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In the previous section, we showed that the molar Gibbs energy of an ideal gas is given by
$$\overline{G}(T, P) = G^{\circ}(T) + RT \ln \frac{P}{P^{\circ}}$$
(22.64)
The pressure $P^{\circ}$ is equal to one bar and $G^{\circ}(T)$ is called the standard molar Gibbs energy. Recall that this equation is derived by... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**22–7.** The Gibbs–Helmholtz Equation Describes the Temperature Dependence of the Gibbs Energy",
"Header 3": "22–8. Fugacity Is a Measure of the No... |
The sum of $\Delta \overline{G}_2$ and $\Delta \overline{G}_3$ gives another expression for $\Delta \overline{G}_1$
$$\Delta \overline{G}_1 = \Delta \overline{G}_2 + \Delta \overline{G}_3 = \int_{P \to 0}^{P} \left( \frac{RT}{P'} - \overline{V} \right) dP'$$
Equating this expression for $\Delta \overline{G}... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**22–7.** The Gibbs–Helmholtz Equation Describes the Temperature Dependence of the Gibbs Energy",
"Header 3": "22–8. Fugacity Is a Measure of the No... |
- **22-1.** The molar enthalpy of vaporization of benzene at its normal boiling point (80.09°C) is $30.72 \text{ kJ} \cdot \text{mol}^{-1}$ . Assuming that $\Delta_{\text{vap}} \overline{H}$ and $\Delta_{\text{vap}} \overline{S}$ stay constant at their values at $80.09^{\circ}\text{C}$ , calculate the value of $... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**Problems**",
"token_count": 1994,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
The density of water at $25^{\circ}$ C is 0.99705 g·mL<sup>-1</sup>.
- **22-10.** Use Equation 22.22 to show that
$$\left(\frac{\partial C_V}{\partial V}\right)_T = T \left(\frac{\partial^2 P}{\partial T^2}\right)_V$$
Show that $(\partial C_V/\partial V)_T = 0$ for an ideal gas and a van der Waals gas, and that ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**Problems**",
"token_count": 1912,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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Show that
$$dH = \left[V - T\left(\frac{\partial V}{\partial T}\right)_{P}\right] dP + C_{P} dT$$
What does this equation tell you about the natural variables of H?
- **22-19.** What are the natural variables of the entropy?
- **22-20.** Experimentally determined entropies are commonly adjusted for nonideality by... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**Problems**",
"token_count": 1958,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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The compressibility factor for ethane at 600 K can be fit to the expression
$$Z = 1.0000 - 0.000612(P/bar) + 2.661 \times 10^{-6}(P/bar)^{2}$$
$$-1.390 \times 10^{-9}(P/bar)^{3} - 1.077 \times 10^{-13}(P/bar)^{4}$$
for 0 .::; *P* jbar .::; 600. Use this expression to determine the fugacity coefficient of ethane as ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**Problems**",
"token_count": 1815,
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You need to use the standard integral
$$\int \frac{dx}{x(a+bx)} = -\frac{1}{a} \ln \frac{a+bx}{x}$$
**22-37.** Show that $\ln \gamma$ for the Redlich-Kwong equation (see Problem 22–36) can be written in the reduced form
$$\begin{split} \ln \gamma &= \frac{0.25992}{\overline{V}_{\rm R} - 0.25992} - \frac{1.2824}... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**Problems**",
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Start with dH = TdS + VdP to derive
$$\left(\frac{\partial H}{\partial V}\right)_T = T \left(\frac{\partial S}{\partial V}\right)_T + V \left(\frac{\partial P}{\partial V}\right)_T$$
and use one of the Maxwell relations for $(\partial S/\partial V)_T$ to obtain
$$\left(\frac{\partial H}{\partial V}\right)_T = T... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**Problems**",
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In this case, *dU* = *dw* = *fdL.* Use the fact that *U* depends upon only the temperature for an ideal rubber band to show that
$$dU = \left(\frac{\partial U}{\partial T}\right)_L dT = f dL \tag{7}$$
The quantity ca *u ;aT)* L is a heat capacity, so Equation 7 becomes
$$C_L dT = f dL (8)$$
Argue now that if a ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**Problems**",
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The relation between all the phases of a substance at various temperatures and pressures can be concisely represented by a phase diagram. In this chapter, we will study the information presented by phase diagrams and the thermodynamic consequences of this information. In particular, we will analyze the temperature and ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**Phase Equilibria**",
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You might recall from general chemistry that we can summarize the solid-liquid-gas behavior of a substance by means of a phase diagram, which indicates under what conditions of pressure and temperature the various states of matter of a substance exist in equilibrium. Figure 23.1 shows the phase diagram of benzene, a ty... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**23–1.** A Phase Diagram Summarizes the Solid–Liquid–Gas Behavior of a Substance",
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The triple point pressure of C02 is 5.11 atm, and so we see that C02 sublimes at one atm. The normal sublimation temperature of C02 is 195 K ( -78°C).
Figure 23.5 shows the phase diagram for water. Water has the unusual property that its melting point decreases with increasing pressure (Figure 23.6). This behavior is... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**23–1.** A Phase Diagram Summarizes the Solid–Liquid–Gas Behavior of a Substance",
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Recall Figure 22.7 where the molar Gibbs energy of benzene is plotted against temperature. As the figure shows, the molar Gibbs energy is a continuous function of temperature, but there is a discontinuity in the slope of $\overline{G}(T)$ versus T at each phase transition. Figure 23.10a is a magnification of a plot o... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**23–2.** The Gibbs Energy of a Substance Has a Close Connection to Its Phase Diagram",
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Consider a system consisting of two phases of a pure substance in equilibrium with each other. For example, we might have water vapor in equilibrium with liquid water. The Gibbs energy of this system is given by G = G<sup>1</sup>+ Gg, where G1 and Gg are the
#### **FIGURE 23.12**
A pl... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**23-3.** The Chemical Potentials of a Pure Substance in Two Phases in Equilibrium Are Equal",
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The molar enthalpy of fusion of benzene at its normal melting point (278.7 K) is 9.95 kJ ·mol-<sup>1</sup> , and ~fus V under the same conditions is 10.3 cm3 -mol- <sup>1</sup> • Thus, *dP ldT* at the normal melting point of benzene is
$$\frac{dP}{dT} = \frac{9950 \text{ J} \cdot \text{mol}^{-1}}{(278.68 \text{ K})(1... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**23-3.** The Chemical Potentials of a Pure Substance in Two Phases in Equilibrium Are Equal",
"token_count": 2037,
"source_pdf": "datasets/websou... |
When we used Equation 23.10 to calculate the variation of the melting points of ice (Example 23-3) and benzene, we assumed that ~trs Hand ~trs *V* do not vary appreciably with pressure. Although this approximation is fairly satisfactory for solid-liquid and solid-solid transitions over a small ~ T, it is not satisfacto... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**23-4.** The Clausius-Clapeyron Equation Gives the Vapor Pressure of a Substance As a Function of Temperature",
"token_count": 2009,
"source_pdf"... |
#### **EXAMPLE 23-6**
Use the Clausius-Clapeyron equation and Equation 23.16 to determine the molar enthalpy of sublimation of ammonia from 146 K to 195 K.
