page_content stringlengths 12 2.63M | metadata unknown |
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Show that the equilibrium condition given by Equation 27.15 also gives
$$\frac{\left[\mathbf{B}\right]_{\mathrm{eq}}}{\left[\mathbf{A}\right]_{\mathrm{eq}}} = \frac{k_1}{k_{-1}}$$
for the reaction described by Equations 27.7 through 27.9.
SOLUTION: Equation 27.15 states that at equilibrium
$$v_1 + v_2 = v_{-1} ... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27-1.** A Mechanism Is a Sequence of Single-Step Chemical Reactions Called Elementary Reactions",
"Header 3": "**EXAMPLE 27-2**"... |
Consider the thermal decomposition of gaseous OClO to form chlorine atoms and oxygen molecules
$$OClO(g) \rightleftharpoons Cl(g) + O_2(g)$$
(27.16)
This reaction occurs by the following two-step mechanism
$$\begin{array}{c}
\text{OClO(g)} & \stackrel{k_1}{\rightleftharpoons} \text{ClOO(g)} \\
\stackrel{k_2}{\rig... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27-3.** When Are Consecutive and Single-Step Reactions Distinguishable?",
"token_count": 1904,
"source_pdf": "datasets/websour... |
Let us first examine the exact solution for [P] (Equation 27.26)
$$[P] = [A]_0 \left\{ 1 + \frac{1}{k_1 - k_2} (k_2 e^{-k_1 t} - k_1 e^{-k_2 t}) \right\}$$
Consider what happens to this expression when k2« k1• First,
$$\frac{1}{k_1 - k_2} \approx \frac{1}{k_1}$$
Second,
$$k_2 e^{-k_1 t} - k_1 e^{-k_2 t} \ap... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27-3.** When Are Consecutive and Single-Step Reactions Distinguishable?",
"token_count": 407,
"source_pdf": "datasets/websourc... |
Reconsider the reaction mechanism
$$A \stackrel{k_1}{\Longrightarrow} I \stackrel{k_2}{\Longrightarrow} P \tag{27.28}$$
for the initial conditions [A] = [A]0 , and [I] <sup>0</sup><sup>=</sup>[P]0 = 0. In Section 27-3, we discussed the time-dependent behavior of the concentration of the reactant, [A], and product, ... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27-4.** The Steady-State Approximation Simplifies Rate Expressions by Assuming that d[l]/dt = 0, where I Is a Reaction Intermediat... |
Recall that the rate law of a complex reaction provides information on how the rate of reaction depends on concentration, but it does not tell us how the reaction occurs. In general chemistry, you may have learned that the rate law for a complex reaction can be derived by combining the rate laws for the individual elem... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27–5.** The Rate Law for a Complex Reaction Does Not Imply a Unique Mechanism",
"token_count": 1997,
"source_pdf": "datasets/w... |
#### EXAMPLE 27-6
In the above discussion, the concentration of $N_2O_2(g)$ satisfies the steady-state approximation if $v_{-1}$ is much larger than both $v_1$ and $v_2$ . The steady-state approximation would also apply if $v_2$ is much larger than both $v_1$ and $v_{-1}$ . What is the predicted rate la... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27–5.** The Rate Law for a Complex Reaction Does Not Imply a Unique Mechanism",
"token_count": 561,
"source_pdf": "datasets/we... |
If the reaction described by
$$CH_3NC(g) \xrightarrow{k_{obs}} CH_3CN(g)$$
(27.47)
is an elementary reaction, then it must obey the rate law
$$\frac{d[\text{CH}_3\text{NC}]}{dt} = -k_{\text{obs}}[\text{CH}_3\text{NC}]$$
(27.48)
A close examination of this and many other supposed unimolecular reactions reveals t... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27–6.** The Lindemann Mechanism Explains How Unimolecular Reactions Occur",
"token_count": 2000,
"source_pdf": "datasets/webso... |
The low-concentration data show that kobs depends linearly on concentration (Equation 27 .58), and data at high concentration show that kobs is independent of concentration (Equation 27.57). In the region between these two limiting behaviors, *<sup>k</sup> <sup>2</sup>*is comparable with k\_1 [M], and neither of the li... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27–6.** The Lindemann Mechanism Explains How Unimolecular Reactions Occur",
"token_count": 200,
"source_pdf": "datasets/websou... |
This example explores the connection between the observed activation parameters for a chemical reaction and the activation parameters of the individual steps of the reaction mechanism. Specifically, suppose that the rate constant for each step of the Lindemann mechanism shows Arrhenius behavior. How are the measured va... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27–6.** The Lindemann Mechanism Explains How Unimolecular Reactions Occur",
"Header 3": "**EXAMPLE** 27-7",
"token_count": 204... |
Then we can substitute these expressions into Equation 27.61 and thereby obtain the predicted rate law of the overall reaction in terms of the concentrations of the reactants and products.
Note that the three terms on the right side of Equation 27.64 are the negatives of the last three terms on the right side of Equa... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27–6.** The Lindemann Mechanism Explains How Unimolecular Reactions Occur",
"Header 3": "**EXAMPLE** 27-7",
"token_count": 877... |
We know that the rate of a reaction can usually be increased by increasing the temperature and that there are practical limitations to the effects of temperature (Section 26-7). For example, reactions in solution are constrained to the temperature range between the melting and boiling point of the solvent. An entirely ... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27-8.** A Catalyst Affects the Mechanism and Activation Energy of a Chemical Reaction",
"token_count": 1942,
"source_pdf": "da... |
One of the most important classes of catalyzed reactions consists of the biological processes that involve enzymes. *Enzymes* are protein molecules that catalyze specific biochemical reactions. Without enzymes, many of the reactions necessary to sustain life would occur at negligible rates and life as we know it would ... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27-9.** The Michaelis-Menten Mechanism Is a Reaction Mechanism for Enzyme Catalysis",
"token_count": 1875,
"source_pdf": "data... |
| Enzyme | Substrate | Turnover number/s <sup>-1</sup> |
|----------------------|---------------|---------------------------------|
| Catalase | $H_2O_2$ | $4.0 \times 10^{7}$ |
| Acetylcholinesterase | Acetylcholine | $1.4 \times 10^{5}$ |
| $\beta$ -Lactama... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27-9.** The Michaelis-Menten Mechanism Is a Reaction Mechanism for Enzyme Catalysis",
"token_count": 779,
"source_pdf": "datas... |
- **27-1.** Give the units of the rate constant for a unimolecular, bimolecular, and termolecular reaction.
