Calculate ?Gº for the following reactions at 25ºC:

Explain what is meant by a spontaneous process. Give two examples each of spontaneous and non-spontaneous processes.

How does the entropy of a system change for each of the following processes? (a) condensing water vapor, (b) forming sucrose crystals from a supersaturated solution, (c) heating hydrogen gas from \(\mathrm{60^\circ C}\) to \(\mathrm{80^\circ C}\), and (d) subliming dry ice.

Referring to the footnote on p. 781, draw the missing distributions in Figure 17.2.

Which of the following processes are spontaneous and which are nonspontaneous? (a) dissolving table salt (NaCl) in hot soup; (b) climbing Mt. Everest; (c) spreading fragrance in a room by removing the cap from a perfume bottle; (d) separating helium and neon from a mixture of the gases

Calculate the standard free-energy changes for the following reactions at \(\mathrm{25^\circ C}\): (a) \(2 \mathrm{CO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{CO}_{2}(g)\) (b) \(3 \mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{O}_{3}(g)\) (c) \(2 \mathrm{NaHCO}_{3}(s) \longrightarrow \mathrm{Na}_{2} \mathrm{CO}_{3}(s)+\mathrm{H}_{2}(\mathrm{O})\)

Consider the gas-phase reaction of \(\mathrm A_2\) (blue) and \(\mathrm B_2\) (orange) to form AB_3. (a) Write a balanced equation for the reaction. (b) What is the sign of \(\Delta \mathrm S\) for the reaction?

Which of the following processes are spontaneous and which are nonspontaneous at a given temperature? (a) \(\mathrm{NaNO}_{3}(s) \stackrel{\mathrm{H}_{2} \mathrm{O}}{\longrightarrow} \mathrm{NaNO}_{3}(a q)\) saturated soln (b) \(\mathrm{NaNO}_{3}(s) \stackrel{\mathrm{H}_{2} \mathrm{O}}{\longrightarrow} \underset{\mathrm{H}_{2} \mathrm{O}}{\longrightarrow} \mathrm{NaNO}_{3}(a q)\) unsaturated soln (c) \(\mathrm{NaNO}_{3}(s) \stackrel{\mathrm{H}_{2} \mathrm{O}}{\longrightarrow} \mathrm{NaNO}_{3}(a q)\) unsaturated soln

Discuss qualitatively the sign of the entropy change expected for each of the following processes: (a) \(\mathrm{I}_{2}(s) \longrightarrow 2 \mathrm{I}(g)\) (b) \(2 \mathrm{Zn}(s)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{ZnO}(s)\) (c) \(\mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{NO}(g)\)

(a) Under what circumstances will an endothermic reaction proceed spontaneously? (b) Explain why, in many reactions in which both the reactant and product species are in the solution phase, \(\Delta H\) often gives a good hint about the spontaneity of a reaction at 298 K.

Define entropy. What are the units of entropy?

Calculate the standard free-energy changes for the following reactions at \(25^\circ \mathrm C\): (a) \(\mathrm{H}_{2}(g)+\mathrm{Br}_{2}(l) \longrightarrow 2 \mathrm{HBr}(g)\) (b) \(2 \mathrm{C}_{2} \mathrm{H}_{6}(g)+7 \mathrm{O}_{2}(g) \longrightarrow 4 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(l)\)

Consider the sublimation of iodine \(\mathrm{(I_2)}\) at \(\mathrm 45^ \circ C\) in a closed flask shown here. If the enthalpy of sublimation is 62.4 kJ/mol, what is the \(\Delta S\) for sublimation?

How does the entropy of a system change for each of the following processes? (a) A solid melts. (b) A liquid freezes. (c) A liquid boils. (d) A vapor is converted to a solid. (e) A vapor condenses to a liquid. (f) A solid sublimes. (g) Urea dissolves in water.

The molar heats of fusion and vaporization of argon are 1.3 kJ/mol and 6.3 kJ/mol, argon’s melting point and boiling point are \(\mathrm{-190^\circ C}\) and \(\mathrm{-186^\circ C}\), respectively. Calculate the entropy changes for fusion and vaporization.

A reaction has a positive \(\Delta H^{\circ}\) and a negative \(\Delta S^{\circ}\). Is the equilibrium constant K for this reaction greater than 1, equal to 1, or less than 1?

Consider the arrangement in Figure 17.1. Because the volume of the two bulbs is the same, the probability of finding a molecule in either bulb is \(\frac{1}{2}\). Calculate the probability of all the molecules ending up in the same bulb if the number is (a) 2, (b) 100, and (c) \(6 \times 10^{23}\). Based on your results, explain why the situation shown in Figure 17.1(b) will not be observed for a macroscopic system. Figure 17.1 (a) A spontaneous process. After the valve is opened, the molecules distribute evenly between the two bulbs. (b) A nonspontaneous process. After the valve is opened, the molecules preferentially gather in one bulb.

Calculate the equilibrium constant (KP) for the reaction \(2 \mathrm{O}_{3}(g) \longrightarrow 3 \mathrm{O}_{2}(g)\) at \(25^\circ \mathrm C\).

State the second law of thermodynamics in words and express it mathematically.

Calculate \(\mathrm{\Delta G^\circ}\) for the following process at \(\mathrm{25^\circ C}\): \(\mathrm{BaF}_{2}(s) \rightleftharpoons \mathrm{Ba}^{2+}(a q)+2 \mathrm{~F}^{-}(a q)\) The \(K_{s p}\) of \(\mathrm{BaF}_{2}\) is \(1.7 \times 10^{-6}\).

State the third law of thermodynamics and explain its usefulness in calculating entropy values.

