Chapter 19

Indicate whether each of the following changes represents an increase or a decrease in entropy in a system, and explain your reasoning: (a) the freezing of ethanol; (b) the sublimation of dry ice; (c) the burning of a rocket fuel.

Arrange the entropy changes of the following processes, all at \(25^{\circ} \mathrm{C},\) in the expected order of increasing \(\Delta S,\) and explain your reasoning: (a) \(\mathrm{H}_{2} \mathrm{O}(1,1 \mathrm{atm}) \longrightarrow \mathrm{H}_{2} \mathrm{O}(\mathrm{g}, 1 \mathrm{atm})\) (b) \(\mathrm{CO}_{2}(\mathrm{s}, 1 \mathrm{atm}) \longrightarrow \mathrm{CO}_{2}(\mathrm{g}, 10 \mathrm{mm} \mathrm{Hg})\) (c) \(\mathrm{H}_{2} \mathrm{O}(1,1 \mathrm{atm}) \longrightarrow \mathrm{H}_{2} \mathrm{O}(\mathrm{g}, 10 \mathrm{mmHg})\)

Use ideas from this chapter to explain this famous remark attributed to Rudolf Clausius (1865)\(:^{\prime \prime} \mathrm{Die}\) Energie der Welt ist konstant; die Entropie der Welt strebt einem Maximum zu." ("The energy of the world is constant; the entropy of the world increases toward a maximum.")

Comment on the difficulties of solving environmental pollution problems from the standpoint of entropy changes associated with the formation of pollutants and with their removal from the environment.

Indicate whether the entropy of the system would increase or decrease in each of the following reactions. If you cannot be certain simply by inspecting the equation, explain why. (a) \(\mathrm{CCl}_{4}(1) \longrightarrow \mathrm{CCl}_{4}(\mathrm{g})\) (b) \(\mathrm{CuSO}_{4} \cdot 3 \mathrm{H}_{2} \mathrm{O}(\mathrm{s})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \longrightarrow\) \(\mathrm{CuSO}_{4} \cdot 5 \mathrm{H}_{2} \mathrm{O}(\mathrm{s})\) (c) \(\mathrm{SO}_{3}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{g}) \longrightarrow \mathrm{SO}_{2}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) (d) \(\mathrm{H}_{2} \mathrm{S}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \longrightarrow \mathrm{H}_{2} \mathrm{O}(\mathrm{g})+\mathrm{SO}_{2}(\mathrm{g})\) (not balanced)

Which substance in each of the following pairs would have the greater entropy? Explain. (a) at \(75^{\circ} \mathrm{C}\) and 1 atm: \(1 \mathrm{mol} \mathrm{H}_{2} \mathrm{O}(1)\) or \(1 \mathrm{mol} \mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) (b) at \(5^{\circ} \mathrm{C}\) and 1 atm: \(50.0 \mathrm{g} \mathrm{Fe}(\mathrm{s})\) or \(0.80 \mathrm{mol} \mathrm{Fe}(\mathrm{s})\) (c) 1 mol \(\mathrm{Br}_{2}\left(1,1 \text { atm }, 8^{\circ} \mathrm{C}\right)\) or \(1 \mathrm{mol} \mathrm{Br}_{2}(\mathrm{s}, 1 \mathrm{atm},\) \(\left.-8^{\circ} \mathrm{C}\right)\) (d) \(0.312 \mathrm{mol} \mathrm{SO}_{2}\left(\mathrm{g}, 0.110 \mathrm{atm}, 32.5^{\circ} \mathrm{C}\right)\) or \(0.284 \mathrm{mol}\) \(\mathrm{O}_{2}\left(\mathrm{g}, 15.0 \mathrm{atm}, 22.3^{\circ} \mathrm{C}\right)\)

For each of the following reactions, indicate whether \(\Delta S\) for the reaction should be positive or negative. If it is not possible to determine the sign of \(\Delta S\) from the information given, indicate why. (a) \(\mathrm{CaO}(\mathrm{s})+\mathrm{H}_{2} \mathrm{O}(\mathrm{l}) \longrightarrow \mathrm{Ca}(\mathrm{OH})_{2}(\mathrm{s})\) (b) \(2 \mathrm{HgO}(\mathrm{s}) \longrightarrow 2 \mathrm{Hg}(1)+\mathrm{O}_{2}(\mathrm{g})\) (c) \(2 \mathrm{NaCl}(1) \longrightarrow 2 \mathrm{Na}(1)+\mathrm{Cl}_{2}(\mathrm{g})\) (d) \(\mathrm{Fe}_{2} \mathrm{O}_{3}(\mathrm{s})+3 \mathrm{CO}(\mathrm{g}) \longrightarrow 2 \mathrm{Fe}(\mathrm{s})+3 \mathrm{CO}_{2}(\mathrm{g})\) (e) \(\operatorname{Si}\left(\text { s) }+2 \mathrm{Cl}_{2}(\mathrm{g}) \longrightarrow \mathrm{SiCl}_{4}(\mathrm{g})\right.\)

Which of the following substances would obey Trouton's rule most closely: HF, \(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{3}\) (toluene), or \(\mathrm{CH}_{3} \mathrm{OH}\) (methanol)? Explain your reasoning.