S 0 L UTI 0 N: According to Equation 23.12
$$\frac{d\ln P}{dT} = \frac{\Delta_{\rm sub}\overline{H}}{RT^2}$$
Using Equation 23.16 for In *P* gives us
$$... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**23-4.** The Clausius-Clapeyron Equation Gives the Vapor Pressure of a Substance As a Function of Temperature",
"token_count": 1999,
"source_pdf"... |
We can now substitute Equation 23.23 into Equation 23.26 to obtain
$$\mu = -k_{\rm B}T \left(\frac{\partial \ln Q}{\partial n}\right)_{\rm V,T} = -RT \left(\frac{\partial \ln Q}{\partial N}\right)_{\rm V,T} \tag{23.27}$$
We have gone from the second term to the third term by multiplying $k_{\rm B}$ and n by the... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**23-4.** The Clausius-Clapeyron Equation Gives the Vapor Pressure of a Substance As a Function of Temperature",
"token_count": 1952,
"source_pdf"... |
Substituting this result into Equation 23.34 gives
$$\begin{split} \mu^{\circ}(T) - E_0 &= -RT \ln \left[ \left( \frac{q^0}{V} \right) \frac{k_{\rm B}T}{P^{\circ}} \right] \\ &= -RT \ln \left[ \left( \frac{q^0}{V} \right) \frac{RT}{N_{\rm A}P^{\circ}} \right] \end{split} \tag{23.36}$$
Problems 949
The partition f... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**23-4.** The Clausius-Clapeyron Equation Gives the Vapor Pressure of a Substance As a Function of Temperature",
"token_count": 999,
"source_pdf":... |
- **23-1.** Sketch the phase diagram for oxygen using the following data: triple point, 54.3 K and 1.14 torr; critical point, 154.6 K and 37 828 torr; normal melting point, -218.4°C; and normal boiling point, -182.9°C. Does oxygen melt under an applied pressure as water does?
- 23-2. Sketch the phase diagram for I<sub>... | {
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"Header 2": "**Problems**",
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| /<br>T<br>K | 1<br>3<br>/<br>l<br>d<br>p<br>m<br>o<br>m<br>-<br>- | 3<br>/<br>l<br>d<br>p<br>g<br>m<br>o<br>m<br>-<br>- | T<br>/<br>K | 1<br>3<br>j<br>l<br>d<br>p<br>m<br>o<br>m<br>·<br>- | 3<br>j<br>l<br>d<br>p<br>g<br>m<br>o<br>m<br>-<br>- |
|-----------------------|---------------------------... | {
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"Header 2": "**Problems**",
"token_count": 1705,
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| T/K | $\Delta_{\mathrm{vap}}\overline{H}/\mathrm{J}\cdot\mathrm{mol}^{-1}$ | T/K | $\Delta_{\mathrm{vap}}\overline{H}/\mathrm{J}\!\cdot\!\mathrm{mol}^{-1}$ |
|-------|----------------------------------------------------------------------|-------|------------------------------------------------------------------... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**Problems**",
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- **23-20.** Determine the value of dT/dP for water at its normal boiling point of 373.15 K given that the molar enthalpy of vaporization is $40.65 \text{ kJ} \cdot \text{mol}^{-1}$ , and the densities of the liquid and vapor are $0.9584 \text{ g} \cdot \text{mL}^{-1}$ and $0.6010 \text{ g} \cdot \text{L}^{-1}$ ,... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**Problems**",
"token_count": 2015,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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Consider the phase change
$$C(graphite) \rightleftharpoons C(diamond)$$
Given that $\Delta_{\rm r}G^\circ/{\rm J\cdot mol}^{-1}=1895+3.363T$ , calculate $\Delta_{\rm r}H^\circ$ and $\Delta_{\rm r}S^\circ$ . Calculate the pressure at which diamond and graphite are in equilibrium with each other at 25°C. Take the... | {
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"Header 2": "**Problems**",
"token_count": 1891,
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To prove *Maxwell's equal-area construction rule*, integrate $(\partial \mu/\partial P)_T = \overline{V}$ by parts along the path bcdef and use the fact that $\mu^1$ (the value of $\mu$ at point f) = $\mu^g$ (the value of $\mu$ at point h) to obtain
$$\mu^{1} - \mu^{g} = P_{0}(\overline{V}^{1} - \overline{V... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**Problems**",
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It has been shown experimentally that
$$\rho^{1} - \rho^{g} \longrightarrow A(T_{c} - T)^{\beta}$$
where fJ = 0.324. Thus, as in the previous problem, although qualitatively correct, the van der Waals theory is not quantitatively correct.
**23-49.** The following data give the temperature, the vapor pressure, and... | {
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"Header 2": "**Problems**",
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For simplicity, we shall discuss gas-phase reactions first. Consider the general gasphase reaction, which is described by the balanced equation
$$\nu_{A}A(g) + \nu_{B}B(g) \rightleftharpoons \nu_{Y}Y(g) + \nu_{Z}Z(g)$$
We define a quantity ~, called the *extent of reaction,* such that the numbers of moles of the re... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-1.** Chemical Equilibrium Results When the Gibbs Energy Is a Minimum with Respect to the Extent of Reaction",
"token_count": 1615,
"source_pd... |
If we assume that all the partial pressures are low enough that we can consider each species to behave ideally, then we can use Equation 23.33 $[\mu_j(T,P)=\mu_j^\circ(T)+RT\ln(P_j/P^\circ)]$ for the $\mu_j(T,P)$ , in which case Equation 24.6 becomes
$$\begin{split} \Delta_{\mathrm{r}}G &= \nu_{\mathrm{Y}}\mu_{\... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-1.** Chemical Equilibrium Results When the Gibbs Energy Is a Minimum with Respect to the Extent of Reaction",
"token_count": 1361,
"source_pd... |
Equation 24.11 says that regardless of the initial pressures of the reactants and products, at equilibrium the ratio of their partial pressures raised to their respective stoichiometric coefficients will be a fixed value at a given temperature. Consider the reaction described by
$$PCl5(g) \rightleftharpoons PCl3(g) +... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-1.** Chemical Equilibrium Results When the Gibbs Energy Is a Minimum with Respect to the Extent of Reaction",
"Header 3": "24–2. An Equilibrium... |
You must always be aware, however, of which reference states are being used in $K_P$ and $K_c$ when converting the numerical value of one to the other.