The \(K_p\) for the reaction \(\mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \rightleftharpoons 2 \mathrm{NH}_{3}(g) \text { is } 4.3 \times 10^{-4}\) at \(375^{\circ} \mathrm{C}\). In one experiment, the initial pressures are: \(P_{\mathrm{H}_{2}}=0.40\) atm, \(P_{\mathrm{N}_{2}}=0.86\) atm, and \(P_{\mathrm{NH}_{3}}=4.4 \times 10^{-3}\) atm. Calculate \(\Delta G\) for the reaction and predict the direction of the net reaction.

For each pair of substances listed here, choose the one having the larger standard entropy value at \(25^\circ \mathrm C\). The same molar amount is used in the comparison. Explain the basis for your choice. (a) Li(s) or Li(l); (b) \(\mathrm{C_2H_5OH}\)(l) or \(\mathrm{CH_3OCH_3}\)(l) (Hint: Which molecule can hydrogen-bond?); (c) Ar(g) or Xe(g); (d) CO(g) or \(\mathrm{CO_2}\)(g); (e) \(\mathrm{O_2}\)(g) or \(\mathrm{O_3}\)(g); (f) \(\mathrm{NO_2}\)(g) or \(\mathrm{N_2O_4}\)(g)

Arrange the following substances (1 mole each) in order of increasing entropy at \(\mathrm{25^\circ C}\): (a) Ne(g), (b) \(\mathrm{SO_2}\)(g), (c) Na(s), (d) NaCl(s), (e) \(\mathrm{H_2}\)(g). Give the reasons for your arrangement.

Using the data in Appendix 3, calculate the standard entropy changes for the following reactions at \(25^{\circ} \mathrm{C}\): (a) \(\mathrm{S}(s)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{SO}_{2}(g)\) (b) \(\mathrm{MgCO}_{3}(s) \longrightarrow \mathrm{MgO}(s)+\mathrm{CO}_{2}(g)\)

Using the data in Appendix 3, calculate the standard entropy changes for the following reactions at \(25^{\circ} \mathrm{C}\): (a) \(\mathrm{H}_{2}(g)+\mathrm{CuO}(s) \longrightarrow \mathrm{Cu}(s)+\mathrm{H}_{2} \mathrm{O}(g) \) (b) \(2 \mathrm{Al}(s)+3 \mathrm{ZnO}(s) \longrightarrow \mathrm{Al}_{2} \mathrm{O}_{3}(s)+3 \mathrm{Zn}(s)\) (c) \(\mathrm{CH}_{4}(g)+2 \mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(l)\)

Without consulting Appendix 3, predict whether the entropy change is positive or negative for each of the following reactions. Give reasons for your predictions. (a) \(2 \mathrm{KClO}_{4}(s) \longrightarrow 2 \mathrm{KClO}_{3}(s)+\mathrm{O}_{2}(g)\) (b) \(\mathrm{H}_{2} \mathrm{O}(g) \longrightarrow \mathrm{H}_{2} \mathrm{O}(l)\) (c) \(2 \mathrm{Na}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow 2 \mathrm{NaOH}(a q)+\mathrm{H}_{2}(g)\) (d) \(\mathrm{N}_{2}(g) \longrightarrow 2 \mathrm{N}(g)\)

State whether the sign of the entropy change expected for each of the following processes will be positive or negative, and explain your predictions. (a) \(\mathrm{PCl}_{3}(l)+\mathrm{Cl}_{2}(g) \longrightarrow \mathrm{PCl}_{5}(s)\) (b) \(2 \mathrm{HgO}(s) \longrightarrow 2 \mathrm{Hg}(l)+\mathrm{O}_{2}(g)\) (c) \(\mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{H}(g)\) (d) \(\mathrm{U}(s)+3 \mathrm{F}_{2}(g) \longrightarrow \mathrm{UF}_{6}(s)\)

Define free energy. What are its units?

Why is it more convenient to predict the direction of a reaction in terms of \(\Delta G_\mathrm{sys}\) instead of \(\Delta S_\mathrm{univ}\)? Under what conditions can \(\Delta G_\mathrm{sys}\) be used to predict the spontaneity of a reaction?

Calculate \(\Delta G^\circ\) for the following reactions at \(25^\circ \mathrm C\): (a) \(\mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{NO}(g)\) (b) \(\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{H}_{2} \mathrm{O}(g)\) (c) \(2 \mathrm{C}_{2} \mathrm{H}_{2}(g)+5 \mathrm{O}_{2}(g) \longrightarrow 4 \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(l)\) (Hint: Look up the standard free energies of formation of the reactants and products in Appendix 3.)

Calculate \(\Delta G^{\circ}\) for the following reactions at \(25^{\circ} \mathrm{C}\): (a) \(2 \mathrm{Mg}(s)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{MgO}(s)\) (b) \(2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{SO}_{3}(g)\) (c) \(2 \mathrm{C}_{2} \mathrm{H}_{6}(g)+7 \mathrm{O}_{2}(g) \longrightarrow 4 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(l)\) See Appendix 3 for the thermodynamic data.

From the values of \(\Delta H\) and \(\Delta S\), predict which of the following reactions would be spontaneous at \(25^{\circ} \mathrm{C}\): Reaction A: \(\Delta H=10.5 \mathrm{~kJ} / \mathrm{mol}\), \(\Delta S=30 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol}\); reaction B: \(\Delta H=1.8 \mathrm{~kJ} / \mathrm{mol}\), \(\Delta S=-113 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol}\). If either of the reactions is nonspontaneous at \(25^{\circ} \mathrm{C}\), at what temperature might it become spontaneous?

Find the temperatures at which reactions with the following \(\Delta H\) and \(\Delta S\) values would become spontaneous: (a) \(\Delta H=-126 \mathrm{~kJ} / \mathrm{mol}, \Delta S=84 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol};\) (b) \(\Delta H=-11.7 \mathrm{~kJ} / \mathrm{mol}, \Delta S=-105 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol}\)

Explain the difference between \(\Delta G\) and \(\Delta G^\circ\).

