day of class, we can see that the countries with the largest solar resource base-- several of them also happen to be countries that are on the path toward development, on the path toward consumption. And solar can really have an impact in those countries to improve the quality of life, and also reduce carbon emissions overall in the world. So what role can you play? I mean, I think important things to keep in mind is, we're really at the beginning of solar. We have a lot of headroom to grow. I think the amount of venture capital investment loss so far due to failed companies is a drop in the bucket compared to GDP. It's a drop in the bucket compared to military investments. It's drop in the bucket to things that we consider important, and especially the quality of life in the world. So it's important to keep that in perspective. We need to maintain momentum in capital innovation culture. That's what we have here. It's growing in other places around the world as well. It's important to foster that growth, and evolve into a global society where we have connections and shared connections with groups around the world, shared interests, and can leverage each other's strengths. The rest of the world is catching up fast with increased competition. That's why isolation won't work in this case. We do need increased R&D efforts on key targets. I think better investments and smarter choices of technology is important. And hopefully, over the course of the class, you're equipped with several of the tools to make those types of decisions yourself. We also need to change the way we innovate. Pooled resources, collaborative efforts, improved industry-university lab relations-- which in the US are somewhat at a precarious state because of differences in priorities between publications and IP protection and so forth. Direct to manufacturing innovations-- instead of always thinking that startup companies are the only route, thinking in terms of how do I get this technology into a large established company in the US? That's a route that we don't think about very often, but could have a huge impact, and certainly has an impact in Germany. The need for a steady predictable market-- all right, I think I might be asking too much in this bullet point. But it would be nice. And certainly policy can play a large role in that. We have enough unpredictability with oversupply and undersupply. We don't need the politicians getting into it as well, and changing their minds every two years. And investment in education as well-- the right type of basic education-- so you can get involved a number of ways. And I encourage you to do so. I thank you for your attention. And I wish you the best of fortunes going forward. Please feel free to call me whenever you have a need. And I look forward to your presentation. So be well. Thanks. [APPLAUSE] |
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