#### EXAMPLE 24-3
The value of $K_p(T)$ (based upon a standard state of one bar) for the reaction described by
$$NH_3(g) \rightleftharpoons \frac{3}{2}H_2(g... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-1.** Chemical Equilibrium Results When the Gibbs Energy Is a Minimum with Respect to the Extent of Reaction",
"Header 3": "24–2. An Equilibrium... |
Notice that combining Equations 24.8 and 24.11 gives a relation between p,j(T), the standard chemical potentials of the reactants and products, and the equilibrium constant, K P. In particular, K P is related to the difference between the standard chemical potentials of the products and reactants. Because a chemical po... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-3.** Standard Gibbs Energies of Formation Can Be Used to Calculate Equilibrium Constants",
"token_count": 981,
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| S<br>b<br>t<br>u<br>s<br>a<br>n<br>c<br>e | F<br>l<br>o<br>r<br>m<br>u<br>a | 1<br>/'<br>f<br>G<br>/<br>k<br>l·<br>l<br>:,<br>o<br>m<br>o<br>- |
|-----------------------------------------------------------------------------------------------... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-3.** Standard Gibbs Energies of Formation Can Be Used to Calculate Equilibrium Constants",
"token_count": 2227,
"source_pdf": "datasets/webso... |
In this section we shall treat a concrete example of the Gibbs energy of a reaction mixture as a function of the extent of reaction. Consider the thermal decomposition of N20 <sup>4</sup> (g) to NO/g) at 298.15 K, which we represent by the equation
$$N_2O_4(g) \rightleftharpoons 2NO_2(g)$$
Suppose we start with one... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-4.** A Plot of the Gibbs Energy of a Reaction Mixture Against the Extent of Reaction Is a Minimum at Equilibrium",
"token_count": 2041,
"sour... |
We can also evaluate $\xi_{eq}$ explicitly by setting Equation 24.23 equal to zero. Using the fact that the last two terms in Equation 24.23 add up to zero, we have
$$\frac{(2)(51.258 \text{ kJ} \cdot \text{mol}^{-1}) - 97.787 \text{ kJ} \cdot \text{mol}^{-1}}{(8.3145 \text{ J} \cdot \text{mol}^{-1} \cdot \text{K... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-4.** A Plot of the Gibbs Energy of a Reaction Mixture Against the Extent of Reaction Is a Minimum at Equilibrium",
"token_count": 349,
"sourc... |
Consider the general reaction described by the equation
$$v_A A(g) + v_A B(g) \rightleftharpoons v_Y Y(g) + v_Z Z(g)$$
Equation 24.7 for this reaction scheme is
$$\Delta_{\rm r} G(T) = \Delta_{\rm r} G^{\circ}(T) + RT \ln \frac{P_{\rm Y}^{\nu_{\rm Y}} P_{\rm Z}^{\nu_{\rm Z}}}{P_{\rm A}^{\nu_{\rm A}} P_{\rm B}^{\n... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24–5.** The Ratio of the Reaction Quotient to the Equilibrium Constant Determines the Direction in Which a Reaction Will Proceed",
"token_count": ... |
It is important to appreciate the difference between $\Delta_r G$ and $\Delta_r G^\circ$ . The superscript $^\circ$ on $\Delta_r G^\circ$ emphasizes that this is the value of $\Delta_r G$ when all the reactants and products are unmixed at partial pressures equal to one bar; $\Delta_r G^\circ$ is the *standar... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24–6.** The Sign of $\\Delta_r G$ And Not That of $\\Delta_r G^{\\circ}$ Determines the Direction of Reaction Spontaneity",
"token_count": 986,
... |
We can use the Gibbs-Helmoltz equation (Equation 22.61)
$$\left(\frac{\partial \Delta G^{\circ}/T}{\partial T}\right)_{P} = -\frac{\Delta H^{\circ}}{T^{2}} \tag{24.28}$$
to derive an equation for the temperature dependence of K *P* ( T). Substitute fl Go ( T) <sup>=</sup> -RTln Kp(T) into Equation 24.28 to obtain ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-7.** The Variation of an Equilibrium Constant with Temperature Is Given by the Van't Hoff Equation",
"token_count": 2031,
"source_pdf": "data... |
Given that $\Delta_f H^\circ[NH_3(g)] = -46.11 \text{ kJ} \cdot \text{mol}^{-1}$ at 300 K and that $K_p = 6.55 \times 10^{-3}$ at 725 K, derive a general expression for the variation of $K_p(T)$ with temperature in the form of Equation 24.34.