Explain why Equation (17.14) is of great importance in chemistry.

Calculate \(K_p\) for the following reaction at \(25^\circ \mathrm C\): \(\mathrm{H}_{2}(g)+\mathrm{I}_{2}(g) \rightleftharpoons 2 \mathrm{HI}(g) \quad \Delta G^{\circ}=2.60 \mathrm{~kJ} / \mathrm{mol}\)

For the autoionization of water at \(25^{\circ} \mathrm{C}\), \(\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{OH}^{-}(a q)\) \(K_{\mathrm{w}}\) is \(1.0 \times 10^{-14}\). What is \(\Delta G^{\circ}\) for the process?

Consider the following reaction at \(25^{\circ} \mathrm{C}\) : \(\mathrm{Fe}(\mathrm{OH})_{2}(s) \rightleftharpoons \mathrm{Fe}^{2+}(a q)+2 \mathrm{OH}^{-}(a q)\) Calculate \(\Delta G^{\circ}\) for the reaction. \(K_{\mathrm{sp}}\) for \(\mathrm{Fe}(\mathrm{OH})_{2}\) is \(1.6 \times 10^{-14}\).

Calculate \(\Delta G^{\circ}\) and \(K_P\) for the following equilibrium reaction at \(25^{\circ} \mathrm{C}\). \(2 \mathrm{H}_2 \mathrm{O}(\mathrm{g}) \rightleftharpoons 2 \mathrm{H}_2(\mathrm{~g})+\mathrm{O}_2(\mathrm{~g})\)

(a) Calculate \(\Delta G^{\circ}\) and \(K_{P}\) for the following equilibrium reaction at \(25^{\circ} \mathrm{C}\). The \(\Delta G_{\mathrm{f}}^{\circ}\) values are 0 for \(\mathrm{Cl}_{2}(g),-286 \mathrm{~kJ} / \mathrm{mol}\) for \(\mathrm{PCl}_{3}(g)\), and \(-325 \mathrm{~kJ} / \mathrm{mol}\) for \(\mathrm{PCl}_{5}(g)\). \(\mathrm{PCl}_{5}(g) \rightleftharpoons \mathrm{PCl}_{3}(g)+\mathrm{Cl}_{2}(g)\) (b) Calculate \(\Delta G\) for the reaction if the partial pressures of the initial mixture are \(P_{\mathrm{PCl}_{5}}=0.0029 \mathrm{~atm}\), \(P_{\mathrm{PCl}_{3}}=0.27 \mathrm{~atm}\), and \(P_{\mathrm{Cl}_{2}}=0.40 \mathrm{~atm}\).

The equilibrium constant \(\left(K_{P}\right)\) for the reaction \(\mathrm{H}_{2}(g)+\mathrm{CO}_{2}(g) \rightleftharpoons \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{CO}(g)\) is 4.40 at 2000 K. (a) Calculate \(\Delta G^{\circ}\) for the reaction. (b) Calculate \(\Delta G\) for the reaction when the partial pressures are \(P_{\mathrm{H}_{2}}=0.25 \mathrm{~atm}\), \(P_{\mathrm{CO}_{2}}=0.78 \mathrm{~atm}\), \(P_{\mathrm{H}_{2} \mathrm{O}}=0.66 \mathrm{~atm}\), and \(P_{\mathrm{CO}}=1.20 \mathrm{~atm}\).

Consider the decomposition of calcium carbonate: \(\mathrm{CaCO}_{3}(s) \rightleftharpoons \mathrm{CaO}(s)+\mathrm{CO}_{2}(g)\) Calculate the pressure in atm of \(\mathrm{CO}_{2}\) in an equilibrium process (a) at \(25^{\circ} \mathrm{C}\) and (b) at \(800^{\circ} \mathrm{C}\). Assume that \(\Delta H^{\circ}=177.8 \mathrm{~kJ} / \mathrm{mol}\) and \(\Delta S^{\circ}=160.5 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol}\) for the temperature range.

The equilibrium constant \(K_{P}\) for the reaction \(\mathrm{CO}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{COCl}_{2}(g)\) is \(5.62 \times 10^{35}\) at \(25^{\circ} \mathrm{C}\). Calculate \(\Delta G_{\mathrm{f}}^{\circ}\) for \(\mathrm{COCl}_{2}\) at \(25^{\circ} \mathrm{C}\).

At \(25^{\circ} \mathrm{C}\), \(\Delta G^{\circ}\) for the process \(\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{H}_{2} \mathrm{O}(g)\) is 8.6 kJ/mol. Calculate the vapor pressure of water at this temperature.

Calculate \(\Delta G^{\circ}\) for the process \(\mathrm{C(diamond)} \longrightarrow \mathrm{C (graphite)}\) Is the formation of graphite from diamond favored at \(25^{\circ} \mathrm{C}\)? If so, why is it that diamonds do not become graphite on standing?

What is a coupled reaction? What is its importance in biological reactions?

What is the role of ATP in biological reactions?

Referring to the metabolic process involving glucose on p. 803, calculate the maximum number of moles of ATP that can be synthesized from ADP from the breakdown of one mole of glucose.

In the metabolism of glucose, the first step is the conversion of glucose to glucose 6-phosphate: \(\begin{array}{r}\text { glucose }+\mathrm{H}_{3} \mathrm{PO}_{4} \longrightarrow \text {glucose 6-phosphate }+\mathrm{H}_{2} \mathrm{O} \\ \Delta G^{\circ}=13.4 \mathrm{~kJ} / \mathrm{mol} \end{array} \) Because \(\Delta G^\circ\) is positive, this reaction does not favor the formation of products. Show how this reaction can be made to proceed by coupling it with the hydrolysis of ATP. Write an equation for the coupled reaction and estimate the equilibrium constant for the coupled process.

