# 10.15: Problems

1. Show that ${\left(\frac{\partial \left({A}/{T}\right)}{\partial T}\right)}_V=-\frac{E}{T^2}$

2. At 60 C, the vapor pressure of water is 19,932 Pa, and the enthalpy of vaporization is 42.482 kJ mol$${}^{-1}$$.

(a) Is the vaporization of water at these conditions impossible, spontaneous, or reversible? What is $$\Delta G$$ for this process?

(b) Estimate $$\Delta G$$ for the vaporization of liquid water at 19,932 Pa and 70 C. Is this process impossible, spontaneous, or reversible?

(c) Estimate $$\Delta G$$ for the vaporization of liquid water at 19,932 Pa and 50 C. Is this process impossible, spontaneous, or reversible?

3. At 298.15 K and 1 bar, the Gibbs free energy of one mole of $$N_2O_4$$ is 4.729 kJ less than the Gibbs free energy of two moles of $$NO_2$$. The enthalpy of one mole of $$N_2O_4$$ is 57.111 kJ less than the enthalpy of two moles of $$NO_2$$. We customarily express these facts by saying that the Gibbs free energy and the enthalpy changes for the reaction $$2\ NO_2\to N_2O_4$$ are $${\Delta }_rG^o\left(298.15\ \mathrm{K}\right)=-4.729\ \mathrm{kJ}$$ and $${\Delta }_rH^o\left(298.15\ \mathrm{K}\right)=-57.111\ \mathrm{kJ}$$. Assume that the enthalpy change for this process is independent of temperature. Estimate the Gibbs free energy change for this reaction at 500 K and 1 bar, $${\Delta }_rG^o\left(500\ \mathrm{K}\right)$$.

4. Over the temperature range $$300\ \mathrm{K}, the Gibbs free energy change for the formation of ammonia from the elements, \({\frac{1}{2}}N_2+{\frac{3}{2\ }} H_2\to NH_3$$, is well approximated by ${\Delta }_fG^o\left(NH_3\right)=a+b\left(T-600\right)+c{\left(T-600\right)}^2+d{\left(T-600\right)}^3$ where $$a=15.824\ \mathrm{kJ}$$, $$b=0.1120\ \mathrm{kJ\ }{\mathrm{K}}^{--\mathrm{1}}$$, $$c=1.316\times {10}^{-5}\ \mathrm{kJ\ }{\mathrm{K}}^{--\mathrm{2}}$$, and $$d=-1.324\times {10}^{-8}\ \mathrm{kJ\ }{\mathrm{K}}^{--\mathrm{3}}$$. Estimate the enthalpy change for this process, $${\Delta }_fH^o\left(NH_3\right)$$, at 600 K.

5. Consider the total differentials for $$S=S\left(P,T\right)$$, $$E=E\left(P,T\right)$$, $$H=H\left(P,T\right)$$, $$A=A\left(P,T\right)$$, and $$G=G\left(P,T\right)$$. Can we ever encounter an undefined integrand when we evaluate the line integral of one of these total differentials between any two points $$\left(P_1T_1\right)$$ and $$\left(P_2T_2\right)$$? (In the next chapter, we find that, because of the third law of thermodynamics, no real system can ever reach the absolute zero of temperature.)

6. Consider the total differentials for $$S=S\left(V,T\right)$$, $$E=E\left(V,T\right)$$, $$H=H\left(V,T\right)$$, $$A=A\left(V,T\right)$$, and $$G=G\left(V,T\right)$$. Can we ever encounter an undefined integrand when we evaluate the line integral of one of these total differentials between any two points $$\left(V_1T_1\right)$$ and $$\left(V_2T_2\right)$$?

7. The normal boiling point of methanol is 337.8 K at 1 atm. The enthalpy of vaporization at the normal boiling point is $${\Delta }_{vap}H=35.21\ \mathrm{kJ}\mathrm{\ }{\mathrm{mol}}^{--\mathrm{1}}$$. Is the process impossible, spontaneous, or reversible? Find $$q$$, $$w$$, $${\Delta }_{vap}E$$, $${\Delta }_{vap}S$$, $${\Delta }_{vap}A$$, $${\Delta }_{vap}G$$ for the vaporization of one mole of methanol at the normal boiling point. Assume that methanol vapor behaves as an ideal gas.