Which of the following changes in a thermodynamic property would you expect to find for the reaction \(\mathrm{Br}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{Br}(\mathrm{g})\) at all temperatures: \((\mathrm{a}) \Delta H<0\) (b) \(\Delta S>0 ;\) (c) \(\Delta G<0 ;\) (d) \(\Delta S<0 ?\) Explain.

If a reaction can be carried out only by electrolysis, which of the following changes in a thermodynamic property must apply: (a) \(\Delta H>0 ;\) (b) \(\Delta S>0\) (c) \(\Delta G=\Delta H ;\) (d) \(\Delta G>0 ?\) Explain.

What values of \(\Delta H, \Delta S,\) and \(\Delta G\) would you expect for the formation of an ideal solution of liquid components? (Is each value positive, negative, or zero?)

Explain why (a) some exothermic reactions do not occur spontaneously, and (b) some reactions in which the entropy of the system increases do not occur spontaneously.

Explain why you would expect a reaction of the type \(\mathrm{AB}(\mathrm{g}) \longrightarrow \mathrm{A}(\mathrm{g})+\mathrm{B}(\mathrm{g})\) always to be spontaneous at high rather than at low temperatures.

From the data given in the following table, determine \(\Delta S^{\circ} \quad\) for the reaction \(\quad \mathrm{NH}_{3}(\mathrm{g})+\mathrm{HCl}(\mathrm{g}) \longrightarrow\) \(\mathrm{NH}_{4} \mathrm{Cl}(\mathrm{s}) .\) All data are at \(298 \mathrm{K}\) $$\begin{array}{lcc} \hline & \Delta H_{f}^{\circ}, \mathrm{kJ} \mathrm{mol}^{-1} & \Delta G_{f,}^{\circ} \mathrm{kJ} \mathrm{mol}^{-1} \\ \hline \mathrm{NH}_{3}(\mathrm{g}) & -46.11 & -16.48 \\ \mathrm{HCl}(\mathrm{g}) & -92.31 & -95.30 \\ \mathrm{NH}_{4} \mathrm{Cl}(\mathrm{s}) & -314.4 & -202.9 \\ \hline \end{array}$$

Use data from Appendix D to determine values of \(\Delta G^{\circ}\) for the following reactions at \(25^{\circ} \mathrm{C}\) (a) \(\mathrm{C}_{2} \mathrm{H}_{2}(\mathrm{g})+2 \mathrm{H}_{2}(\mathrm{g}) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{6}(\mathrm{g})\) (b) \(2 \mathrm{SO}_{3}(\mathrm{g}) \longrightarrow 2 \mathrm{SO}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g})\) (c) \(\mathrm{Fe}_{3} \mathrm{O}_{4}(\mathrm{s})+4 \mathrm{H}_{2}(\mathrm{g}) \longrightarrow 3 \mathrm{Fe}(\mathrm{s})+4 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) (d) \(2 \mathrm{Al}(\mathrm{s})+6 \mathrm{H}^{+}(\mathrm{aq}) \longrightarrow 2 \mathrm{Al}^{3+}(\mathrm{aq})+3 \mathrm{H}_{2}(\mathrm{g})\)

At \(298 \mathrm{K},\) for the reaction \(2 \mathrm{PCl}_{3}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \stackrel{\text { Lest }}{\longrightarrow}\) \(2 \mathrm{POCl}_{3}(1), \Delta H^{\circ}=-620.2 \mathrm{kJ}\) and the standard molar entropies are \(\mathrm{PCl}_{3}(\mathrm{g}), 311.8 \mathrm{JK}^{-1} ; \mathrm{O}_{2}(\mathrm{g}), 205.1 \mathrm{JK}^{-1}\) and \(\mathrm{POCl}_{3}(1), 222.4 \mathrm{JK}^{-1} .\) Determine (a) \(\Delta G^{\circ}\) at \(298 \mathrm{K}\) and (b) whether the reaction proceeds spontaneously in the forward or the reverse direction when reactants and products are in their standard states.

At \(298 \mathrm{K},\) for the reaction \(2 \mathrm{H}^{+}(\mathrm{aq})+2 \mathrm{Br}^{-}(\mathrm{aq})+\) \(2 \mathrm{NO}_{2}(\mathrm{g}) \longrightarrow \mathrm{Br}_{2}(1)+2 \mathrm{HNO}_{2}(\mathrm{aq}), \Delta H^{\circ}=-61.6 \mathrm{kJ}\) and the standard molar entropies are \(\mathrm{H}^{+}(\mathrm{aq}), 0 \mathrm{JK}^{-1}\) \(\mathrm{Br}^{-}(\mathrm{aq}), 82.4 \mathrm{JK}^{-1} ; \mathrm{NO}_{2}(\mathrm{g}), 240.1 \mathrm{JK}^{-1} ; \mathrm{Br}_{2}(1), 152.2\) \(\mathrm{J} \mathrm{K}^{-1} ; \mathrm{HNO}_{2}(\mathrm{aq}), 135.6 \mathrm{JK}^{-1} .\) Determine (a) \(\Delta G^{\circ}\) at 298 K and (b) whether the reaction proceeds spontaneously in the forward or the reverse direction when reactants and products are in their standard states.