Explain the following nursery rhyme in terms of the second law of thermodynamics. Humpty Dumpty sat on a wall; Humpty Dumpty had a great fall. All the King's horses and all the King's men Couldn't put Humpty together again.

Calculate \(\Delta G\) for the reaction \(\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{OH}^{-}(a q)\) at \(25^{\circ} \mathrm{C}\) for the following conditions: (a) \(\left[\mathrm{H}^{+}\right]=1.0 \times 10^{-7} \mathrm{M},\left[\mathrm{OH}^{-}\right]=1.0 \times 10^{-7}\) M (b) \(\left[\mathrm{H}^{+}\right]=1.0 \times 10^{-3} \mathrm{M},\left[\mathrm{OH}^{-}\right]=1.0 \times 10^{-4}\) M (c) \(\left[\mathrm{H}^{+}\right]=1.0 \times 10^{-12} \mathrm{M},\left[\mathrm{OH}^{-}\right]=2.0 \times 10^{-8}\) M (d) \(\left[\mathrm{H}^{+}\right]=3.5 \mathrm{M},\left[\mathrm{OH}^{-}\right]=4.8 \times 10^{-4}\) M

Calculate the \(\Delta S_{\text {soln }}^{\circ}\) for the following processes: (a) \(\mathrm{NH}_{4} \mathrm{NO}_{3}(s) \longrightarrow \mathrm{NH}_{4}^{+}(a q)+\mathrm{NO}_{3}^{-}(a q)\) and (b) \(\mathrm{FeCl}_{3}(s) \longrightarrow \mathrm{Fe}^{3+}(a q)+3 \mathrm{Cl}^{-}(a q)\). Give a qualitative explanation for the signs.

The following reaction is spontaneous at a certain temperature T. Predict the sign of \(\Delta S_{\text {surr }}\).

Which of the following thermodynamic functions are associated only with the first law of thermodynamics: S, U, G, and H?

A student placed 1 g of each of three compounds A, B, and C in a container and found that after 1 week no change had occurred. Offer some possible explanations for the fact that no reactions took place. Assume that A, B, and C are totally miscible liquids.

Use the data in Appendix 3 to calculate the equilibrium constant for the reaction \(\mathrm{AgI}(s) \rightleftharpoons \mathrm{Ag}^{+}(a q)+ \mathrm{I}^{-}(a q)\) at \(25^{\circ} \mathrm{C}\). Compare your result with the \(K_{\mathrm{sp}}\) value in Table 16.2.

Predict the signs of \(\Delta H, \Delta S\), and \(\Delta G\) of the system for the following processes at 1 atm: (a) ammonia melts at \(-60^{\circ} \mathrm{C}\), (b) ammonia melts at \(-77.7^{\circ} \mathrm{C}\), (c) ammonia melts at \(-100^{\circ} \mathrm{C}\). (The normal melting point of ammonia is \(-77.7^{\circ} \mathrm{C}\).)

Consider the following facts: Water freezes spontaneously at \(\mathrm{-5^\circ C}\) and 1 atm, and ice has a more ordered structure than liquid water. Explain how a spontaneous process can lead to a decrease in entropy.

Ammonium nitrate \(\mathrm{(NH_4NO_3)}\) dissolves spontaneously and endothermically in water. What can you deduce about the sign of \(\Delta S\) for the solution process?

Calculate the equilibrium pressure of \(\mathrm{CO_2}\) due to the decomposition of barium carbonate \(\mathrm{(BaCO_3)}\) at \(\mathrm{25^\circ C}\).

(a) Trouton's rule states that the ratio of the molar heat of vaporization of a liquid \((\Delta H_\mathrm{vap})\) to its boiling point in kelvins is approximately \(\mathrm{90~ J/K \cdot mol}\). Use the following data to show that this is the case and explain why Trouton's rule holds true: (b) Use the values in Table 11.6 to calculate the same ratio for ethanol and water. Explain why Trouton's rule does not apply to these two substances as well as it does to other liquids.

Referring to Problem 17.48, explain why the ratio is considerably smaller than \(\mathrm{90~ J/K \cdot mol}\) for liquid HF.

Carbon monoxide (CO) and nitric oxide (NO) are polluting gases contained in automobile exhaust. Under suitable conditions, these gases can be made to react to form nitrogen \(\left(\mathrm{N}_{2}\right)\) and the less harmful carbon dioxide \(\left(\mathrm{CO}_{2}\right)\). (a) Write an equation for this reaction. (b) Identify the oxidizing and reducing agents. (c) Calculate the \(K_{P}\) for the reaction at \(25^{\circ} \mathrm{C}\). (d) Under normal atmospheric conditions, the partial pressures are \(P_{\mathrm{N}_{2}}=0.80 \mathrm{~atm}\), \(P_{\mathrm{CO}_{2}}=3.0 \times 10^{-4}~\mathrm{atm}\), \(P_{\mathrm{CO}}=5.0 \times 10^{-5} \mathrm{~atm}\) and \(P_{\mathrm{NO}}=5.0 \times 10^{-7}~\mathrm {atm}\). Calculate \(Q_{P}\) and predict the direction toward which the reaction will proceed. (e) Will raising the temperature favor the formation of \(\mathrm{N}_{2}\) and \(\mathrm{CO}_{2}\)?