8. For $$S=S\left(P,V\right)$$, we obtain ${\left(\frac{\partial S}{\partial V}\right)}_P=\frac{C_P}{T}{\left(\frac{\partial T}{\partial V}\right)}_P$ For $$S=S\left(P,T\right)$$, we obtain ${\left(\frac{\partial S}{\partial T}\right)}_P=\frac{C_P}{T}$ For temperatures near 4 C and at a pressure of 1 atm, the molar volume of water is given by $\overline{V}={\overline{V}}_4+a{\left(T-277.15\right)}^2$ where $${\overline{V}}_4=1.801575\times {10}^{-6}\ {\mathrm{m}}^{\mathrm{3}}\mathrm{\ }{\mathrm{mol}}^{--\mathrm{1}}$$ and $$a=1.45\times {10}^{-11}\ {\mathrm{m}}^{\mathrm{3}}\mathrm{\ }{\mathrm{K}}^{-\mathrm{1}}$$. The heat capacity of liquid water is 75.49 J mol$${}^{-1}$$.

(a) Using $${\left({\partial S}/{\partial T}\right)}_P$$, calculate the entropy change when one mole of water is warmed from 2 C to 6 C while the pressure is constant at 1 atm.

(b) Repeat the calculation in (a), for warming the water from 4 C to 6 C.

(c) Can we calculate the entropy change when one mole of water is warmed from 2 C to 6 C using $$\left(\partial S/\partial V\right)_P$$? Why, or why not? The required integral can be transformed to $\int \frac{A\ du}{u^{1/2}+Bu}=\left(\frac{2A}{B}\right) \ln \left(1+\beta u^{1/2}\right) +C$

where $$C$$ is an arbitrary constant.

(d) Using $${\left({\partial S}/{\partial V}\right)}_P$$, calculate the entropy change when one mole of water is warmed from 4 C to 6 C. Compare this result to the value obtained in (b).

9. For an ideal gas, show that $${\left({\partial E}/{\partial V}\right)}_T$$, $${\left({\partial E}/{\partial P}\right)}_T$$, $${\left({\partial H}/{\partial V}\right)}_T$$, $${\left({\partial H}/{\partial P}\right)}_T$$, $${\left({\partial C_V}/{\partial V}\right)}_T$$, and $${\left({\partial C_P}/{\partial P}\right)}_T$$

are all zero.

10. Find $${\left({\partial E}/{\partial P}\right)}_T$$ for a gas that obeys the virial equation of state $$P\left[\overline{V}-B\left(T\right)\right]=RT$$, in which $$B(T$$) is a function of temperature.

11. Derive the following relationships for an ideal gas:

(a) $$dE=C_VdT$$

(b) $$dS=\left({C_V}/{T}\right)dT+\left({R}/{V}\right)dV$$

(c) $$dS=\left({C_P}/{T}\right)dT-\left({R}/{P}\right)dP$$

12. Derive the following relationships for a gas that obeys the virial equation, $$P\left[\overline{V}-B\left(T\right)\right]=RT$$, where $$B\left(T\right)$$ is a function of temperature:

(a) $d\overline{E}=C_VdT-\left[\frac{RT}{\overline{V}-B}+\frac{RT}{\overline{V}-B}\left(\frac{dB}{dT}\right)-\frac{RT^2}{{\left(\overline{V}-B\right)}^2}\left(\frac{dB}{dT}\right)\right]d\overline{V}$ (b) $d\overline{E}=\left[C_P-R-P\left(\frac{dB}{dT}\right)\right]dT-T\left(\frac{dB}{dT}\right)dP$ (c) $d\overline{S}=\frac{C_V}{T}dT+\left[\frac{R}{\overline{V}-B}+\frac{RT}{{\left(\overline{V}-B\right)}^2}\left(\frac{dB}{dT}\right)\right]d\overline{V}$ (d) $\ d\overline{S}=\frac{C_P}{T}dT-\left[\frac{R}{P}+\left(\frac{dB}{dT}\right)\right]dP$