The following standard Gibbs energy changes are given for \(25^{\circ} \mathrm{C}\) (1) \(\mathrm{N}_{2}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{NH}_{3}(\mathrm{g})\) \(\Delta G^{\circ}=-33.0 \mathrm{kJ}\) (2) \(4 \mathrm{NH}_{3}(\mathrm{g})+5 \mathrm{O}_{2}(\mathrm{g}) \longrightarrow 4 \mathrm{NO}(\mathrm{g})+6 \mathrm{H}_{2} \mathrm{O}(1)\) \(\Delta G^{\circ}=-1010.5 \mathrm{kJ}\) (3) \(\mathrm{N}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{NO}(\mathrm{g})\) \(\Delta G^{\circ}=+173.1 \mathrm{kJ}\) (4) \(\mathrm{N}_{2}(\mathrm{g})+2 \mathrm{O}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{NO}_{2}(\mathrm{g})\) \(\Delta G^{\circ}=+102.6 \mathrm{kJ}\) (5) \(2 \mathrm{N}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{N}_{2} \mathrm{O}(\mathrm{g})\) \(\Delta G^{\circ}=+208.4 \mathrm{kJ}\) Combine the preceding equations, as necessary, to obtain \(\Delta G^{\circ}\) values for each of the following reactions. (a) \(\mathrm{N}_{2} \mathrm{O}(\mathrm{g})+\frac{3}{2} \mathrm{O}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{NO}_{2}(\mathrm{g}) \quad \Delta G^{\circ}=?\) (b) \(2 \mathrm{H}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(1) \quad \Delta G^{\circ}=?\) (c) \(2 \mathrm{NH}_{3}(\mathrm{g})+2 \mathrm{O}_{2}(\mathrm{g}) \longrightarrow \mathrm{N}_{2} \mathrm{O}(\mathrm{g})+3 \mathrm{H}_{2} \mathrm{O}(1)\) \(\Delta G^{\circ}=?\) Of reactions (a), (b), and (c), which would tend to go to completion at \(25^{\circ} \mathrm{C}\), and which would reach an equilibrium condition with significant amounts of all reactants and products present?

The following standard Gibbs energy changes are given for \(25^{\circ} \mathrm{C}\) (1) \(\mathrm{SO}_{2}(\mathrm{g})+3 \mathrm{CO}(\mathrm{g}) \longrightarrow \operatorname{COS}(\mathrm{g})+2 \mathrm{CO}_{2}(\mathrm{g})\) \(\Delta G^{\circ}=-246.4 \mathrm{kJ}\) (2) \(\mathrm{CS}_{2}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \longrightarrow \operatorname{COS}(\mathrm{g})+\mathrm{H}_{2} \mathrm{S}(\mathrm{g})\) \(\Delta G^{\circ}=-41.5 \mathrm{kJ}\) (3) \(\mathrm{CO}(\mathrm{g})+\mathrm{H}_{2} \mathrm{S}(\mathrm{g}) \longrightarrow \operatorname{COS}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{g})\) \(\Delta G^{\circ}=+1.4 \mathrm{kJ}\) (4) \(\mathrm{CO}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \longrightarrow \mathrm{CO}_{2}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{g})\) \(\Delta G^{\circ}=-28.6 \mathrm{kJ}\) Combine the preceding equations, as necessary, to obtain \(\Delta G^{\circ}\) values for the following reactions. (a) \(\operatorname{COS}(\mathrm{g})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \longrightarrow\) \(\begin{aligned} \mathrm{SO}_{2}(\mathrm{g})+\mathrm{CO}(\mathrm{g})+2 \mathrm{H}_{2}(\mathrm{g}) & \Delta G^{\circ}=? \end{aligned}\) (b) \(\cos (g)+3 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \longrightarrow\) \(\mathrm{SO}_{2}(\mathrm{g})+\mathrm{CO}_{2}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{g}) \quad \Delta G^{\circ}=?\) \(\left.+\quad \mathrm{H}_{\mathrm{O}} \mathrm{C}(\mathrm{d})=\mathrm{CO}_{-}^{\circ} \mathrm{G}\right)+\mathrm{H}_{-}^{-} \mathrm{S}(\mathrm{q})\) (c) \(\cos (\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \longrightarrow \mathrm{CO}_{2}(\mathrm{g})+\mathrm{H}_{2} \mathrm{S}(\mathrm{g})\) \(\Delta G^{\circ}=?\) Of reactions (a), (b), and (c), which is spontaneous in the forward direction when reactants and products are present in their standard states?

For one of the following reactions, \(K_{c} K_{p}=K .\) Identify that reaction. For the other two reactions, what is the relationship between \(K_{c}, \bar{K}_{\mathrm{p}},\) and \(K ?\) Explain. (a) \(2 \mathrm{SO}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \rightleftharpoons 2 \mathrm{SO}_{3}(\mathrm{g})\) (b) \(\mathrm{HI}(\mathrm{g}) \rightleftharpoons \frac{1}{2} \mathrm{H}_{2}(\mathrm{g})+\frac{1}{2} \mathrm{I}_{2}(\mathrm{g})\) (c) \(\mathrm{NH}_{4} \mathrm{HCO}_{3}(\mathrm{s}) \rightleftharpoons \mathrm{NH}_{3}(\mathrm{g})+\mathrm{CO}_{2}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(1)\)