For reactions carried out under standard-state conditions, Equation (17.10) takes the form \(\Delta G^{\circ}=\Delta H^{\circ}-T \Delta S^{\circ}\). (a) Assuming \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) are independent of temperature, derive the equation \(\ln \frac{K_{2}}{K_{1}}=\frac{\Delta H^{\circ}}{R}\left(\frac{T_{2}-T_{1}}{T_{1} T_{2}}\right)\) where \(K_{1}\) and \(K_{2}\) are the equilibrium constants at \(T_{1}\) and \(T_{2}\), respectively. (b) Given that at \(25^{\circ} \mathrm{C}\) \(K_{\mathrm{c}}\) is \(4.63 \times 10^{-3}\) for the reaction \(\mathrm{N}_{2} \mathrm{O}_{4}(g) \rightleftharpoons 2 \mathrm{NO}_{2}(g) \quad \Delta H^{\circ}=58.0 \mathrm{~kJ} / \mathrm{mol}\) calculate the equilibrium constant at \(65^{\circ} \mathrm{C}\).

Use the thermodynamic data in Appendix 3 to calculate the \(K_\mathrm{sp}\) of AgCl.

Consider the reaction \(\mathrm{A} \longrightarrow \mathrm{B}+\mathrm{C}\) at 298 K. Given that the forward rate constant \(\left(k_{\mathrm{f}}\right)\) is \(0.46 \mathrm{~s}^{-1}\) and the reverse rate constant \(\left(k_{\mathrm{r}}\right)\) is \(1.5 \times 10^{-2} / M \cdot \mathrm{s}\), calculate \(\Delta G^{\circ}\) of the reaction.

The \(K_{\text {sp}}\) of AgCl is given in Table 16.2. What is its value at \(60^{\circ} \mathrm{C}\)? [Hint: You need the result of Problem 17.51(a) and the data in Appendix 3 to calculate \(\Delta H^{\circ}\).]

Under what conditions does a substance have a standard entropy of zero? Can a substance ever have a negative standard entropy?

Water gas, a mixture of \(\mathrm H_2\) and CO, is a fuel made by reacting steam with red-hot coke (a by-product of coal distillation): \(\mathrm{H}_{2} \mathrm{O}(g)+\mathrm{C}(s) \rightleftharpoons \mathrm{CO}(g)+\mathrm{H}_{2}(g)\) From the data in Appendix 3, estimate the temperature at which the reaction begins to favor the formation of products.

Consider the following Brønstead acid-base reaction at \(25^{\circ} \mathrm{C}\) : \(\mathrm{HF}(a q)+\mathrm{Cl}^{-}(a q) \rightleftharpoons \mathrm{HCl}(a q)+\mathrm{F}^{-}(a q)\) (a) Predict whether K will be greater or smaller than unity. (b) Does \(\Delta S^{\circ}\) or \(\Delta H^{\circ}\) make a greater contribution to \(\Delta G^{\circ}\) ? (c) Is \(\Delta H^{\circ}\) likely to be positive or negative?

Crystallization of sodium acetate from a supersaturated solution occurs spontaneously (see p. 521). What can you deduce about the signs of \(\Delta S\) and \(\Delta H\)?

Consider the thermal decomposition of \(\mathrm{CaCO}_{3}\): \(\mathrm{CaCO}_{3}(s) \rightleftharpoons \mathrm{CaO}(s)+\mathrm{CO}_{2}(g)\) The equilibrium vapor pressures of \(\mathrm{CO}_{2}\) are 22.6 mmHg at \(700^{\circ} \mathrm{C}\) and 1829 mmHg at \(950^{\circ} \mathrm{C}\). Calculate the standard enthalpy of the reaction. [Hint: See Problem 17.51(a).]

A certain reaction is spontaneous at \(72^{\circ} \mathrm{C}\). If the enthalpy change for the reaction is 19 kJ/mol, what is the minimum value of \(\Delta S\) (in \(\mathrm{J} / \mathrm{K} \cdot \mathrm{mol}\)) for the reaction?

Predict whether the entropy change is positive or negative for each of these reactions: (a) \(\mathrm{Zn}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{ZnCl}_{2}(a q)+\mathrm{H}_{2}(g)\) (b) \(\mathrm{O}(g)+\mathrm{O}(g) \longrightarrow \mathrm{O}_{2}(g)\) (c) \(\mathrm{NH}_{4} \mathrm{NO}_{3}(s) \longrightarrow \mathrm{N}_{2} \mathrm{O}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)\) (d) \(2 \mathrm{H}_{2} \mathrm{O}_{2}(l) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{O}_{2}(g)\)

The reaction \(\mathrm{NH}_{3}(g)+\mathrm{HCl}(g) \longrightarrow \mathrm{NH}_{4} \mathrm{Cl}(s)\) proceeds spontaneously at \(25^{\circ} \mathrm{C}\) even though there is a decrease in the number of microstates of the system (gases are converted to a solid). Explain.

Use the following data to determine the normal boiling point, in kelvins, of mercury. What assumptions must you make in order to do the calculation? Hg(l): \(\Delta H_{\mathrm{f}}^{\circ}=0\) (by definition) \(S^{\circ}=77.4 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol}\) Hg(g): \(\Delta H_{\mathrm{f}}^{\circ}=60.78 \ \mathrm{kJ} / \mathrm{mol}\) \(S^{\circ}=174.7 \ \mathrm{J} / \mathrm{K} \cdot \mathrm{mol}\)

The molar heat of vaporization of ethanol is 39.3 kJ/mol and the boiling point of ethanol is \(78.3^{\circ} \mathrm{C}\). Calculate \(\Delta S\) for the vaporization of 0.50 mol ethanol.

A certain reaction is known to have a \(\Delta G^{\circ}\) value of -122 kJ/mol. Will the reaction necessarily occur if the reactants are mixed together?

In the Mond process for the purification of nickel, carbon monoxide is reacted with heated nickel to produce \(\mathrm{Ni}(\mathrm{CO})_{4}\), which is a gas and can therefore be separated from solid impurities: \(\mathrm{Ni}(s)+4 \mathrm{CO}(g) \rightleftharpoons \mathrm{Ni}(\mathrm{CO})_{4}(g)\) Given that the standard free energies of formation of CO(g) and \(\mathrm{Ni}(\mathrm{CO})_{4}(g)\) are - 137.3 kJ/mol and -587.4 kJ/mol, respectively, calculate the equilibrium constant of the reaction at \(80^{\circ} \mathrm{C}\). Assume that \(\Delta G_{\mathrm{f}}^{\circ}\) is temperature independent.