SOLUTION: We first use Equation 24.32
$$\Delta_{\rm r} H^\circ(T_2)... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-7.** The Variation of an Equilibrium Constant with Temperature Is Given by the Van't Hoff Equation",
"token_count": 1140,
"source_pdf": "data... |
An important chemical application of statistical thermodynamics is the calculation of equilibrium constants in terms of molecular parameters. Consider the general homogeneous gas-phase chemical reaction
$$\nu_{A}A(g) + \nu_{B}B(g) \rightleftharpoons \nu_{Y}Y(g) + \nu_{Z}Z(g)$$
in a reaction vessel at fixed volume a... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-7.** The Variation of an Equilibrium Constant with Temperature Is Given by the Van't Hoff Equation",
"Header 3": "**24-8.** We Can Calculate Eq... |
We shall calculate the equilibrium constant for the reaction
$$H_2(g) + I_2(g) \rightleftharpoons 2 HI(g)$$
from 500 K to 1000 K. The equilibrium constant is given by
$$K(T) = \frac{(q_{\rm HI}/V)^2}{(q_{\rm H_2}/V)(q_{\rm I_2}/V)} = \frac{q_{\rm HI}^2}{q_{\rm H_2}q_{\rm I_2}}$$
(24.40)
Using Equation 18.39 for... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-7.** The Variation of an Equilibrium Constant with Temperature Is Given by the Van't Hoff Equation",
"Header 3": "A. A Chemical Reaction Involv... |
**TABLE 24.3** The logarithm of the equilibrium constant for the reaction H<sup>2</sup> (g) + ~ <sup>0</sup> <sup>2</sup> (g) ¢ *HzD* (g)
| T<br>/<br>K | l<br>K<br>P<br>(<br>l<br>)<br>n<br>c<br>a<br>c | I<br>K<br>P<br>(<br>)<br>n<br>e<br>x<br>p |
|-------------|------------------------------------------------|----... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-7.** The Variation of an Equilibrium Constant with Temperature Is Given by the Van't Hoff Equation",
"Header 3": "A. A Chemical Reaction Involv... |
In the previous section we have seen that the rigid rotator-harmonic oscillator approximation can be used to calculate equilibrium constants in reasonably good agreement with experiment, and because of the simplicity of the model, the calculations involved are not extensive. If greater accuracy is desired, however, one... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-9.** Molecular Partition Functions and Related Thermodynamic Data Are Extensively Tabulated",
"token_count": 364,
"source_pdf": "datasets/web... |
Consequently, Go(T) and *Ho(T)* are expressed relative to that value, as expressed
| | | | | | ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-9.** Molecular Partition Functions and Related Thermodynamic Data Are Extensively Tabulated",
"token_count": 1711,
"source_pdf": "datasets/we... |
| -1G°- H°(T,)/√ | IMPINITE<br>10 223.211 | | | | ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-9.** Molecular Partition Functions and Related Thermodynamic Data Are Extensively Tabulated",
"token_count": 1089,
"source_pdf": "datasets/we... |
| ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-9.** Molecular Partition Functions and Related Thermodynamic Data Are Extensively Tabulated",
"token_count": 1956,
"source_pdf": "datasets/we... |
| Heat Capacity and Satzopy<br>The thermoderant functions differ from thems of the 1885 JAMAF table (1) in being taken directly form the later and more | complete work of Maar treated in detail the contribution of the highly anharmonic out-of-plane vibrational gode, | inclosing its market Cooperate Cooperate State Coop... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-9.** Molecular Partition Functions and Related Thermodynamic Data Are Extensively Tabulated",
"token_count": 2011,
"source_pdf": "datasets/we... |
If we substitute Equation 24.46 into Equation 17.41, then we obtain
$$U = \langle E \rangle = Nk_{\rm B}T^2 \left(\frac{\partial \ln q}{\partial T}\right)_{V}$$
$$= N\varepsilon_0 + Nk_{\rm B}T^2 \left(\frac{\partial \ln q^0}{\partial T}\right)_{V}$$
(24.47)
For one mole of an ideal gas, $\overline{H} = H^{\circ}(... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-9.** Molecular Partition Functions and Related Thermodynamic Data Are Extensively Tabulated",
"token_count": 1927,
"source_pdf": "datasets/we... |
Because $G^{\circ} = \mu^{\circ}$ for a pure substance, we can write Equation 24.50 as
$$G^{\circ} - H_0^{\circ} = -RT \ln \left\{ \left( \frac{q^0}{V} \right) \frac{RT}{N_{\scriptscriptstyle \Delta} P^{\circ}} \right\}$$
(24.51)
It is easy to show that $G^{\circ} \to H_0^{\circ}$ as $T \to 0$ (because $T ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-9.** Molecular Partition Functions and Related Thermodynamic Data Are Extensively Tabulated",
"token_count": 1850,
"source_pdf": "datasets/we... |
These values correspond to the equations
$$\frac{1}{2} \operatorname{H}_{2}(g) \rightleftharpoons \operatorname{H}(g) \quad \Delta_{f} H^{\circ}(0 \text{ K}) = 216.035 \text{ kJ} \cdot \text{mol}^{-1}$$
(2)
and
$$\frac{1}{2} N_2(g) \rightleftharpoons N(g) \qquad \Delta_f H^\circ(0 \text{ K}) = 470.82 \text{ kJ} \... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-9.** Molecular Partition Functions and Related Thermodynamic Data Are Extensively Tabulated",
"token_count": 880,
"source_pdf": "datasets/web... |
Up to this point in this chapter, we have discussed equilibria in systems of ideal gases only. In this section, we shall discuss equilibria in systems of nonideal gases. In Section 22-8 we introduced the idea of fugacity through the equation
$$\mu(T, P) = \mu^{\circ}(T) + RT \ln \frac{f}{f^{\circ}}$$
(24.53)
where ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24-10.** Equilibrium Constants for Real Gases Are Expressed in Terms of Partial Fugacities",
"token_count": 1957,
"source_pdf": "datasets/websou... |
In the previous section we discussed the condition of equilibrium for a reaction system consisting of real gases. The central result was the introduction of $K_f$ , in which the equilibrium constant is expressed in terms of partial fugacites. In this section we shall derive a similar expression for general equilibrium... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24–11.** Thermodynamic Equilibrium Constants Are Expressed in Terms of Activities",
"token_count": 1681,
"source_pdf": "datasets/websources/bioc... |
To calculate the activity, we start with
$$\left(\frac{\partial \mu}{\partial P}\right)_T = \overline{V} \tag{24.66}$$
and the constant-temperature derivative of Equation 24.59
$$d\mu = RTd \ln a \qquad \text{(constant } T\text{)} \tag{24.67}$$
If we write Equation 24.66 as
$$d\mu = \overline{V}dP \qquad \tex... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24–11.** Thermodynamic Equilibrium Constants Are Expressed in Terms of Activities",
"token_count": 1363,
"source_pdf": "datasets/websources/bioc... |
**24-1.** Express the concentrations of each species in the following chemical equations in terms of the extent of reaction, $\xi$ . The initial conditions are given under each equation.