- 27-2. Determine the rate law for the following reaction
$$F(g) + D_2(g) \stackrel{k}{\Longrightarrow} FD(g) + D(g)$$
Give the units of k. Determine the molecularity of this reaction.
27-3. Determine the r... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27-9.** The Michaelis-Menten Mechanism Is a Reaction Mechanism for Enzyme Catalysis",
"Header 3": "**Problems**",
"token_count... |
$$\begin{array}{c} \operatorname{Cl}_2(g) + \operatorname{M}(g) & \stackrel{\stackrel{k_1}{\Longleftrightarrow}}{\rightleftharpoons} 2\operatorname{Cl}(g) + \operatorname{M}(g) & \text{(fast equilibrium)} \\ \\ \operatorname{Cl}(g) + \operatorname{CO}(g) + \operatorname{M}(g) & \stackrel{k_2}{\rightleftharpoons} \ope... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27-9.** The Michaelis-Menten Mechanism Is a Reaction Mechanism for Enzyme Catalysis",
"Header 3": "**Problems**",
"token_count... |
An alternative mechanism for the chemical reaction
$$Cl_2(g) + CO(g) \xrightarrow{k_{obs}} Cl_2CO(g)$$
(see Problem 27-12) is
$$Cl_{2}(g) + M(g) \stackrel{k_{1}}{\rightleftharpoons} 2Cl(g) + M(g) \qquad \text{(fast equilibrium)}$$
$$Cl(g) + Cl_{2}(g) \stackrel{k_{2}}{\rightleftharpoons} Cl_{3}(g) \qquad \text{(... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27-9.** The Michaelis-Menten Mechanism Is a Reaction Mechanism for Enzyme Catalysis",
"Header 3": "**Problems**",
"token_count... |
Show that $\gamma$ for the reaction mechanism and rate law given in Problem 27–25 is
$$\gamma = k_3 \left(\frac{1}{k_1 k_4}\right)^{1/2} [\text{CH}_3 \text{CHO}]^{-1/2}$$
**27-27.** Show that the chain length $\gamma$ (see Problem 27–26) for the reaction mechanism and the rate law given in Problem 27–24 is
$$... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27-9.** The Michaelis-Menten Mechanism Is a Reaction Mechanism for Enzyme Catalysis",
"Header 3": "**Problems**",
"token_count... |
Show that if the steady-state assumption is applied toES, then
$$[ES] = \frac{[E][S]}{K_m}$$
where *Km* is the Michaelis constant, *Km* = (k\_ <sup>1</sup><sup>+</sup>*k3)/* <sup>k</sup>1• Now show that material balance for the enzyme gives
$$\left[\mathbf{E}\right]_0 = \left[\mathbf{E}\right] + \frac{\left[\math... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27-9.** The Michaelis-Menten Mechanism Is a Reaction Mechanism for Enzyme Catalysis",
"Header 3": "**Problems**",
"token_count... |
Which gas is the better catalyst for this reaction?
| T<br>j<br>K | N<br>e<br>1<br>2<br>k<br>6<br>j<br>l<br>d<br>m<br>o<br>m<br>s<br>b<br>-<br>-<br>-<br>-<br>o<br>s | C<br>C<br>1<br>4<br>2<br>1<br>6<br>k<br>b<br>l<br>d<br>J<br>o<br>m<br>o<br>m<br>s<br>·<br>-<br>-<br>- |
|-------------|--------------------------------... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27-9.** The Michaelis-Menten Mechanism Is a Reaction Mechanism for Enzyme Catalysis",
"Header 3": "**Problems**",
"token_count... |
Show that this use of the steady-state approximation gives
$$[O] = \frac{k_1[H][O_2]}{k_2[H_2]}$$
and $[OH] = \frac{2k_1[H][O_2]}{k_3[H_2]}$
Use these results and your rate expression for [H] to show that
$$\frac{d[H]}{dt} = I_0 + (2k_1[O_2] - k_4[O_2][M])[H]$$
27-50. Consider the result of Problem 27-49. The ... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**27-9.** The Michaelis-Menten Mechanism Is a Reaction Mechanism for Enzyme Catalysis",
"Header 3": "**Problems**",
"token_count... |
Bimolecular gas-phase reactions are among the simplest elementary kinetic processes that occur in nature. In this chapter, we will examine some of the current models that are used to describe the molecular aspects of bimolecular gas-phase reactions. First, we will modify the collision theory presented in Chapter 25 and... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**Gas-Phase Reaction Dynamics**",
"token_count": 2026,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
As two molecules collide, the valence electrons of the two molecules repel one another, so no reaction can occur unless the relative speed is sufficient to overcome this repulsive force. We make our first improvement to collision theory by taking into account the dependence of the reaction rate on the relative speed, o... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**Gas-Phase Reaction Dynamics**",
"token_count": 1966,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
The simple energy-dependent reaction cross section given by Equation 28.12 is not realistic. To see why this is so, consider the following two collision geometries
where the arrows indicate the direction from which the molecules approach the point of collision. When the relative colli... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**Gas-Phase Reaction Dynamics**",
"Header 3": "28–2. A Reaction Cross Section Depends Upon the Impact Parameter",
"token_count":... |
The experimental pre-exponential factors are compared with those calculated using hard-sphere collision theory.