\(\mathrm{H}_{2}(\mathrm{g})\) can be prepared by passing steam over hot iron: \(3 \mathrm{Fe}(\mathrm{s})+4 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \rightleftharpoons \mathrm{Fe}_{3} \mathrm{O}_{4}(\mathrm{s})+4 \mathrm{H}_{2}(\mathrm{g})\) (a) Write an expression for the thermodynamic equilibrium constant for this reaction. (b) Explain why the partial pressure of \(\mathrm{H}_{2}(\mathrm{g})\) is independent of the amounts of \(\mathrm{Fe}(\mathrm{s})\) and \(\mathrm{Fe}_{3} \mathrm{O}_{4}(\mathrm{s})\) present. (c) Can we conclude that the production of \(\mathrm{H}_{2}(\mathrm{g})\) from \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) could be accomplished regardless of the proportions of \(\mathrm{Fe}(\mathrm{s})\) and \(\mathrm{Fe}_{3} \mathrm{O}_{4}(\mathrm{s})\) present? Explain.

Calculate the equilibrium constant and Gibbs energy for the reaction \(\mathrm{CO}(\mathrm{g})+2 \mathrm{H}_{2}(\mathrm{g}) \longrightarrow \mathrm{CH}_{3} \mathrm{OH}(\mathrm{g})\) at \(483 \mathrm{K}\) by using the data tables from Appendix D. Are the values determined here different from or the same as those in exercise \(35 ?\) Explain.

Use thermodynamic data at \(298 \mathrm{K}\) to decide in which direction the reaction $$2 \mathrm{SO}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \rightleftharpoons 2 \mathrm{SO}_{3}(\mathrm{g})$$ is spontaneous when the partial pressures of \(\mathrm{SO}_{2}, \mathrm{O}_{2},\) and \(\mathrm{SO}_{3}\) are \(1.0 \times 10^{-4}, 0.20,\) and \(0.10 \mathrm{atm}\) respectively.

The standard Gibbs energy change for the reaction \(\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\mathrm{l}) \rightleftharpoons$$$ \mathrm{CH}_{3} \mathrm{CO}_{2}^{-}(\mathrm{aq})+\mathrm{H}_{3} \mathrm{O}^{+}(\mathrm{aq})$$is \)27.07 \mathrm{kJmol}^{-1}\( at 298 K. Use this thermodynamic quantity to decide in which direction the reaction is spontaneous when the concentrations of \)\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}(\mathrm{aq}), \mathrm{CH}_{3} \mathrm{CO}_{2}^{-}(\mathrm{aq}),\( and \)\mathrm{H}_{3} \mathrm{O}^{+}(\mathrm{aq})\( are \)0.10 \mathrm{M}, 1.0 \times 10^{-3} \mathrm{M},\( and \)1.0 \times 10^{-3} \mathrm{M},$ respectively.

The standard Gibbs energy change for the reaction $$\mathrm{NH}_{3}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(1) \rightleftharpoons \mathrm{NH}_{4}^{+}(\mathrm{aq})+\mathrm{OH}^{-}(\mathrm{aq})$$ is \(29.05 \mathrm{kJ} \mathrm{mol}^{-1}\) at \(298 \mathrm{K}\). Use this thermodynamic quantity to decide in which direction the reaction is spontaneous when the concentrations of \(\mathrm{NH}_{3}(\mathrm{aq})\) \(\mathrm{NH}_{4}^{+}(\mathrm{aq}),\) and \(\mathrm{OH}^{-}(\mathrm{aq})\) are \(0.10 \mathrm{M}, 1.0 \times 10^{-3} \mathrm{M}\) and \(1.0 \times 10^{-3} \mathrm{M},\) respectively.

For the reaction \(2 \mathrm{NO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{NO}_{2}(\mathrm{g})\) all but one of the following equations is correct. Which is incorrect, and why? (a) \(K=K_{\mathrm{p}} ;\) (b) \(\Delta S^{\circ}=\) \(\left(\Delta G^{\circ}-\Delta H^{\circ}\right) / T ;\left(\text { c) } K_{\mathrm{p}}=e^{-\Delta G^{\circ} / R T} ;(\mathrm{d}) \Delta G=\Delta G^{\circ}+\right.\) \(R T \ln Q\).

Why is \(\Delta G^{\circ}\) such an important property of a chemical reaction, even though the reaction is generally carried out under nonstandard conditions?

At \(1000 \mathrm{K},\) an equilibrium mixture in the reaction \(\mathrm{CO}_{2}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{g}) \rightleftharpoons \mathrm{CO}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \quad\) contains \(0.276 \mathrm{mol} \quad \mathrm{H}_{2}, 0.276 \mathrm{mol} \mathrm{CO}_{2}, \quad 0.224 \mathrm{mol} \mathrm{CO}, \quad\) and \(0.224 \mathrm{mol} \mathrm{H}_{2} \mathrm{O}\) (a) What is \(K_{\mathrm{p}}\) at \(1000 \mathrm{K} ?\) (b) Calculate \(\Delta G^{\circ}\) at \(1000 \mathrm{K}\). (c) In which direction would a spontaneous reaction occur if the following were brought together at 1000 K: \(0.0750 \mathrm{mol} \mathrm{CO}_{2}, 0.095 \mathrm{mol} \mathrm{H}_{2}, 0.0340 \mathrm{mol} \mathrm{CO},\) and \(0.0650 \mathrm{mol} \mathrm{H}_{2} \mathrm{O} ?\)

For the reaction \(2 \mathrm{SO}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \rightleftharpoons 2 \mathrm{SO}_{3}(\mathrm{g})\) \(K_{\mathrm{c}}=2.8 \times 10^{2}\) at \(1000 \mathrm{K}\) (a) What is \(\Delta G^{\circ}\) at \(1000 \mathrm{K} ?\left[\text { Hint: What is } \mathrm{K}_{\mathrm{p}} ?\right]\) (b) If \(0.40 \mathrm{mol} \mathrm{SO}_{2}, 0.18 \mathrm{mol} \mathrm{O}_{2},\) and \(0.72 \mathrm{mol} \mathrm{SO}_{3}\) are mixed in a 2.50 L flask at \(1000 \mathrm{K}\), in what direction will a net reaction occur?