Calculate \(\Delta G^{\circ}\) and \(K_{p}\) for the following processes at \(25^{\circ} \mathrm{C}\): (a) \(\mathrm{H}_{2}(g)+\mathrm{Br}_{2}(l) \rightleftharpoons 2 \mathrm{HBr}(g)\) (b) \(\frac{1}{2} \mathrm{H}_{2}(g)+\frac{1}{2} \mathrm{Br}_{2}(l) \rightleftharpoons \operatorname{HBr}(g)\) Account for the differences in \(\Delta G^{\circ}\) and \(K_{p}\) obtained for (a) and (b).

Calculate the pressure of \(\mathrm{O}_{2}\) (in atm) over a sample of NiO at \(25^{\circ} \mathrm{C}\) if \(\Delta G^{\circ}\) = 212 kJ/mol for the reaction \(\mathrm{NiO}(s) \rightleftharpoons \mathrm{Ni}(s)+\frac{1}{2} \mathrm{O}_{2}(g)\)

Comment on the statement: "Just talking about entropy increases its value in the universe."

For a reaction with a negative \(\Delta G^{\circ}\) value, which of the following statements is false? (a) The equilibrium constant K is greater than one, (b) the reaction is spontaneous when all the reactants and products are in their standard states, and (c) the reaction is always exothermic.

Consider the reaction \(\mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{NO}(g)\) Given that \(\Delta G^{\circ}\) for the reaction at \(25^{\circ} \mathrm{C}\) is 173.4 kJ/ mol, (a) calculate the standard free energy of formation of NO, and (b) calculate \(K_{P}\) of the reaction. (c) One of the starting substances in smog formation is NO. Assuming that the temperature in a running automobile engine is \(1100^{\circ} \mathrm{C}\), estimate \(K_{P}\) for the above reaction. (d) As farmers know, lightning helps to produce a better crop. Why?

Heating copper(II) oxide at \(400^{\circ} \mathrm{C}\) does not produce any appreciable amount of Cu: \(\mathrm{CuO}(s)\rightleftharpoons\mathrm{Cu}(s)+\frac{1}{2}\mathrm{O}_2(g)\quad\ \ \ \Delta G^{\circ}=127.2\mathrm{kJ}/\mathrm{mol}\) However, if this reaction is coupled to the conversion of graphite to carbon monoxide, it becomes spontaneous. Write an equation for the coupled process and calculate the equilibrium constant for the coupled reaction.

The internal engine of a 1200-kg car is designed to run on octane \(\left(\mathrm{C}_{8} \mathrm{H}_{18}\right)\), whose enthalpy of combustion is 5510 kJ/mol. If the car is moving up a slope, calculate the maximum height (in meters) to which the car can be driven on 1.0 gallon of the fuel. Assume that the engine cylinder temperature is \(2200^{\circ} \mathrm{C}\) and the exit temperature is \(760^{\circ} \mathrm{C}\), and neglect all forms of friction. The mass of 1 gallon of fuel is 3.1 kg. [Hint: See the Chemistry in Action essay on p. 792. The work done in moving the car over a vertical distance is mgh, where \(m\) is the mass of the car in kg, \(g\) the acceleration due to gravity \(\left(9.81 \ \mathrm{m} / \mathrm{s}^{2}\right)\), and \(h\) the height in meters.]

Consider the decomposition of magnesium carbonate: \(\mathrm{MgCO}_{3}(s) \rightleftharpoons \mathrm{MgO}(s)+\mathrm{CO}_{2}(g)\) Calculate the temperature at which the decomposition begins to favor products. Assume that both \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) are independent of temperature.

(a) Over the years there have been numerous claims about "perpetual motion machines," machines that will produce useful work with no input of energy. Explain why the first law of thermodynamics prohibits the possibility of such a machine existing. (b) Another kind of machine, sometimes called a "perpetual motion of the second kind,” operates as follows. Suppose an ocean liner sails by scooping up water from the ocean and then extracting heat from the water, converting the heat to electric power to run the ship, and dumping the water back into the ocean. This process does not violate the first law of thermodynamics, for no energy is created—energy from the ocean is just converted to electrical energy. Show that the second law of thermodynamics prohibits the existence of such a machine.

The activity series in Section 4.4 shows that reaction (a) is spontaneous while reaction (b) is nonspontaneous at \(25^{\circ} \mathrm{C}\): (a) \(\mathrm{Fe}(s)+2 \mathrm{H}^{+} \longrightarrow \mathrm{Fe}^{2+}(a q)+\mathrm{H}_{2}(g)\) (b) \(\mathrm{Cu}(s)+2 \mathrm{H}^{+} \longrightarrow \mathrm{Cu}^{2+}(a q)+\mathrm{H}_{2}(g)\) Use the data in Appendix 3 to calculate the equilibrium constant for these reactions and hence confirm that the activity series is correct.

The rate constant for the elementary reaction \(2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{NO}_{2}(g)\) is \(7.1 \times 10^{9} / \mathrm{M}^{2} \cdot \mathrm{s}\) at \(25^{\circ} \mathrm{C}\). What is the rate constant for the reverse reaction at the same temperature?

The following reaction is the cause of sulfur deposits formed at volcanic sites (see p. 913): \(2\mathrm{H}_2\mathrm{S}(g)+\mathrm{SO}_2(g)\rightleftharpoons3\mathrm{S}(s)+2\mathrm{H}_2\mathrm{O}(g)\) It may also be used to remove \(\mathrm{SO}_{2}\) from powerplant stack gases. (a) Identify the type of redox reaction it is. (b) Calculate the equilibrium constant \(\left(K_{P}\right)\) at \(25^{\circ} \mathrm{C}\) and comment on whether this method is feasible for removing \(\mathrm{SO}_{2}\). (c) Would this procedure become more or less effective at a higher temperature?