| | 3<br>A<br>j<br>d<br>m<br>· | 1<br>1<br>l<br>m<br>o<br>s<br>-... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**Gas-Phase Reaction Dynamics**",
"Header 3": "28–2. A Reaction Cross Section Depends Upon the Impact Parameter",
"token_count":... |
The data in Table 28.1 show that hard-sphere collision theory does not accurately account for the magnitude of the Arrhenius A factor. One of the fundamental flaws of this model is the assumption that every collision of sufficient energy is reactive. In addition to an energy requirement, the reacting molecules may need... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**28-3.** The Rate Constant for a Gas-Phase Chemical Reaction May Depend on the Orientations of the Colliding Molecules",
"token_c... |
The reaction cross section for many gas-phase reactions is dependent on the internal energy of the reacting molecules. Consider the data plotted in Figure 28.4. In this figure, the reaction cross section for the reaction of the hydrogen molecular ion with atomic helium
$$H_2^+(g) + He(g) \Longrightarrow HeH^+(g) + H(... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**28–4.** The Internal Energy of the Reactants Can Affect the Cross Section of a Reaction",
"token_count": 705,
"source_pdf": "d... |
Consider the collision and subsequent scattering process for the bimolecular reaction
$$A(g) + B(g) \Longrightarrow C(g) + D(g)$$
For simplicity, we will assume there are no intermolecular forces between the separated reactants and separated products. Before the collision, molecules A and B are traveling with a vel... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**28-5.** A Reactive Collision Can Be Described in a Center-of-Mass Coordinate System",
"token_count": 1939,
"source_pdf": "data... |
Linear momentum must be conserved in the collision, so
$$m_{\mathrm{A}}\mathbf{u}_{\mathrm{A}} + m_{\mathrm{B}}\mathbf{u}_{\mathrm{B}} = m_{\mathrm{C}}\mathbf{u}_{\mathrm{C}} + m_{\mathrm{D}}\mathbf{u}_{\mathrm{D}} \tag{28.23}$$
Using Equation 28.23, we see that Equations 28.21 and 28.17 are the same, confirming th... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**28-5.** A Reactive Collision Can Be Described in a Center-of-Mass Coordinate System",
"token_count": 1649,
"source_pdf": "data... |
One of the most important experimental techniques used to study the molecular dynamics of bimolecular gas-phase reactions is the *crossed molecular beam method*. The basic design of a crossed molecular beam apparatus is shown in Figure 28.6a. The experimental device is designed to cross a beam of A molecules with a bea... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**28–6.** Reactive Collisions Can Be Studied Using Crossed Molecular Beam Machines",
"token_count": 959,
"source_pdf": "datasets... |
In this and several of the following sections, we will be concerned with the reaction
$$F(g) + D_2(g) \Longrightarrow DF(g) + D(g)$$
(28.25)
Figure 28.8 shows a one-dimensional energy diagram for this reaction. The energy diagram reflects only the changes in potential energy. Diagrams that indicate how the potentia... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**28–7.** The Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Produce Vibrationally Excited DF(g) Molecules",
"token_count"... |
We will now examine the crossed molecular beam data for the reaction described by $F(g) + D_2(v = 0) \Longrightarrow DF(v) + D(g)$ for the case in which the relative translational energy of the reactants is $7.62 \text{ kJ} \cdot \text{mol}^{-1}$ . In Example 28–6, we found that for this value of the relative transl... | {
"Header 1": "**28–8.** The Velocity and Angular Distribution of the Products of a Reactive Collision Provide a Molecular Picture of the Chemical Reaction",
"token_count": 2029,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Thus, we can determine the relative populations of each vibrational state for all possible scattering angles. Rather than in a threedimensional picture that depicts all the reaction trajectories, the data are commonly represented in a two-dimensional polar contour plot. Figure 28.11 shows the contour plot for the react... | {
"Header 1": "**28–8.** The Velocity and Angular Distribution of the Products of a Reactive Collision Provide a Molecular Picture of the Chemical Reaction",
"token_count": 2035,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
The relative populations of the first five vibrational states deserve a bit more attention. Note the lack of contour lines between the dashed circles for v=0 and v=1. This result means that no product molecules are formed in the ground vibrational state. The populations determined from the contour diagram are given i... | {
"Header 1": "**28–8.** The Velocity and Angular Distribution of the Products of a Reactive Collision Provide a Molecular Picture of the Chemical Reaction",
"token_count": 1929,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
In Chapter 9, we learned that the potential energy of a diatomic molecule depends on only the distance between the two bonded atoms. Thus, the potential-energy surface for a diatomic molecule such as $D_2(g)$ or DF(g) can be plotted in two dimensions by plotting the potential energy as a function of the bond length. ... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"token_count": 2044,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
**28-1.** Calculate the hard-sphere collision theory rate constant for the reaction
$$NO(g) + Cl_2(g) \Longrightarrow NOCl(g) + Cl(g)$$
at 300 K. The collision diameters of NO and $\text{Cl}_2$ are 370 pm and 540 pm, respectively. The Arrhenius parameters for the reaction are $A = 3.981 \times 10^9 \, \text{dm}^... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**Problems**",
"token_count": 1969,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
**28-15.** The peak speed of the molecules in a supersonic molecular beam of a carrier gas is well approximated by
$$u_{\text{peak}} = \left(\frac{2RT}{M}\right)^{1/2} \left(\frac{\gamma}{\gamma - 1}\right)^{1/2}$$
where T is the temperature of the source chamber of the gas mixture, M is the molar mass of the car... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**Problems**",
"token_count": 1995,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Using the data given in Table 13.2, estimate the minimum value of the relative speed of the reactants so that the following reactions occur:
$$HCl(v = 0) + Br(g) \Longrightarrow HBr(v = 0) + Cl(g)$$
and
$$HCl(v = 1) + Br(g) \Longrightarrow HBr(v = 0) + Cl(g)$$
- 28-27. Do the values of the radii of the dashed c... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**Problems**",
"token_count": 2031,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
The energy spacing betwen contour lines is $38.6~\text{kJ}\cdot\text{mol}^{-1}$ . Label the location of the oxygen atom in the reactant (OClO) and product (ClOO) molecules. Draw the minimum energy path for the isomerization reaction. Which isomer is more stable? Estimate the range for the height of the activation barr... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**Problems**",
"token_count": 1272,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
In this chapter, we examine some of the modem topics in solid-state chemistry. In the first half, we discuss the structure of crystals. We will learn that X-ray diffraction can be used to determine the structure of atomic and molecular crystals. We will show that the X-ray diffraction pattern of a crystal reflects the ... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**Solids and Surface Chemistry**",
"token_count": 350,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Figure 29.1 shows the arrangement of the atoms in a crystal of copper. From this arrangement, we can see that the crystal possesses a periodic structure, and we should take advantage of this periodicity to describe its structure. We define a *unit cell* to be the smallest collection of atoms (or molecules) in the cryst... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29-1.** The Unit Cell Is the Fundamental Building Block of a Crystal",
"token_count": 2036,
"source_pdf": "datasets/websources/biochem/F814... |
The 14 Bravais lattices are organized into seven classes (triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, trigonal, and cubic) by the general geometric features of the lengths of the three sides of the parallelepiped and the angles between the **a**, **b**, and **c** axes of the unit cells.