13. One mole of a monatomic ideal gas ($$C_V={3R}/{2}$$), originally at 10 bar and 300 K (state A), undergoes an adiabatic free expansion against a constant applied pressure of 1 bar to reach state B. Thereafter the gas is warmed reversibly at constant volume back to 300 K, reaching state C. Finally, the warmed gas is compressed reversibly and isothermally to the original pressure. What is are the temperature and volume in state B, following the original adiabatic free expansion? Find $$q$$, $$w$$, $$\Delta E$$, $$\Delta H$$, and $$\Delta S$$ for each of the steps and for the cycle A$$\mathrm{\to }$$B$$\mathrm{\to }$$C$$\mathrm{\to }$$A.

14. As in problem 13, one mole of a monatomic ideal gas ($$C_V={3R}/{2}$$), originally at 10 bar and 300 K (state A), undergoes an adiabatic free expansion against a constant applied pressure of 1 bar to reach state B. The gas is then returned to its original state in a different two-step process. From state A a reversible constant-pressure warming takes the gas to state D at the original temperature of 300 K. The gas is then returned to state A by an isothermal compression to the original volume. What are the temperature and volume after the constant-pressure warming step? Find $$q$$, $$w$$, $$\Delta E$$, $$\Delta H$$, and $$\Delta S$$ for each of the steps and for the cycle A$$\mathrm{\to }$$B$$\mathrm{\to }$$D$$\mathrm{\to }$$A.

15. As in problem 13, one mole of a monatomic ideal gas ($$C_V={3R}/{2}$$), originally at 10 bar and 300 K (state A), undergoes an adiabatic free expansion against a constant applied pressure of 1 bar to reach state B. Now consider a reversible adiabatic expansion from the same initial state, A, that reaches the same temperature as the gas in state B. Call this state F. Find $$q$$, $$w$$, $$\Delta E$$, $$\Delta H$$, and $$\Delta S$$ for the step A$$\mathrm{\to }$$F. Find $$q$$, $$w$$, $$\Delta E$$, $$\Delta H$$, and $$\Delta S$$ for reversible isothermal expansion from state F to state B. What are $$q$$, $$w$$, $$\Delta E$$, $$\Delta H$$, $$\Delta S$$, and $$\Delta \hat{S}$$ for the cycle A$$\mathrm{\to }$$F$$\mathrm{\to }$$B$$\to$$A. Does this cycle violate the machine-based statement of the second law?

16. One mole of carbon dioxide, originally at 10 bar and 300 K, is taken around the cycle in problem 13. Find the energy and entropy changes for the steps in this cycle using the ideal gas equation and the temperature-dependent heat capacity. The constant-volume heat capacity is $$C_V=14.7+0.046\times T$$. Find $$q$$, $$w$$, $$\Delta E$$, $$\Delta H$$, and $$\Delta S$$ for each of the steps and for the cycle when $$CO_2$$ is taken around the cycle A$$\mathrm{\to }$$B$$\mathrm{\to }$$C$$\mathrm{\to }$$A.

17. Ten moles of a monatomic ideal gas, initially occupying a volume of 30 L at 25 C, is expanded against a constant applied pressure of 2 bar. The final temperature is 25 C.

(a) What is the initial pressure? The final volume?

(b) Is this process impossible, spontaneous, or reversible?

(c) Find $$q$$, $$w$$, $$\Delta E$$, $$\Delta H$$, $$\Delta A$$, $$\Delta S$$, and $$\Delta G$$ for this process.

18. One mole of $$CO_2$$, originally at 1.00 bar and 300 K, expands adiabatically against a constant applied pressure of 0.200 bar. Assume that $$CO_2$$ behaves as an ideal gas with constant heat capacity, $$C_V=28.5\ \ \mathrm{J\ }{\mathrm{mol}}^{--\mathrm{1}}\mathrm{\ }{\mathrm{K}}^{--\mathrm{1}}$$.

(a) For the spontaneous expansion, we have $$dE=C_VdT-P_{applied}dV$$. Find the final temperature and volume for this spontaneous expansion. What is $$\Delta E$$ for this process?

(b) Find the volume and pressure after the gas is compressed adiabatically and reversibly to the original temperature of 300 K. What are $$\Delta S$$ and $$\Delta E$$ for this step?