For the following equilibrium reactions, calculate \(\Delta G^{\circ}\) at the indicated temperature. [Hint: How is each equilibrium constant related to a thermodynamic equilibrium constant, \(K ?]\) (a) \(\mathrm{H}_{2}(\mathrm{g})+\mathrm{I}_{2}(\mathrm{g}) \rightleftharpoons 2 \mathrm{HI}(\mathrm{g}) \quad K_{\mathrm{c}}=50.2\) at \(445^{\circ} \mathrm{C}\) (b) \(\mathrm{N}_{2} \mathrm{O}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g}) \rightleftharpoons 2 \mathrm{NO}(\mathrm{g})\) \(K_{c}=1.7 \times 10^{-13} \mathrm{at} 25^{\circ} \mathrm{C}\) (c) \(\mathrm{N}_{2} \mathrm{O}_{4}(\mathrm{g}) \rightleftharpoons 2 \mathrm{NO}_{2}(\mathrm{g})\) \(K_{c}=4.61 \times 10^{-3}\) at \(25^{\circ} \mathrm{C}\) (d) \(2 \mathrm{Fe}^{3+}(\mathrm{aq})+\mathrm{Hg}_{2}^{2+}(\mathrm{aq}) \rightleftharpoons\) \(2 \mathrm{Fe}^{2+}(\mathrm{aq})+2 \mathrm{Hg}^{2+}(\mathrm{aq})\) \(K_{\mathrm{c}}=9.14 \times 10^{-6} \mathrm{at} 25^{\circ} \mathrm{C}\)

Two equations can be written for the dissolution of \(\mathrm{Mg}(\mathrm{OH})_{2}(\mathrm{s})\) in acidic solution. $$\begin{aligned} \mathrm{Mg}(\mathrm{OH})_{2}(\mathrm{s})+2 \mathrm{H}^{+}(\mathrm{aq}) \rightleftharpoons & \mathrm{Mg}^{2+}(\mathrm{aq})+2 \mathrm{H}_{2} \mathrm{O}(1) \\ & \Delta G^{\circ}=-95.5 \mathrm{kJ} \mathrm{mol}^{-1} \\ (c) Will the solubilities of \(\mathrm{Mg}(\mathrm{OH})_{2}(\mathrm{s})\) in a buffer solution at \(\mathrm{pH}=8.5\) depend on which of the two equations is used as the basis of the calculation? Explain. \frac{1}{2} \mathrm{Mg}(\mathrm{OH})_{2}(\mathrm{s})+\mathrm{H}^{+}(\mathrm{aq}) \rightleftharpoons & \frac{1}{2} \mathrm{Mg}^{2+}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(1) \\ & \Delta G^{\circ}=-47.8 \mathrm{kJ} \mathrm{mol}^{-1} \end{aligned}$$ (a) Explain why these two equations have different \(\Delta G^{\circ}\) values. (b) Will \(K\) for these two equations be the same or different? Explain.

At \(298 \mathrm{K}, \Delta G_{\mathrm{f}}^{\mathrm{p}}[\mathrm{CO}(\mathrm{g})]=-137.2 \mathrm{kJ} / \mathrm{mol}\) and \(K_{\mathrm{p}}=\) \(6.5 \times 10^{11}\) for the reaction \(\mathrm{CO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{g}) \rightleftharpoons\) \(\mathrm{COCl}_{2}(\mathrm{g}) . \quad\) Use these data to determine \(\Delta G_{f}\left[\mathrm{COCl}_{2}(\mathrm{g})\right],\) and compare your result with the value in Appendix D.

Use thermodynamic data from Appendix D to calculate values of \(K_{\mathrm{sp}}\) for the following sparingly soluble solutes: (a) \(\operatorname{AgBr} ;\) (b) \(\operatorname{CaSO}_{4} ;\) (c) \(\operatorname{Fe}(\text { OH })_{3}\). [Hint: Begin by writing solubility equilibrium expressions.

To establish the law of conservation of mass, Lavoisier carefully studied the decomposition of mercury(II) oxide: $$\mathrm{HgO}(\mathrm{s}) \longrightarrow \mathrm{Hg}(1)+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g})$$ At \(25^{\circ} \mathrm{C}, \Delta H^{\circ}=+90.83 \mathrm{kJ}\) and \(\Delta G^{\circ}=+58.54 \mathrm{kJ}\) (a) Show that the partial pressure of \(\mathrm{O}_{2}(\mathrm{g})\) in equilibrium with \(\mathrm{HgO}(\mathrm{s})\) and \(\mathrm{Hg}(\mathrm{l})\) at \(25^{\circ} \mathrm{C}\) is extremely low. (b) What conditions do you suppose Lavoisier used to obtain significant quantities of oxygen?