Figure 29.3c s... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29-1.** The Unit Cell Is the Fundamental Building Block of a Crystal",
"token_count": 884,
"source_pdf": "datasets/websources/biochem/F814B... |
The coordinates of the atoms contained in a unit cell are expressed in units of a, b, and c, the lengths of three edges of the unit cell. For example, consider the primitive cube unit cell (Figure 29.5). If we take the lattice point at the bottom left corner to be the origin of the crystal coordinate system, the coordi... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29–2.** The Orientation of a Lattice Plane Is Described by its Miller Indices",
"token_count": 1979,
"source_pdf": "datasets/websources/bio... |
The structure of a crystal can be determined using the technique of X-ray diffraction. X-rays are generated by bombarding a metal target (often copper) with high-energy electrons inside a vacuum tube. Collisions between the high-energy electrons and copper atoms generate electronically excited copper cations, which the... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29-3.** The Spacing Between Lattice Planes Can Be Determined from X-Ray Diffraction Measurements",
"token_count": 2019,
"source_pdf": "data... |
The distance between the detected spots corresponding to diffraction from the 000 and 100 planes of the crystal is 2.25 em. The A = 154.433 pm line of copper is used as the X-ray source. What is the length of the unit cell along the a axis?
S 0 L UTI 0 N : The geometry of this experiment is shown below.
![](_page_1... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29-3.** The Spacing Between Lattice Planes Can Be Determined from X-Ray Diffraction Measurements",
"token_count": 1512,
"source_pdf": "data... |
X-rays are scattered by the electrons in crystals. Because both the number of electrons and the size of the atomic orbitals vary from atom to atom, different atoms have different scattering efficiencies. The *scattering factor* of an atom, f, is defined by
$$f = 4\pi \int_0^\infty \rho(r) \frac{\sin kr}{kr} r^2 dr \t... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29–4.** The Total Scattering Intensity Is Related to the Periodic Structure of the Electron Density in the Crystal",
"token_count": 2037,
"... |
The coordinates *x., y.,* and *z.* are commonly expressed in units of *a,* b, and *c,* the lengths *1 1 1* of the unit cells, in which case Equation 29.23 can be written as
$$F(hkl) = \sum_{i} f_{j} e^{2\pi i (hx'_{j} + ky'_{j} + lz'_{j})}$$
(29.24)
where *x1 '* = *x.ja, y'* = *y.jb,* and *z'* = *z.jc.* The quanti... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29–4.** The Total Scattering Intensity Is Related to the Periodic Structure of the Electron Density in the Crystal",
"token_count": 1899,
"... |
In the previous section, we modeled the crystal as a set of atoms located at points (x., y., z.) of the unit cell. We then defined the structure factor in terms of the scattering *1 1 1* ofX-rays from atoms located at each of the positions in the unit cell. In both atomic and molecular crystals, the electron density i... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29-5.** The Structure Factor and the Electron Density Are Related by a Fourier Transform",
"token_count": 1104,
"source_pdf": "datasets/web... |
In 1834, the English chemist Michael Faraday suggested that the first step of a surfacecatalyzed reaction was the sticking of the reactant molecule to the solid surface. Originally, it was believed that the main effect of the surface is to produce a local reactant concentration that is much higher than in the gas phase... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29-5.** The Structure Factor and the Electron Density Are Related by a Fourier Transform",
"Header 3": "**29-6.** A Gas Molecule Can Physisor... |
A plot of surface coverage as a function of gas pressure at constant temperature is called an *adsorption isotherm.* In this section, we will learn that adsorption isotherms can be used to determine the equilibrium constant for the adsorption-desorption reaction, the concentration of surface sites available for adsorpt... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29-7.** Isotherms Are Plots of Surface Coverage as a Function of Gas Pressure at Constant Temperature",
"token_count": 1947,
"source_pdf": ... |
If the mica substrate were a 0.010-m square, the concentration of surface sites would be
$$\sigma_0 = \frac{1.06 \times 10^{18} \text{ molecules}}{(0.010 \text{ m})^2} = 1.06 \times 10^{22} \text{ m}^{-2}$$
Figure 29.22 shows that experimental data for the adsorption of oxygen and carbon monoxide on a silica surfac... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29-7.** Isotherms Are Plots of Surface Coverage as a Function of Gas Pressure at Constant Temperature",
"token_count": 1714,
"source_pdf": ... |
Consider the surface catalysis of the first-order gas-phase reaction
$$A(g) \stackrel{k_{obs}}{\longrightarrow} B(g)$$
such that the observed rate law is given by
$$\frac{d[B]}{dt} = k_{\text{obs}} P_{A} \tag{29.39}$$
We will propose that this reaction occurs by the following two-step mechanism
$$A(g) \stackr... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29–8.** The Langmuir Adsorption Isotherm Can Be Used to Derive Rate Laws for Surface–Catalyzed Gas–Phase Reactions",
"token_count": 2003,
"... |
Plot the predictions of the two models, Equations 29.45 and 29.46 as a function of the partial pressure of CO(g) at a fixed pressure of 0 <sup>2</sup> (g).
S 0 L UTI 0 N : Let us first consider the limiting expressions for the rate laws when *Pco* « *P*0and when *Pco* » *P*<sup>0</sup>. 2 2
(a) The Langmuir-Hinshel... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29–8.** The Langmuir Adsorption Isotherm Can Be Used to Derive Rate Laws for Surface–Catalyzed Gas–Phase Reactions",
"token_count": 2003,
"... |
One of the most thoroughly studied surface-catalyzed reactions is the synthesis of NH3 (g) from H2 (g) and N 2 (g),
$$3 H_2(g) + N_2(g) \longrightarrow 2 NH_3(g)$$
For this reaction to take place, the N2 bond must be broken; thus, the activation barrier is on the order of the dissociation energy of N2 , which is 94... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29-10.** The Reaction Between H2(g) and N2(g) to Produce NH3(g) Can Be Surface Catalyzed",
"token_count": 946,
"source_pdf": "datasets/webs... |
- 29-1. Polonium is the only metal that exists as a simple cubic lattice. Given that the length of a side of the unit cell of polonium is 334.7 pm at 2SOC, calculate the density of polonium.
- 29-2. Consider the packing of hard spheres of radius *R* in a primitive cubic lattice, a facecentered cubic lattice, and a body... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29-10.** The Reaction Between H2(g) and N2(g) to Produce NH3(g) Can Be Surface Catalyzed",
"Header 3": "Problems",
"token_count": 1888,
"... |
What is the distance between the diffraction spots from the 001 and 002 planes on the face of the detector for (a) the $\lambda = 154.433$ -pm line of copper, and (b) the $\lambda = 70.926$ -pm line of a molybdenum X-ray source? Which X-ray source gives you the best spatial resolution between the diffraction spots?
-... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29-10.** The Reaction Between H2(g) and N2(g) to Produce NH3(g) Can Be Surface Catalyzed",
"Header 3": "Problems",
"token_count": 2040,
"... |
Show that there will be observed reflections for a primitive unit cell for all integer values of *h*, *k*, and *l* and reflections for a face-centered-cubic unit cell only if *h*, *k*, and *l* are either all even or all odd.
- **29-38.** Use the results of the previous problem and Example 29–9 to verify the entries in ... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29-10.** The Reaction Between H2(g) and N2(g) to Produce NH3(g) Can Be Surface Catalyzed",
"Header 3": "Problems",
"token_count": 1973,
"... |
Now show that the only values of $\phi$ that satisfy the above relation are 360° (n=1), 180° (n=2), 120° (n=3), 90° (n=4), and 60° (n=6), corresponding to N=2,-2,-1,0, and 1, respectively.
**29-44.** The von Laue equations are often expressed in vector notation. The following figure illustrates the X-ray scattering... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29-10.** The Reaction Between H2(g) and N2(g) to Produce NH3(g) Can Be Surface Catalyzed",
"Header 3": "Problems",
"token_count": 2036,
"... |
Use the Langmuir adsorption isotherm (Equation 29.35) to obtain an expression for the reaction rate in terms of *Kc* and [A]. Under what conditions will the reaction be first order in the concentration of A?
29-55. Consider a surface-catalyzed bimolecular reaction between molecules A and B that has a rate law of the ... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29-10.** The Reaction Between H2(g) and N2(g) to Produce NH3(g) Can Be Surface Catalyzed",
"Header 3": "Problems",
"token_count": 2037,
"... |
**29-68.** Multilayer physisorption is often described by the *BET adsorption isotherm*
$$\frac{P}{V(P^* - P)} = \frac{1}{cV_{\rm m}} + \frac{(c - 1)P}{V_{\rm m}cP^*}$$
where P\* is the vapor pressure of the adsorbate at the temperature of the experiment, *Vm* is the volume corresponding to a monolayer of coverag... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "**29-10.** The Reaction Between H2(g) and N2(g) to Produce NH3(g) Can Be Surface Catalyzed",
"Header 3": "Problems",
"token_count": 1014,
"... |
#### Chapter 1
- **1-1.** $\nu = 1.50 \times 10^{15} \,\mathrm{Hz}$ $\tilde{\nu} = 5.00 \times 10^4 \,\mathrm{cm}^{-1}, \ E = 9.93 \times 10^{-19} \,\mathrm{J}$
- **1-2.** $v = 3 \times 10^{13} \text{ Hz}, \lambda =$ $1 \times 10^{-5} \text{ m}, E = 2 \times 10^{-20} \text{ J}$
- **1-3.** $v = 2.0 \times 10^{10}... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"token_count": 1823,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
- **2-1.** (a) $y(x) = c_1 e^{3x} + c_2 e^x$
- (b) $y(x) = c_1 + c_2 e^{-6x}$ (c) $y(x) = c_1 e^{-3x}$
- (d) $y(x) = c_1 e^{(-1+\sqrt{2})x} + c_2 e^{(-1-\sqrt{2})x}$
- (e) $y(x) = c_1 e^{2x} + c_2 e^x$
- **2-2.** (a) $y(x) = 2e^{2x}$
- (b) $y(x) = -3e^{2x} + 2e^{3x}$ (c) $y(x) = 2e^{2x}$
- **2-4.** (a) $x(t)... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "Chapter 2",
"token_count": 2028,
"source_pdf": "datasets/websources/biochem/F814BC59158753... |
(a) commutes (b) does not commute(c) does not commute (d) does not commute
- **4-15.** $\hat{P}$ and $\hat{Q}$ must commute
**4-16.** (a)
$$[\hat{A}, \hat{B}] = \hat{A}\hat{B}f - \hat{B}\hat{A}f = 2\frac{d}{dx}$$
(b) $[\hat{A}, \hat{B}] = 2$ (c) $[\hat{A}, \hat{B}]f = -f(0)$
(d)
$$[\hat{A}, \hat{B}] = 4x \... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "Chapter 2",
"token_count": 643,
"source_pdf": "datasets/websources/biochem/F814BC591587538... |
**5-3.** the period, which is the time it takes to undergo one cycle, $= 2\pi/\omega = 1/\nu$
**5-7.** 9.104 432 × $10^{-31}$ kg; 0.05%
**5-9.** 479 $N \cdot m^{-1}$
**5-10.** $1.81 \times 10^{10} \text{ m}^{-1}$
**5-11.** V(x) =
$$D[\beta^{2}x^{2} - \beta^{3}x^{3} + \frac{7}{12}\beta^{4}x^{4} + O(x^{5})... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "Chapter 5",
"token_count": 1296,
"source_pdf": "datasets/websources/biochem/F814BC59158753... |
**E-1.** 5, 5, 5
**E-2.** -5, -5
**E-3.** 0, 0
**E-4.** $x^4 - 3x^2 = 0$ ; $x = 0, 0, \pm \sqrt{3}$
**E-5.** $x^4 - 4x^2 = 0$ ; $x = 0, 0, \pm 2$
**E-6.**
$$\cos^2 \theta + \sin^2 \theta = 1$$
**E-7.** $(\frac{9}{5}, \frac{1}{5})$
**E-8.** $(1, 3, -4)$
#### Chapter 7
**7-3.**
$$\beta = (k\mu/7\hba... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "MathChapter E",
"token_count": 1666,
"source_pdf": "datasets/websources/biochem/F814BC5915... |
**8-25.**
$$E(\text{triplet}) = -2.1735E_{\text{h}} = -477\ 100\ \text{cm}^{-1}$$
$$E(\text{singlet}) = -2.0857E_{\text{h}} = -457\ 800\ \text{cm}^{-1}$$
$E(\text{ground state}) = -2.7500E_{\text{h}} = -603\ 555\ \text{cm}^{-1}$
$E(\text{triplet} \rightarrow \text{ground state}) = 126 500 \text{ cm}^{-1}$ $E(\... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "MathChapter E",
"token_count": 1647,
"source_pdf": "datasets/websources/biochem/F814BC5915... |
<sup>3</sup>Π; <sup>1</sup>Π
**9-33.** $-\frac{1}{2}$ $E_{\rm h}$ per atom, 0.625
**9-34.** $19.4 \times 10^{-30} \text{ C} \cdot \text{m}$
**9-35.** $19.4 \times 10^{-30} \text{ C} \cdot \text{m}$
**9-36.** $2.55 \times 10^{-29} \text{ C} \cdot \text{m}, 76.0\%$
**9-38.** 0.181*e*
**9-39.** 0.43*e* ... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "MathChapter E",
"token_count": 1742,
"source_pdf": "datasets/websources/biochem/F814BC5915... |
(b) for hexatriene
$$\begin{split} E_1 &= \alpha + 1.802\beta \ E_2 = \alpha + 1.247\beta \\ E_3 &= \alpha + 0.4450\beta \ E_4 = \\ \alpha - 0.4450\beta \ E_5 = \alpha - 1.247\beta \\ E_6 &= \alpha - 1.802\beta; \ E_{\rm deloc} = 0.9880\beta; \\ E_{\rm deloc} \ {\rm per \ carbon \ atom} = 0.1647 \ {\rm for \ octatetr... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "MathChapter E",