$$\begin{array}{cccccccccccccccccccccccccccccccccccc$$
**24-2.** Write out the equilibrium-constant expression for the reaction ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24–11.** Thermodynamic Equilibrium Constants Are Expressed in Terms of Activities",
"Header 3": "**Problems**",
"token_count": 1846,
"source_p... |
**24-16.** Consider the two equations
(1)
$$CO(g) + H_2O(g) \rightleftharpoons CO_2(g) + H_2(g)$$
$K_1$
(2)
$$CH_4(g) + H_2O(g) \rightleftharpoons CO(g) + 3H_2(g)$$
$K_2$
Show that *K3* = *K* 1*K2* for the sum of these two equations
(3)
$$CH_4(g) + 2H_2O(g) \rightleftharpoons CO_2(g) + 4H_2(g)$$
$K_2$
How d... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24–11.** Thermodynamic Equilibrium Constants Are Expressed in Terms of Activities",
"Header 3": "**Problems**",
"token_count": 1898,
"source_p... |
Is the reaction described by the equation
$$H_2(g) + CO_2(g) \rightleftharpoons CO(g) + H_2O(g)$$
$K_P = 1.59$
at equilibrium under these conditions? If not, in what direction will the reaction proceed to attain equilibrium?
**24-22.** Given that $K_P = 2.21 \times 10^4$ at 25°C for the equation
$$2 H_2(g) + ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24–11.** Thermodynamic Equilibrium Constants Are Expressed in Terms of Activities",
"Header 3": "**Problems**",
"token_count": 1990,
"source_p... |
**24-33.** Show that
$$\mu = -RT \ln \frac{q(V, T)}{N}$$
if
$$Q(N, V, T) = \frac{[q(V, T)]^N}{N!}$$
- **24-34.** Use Equation 24.40 to calculate K(T) at 750 K for the reaction described by H2 (g) + 1<sup>2</sup> (g) ~ 2HI(g). Use the molecular parameters given in Table 18.2. Compare your value to the one give... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24–11.** Thermodynamic Equilibrium Constants Are Expressed in Terms of Activities",
"Header 3": "**Problems**",
"token_count": 1965,
"source_p... |
**24-49.** The JANAF tables give the following data for Ar(g) at 298.15 K and one bar:
$$-\frac{G^{\circ} - H^{\circ}(298.15 \text{ K})}{T} = 154.845 \text{ J} \cdot \text{mol}^{-1} \cdot \text{K}^{-1}$$
and
$$H^{\circ}(0 \text{ K}) - H^{\circ}(298.15 \text{ K}) = -6.197 \text{ kJ} \cdot \text{mol}^{-1}$$
Use... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24–11.** Thermodynamic Equilibrium Constants Are Expressed in Terms of Activities",
"Header 3": "**Problems**",
"token_count": 1941,
"source_p... |
Given that
$$C_P^{\circ}[O_2(g)]/R = 4.8919 - \frac{829.931 \text{ K}}{T} - \frac{127962 \text{ K}^2}{T^2}$$
$C_P^{\circ}[Hg(g)]/R = 2.500$
$$C_P^{\circ}[\text{HgO(s, red)}]/R = 5.2995$$
in the interval298 K < *T* < 750 K, calculate ~,H , ~,S , and ~,Go at 298 K.
**24-60.** Consider the dissociation of Ag2 0(... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**24–11.** Thermodynamic Equilibrium Constants Are Expressed in Terms of Activities",
"Header 3": "**Problems**",
"token_count": 1172,
"source_p... |
The fact that *all* gases obey the ideal gas equation when the pressure is sufficiently low implies that the form of the equation is independent of the nature of the gas itself. In this chapter we shall introduce a simple model of gases in which we picture the molecules of a gas to be in constant, incessant motion, col... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**The Kinetic Theory of Gases**",
"token_count": 278,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
The pressure that a gas exerts on the walls of its container is due to the collisions that the particles of the gas make with the walls. Let's consider one of the molecules of the gas (call it molecule 1) as it moves throughout its container, as illustrated in Figure 25 .1. We have assumed that the container is a recta... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**25-1.** The Average Translational Kinetic Energy of the Molecules in a Gas Is Directly Proportional to the Kelvin Temperature",
"token_count": 198... |
If we use the value $R=8.314~{\rm J\cdot mol^{-1}\cdot K^{-1}}$ and be sure to express the molar mass in units of kg·mol<sup>-1</sup>, then $u_{\rm rms}$ will have units of ${\rm m\cdot s^{-1}}$ . Therefore
$$u_{\text{rms}} = \left(\frac{3 \times 8.314 \text{ J} \cdot \text{mol}^{-1} \cdot \text{K}^{-1} \times 2... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**25-1.** The Average Translational Kinetic Energy of the Molecules in a Gas Is Directly Proportional to the Kelvin Temperature",
"token_count": 132... |
As we implied in the previous section, all molecules in a gas do not have the same speed. Experimentally, the molecular speeds in a gas are described by the curves in Figure 25.2, where the distribution of molecular speeds is plotted against u. Notice that a greater fraction of molecules has higher speeds as the temper... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**25–2.** The Distribution of the Components of Molecular Speeds Are Described by a Gaussian Distribution",
"token_count": 1788,
"source_pdf": "da... |
In terms of $f(u_x)$ , the average of $u_x^2$ is given by (MathChapter B)
$$\langle u_x^2 \rangle = \frac{RT}{M} = \int_{-\infty}^{\infty} u_x^2 f(u_x) du_x = \left(\frac{\gamma}{\pi}\right)^{1/2} \int_{-\infty}^{\infty} u_x^2 e^{-\gamma u_x^2} du_x$$
(25.30)
Notice once again that the integrand in Equation 25.3... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**25–2.** The Distribution of the Components of Molecular Speeds Are Described by a Gaussian Distribution",
"token_count": 2028,
"source_pdf": "da... |
Substituting the relation $u_x = c(v - v_0)/v_0$ from Equation 25.36 into Equation 25.33 gives
$$I(\nu) \propto e^{-mc^2(\nu - \nu_0)^2/2\nu_0^2 k_{\rm B}T}$$
(25.37)
for the observed shape of the spectral line. The form of I(v) is that of a Gaussian curve centered at $v_0$ with a variance given by (see MathCha... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**25–2.** The Distribution of the Components of Molecular Speeds Are Described by a Gaussian Distribution",
"token_count": 339,
"source_pdf": "dat... |
So far we have derived the probability distribution for a given component of the molecular velocity. Because a homogeneous gas is isotropic, the direction in which a molecule moves has no physical consequence on the properties of the gas; only the magnitude of **u,** the speed, is relevant. Therefore, in this section w... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**25-3.** The Distribution of Molecular Speeds Is Given by the Maxwell-Boltzmann Distribution",
"token_count": 1987,
"source_pdf": "datasets/webso... |
$$\frac{dF(u)}{du} = 4\pi \left(\frac{m}{2\pi k_{\rm B}T}\right)^{3/2} \left[2u - \frac{mu^3}{k_{\rm B}T}\right] e^{-mu^2/2k_{\rm B}T} = 0$$