What must be the temperature if the following reaction has \(\Delta G^{\circ}=-45.5 \mathrm{kJ}, \Delta H^{\circ}=-24.8 \mathrm{kJ},\) and \(\Delta S^{\circ}=15.2 \mathrm{JK}^{-1} ?\) $$\mathrm{Fe}_{2} \mathrm{O}_{3}(\mathrm{s})+3 \mathrm{CO}(\mathrm{g}) \longrightarrow 2 \mathrm{Fe}(\mathrm{s})+3 \mathrm{CO}_{2}(\mathrm{g})$$

The following equilibrium constants have been determined for the reaction \(\mathrm{H}_{2}(\mathrm{g})+\mathrm{I}_{2}(\mathrm{g}) \rightleftharpoons 2 \mathrm{HI}(\mathrm{g})\) \(K_{\mathrm{p}}=50.0\) at \(448^{\circ} \mathrm{C}\) and 66.9 at \(350^{\circ} \mathrm{C} .\) Use these data to estimate \(\Delta H^{\circ}\) for the reaction.

Sodium carbonate, an important chemical used in the production of glass, is made from sodium hydrogen carbonate by the reaction \(2 \mathrm{NaHCO}_{3}(\mathrm{s}) \rightleftharpoons \mathrm{Na}_{2} \mathrm{CO}_{3}(\mathrm{s})+\mathrm{CO}_{2}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) Data for the temperature variation of \(K_{\mathrm{p}}\) for this reaction are \(K_{\mathrm{p}}=1.66 \times 10^{-5}\) at \(30^{\circ} \mathrm{C} ; 3.90 \times 10^{-4} \mathrm{at}\) \(50^{\circ} \mathrm{C} ; 6.27 \times 10^{-3}\) at \(70^{\circ} \mathrm{C} ;\) and \(2.31 \times 10^{-1}\) at \(100^{\circ} \mathrm{C}\) (a) Plot a graph similar to Figure \(19-12,\) and determine \(\Delta H^{\circ}\) for the reaction. (b) Calculate the temperature at which the total gas pressure above a mixture of \(\mathrm{NaHCO}_{3}(\mathrm{s})\) and \(\mathrm{Na}_{2} \mathrm{CO}_{3}(\mathrm{s})\) is \(2.00 \mathrm{atm}\).

Titanium is obtained by the reduction of \(\mathrm{TiCl}_{4}(1)\) which in turn is produced from the mineral rutile \(\left(\mathrm{TiO}_{2}\right)\) (a) With data from Appendix D, determine \(\Delta G^{\circ}\) at 298 K for this reaction. $$\mathrm{TiO}_{2}(\mathrm{s})+2 \mathrm{Cl}_{2}(\mathrm{g}) \longrightarrow \mathrm{TiCl}_{4}(1)+\mathrm{O}_{2}(\mathrm{g})$$ (b) Show that the conversion of \(\mathrm{TiO}_{2}(\mathrm{s})\) to \(\mathrm{TiCl}_{4}(1)\) with reactants and products in their standard states, is spontaneous at \(298 \mathrm{K}\) if the reaction in (a) is coupled with the reaction $$2 \mathrm{CO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{CO}_{2}(\mathrm{g})$$

Following are some standard Gibbs energies of formation, \(\Delta G_{f}^{2},\) per mole of metal oxide at \(1000 \mathrm{K}: \mathrm{NiO},\) \(-115 \mathrm{kJ} ; \mathrm{MnO},-280 \mathrm{kJ} ; \mathrm{TiO}_{2},-630 \mathrm{kJ} .\) The standard Gibbs energy of formation of \(\mathrm{CO}\) at \(1000 \mathrm{K}\) is \(-250 \mathrm{kJ}\) per mol CO. Use the method of coupled reactions (page 851 ) to determine which of these metal oxides can be reduced to the metal by a spontaneous reaction with carbon at \(1000 \mathrm{K}\) and with all reactants and products in their standard states.

In biochemical reactions the phosphorylation of amino acids is an important step. Consider the following two reactions and determine whether the phosphorylation of arginine with ATP is spontaneous. $$\begin{array}{c} \mathrm{ATP}+\mathrm{H}_{2} \mathrm{O} \longrightarrow \mathrm{ADP}+\mathrm{P} \quad \Delta G^{\circ \prime}=-31.5 \mathrm{kJ} \mathrm{mol}^{-1} \\ \text {arginine }+\mathrm{P} \longrightarrow \text { phosphorarginine }+\mathrm{H}_{2} \mathrm{O} \\ \Delta G^{\circ \prime}=33.2 \mathrm{kJ} \mathrm{mol}^{-1} \end{array}$$

The synthesis of glutamine from glutamic acid is given by Glu \(^{-}+\mathrm{NH}_{4}^{+} \longrightarrow \mathrm{Gln}+\mathrm{H}_{2} \mathrm{O}\). The Gibbs energy for this reaction at \(\mathrm{pH}=7\) and \(T=310 \mathrm{K}\) is \(\Delta G^{\circ \prime}=14.8 \mathrm{kJ} \mathrm{mol}^{-1} .\) Will this reaction be sponta- neous if coupled with the hydrolysis of ATP? \(\mathrm{ATP}+\mathrm{H}_{2} \mathrm{O} \longrightarrow \mathrm{ADP}+\mathrm{P}\) $$\Delta G^{\circ \prime}=-31.5 \mathrm{kJ} \mathrm{mol}-1$$