"token_count": 1869,
"source_pdf": "datasets/websources/biochem/F814BC5915... |
**12-3.** $E, C_3, 3C_2, \sigma_h, 3\sigma_v, S_3$
**12-7.** $\hat{C}_4^3$ is a counter-clockwise rotation by 270° and $\hat{C}_4^{-1}$ is a clockwise rotation by 90°.
**12-9.** 16
**12-10.** 24
**12-11.** $\hat{C}_{2}, \hat{\sigma}'_{v}, \hat{\sigma}_{v}$
**12-12.** $\hat{\sigma}'_v, \hat{\sigma}_v$ ... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "Chapter 12",
"token_count": 1367,
"source_pdf": "datasets/websources/biochem/F814BC5915875... |
**12-36.** $\hat{E}f_{\text{even}}(x) = f_{\text{even}}, \hat{\sigma}f_{\text{even}}(x) = f_{\text{even}};$ $\hat{E}f_{\text{odd}}(x) = f_{\text{odd}}(x), \hat{\sigma}f_{\text{odd}}(x) = -f_{\text{odd}}(x)$
**12-37.** $\Gamma = 4 \ 1 - 2 - 4 - 1 \ 2,$ $\Gamma = 2A_2^{"} + E^{"};$
$$\begin{split} \phi_1 &= (\p... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "Chapter 12",
"token_count": 2436,
"source_pdf": "datasets/websources/biochem/F814BC5915875... |
**13-1.** 127 pm
**13-25.** 2558.539 cm<sup>-1</sup>; 5026.642 cm<sup>-1</sup>;
```
13-2. 305 pm
13-3. 1.96 \times 10^{11} \text{ s}^{-1}, 1.96 \times 10^{5} \text{ MHz},
6.54~\rm{cm^{-1}}
13-4. 1.36 \times 10^{12} revolution·s<sup>-1</sup>
13-5. 2\tilde{B} = 2.96 \text{ cm}^{-1}
13-6. 1.896 \times 10^{-46} \text{ ... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "Chapter 13",
"token_count": 2052,
"source_pdf": "datasets/websources/biochem/F814BC5915875... |
$\lambda^2 - \lambda(2\cos^2\theta + 8\sin^2\theta + 8\cos^2\theta +$ $2\sin^2\theta$ ) + $16\cos^4\theta$ + $68\cos^2\theta\sin^2\theta$ + $16\sin^4\theta - 36\sin^2\theta\cos^2\theta =$ $\lambda^2 - 10\lambda + 16(\cos^2\theta + \sin^2\theta)^2 =$ $\lambda^2 - 10\lambda + 16 = 0; \lambda = 2, 8$ 13-42. $\sq... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "Chapter 13",
"token_count": 1269,
"source_pdf": "datasets/websources/biochem/F814BC5915875... |
15-1. fluorescence
**15-3.** $J^{-1} \cdot m^2 \cdot s^{-1}, J^{-1} \cdot m^4 \cdot s^{-1}$
**15-9.** $1.1 \times 10^{-29} \text{ C} \cdot \text{m}$
**15-10.** $9.98 \times 10^{19} \text{ J}^{-1} \cdot \text{m}^2 \cdot \text{s}^{-2}$ ; $7.68 \times 10^{-30} \text{ C} \cdot \text{m}$
**15-12.** $(A_{32} + A... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "Chapter 15",
"token_count": 2022,
"source_pdf": "datasets/websources/biochem/F814BC5915875... |
**16-23.** RK: $20.13 \text{ mol} \cdot \text{L}^{-1}$ , $5.148 \text{ mol} \cdot \text{L}^{-1}$ ; PR: $23.61 \text{ mol} \cdot \text{L}^{-1}$ , $5.564 \text{ mol} \cdot \text{L}^{-1}$
**16-24.** vdW: $4.786 \text{ mol} \cdot \text{L}^{-1}$ , $0.5741 \text{ mol} \cdot \text{L}^{-1}$ ;
$RK{:}\;6.823\;mol{\cd... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "Chapter 15",
"token_count": 1717,
"source_pdf": "datasets/websources/biochem/F814BC5915875... |
$$[x^{2n+3}/(2n+3)!]/[x^{2n+1}/(2n+1)!] \longrightarrow x^2/n^2$$
#### Chapter 18
**18-4.**
$$f_2(T = 300\text{K}) = 4.8 \times 10^{-36}$$
;
$f_2(T = 1000\text{K}) = 2.5 \times 10^{-11}$ ;
$f_2(T = 2000\text{K}) = 5.0 \times 10^{-6}$
**18-5.**
$$f_2(T = 300\text{K}) = 9.0 \times 10^{-32}$$
; $f_2(T = 1000\text... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "Chapter 15",
"token_count": 1615,
"source_pdf": "datasets/websources/biochem/F814BC5915875... |
15 bar; 3000 J
**19-3.** 28.7 bar; 3.60 J
**19-4.** 4.01 kJ
**19-5.** -1.73 kJ
**19-6.** 11.4 kJ
**19-7.** +325 J; +309 J; they differ because w is a path function
**19-9.** −3.93 kJ·mol<sup>-1</sup>
**19-10.** $-3.92 \text{ kJ} \cdot \text{mol}^{-1}$
**19-12.**
$$V_1 = 11.35 \text{ L}$$
; $V_2 = 22.7... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "Chapter 15",
"token_count": 1364,
"source_pdf": "datasets/websources/biochem/F814BC5915875... |
0.0194 in Table J.1
#### Chapter 20
**20-2.**
$$dz/y$$
$$\begin{aligned} \mathbf{20-6.} & \quad q_{\text{rev}} = \int_{T_1}^{T_4} \overline{C}_V(T) dT + \int_{T_4}^{T_1} \overline{C}_V(T) dT \\ - \int_{\overline{V}}^{\overline{V}_2} P_2 d\overline{V}; \, \Delta \overline{S} = R \ln \frac{V_2}{V_1} \end{aligned}$$... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "Chapter 15",
"token_count": 1282,
"source_pdf": "datasets/websources/biochem/F814BC5915875... |