**TABLE 25.2** Some integrals that occur frequently in the kinetic theory of gases.
$$\int_{0}^{\infty} x^{2n} e^{-\alpha x^{2}} dx = \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**25-3.** The Distribution of Molecular Speeds Is Given by the Maxwell-Boltzmann Distribution",
"token_count": 1307,
"source_pdf": "datasets/webso... |
In this section we shall derive an expression for the frequency of collisions that the molecules of a gas make with the walls of its container. Such a quantity is central to the theory of the rates of surface reactions. The geometry that we shall use to derive the desired equation is shown in Figure 25.5. Figure 25.5 s... | {
"Header 1": "**25–4.** The Frequency of Collisions that a Gas Makes with a Wall Is Proportional to its Number Density and to the Average Molecular Speed",
"token_count": 1836,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
The pressure exerted on the wall by those molecules whose speed is between u and u + du and whose direction lies in the solid angle $\sin\theta d\theta d\phi$ is equal to the product of the momentum change per collision and the frequency of collisions per unit area (Equation 25.46)
$$dP = (2mu\cos\theta)dz_{\text{c... | {
"Header 1": "**25–4.** The Frequency of Collisions that a Gas Makes with a Wall Is Proportional to its Number Density and to the Average Molecular Speed",
"token_count": 480,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
The Maxwell-Boltzmann distribution has been verified experimentally in a number of different experiments, but one of the most straightforward is due to Kusch and his coworkers at Columbia University in the 1950s. Their experimental set-up, sketched in Figure 25.8, consists of a furnace with a very small hole that allow... | {
"Header 1": "**25–4.** The Frequency of Collisions that a Gas Makes with a Wall Is Proportional to its Number Density and to the Average Molecular Speed",
"Header 2": "**25–5.** The Maxwell–Boltzmann Distribution Has Been Verified Experimentally",
"token_count": 468,
"source_pdf": "datasets/websources/biochem... |
When we discuss the theory of the rates of gas-phase chemical reactions in Chapter 28, we shall need to know about the frequency of collisions between the molecules in a gas. First let's consider the frequency of collisions of a single gas-phase molecule. As usual, we shall treat the molecules as hard spheres of diamet... | {
"Header 1": "**25–4.** The Frequency of Collisions that a Gas Makes with a Wall Is Proportional to its Number Density and to the Average Molecular Speed",
"Header 2": "**25–5.** The Maxwell–Boltzmann Distribution Has Been Verified Experimentally",
"Header 3": "**25-6.** The Mean Free Path Is the Average Distanc... |
Use Equation 25.50 to calculate the collision frequency of a single nitrogen molecule in nitrogen at 25oC and one bar.
SOLUTION: According to Table 25.3, 1J = 0.450 x 10-18 m2 for nitrogen. The number density of nitrogen at 25oC and one bar was calculated in Example 25-6 to be p = 2.43 x 1025 m-3 and the average spee... | {
"Header 1": "**25–4.** The Frequency of Collisions that a Gas Makes with a Wall Is Proportional to its Number Density and to the Average Molecular Speed",
"Header 2": "**25–5.** The Maxwell–Boltzmann Distribution Has Been Verified Experimentally",
"Header 3": "**EXAMPLE** 25-7",
"token_count": 2034,
"source... |
Thus, one half of the molecules will be scattered before they travel 70% of the mean-free-path.
Figure 25.12 shows the probability that a molecule collides before it travels a distance x plotted against x/l.
One other quantity that we shall introduce in this section is the total collision frequency per unit volume,... | {
"Header 1": "**25–4.** The Frequency of Collisions that a Gas Makes with a Wall Is Proportional to its Number Density and to the Average Molecular Speed",
"Header 2": "**25–5.** The Maxwell–Boltzmann Distribution Has Been Verified Experimentally",
"Header 3": "**EXAMPLE** 25-7",
"token_count": 1033,
"source... |
In Example 25–10, we calculated that the number of collisions between molecules at one bar and $20^{\circ}\text{C}$ is about $6 \times 10^{28} \text{ s}^{-1} \cdot \text{cm}^{-3}$ , or about $10^{8} \text{ mol} \cdot \text{dm}^{-3} \cdot \text{s}^{-1}$ . Consider now a gas-phase chemical reaction $A + B \to \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.",
"token_count": 1644,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
- **25-1.** Calculate the average translational energy of one mole of ethane at 400 K, assuming ideal behavior. Compare your result to $\overline{U}^{id}$ for ethane at 400 K given in Figure 22.3.
- **25-2.** Calculate the root-mean-square speed of a nitrogen molecule at 200 K, 300 K, 500 K, and 1000 K.
- **25-3.** I... | {
"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": "**Problems**",
"token_count": 1966,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Take the mass of the moon to be $7.35 \times 10^{22}$ kg and its radius to be $1.74 \times 10^6$ m.
- **25-20.** Show that the variance of Equation 25.37 is given by $\sigma^2 = v_0^2 k_B T/mc^2$ . Calcuate $\sigma$ for the $3p^2 P_{3/2}$ to $3s^2 S_{1/2}$ transition in atomic sodium vapor (see Figure 8.4) a... | {
"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": "**Problems**",
"token_count": 2027,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
- **25-44.** Calculate the frequency of nitrogen—oxygen collisions per dm<sup>3</sup> in air at the conditions given in Problem 25–40. Assume in this case that 80% of the molecules are nitrogen molecules.