Consider the vaporization of water: \(\mathrm{H}_{2} \mathrm{O}(1) \longrightarrow\) \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) at \(100^{\circ} \mathrm{C},\) with \(\mathrm{H}_{2} \mathrm{O}(1)\) in its standard state, but with the partial pressure of \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) at \(2.0 \mathrm{atm}\) Which of the following statements about this vaporization at \(100^{\circ} \mathrm{C}\) are true? (a) \(\Delta G^{\circ}=0,\) (b) \(\Delta G=0\) (c) \(\Delta G^{\circ}>0,\) (d) \(\Delta G>0 ?\) Explain.

The following table shows the enthalpies and Gibbs energies of formation of three metal oxides at \(25^{\circ} \mathrm{C}\). (a) Which of these oxides can be most readily decomposed to the free metal and \(\mathrm{O}_{2}(\mathrm{g}) ?\) (b) For the oxide that is most easily decomposed, to what temperature must it be heated to produce \(\mathrm{O}_{2}(\mathrm{g})\) at 1.00 atm pressure? $$\begin{array}{lll} \hline & \Delta \mathrm{H}_{7}, \mathrm{kJ} \mathrm{mol}^{-1} & \Delta \mathrm{G}_{7}, \mathrm{kJ} \mathrm{mol}^{-1} \\ \hline \mathrm{PbO}(\mathrm{red}) & -219.0 & -188.9 \\ \mathrm{Ag}_{2} \mathrm{O} & -31.05 & -11.20 \\ \mathrm{ZnO} & -348.3 & -318.3 \\ \hline \end{array}$$

The following data are given for the two solid forms of \(\mathrm{HgI}_{2}\) at \(298 \mathrm{K}\). $$\begin{array}{llll} \hline & \Delta H_{f}^{\circ} & \Delta G_{f,}^{\circ} & S^{\circ} \\ & \text { kJ mol }^{-1} & \text {kJ mol }^{-1} & \text {J mol }^{-1} \text {K }^{-1} \\ \hline \mathrm{HgI}_{2} \text { (red) } & -105.4 & -101.7 & 180 \\ \mathrm{Hg} \mathrm{I}_{2} \text { (yellow) } & -102.9 & (?) & (?) \\ \hline \end{array}$$ Estimate values for the two missing entries. To do this, assume that for the transition \(\mathrm{HgI}_{2}(\mathrm{red}) \longrightarrow\) \(\mathrm{HgI}_{2}(\text { yellow }),\) the values of \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) at \(25^{\circ} \mathrm{C}\) have the same values that they do at the equilibrium temperature of \(127^{\circ} \mathrm{C}\).

For the dissociation of \(\mathrm{CaCO}_{3}(\mathrm{s})\) at \(25^{\circ} \mathrm{C}, \mathrm{CaCO}_{3}(\mathrm{s})\) \(\rightleftharpoons \mathrm{CaO}(\mathrm{s})+\mathrm{CO}_{2}(\mathrm{g}) \Delta G^{\circ}=+131 \mathrm{kJ} \mathrm{mol}^{-1} .\) A sample of pure \(\mathrm{CaCO}_{3}(\mathrm{s})\) is placed in a flask and connected to an ultrahigh vacuum system capable of reducing the pressure to \(10^{-9} \mathrm{mmHg}\) (a) Would \(\mathrm{CO}_{2}(\mathrm{g})\) produced by the decomposition of \(\mathrm{CaCO}_{3}(\mathrm{s})\) at \(25^{\circ} \mathrm{C}\) be detectable in the vacuum system at \(25^{\circ} \mathrm{C} ?\) (b) What additional information do you need to determine \(P_{\mathrm{CO}_{2}}\) as a function of temperature? (c) With necessary data from Appendix D, determine the minimum temperature to which \(\mathrm{CaCO}_{3}(\mathrm{s})\) would have to be heated for \(\mathrm{CO}_{2}(\mathrm{g})\) to become detectable in the vacuum system.

The standard molar entropy of solid hydrazine at its melting point of \(1.53^{\circ} \mathrm{C}\) is \(67.15 \mathrm{Jmol}^{-1} \mathrm{K}^{-1}\). The enthalpy of fusion is \(12.66 \mathrm{kJmol}^{-1} .\) For \(\mathrm{N}_{2} \mathrm{H}_{4}(1)\) in the interval from \(1.53^{\circ} \mathrm{C}\) to \(298.15 \mathrm{K}\), the molar heat capacity at constant pressure is given by the expression \(C_{p}=97.78+0.0586(T-280) .\) Determine the standard molar entropy of \(\mathrm{N}_{2} \mathrm{H}_{4}(\mathrm{l})\) at \(298.15 \mathrm{K}\). [Hint: The heat absorbed to produce an infinitesimal change in the temperature of a substance is \(d q_{\mathrm{rev}}=C_{p} d T\).