(c) Find $$\Delta E$$ when the gas in the final state of part (b) is compressed isothermally to the original volume. What is $$\Delta S$$ for this step?

(d) What are $$\Delta E$$ and $$\Delta S$$ for the cycle comprised of the spontaneous expansion of part (a), the adiabatic compression of part (b), and the isothermal compression of part (c)?

(e) What are $$\Delta S$$, $$\Delta \hat{S}$$, and $$\Delta S_{universe}$$ for the spontaneous expansion?

19. Consider the energy surface depicted in Figure 1. As sketched, $$E$$ increases monotonically as $$S$$ increases. $$E$$ decreases monotonically as $$V$$ increases. Could the energy surface decrease as $$S$$ increases or increase as $$V$$ increases?

20. At 298.15 K, the vapor pressure of water is $$3.169\times {10}^{-3}\ \mathrm{Pa}$$. Some thermodynamic properties for liquid and gaseous water at this temperature and pressure are given in the table below.

 liquid gas $$\overline{G},\ \mathrm{kJ\ }{\mathrm{mol}}^{--\mathrm{1}}$$ – 237.1 – 237.1 $$\overline{S},\ \mathrm{J\ }{\mathrm{mol}}^{--\mathrm{1}}\mathrm{\ }{\mathrm{K}}^{--\mathrm{1}}$$ 70.0 217.5 $$\overline{E},\ \mathrm{kJ\ }{\mathrm{mol}}^{--\mathrm{1}}$$ – 285.5 – 245.1 $$C_P,\ \mathrm{J\ }{\mathrm{mol}}^{--\mathrm{1}}\mathrm{\ }{\mathrm{K}}^{--\mathrm{1}}$$ 75.3 33.6 $$C_V,\ \mathrm{J\ }{\mathrm{mol}}^{--\mathrm{1}}\mathrm{\ }{\mathrm{K}}^{--\mathrm{1}}$$ 67.0 25.3
1. Find $${\Delta }_{vap}\overline{G}$$, $${\Delta }_{vap}\overline{S}$$, $${\Delta }_{vap}\overline{E}$$ for water at this temperature and pressure. Is this process reversible, spontaneous, or impossible?

(b) Sketch $$\overline{G}\left(\ell \right)$$ and $$\overline{G}\left(g\right)$$ vs. $$T$$ for $$288.15. What path is followed when one mole of water at 288.15 K and \(3.169\times {10}^3\ \mathrm{Pa}$$ goes reversibly to 308.15 K at the same pressure?

(c) On the graph of part (b), indicate the transition in which superheated liquid water at 300 K and $$3.169\times {10}^3\ \mathrm{Pa}$$ goes to gaseous water at 300 K and the same pressure. Is this process spontaneous, reversible, or impossible? Is $$\Delta \overline{G}$$ for this process positive, zero, or negative?

(d) Sketch $$\overline{E}\left(\ell \right)$$ and $$\overline{E}\left(g\right)$$ vs. $$T$$ for $$288.15. What path is followed when one mole of water at 288.15 K and \(3.169\times {10}^3\ \mathrm{Pa}$$ goes reversibly to 308.15 K at the same pressure?

(e) On the graph of part (b), indicate the transition in which superheated liquid water at 300 K and $$3.169\times {10}^3\ \mathrm{Pa}$$ goes to gaseous water at 300 K and the same pressure. Is $$\Delta \overline{E}$$ for this process positive, zero, or negative?

21. At 273.15 K and 1 bar, the enthalpy of fusion of ice is $$6010\ \mathrm{J\ }{\mathrm{mol}}^{--\mathrm{1}}$$. Estimate the Gibbs free energy change for the fusion of ice at 283.15 K and 1 bar.

Notes

$${}^{1}$$ J. R. Roebuck and H. Osterberg, The Joule-Thomson Effect in Nitrogen, Phys. Rev., Vol. 48, pp 450-457 (1935).

$${}^{2}$$ See T. L. Hill, An Introduction to Statistical Thermodynamics, Addison-Wesley Publishing Co., Reeding, MA, 1960, pp 266-268.