- 25-45. Use Equation 25.58 to show that
$$\langle u_{r} \rangle = (\langle u_{A} \rangle^{2} + \langle u_{B} \... | {
"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": "**Problems**",
"token_count": 1746,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
This chapter begins our study of the area of physical chemistry known as chemical kinetics. Our development of chemical kinetics will differ from our presentation of quantum mechanics and thermodynamics. In developing quantum mechanics, we started with a small set of postulates, and classical thermodynamics is built up... | {
"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": "**Chemical Kinetics 1: Rate Laws**",
"token_count": 368,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Consider the general chemical reaction described by
$$\nu_{A}A + \nu_{B}B \longrightarrow \nu_{Y}Y + \nu_{Z}Z$$
(26.1)
Recall that we defined the extent of reaction, $\xi$ , in Chapter 24 such that
$$\begin{split} n_{\rm A}(t) &= n_{\rm A}(0) - \nu_{\rm A} \xi(t) & n_{\rm B}(t) = n_{\rm B}(0) - \nu_{\rm B} \xi(t... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**26–1.** The Time Dependence of a Chemical Reaction Is Described by a Rate Law",
"token_count": 1540,
"source_pdf": "datasets/w... |
**TABLE 26.1** Examples of gas-phase chemical reactions and their corresponding rate laws
| C<br>h<br>i<br>l<br>i<br>t<br>e<br>m<br>c<br>a<br>r<br>e<br>a<br>c<br>o<br>n | R<br>l<br>t<br>a<br>e<br>a<br>w |
|----... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**26–1.** The Time Dependence of a Chemical Reaction Is Described by a Rate Law",
"token_count": 1507,
"source_pdf": "datasets/w... |
In this section, we will examine two experimental techniques that chemists use to determine rate laws. For discussion purposes, we consider the general chemical equation
$$\nu_{A} A + \nu_{B} B \longrightarrow \nu_{Y} Y + \nu_{Z} Z$$
(26.12)
and assume that the rate law has the form
$$v = k[A]^{m_A}[B]^{m_B} (26.... | {
"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": "**26–1.** The Time Dependence of a Chemical Reaction Is Described by a Rate Law",
"Header 3": "**26–2.** Rate Laws Must Be Determi... |
Using an expression analogous to Equation 26.19 for $m_{\rm F_2}$ , we find
$$\begin{split} m_{\text{NO}_2} &= \frac{\ln \frac{1.36 \times 10^{-3} \text{ mol} \cdot \text{dm}^{-3} \cdot \text{s}^{-1}}{6.12 \times 10^{-4} \text{ mol} \cdot \text{dm}^{-3} \cdot \text{s}^{-1}}}{\ln \frac{1.72 \text{ mol} \cdot \text{dm... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**26–1.** The Time Dependence of a Chemical Reaction Is Described by a Rate Law",
"Header 3": "**26–2.** Rate Laws Must Be Determi... |
Consider the reactions given by Equation 26.20, where A and B denote the reactants:
$$A + B \longrightarrow products$$
(26.20)
This chemical equation does not tell us anything about its rate law. Suppose the rate law is first order in [A]. Then
$$v(t) = -\frac{d[A]}{dt} = k[A]$$
(26.21)
If the concentration of ... | {
"Header 1": "**25–7.** The Rate of a Gas–Phase Chemical Reaction Depends Upon the Rate of Collisions in Which the Relative Kinetic Energy Exceeds Some Critical Value.",
"Header 2": "**26–3.** First-Order Reactions Show an Exponential Decay of Reactant Concentration with Time",
"token_count": 1815,
"source_pdf... |
S 0 L UTI 0 N: The rate of formation of NO is given by the rate law
$$v = \frac{1}{2} \frac{d[NO]}{dt} = k[N_2 O_2]$$
(1)
The rate law for the disappearance ofN20 <sup>2</sup>is first order, so we can use Equation 26.23 to describe [N2 0 <sup>2</sup>], which allows us to rewrite Equation 1 as
$$\frac{d[\text{NO... | {
"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": "**26–3.** First-Order Reactions Show an Exponential Decay of Reactant Concentration with Time",
"token_count": 819,
"source_pdf"... |
What are the time dependencies of concentrations for rate laws that are not first order? Do the concentrations of the reactants still decay exponentially with time? If the reactant concentrations were to decay exponentially regardless of order, experimental measurements of concentration as a function of time would not ... | {
"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": "**26-4.** The Rate Laws for Different Reaction Orders Predict Different Behaviors for Time-Dependent Reactant Concentrations",
"to... |
| Reaction | $k/\mathrm{dm}^3 \cdot \mathrm{mol}^{-1} \cdot \mathrm{s}^-$ |
|----------------------------------------------------------------------------|--------------------------------------------------------------|
| $2 \operatorname{HI}(g) \to \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": "**26-4.** The Rate Laws for Different Reaction Orders Predict Different Behaviors for Time-Dependent Reactant Concentrations",
"to... |
#### **FIGURE 26.5**
The time dependence of $[A]/[A]_0$ (solid line) and $[B]/[A]_0$ (dashed line) for the reversible reaction given by Equation 26.34 subject to the inital conditions $[A] = [A]_0$ and [B] = 0 at time t = 0. The rate constants for the forward and reverse reacti... | {
"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": "**26-4.** The Rate Laws for Different Reaction Orders Predict Different Behaviors for Time-Dependent Reactant Concentrations",
"to... |
In Section 26–2, we discussed two experimental techniques used to determine the rate law for a chemical reaction provided that the half-life is long compared with the time needed to mix the two reactants. The same constraint applies to the study of reversible reactions. If equilibrium is achieved faster than the reacta... | {
"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": "**26–6.** The Rate Constants of a Reversible Reaction Can Be Determined Using Relaxation Methods",
"token_count": 1853,
"source_... |
Furthermore, $[A]_{2,eq}$ and $[B]_{2,eq}$ satisfy Equation 26.42 $(k_1[A]_{2,eq} = k_{-1}[B]_{2,eq})$ , so Equation 26.47 becomes
$$\frac{d\Delta[B]}{dt} = -(k_1 + k_{-1})\Delta[B]$$
(26.48)
Integrating Equation 26.48 subject to the condition that $[B] = [B]_{1,eq}$ at t = 0, or that $\Delta[B]$ at time 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": "**26–6.** The Rate Constants of a Reversible Reaction Can Be Determined Using Relaxation Methods",
"token_count": 2033,
"source_... |
SOLUTION: If we assume that both the forward and reverse reactions are first order in their respective reactants, the relaxation time is given by Equation 26.54 or
$$\tau = \frac{1}{k_1([\mathrm{H}^+]_{2,\mathrm{eq}} + [\mathrm{C_6H_5COO}^-]_{2,\mathrm{eq}}) + k_{-1}}$$
(1)
From Table 26.7, $k_1=3.5\times 10^{10... | {
"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": "**26–6.** The Rate Constants of a Reversible Reaction Can Be Determined Using Relaxation Methods",
"token_count": 520,
"source_p... |
The rates of chemical reactions almost always depend strongly upon the temperature. Figure 26.7 shows the temperature dependence of the rates of several types of reactions. The temperature dependence shown in (a) is the most common, and is the one we will discuss in detail. The other two curves in the figure illustrate... | {
"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": "**26–6.** The Rate Constants of a Reversible Reaction Can Be Determined Using Relaxation Methods",
"Header 3": "26–7. Rate Constan... |
In this section, we will briefly discuss a theory of reaction rates called *activated-complex theory* or *transition-state theory*. This theory was developed in the 1930s, principally by Henry Eyring, and focuses on the transient species in the vicinity of the top of the activation barrier to reaction. This species 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": "**26–8.** Transition-State Theory Can Be Used to Estimate Reaction Rate Constants",
"token_count": 1821,
"source_pdf": "datasets... |
Solving Equation 26.71 for $K_c^{\ddagger}$ and substituting the result into Equation 26.70 gives us
$$k(T) = \frac{k_{\rm B}T}{hc^{\circ}} e^{-\Delta^{\ddagger}G^{\circ}/RT}$$
(26.72)
We can express $\Delta^{\ddagger}G^{\circ}$ in terms of $\Delta^{\ddagger}H^{\circ}$ , the standard enthalpy of activation, an... | {
"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": "**26–8.** Transition-State Theory Can Be Used to Estimate Reaction Rate Constants",
"token_count": 1435,
"source_pdf": "datasets... |
**26-1.** For each of the following chemical reactions, calculate the equilibrium extent of reaction at 298.15 K and one bar. (See Section 24–4.)