Use the following data to estimate the standard molar entropy of gaseous benzene at \(298.15 \mathrm{K} ;\) that is, \(S^{\circ}\left[\mathrm{C}_{6} \mathrm{H}_{6}(\mathrm{g}, 1 \mathrm{atm})\right] .\) For \(\mathrm{C}_{6} \mathrm{H}_{6}(\mathrm{s}, 1 \mathrm{atm})\) at its melting point of \(5.53^{\circ} \mathrm{C}, S^{\circ}\) is \(128.82 \mathrm{Jmol}^{-1} \mathrm{K}^{-1}\). The enthalpy of fusion is \(9.866 \mathrm{kJ} \mathrm{mol}^{-1} .\) From the melting point to 298.15 K, the average heat capacity of liquid benzene is \(134.0 \mathrm{JK}^{-1} \mathrm{mol}^{-1} .\) The enthalpy of vaporization of \(\mathrm{C}_{6} \mathrm{H}_{6}(1)\) at \(298.15 \mathrm{K}\) is \(33.85 \mathrm{kJ} \mathrm{mol}^{-1},\) and in the vapor- ization, \(\mathrm{C}_{6} \mathrm{H}_{6}(\mathrm{g})\) is produced at a pressure of 95.13 Torr. Imagine that this vapor could be compressed to 1 atm pressure without condensing and while behaving as an ideal gas. Calculate \(S^{\circ}\left[\mathrm{C}_{6} \mathrm{H}_{6}(\mathrm{g}, 1 \text { atm) }] .[ \text { Hint: Refer to }\right.\) Exercise \(88,\) and note the following: For infinitesimal quantities, \(d S=d q / d T ;\) for the compression of an ideal gas, \(d q=-d w ;\) and for pressure-volume work, \(d w=-P d V\).

On page 822 the terms states and microstates were introduced. Consider a system that has four states (i.e., energy levels), with energy \(\varepsilon=0,1,2,\) and 3 energy units, and three particles labeled \(A, B,\) and \(C .\) The total energy of the system, in energy units, is 3 . How many microstates can be generated?

A tabulation of more precise thermodynamic data than are presented in Appendix D lists the following values for \(\mathrm{H}_{2} \mathrm{O}(\mathrm{l})\) and \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) at \(298.15 \mathrm{K},\) at a standard state pressure of 1 bar. $$\begin{array}{llll} \hline & \Delta H_{f}^{\circ}, & \Delta G_{f,}^{\circ} & S_{\prime}^{\circ} \\\ & \text { kJ mol }^{-1} & \text {kJ mol }^{-1} & \text {J mol }^{-1} \text {K }^{-1} \\ \hline \mathrm{H}_{2} \mathrm{O}(1) & -285.830 & -237.129 & 69.91 \\ \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) & -241.818 & -228.572 & 188.825 \\ \hline \end{array}$$ (a) Use these data to determine, in two different ways, \(\Delta G^{\circ}\) at \(298.15 \mathrm{K}\) for the vaporization: \(\mathrm{H}_{2} \mathrm{O}(1,1 \mathrm{bar}) \rightleftharpoons \mathrm{H}_{2} \mathrm{O}(\mathrm{g}, 1 \mathrm{bar}) .\) The value you obtain will differ slightly from that on page 838 because here, the standard state pressure is 1 bar, and there, it is 1 atm. (b) Use the result of part (a) to obtain the value of \(K\) for this vaporization and, hence, the vapor pressure of water at \(298.15 \mathrm{K}\) (c) The vapor pressure in part (b) is in the unit bar. Convert the pressure to millimeters of mercury. (d) Start with the value \(\Delta G^{\circ}=8.590 \mathrm{kJ}\), given on page 838 and calculate the vapor pressure of water at 298.15 K in a fashion similar to that in parts (b) and (c). In this way, demonstrate that the results obtained in a thermodynamic calculation do not depend on the convention we choose for the standard state pressure, as long as we use standard state thermodynamic data consistent with that choice.

In a heat engine, heat \(\left(q_{\mathrm{h}}\right)\) is absorbed by a working substance (such as water) at a high temperature \(\left(T_{\mathrm{h}}\right)\) Part of this heat is converted to work \((w),\) and the rest \(\left(q_{1}\right)\) is released to the surroundings at the lower temperature ( \(T_{1}\) ). The efficiency of a heat engine is the ratio \(w / q_{\mathrm{h}}\). The second law of thermodynamics establishes the following equation for the maximum efficiency of a heat engine, expressed on a percentage basis. $$\text { efficiency }=\frac{w}{q_{\mathrm{h}}} \times 100 \%=\frac{T_{\mathrm{h}}-T_{1}}{T_{\mathrm{h}}} \times 100 \%$$ In a particular electric power plant, the steam leaving a steam turbine is condensed to liquid water at \(41^{\circ} \mathrm{C}\left(T_{1}\right)\) and the water is returned to the boiler to be regenerated as steam. If the system operates at \(36 \%\) efficiency, (a) What is the minimum temperature of the steam \(\left[\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\right]\) used in the plant? (b) Why is the actual steam temperature probably higher than that calculated in part (a)? (c) Assume that at \(T_{\mathrm{h}}\) the \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) is in equilibrium with \(\mathrm{H}_{2} \mathrm{O}(1) .\) Estimate the steam pressure at the temperature calculated in part (a). (d) Is it possible to devise a heat engine with greater than 100 percent efficiency? With 100 percent efficiency? Explain.