| S<br>b<br>t<br>u<br>s<br>a<br>n<br>c<br>e | S<br>f<br>1<br>1<br>J<br>K<br>l<br>m<br>o<br>-<br>-<br>-<br>-<br>a<br>p<br>v<br>~ |
|----------------------------------------------------------------------------------------|-----------------------------------------------------... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "Chapter 15",
"token_count": 2035,
"source_pdf": "datasets/websources/biochem/F814BC5915875... |
~ G(80.09°C) = o; vap
$$\Delta_{\text{vap}} \overline{G}(75.0^{\circ}\text{C}) = 0.43 \text{ kJ} \cdot \text{mol}^{-1};$$
$\Delta_{\text{mol}} \overline{G}(85.0^{\circ}\text{C}) = -0.44 \text{ kJ} \cdot \text{mol}^{-1};$
vap 22-2. ~vap G(80.09°C) = 0;
$$\Delta_{\text{vap}}\overline{G}(75.0^{\circ}\text{C}) = 43... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "Chapter 15",
"token_count": 900,
"source_pdf": "datasets/websources/biochem/F814BC59158753... |
$$\begin{array}{c|cccc} Gas & Ar & N_2 & CO_2 \\ \hline T_i(theor.)/K & 791 & 634 & 1310 \\ T_i(exp.)/K & 794 & 621 & 1500 \\ Percent Difference & 0.378 & 2.09 & 12.7 \\ \end{array}$$
22-53. Ar: 42.6 K, N2: 25.7 K, C02: 129 K. 22-55. -19.7 J.K- <sup>1</sup> -mol- <sup>1</sup>versus
-19.1 J-K- <sup>1</sup> -mol- <su... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "22-52.",
"token_count": 1549,
"source_pdf": "datasets/websources/biochem/F814BC59158753848... |
**24-23.**
$$\Delta_{r}H^{\circ} = 6.91 \text{ kJ} \cdot \text{mol}^{-1}$$
**24-24.**
$$K_P = 1.35$$
**24-25.**
$$K_p = 14.9$$
**24-26.**
$$\Delta_r H^\circ = 99.6 \text{ kJ} \cdot \text{mol}^{-1}$$
**24-27.**
$$\Delta_{\rm r} H^{\circ} = 12.02 \text{ kJ} \cdot \text{mol}^{-1}$$
**24-29.**
$$\Delta_r G^\cir... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "22-52.",
"token_count": 1251,
"source_pdf": "datasets/websources/biochem/F814BC59158753848... |
| T/K | $K_P(JANAF)$ | $K_p$ (Problem 24-39) |
|------|-----------------------|-----------------------|
| 800 | $3.05 \times 10^{-5}$ | $3.14 \times 10^{-5}$ |
| 900 | $4.26\times10^{-4}$ | $4.08 \times 10^{-4}$ |
| 1000 | $3.08 \times 10^{-3}$ | $3.19 \times 10^{-3}$ |
| 1100 | $1.66\times10^{-2}$ | ... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "22-52.",
"token_count": 1725,
"source_pdf": "datasets/websources/biochem/F814BC59158753848... |
**25-31.**
$$1.6 \times 10^{-11}$$
s to cover 1.0% of one square meter
**25-32.**
$$5.1 \times 10^{17}$$
**25-35.** (a)
$$z_A = 1.32 \times 10^7 \text{ s}^{-1}$$
(b) $z_A = 9.89 \times 10^9 \text{ s}^{-1}$
**25-36.** (a)
$$t = 1.88 \times 10^{-7} \text{ s}$$
(b) $t = 2.51 \times 10^{-10} \text{ s}$
**25-3... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "22-52.",
"token_count": 916,
"source_pdf": "datasets/websources/biochem/F814BC591587538482... |
**26-2.**
$$d[N_2O]/dt = -1.23 \times 10^{-5} \text{ mol·dm}^{-3} \cdot \text{s}^{-1},$$
$d[N_2]/dt = 1.23 \times 10^{-5} \text{ mol·dm}^{-3} \cdot \text{s}^{-1},$
$d[O_2]/dt = 6.16 \times 10^{-6} \text{ mol} \cdot \text{dm}^{-3} \cdot \text{s}^{-1}$
**26-3.**
$$1.64 \times 10^{-5} \text{ mol} \cdot \text{s}^{-... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "22-52.",
"token_count": 1957,
"source_pdf": "datasets/websources/biochem/F814BC59158753848... |
**27-15.** fast equilibrium for step 1; $k_{\text{obs}} = k_2 k_1 / k_{-1}$
**27-16.**
$$\frac{d[O_2]}{dt} = \frac{k_2 k_1}{k_1} \frac{[C_6 H_5 CO_3 H]^2}{[H^+]}$$
- **27-17.** Assume steady-state for $N_2O$ . $k_{obs} = k_1$
- **27-18.** fast equilibrium step 1; steady state for N<sub>2</sub>O; $k_{\text{obs... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "22-52.",
"token_count": 1817,
"source_pdf": "datasets/websources/biochem/F814BC59158753848... |
| v | $u_{\rm r}^{\prime}/10^3{\rm m\cdot s^{-1}}$ | $ \mathbf{u}_{\mathrm{HCl}} - \mathbf{u}_{\mathrm{cm}} /10^{3} \mathrm{m \cdot s^{-1}}$ |
|---|----------------------------------------------|-----------------------------------------------------------------------------------------|
| 0 | 2.437 ... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "22-52.",
"token_count": 1984,
"source_pdf": "datasets/websources/biochem/F814BC59158753848... |
**29-54.** first order if K)A] « **1**
1/2 **29-59.** *k* = *k3 KH Kc* H , *K* = *Kc* H , C2H4 2 2 **4 2 4**
adsorbs more extensively than H2to the surface.