Describe two ways that you could measure \(\Delta G^{\circ}\) of a reaction.

The following reaction represents the removal of ozone in the stratosphere: \(2 \mathrm{O}_{3}(g) \rightleftharpoons 3 \mathrm{O}_{2}(g)\) Calculate the equilibrium constant \(\left(K_{P}\right)\) for the reaction. In view of the magnitude of the equilibrium constant, explain why this reaction is not considered a major cause of ozone depletion in the absence of man-made pollutants such as the nitrogen oxides and CFCs. Assume the temperature of the stratosphere to be \(-30^{\circ} \mathrm{C}\) and \(\Delta G_{\mathrm{f}}^{\circ}\) to be temperature independent.

A 74.6-g ice cube floats in the Arctic Sea. The temperature and pressure of the system and surroundings are at 1 atm and \(0^{\circ} \mathrm{C}\). Calculate \(\Delta S_{\text {sys }}\), \(\Delta S_{\text {surr }}\) and \(\Delta S_{\text {univ }}\) for the melting of the ice cube. What can you conclude about the nature of the process from the value of \(\Delta S_{\text {univ }}\)? (The molar heat of fusion of water is 6.01 kJ/mol.)

Comment on the feasibility of extracting copper from its ore chalcocite \(\left(\mathrm{Cu}_{2} \mathrm{S}\right)\) by heating: \(\mathrm{Cu}_{2} \mathrm{S}(s) \longrightarrow 2 \mathrm{Cu}(s)+\mathrm{S}(s)\) Calculate the \(\Delta G^{\circ}\) for the overall reaction if the above process is coupled to the conversion of sulfur to sulfur dioxide. Given that \(\Delta G_{\mathrm{f}}^{\circ}\left(\mathrm{Cu}_{2} \mathrm{S}\right)\) = -86.1 kJ/mol.

Active transport is the process in which a substance is transferred from a region of lower concentration to one of higher concentration. This is a nonspontaneous process and must be coupled to a spontaneous process, such as the hydrolysis of ATP. The concentrations of \(\mathrm{K}^{+}\) ions in the blood plasma and in nerve cells are 15 mM and 400 mM, respectively \(\left(1 \mathrm{m} M=1 \times 10^{-3} M\right)\). Use Equation (17.13) to calculate \(\Delta G\) for the process at the physiological temperature of \(37^{\circ} \mathrm{C}\): \(\mathrm{K}^{+}(15 \mathrm{m} M) \longrightarrow \mathrm{K}^{+}(400 \mathrm{m} M)\) In this calculation, the \(\Delta G^{\circ}\) term can be set to zero. What is the justification for this step?

Large quantities of hydrogen are needed for the synthesis of ammonia. One preparation of hydrogen involves the reaction between carbon monoxide and steam at \(300^{\circ} \mathrm{C}\) in the presence of a copper-zinc catalyst: \(\mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons \mathrm{CO}_{2}(g)+\mathrm{H}_{2}(g)\) Calculate the equilibrium constant \(\left(K_{P}\right)\) for the reaction and the temperature at which the reaction favors the formation of CO and \(\mathrm{H}_{2} \mathrm{O}\). Will a larger \(K_{P}\) be attained at the same temperature if a more efficient catalyst is used?

Shown here are the thermodynamic data for ethanol: Calculate the vapor pressure of ethanol at \(25^{\circ} \mathrm{C}\). Assume the thermodynamic values are independent of temperature.

The reaction shown here is spontaneous at a certain temperature \(T\). What is the sign of \(\Delta S_{\text {surr }}\)?

Consider two carboxylic acids (acids that contain the -COOH group): \(\mathrm{CH}_{3} \mathrm{COOH}\) (acetic acid, \(K_{\mathrm{a}}=1.8 \times 10^{-5}\)) and \(\mathrm{CH}_{2} \mathrm{ClCOOH}\) (chloroacetic acid, \(K_{\mathrm{a}}=1.4 \times 10^{-3}\)). (a) Calculate \(\Delta G^{\circ}\) for the ionization of these acids at \(25^{\circ} \mathrm{C}\). (b) From the equation \(\Delta G^{\circ}=\Delta H^{\circ}-T \Delta S^{\circ}\), we see that the contributions to the \(\Delta G^{\circ}\) term are an enthalpy term \(\left(\Delta H^{\circ}\right)\) and a temperature times entropy term \(\left(T \Delta S^{\circ}\right)\). These contributions are listed below for the two acids: Which is the dominant term in determining the value of \(\Delta G^{\circ}\) (and hence \(K_{\mathrm{a}}\), of the acid)? (c) What processes contribute to \(\Delta H^{\circ}\)? (Consider the ionization of the acids as a \(\text { Brønsted }\) acid-base reaction.) (d) Explain why the \(T \Delta S^{\circ}\) term is more negative for \(\mathrm{CH}_{3} \mathrm{COOH}\).

Many hydrocarbons exist as structural isomers, which are compounds that have the same molecular formula but different structures. For example, both butane and isobutane have the same molecular formula of \(\mathrm{C}_{4} \mathrm{H}_{10}\) (see Problem 11.19). Calculate the mole percent of these molecules in an equilibrium mixture at \({25^{\circ} \mathrm{C}}\), given that the standard free energy of formation of butane is -15.9 kJ/mol and that of isobutane is -18.0 kJ/mol. Does your result support the notion that straight-chain hydrocarbons (that is, hydrocarbons in which the \(C\) atoms are joined along a line) are less stable than branch-chain hydrocarbons?