(a)
$$\mathrm{H_2(g)} + \mathrm{Cl_2(g)} \rightleftharpoons 2\,\mathrm{HCl(g)}$$
$\Delta_\mathrm{r}G^\circ = -190.54\,\mathrm{kJ}\cdot\mathrm{mol}^{-1}$
Initial amounts: ... | {
"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": "**26–8.** Transition-State Theory Can Be Used to Estimate Reaction Rate Constants",
"Header 3": "**Problems**",
"token_count": 1... |
| 3<br>l<br>d<br>[<br>o<br>c<br>n<br>;<br>m<br>o<br>m<br>-<br>- | 3<br>l<br>d<br>(<br>]<br>/<br>r<br>m<br>o<br>m<br>-<br>- | 3<br>l<br>d<br>[<br>O<br>H<br>]<br>/<br>m<br>o<br>m<br>-<br>-<br>- | 3<br>1<br>j<br>l<br>d<br>m<br>o<br>m<br>s<br>v<br>·<br>-<br>-<br>-<br>0 |
|-------------------------------------------------... | {
"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": "**26–8.** Transition-State Theory Can Be Used to Estimate Reaction Rate Constants",
"Header 3": "**Problems**",
"token_count": 2... |
Potassium-40 decays by two different paths
$$^{40}_{19}\text{K} \longrightarrow ^{40}_{20}\text{Ca} + ^{0}_{-1}e$$
(89.3%)
$^{40}_{19}\text{K} \longrightarrow ^{40}_{18}\text{Ar} + ^{0}_{1}e$ (10.7%)
The overall half-life for the decay of potassium—40 is $1.3 \times 10^9$ y. Estimate the age of sedimentary rocks... | {
"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": "**26–8.** Transition-State Theory Can Be Used to Estimate Reaction Rate Constants",
"Header 3": "**Problems**",
"token_count": 1... |
Show that $c = k_1[A]_0$ and that
$$(k_1 + k_{-1})[A] - k_{-1}[A]_0 = k_1[A]_0 e^{-(k_1 + k_{-1})t}$$
(1)
Now let $t \to \infty$ and show that
$[A]_0 = \frac{(k_1 + k_{-1})[A]_{eq}}{k_{-1}}$
and
$$[A]_0 - [A]_{eq} = \frac{k_1 [A]_{eq}}{k_{-1}} = \frac{k_1 [A]_0}{k_1 + k_{-1}}$$
Use these results in 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": "**26–8.** Transition-State Theory Can Be Used to Estimate Reaction Rate Constants",
"Header 3": "**Problems**",
"token_count": 2... |
**26-42.** The Arrhenius parameters for the reaction described by
$$HO_2(g) + OH(g) \longrightarrow H_2O(g) + O_2(g)$$
are *A=* 5.01 x 1010 dm3 ·mol- 1 ·s-1 and *E* = 4.18 kJ·mol- <sup>1</sup> • Determine the value of the a rate constant for this reaction at 298 K.
- **26-43.** At what temperature will the reac... | {
"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": "**26–8.** Transition-State Theory Can Be Used to Estimate Reaction Rate Constants",
"Header 3": "**Problems**",
"token_count": 2... |
from the University of California at Berkeley in 1972, he joined Rowland as a postdoctoral fellow at UC Irvine and worked with him on the studies of chlorine and ftuorochloromethane in the atmosphere. In 1989, he joined the faculty of The Massachusetts Institute of Technology, where he currently remains. Molina donated... | {
"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": "**26–8.** Transition-State Theory Can Be Used to Estimate Reaction Rate Constants",
"Header 3": "**Problems**",
"token_count": 2... |
Many chemical reactions involve reaction intermediates, and the overall kinetic process can be written as
reactants ---+ intermediates ---+ products
As an example of this type of reaction, consider the chemical reaction given by
$$NO_2(g) + CO(g) \xrightarrow{k_{obs}} NO(g) + CO_2(g)$$
(27.1)
This reaction does... | {
"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",
"token_count": 2017,
"source... |
Therefore, at equilibrium,
$$v_1 = k_1[A]_{eq}[C]_{eq} = v_{-1} = k_{-1}[B]_{eq}[C]_{eq}$$
(27.10)
and
$$v_2 = k_2[A]_{eq} = v_{-2} = k_{-2}[B]_{eq}$$
(27.11)
The equilibrium conditions given by Equations 27.10 and 27.11 become
$$\frac{[B]_{eq}}{[A]_{eq}} = K_c = \frac{k_1}{k_{-1}}$$
(27.12)
and
$$\frac{[... | {
"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",
"token_count": 420,
"source_... |
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