**29-60.** Langmuir-Hinshelwood mechanism, <sup>k</sup>1/2 *K* = KNH ; *k* <sup>=</sup><sup>3</sup> KNH *K*<sup>0</sup>, NH3 adsords 3 3 2 to the surface m... | {
"Header 1": "**28–10.** The Potential-Energy Surface for the Reaction $F(g) + D_2(g) \\Rightarrow DF(g) + D(g)$ Can Be Calculated Using Ouantum Mechanics",
"Header 2": "Answers to the Numerical Problems",
"Header 3": "22-52.",
"token_count": 350,
"source_pdf": "datasets/websources/biochem/F814BC591587538482... |
| | Preface | 6 |
|-----|-------------------------------------------|----|
| 1 | Ecology and ecosystems | 7 |
| 1.1 | Ecology and biosystems | 7 |
| 1.2 | Ecosystem concept | 8 |
| 2 | Energy flow in ecosystems ... | {
"Header 1": "CONTENTS",
"token_count": 766,
"source_pdf": "datasets/websources/biochem/General_Ecology.pdf"
} |
This book is written to meet the need for a concise textbook of ecology. The book describes the basic features of the modern ecology and is addressed to college students without special biological knowledge. The book can be used in high schools, technical colleges and other places of study where ecology forms part of t... | {
"Header 1": "PREFACE",
"token_count": 279,
"source_pdf": "datasets/websources/biochem/General_Ecology.pdf"
} |
The word ecology is derived from the Greek word "oikos" meaning house, combined with "logy" meaning the learning of ecology and can therefore be translated to "the learning of nature's household", though the word ecology is not of ancient Greek origin. The term ecology was first used in 1869 by the German biologist Ern... | {
"Header 1": "1 ECOLOGY AND ECOSYSTEMS",
"token_count": 208,
"source_pdf": "datasets/websources/biochem/General_Ecology.pdf"
} |
It has become customary to define ecology as "the science of biological systems above the organism level". This definition and delimitation of modern ecology is illustrated in Fig. 1, showing the different levels of organization (cell, organ, individual, population, community) and how these living (biotic) components i... | {
"Header 1": "1.1 ECOLOGY AND BIOSYSTEMS",
"token_count": 400,
"source_pdf": "datasets/websources/biochem/General_Ecology.pdf"
} |
An ideal ecosystem is a closed – but not isolated – biosystem with all of its biotic and abiotic components flowing through with energy. You can imagine an ideal ecosystem as an illuminated aquarium, see Fig. 2.
Fig. 2. Simplified model of an aquatic ecosystem. An ecosystem has a number ... | {
"Header 1": "1.2 ECOSYSTEM CONCEPT",
"token_count": 472,
"source_pdf": "datasets/websources/biochem/General_Ecology.pdf"
} |
Solar radiation consists of electromagnetic waves which are created when hydrogen nuclei in the sun fusion at very high temperatures. The sunbeam spectrum is very wide, but almost all the radiation energy is in the visible light between the ultraviolet and the infrared spectrum, see Fig. 3.
... | {
"Header 1": "2 ENERGY FLOW IN ECOSYSTEMS",
"Header 2": "2.1 SOLAR RADIATION AND GLOBAL ENERGY BALANCE",
"token_count": 1379,
"source_pdf": "datasets/websources/biochem/General_Ecology.pdf"
} |
The fraction of light energy that is absorbed by the green plants is by photosynthesis converted to chemically bound energy in the produced organic matter. The photosynthesis process takes place in the chloroplasts of the plants and can be described by the following reaction:
$$CO_2 + 2H_2O^* \rightarrow (CH_2O) + O_... | {
"Header 1": "2 ENERGY FLOW IN ECOSYSTEMS",
"Header 2": "2.2 PRIMARY PRODUCTION AND PRODUCTIVITY",
"token_count": 613,
"source_pdf": "datasets/websources/biochem/General_Ecology.pdf"
} |
The transfer of energy from plants to animals is done through a number of links called a food chain. Organisms located at the same step in relation to the food source are said to be on the same trophic level. The trophic classification is based on function – not on species that often can be placed on multiple trophic l... | {
"Header 1": "2.3 FOOD CHAINS",
"token_count": 581,
"source_pdf": "datasets/websources/biochem/General_Ecology.pdf"
} |
Bioenergetics is concerned with examining how the living organisms absorbs food, digests and distributes consumed or synthesized substance and energy for maintenance (metabolism = respiration) and production (new cells, storage, reproduction). The flow of energy through a population can be determined from knowledge of ... | {
"Header 1": "2.4 BIOENERGETICS",
"token_count": 1384,
"source_pdf": "datasets/websources/biochem/General_Ecology.pdf"
} |
Figure 11 shows with realistic figures the flow of energy through three trophic levels in a theoretical grazing food chain. The simplified diagram shows how energy (as heat and detritus) is lost in and between each link in the food chain. The relationships between energy flows both within and between trophic levels hav... | {
"Header 1": "2.5 ECOLOGICAL EFFICIENCIES",
"token_count": 1308,
"source_pdf": "datasets/websources/biochem/General_Ecology.pdf"
} |
Chemical substances (pollutants) that are difficult to decompose (persistent) and at the same time fat-soluble, tend to accumulate (biomagnify) in the food chains because these substances may effectively be transferred from link to link in the food chain. At the same time there is a combustion of biomass which becomes ... | {
"Header 1": "2.6 BIOMAGNIFICATION OF POLLUTANTS",
"token_count": 752,
"source_pdf": "datasets/websources/biochem/General_Ecology.pdf"
} |
In the biological evolution, only certain elements have been used as the atomic building blocks in the living cell. Of the 92 naturally occurring elements, it is now believed that only about 24 are involved in the life processes. The molecular building blocks of the cell, namely proteins, carbohydrates and fats are mad... | {
"Header 1": "3.1 SEDIMENTARY AND GASEOUS NUTRIENT CYCLES",
"token_count": 680,
"source_pdf": "datasets/websources/biochem/General_Ecology.pdf"
} |
The carbon compounds in the biosphere are all the time being formed, transformed and decomposed, see Fig. 16 [12, 13]. This dynamic state is maintained by the autotrophic and heterotrophic organisms. The autotrophic organisms (i. e. the green plants and the chemoand photoautotrophic bacteria) produce organic carbon com... | {
"Header 1": "3.2 CARBON CYCLE",
"token_count": 1894,
"source_pdf": "datasets/websources/biochem/General_Ecology.pdf"
} |
Since 1958 reliable measurements have been made of the atmospheric carbon dioxide content. Fig. 17 shows that the content of carbon dioxide is increasing. It is not currently known to what extent this increase in atmospheric carbon dioxide content will be able to change the world's climate. But there is reason for conc... | {
"Header 1": "3.2.1 INCREASED GREENHOUSE EFFECT",
"token_count": 539,
"source_pdf": "datasets/websources/biochem/General_Ecology.pdf"
} |
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