Use the thermodynamic data in Appendix 3 to determine the normal boiling point of liquid bromine. Assume the values are independent of temperature.

In each of the following reactions, there is one species for which the standard entropy value is not listed in Appendix 3. Determine the \(S^{\circ}\) for that species. (a) The \(\Delta S_{\mathrm{rxn}}^{\circ}\) for the reaction \(\mathrm{Na}(s) \longrightarrow \mathrm{Na}(l)\) is \(48.64 \ \mathrm{J} / \mathrm{K} \cdot \mathrm{mol}\). (b) The \(\Delta S_{\mathrm{rxn}}^{\circ}\) for the reaction 2S(monoclinic) + \(\mathrm{Cl}_{2}(g) \longrightarrow \mathrm{S}_{2} \mathrm{Cl}_{2}(g)\) is \(43.4 \ \mathrm{J} / \mathrm{K} \cdot \mathrm{mol}\). (c) The \(\Delta S_{\text {run }}^{\circ}\) for the reaction \(\mathrm{FeCl}_{2}(s) \longrightarrow \mathrm{Fe}^{2+}(a q)+2 \mathrm{Cl}^{-}(a q)\) is \(-118.3 \ \mathrm{J} / \mathrm{K} \cdot \mathrm{mol}\).

A rubber band is stretched vertically by attaching a weight to one end and holding the other end by hand. On heating the rubber band with a hot-air blower, it is observed to shrink slightly in length. Give a thermodynamic analysis for this behavior. (Hint: See the Chemistry in Action essay on p. 803.)

One of the steps in the extraction of iron from its ore (FeO) is the reduction of iron(II) oxide by carbon monoxide at \(900^{\circ} \mathrm{C}\): \(\mathrm{FeO}(s)+\mathrm{CO}(g) \rightleftharpoons \mathrm{Fe}(s)+\mathrm{CO}_{2}(g)\) If CO is allowed to react with an excess of FeO, calculate the mole fractions of CO and \(\mathrm{CO}_{2}\) at equilibrium. State any assumptions.

Derive the following equation \(\Delta G=R T \ln (Q / K)\) where \(Q\) is the reaction quotient and describe how you would use it to predict the spontaneity of a reaction.

The sublimation of carbon dioxide at \(-78^{\circ} \mathrm{C}\) is \(\mathrm{CO}_{2}(s) \longrightarrow \mathrm{CO}_{2}(g) \quad \Delta H_{\text {sub }}=62.4 \ \mathrm{kJ} / \mathrm{mol}\) Calculate \(\Delta S_{\text {sub }}\) when 84.8 \(g\) of \(\mathrm{CO}_{2}\) sublimes at this temperature.

Entropy has sometimes been described as "time's arrow" because it is the property that determines the forward direction of time. Explain.

Referring to Figure 17.1, we see that the probability of finding all 100 molecules in the same bulb is \(8 \times 10^{-31}\). Assuming that the age of the universe is 13 billion years, calculate the time in seconds during which this event can be observed.

A student looked up the \(\Delta G_{\mathrm{f}}^{\circ}\), \(\Delta H_{\mathrm{f}}^{\circ}\), and \(S^{\circ}\) values for \(\mathrm{CO}_{2}\) in Appendix 3. Plugging these values into Equation (17.10), he found that \(\Delta G_{\mathrm{f}}^{\circ} \neq \Delta H_{\mathrm{f}}^{\circ}-T S^{\circ}\) at 298 K. What is wrong with his approach?

Consider the following reaction at 298 K: \(2 \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(l) \ \ \ \ \ \quad \Delta H^{\circ}=-571.6 \ \mathrm{kJ} / \mathrm{mol}\) Calculate \(\Delta S_{\text {sys }}\), \(\Delta S_{\text {surr }}\), and \(\Delta S_{\text {univ }}\) for the reaction.

As an approximation, we can assume that proteins exist either in the native (or physiologically functioning) state and the denatured state \(\text { native } \rightleftharpoons \text { denatured }\) The standard molar enthalpy and entropy of the denaturation of a certain protein are 512 kJ/mol and \(1.60 \ \mathrm{kJ} / \mathrm{K} \cdot \mathrm{mol}\), respectively. Comment on the signs and magnitudes of these quantities, and calculate the temperature at which the process favors the denatured state.

Which of the following are not state functions: \(S\), \(H\), \(q\), \(w\), \(T\)?

Which of the following is not accompanied by an increase in the entropy of the system? (a) mixing of two gases at the same temperature and pressure, (b) mixing of ethanol and water, (c) discharging a battery, (d) expansion of a gas followed by compression to its original temperature, pressure, and volume.

Hydrogenation reactions (for example, the process of converting \(\mathrm{C}=\mathrm{C}\) bonds to \(\mathrm{C}-\mathrm{C}\) bonds in food industry) are facilitated by the use of a transition metal catalyst, such as \(Ni\) or \(Pt\). The initial step is the adsorption, or binding, of hydrogen gas onto the metal surface. Predict the signs of \(\Delta H\), \(\Delta S\), and \(\Delta G\) when hydrogen gas is adsorbed onto the surface of \(Ni\) metal.

Give a detailed example of each of the following, with an explanation: (a) a thermodynamically spontaneous process; (b) a process that would violate the first law of thermodynamics; (c) a process that would violate the second law of thermodynamics; (d) an irreversible process; (e) an equilibrium process.

At 0 \(K\), the entropy of carbon monoxide crystal is not zero but has a value of \(4.2 \ \mathrm{J} / \mathrm{K} \cdot \mathrm{mol}\), called the residual entropy. According to the third law of thermodynamics, this means that the crystal does not have a perfect arrangement of the CO molecules, (a) What would be the residual entropy if the arrangement were totally random? (b) Comment on the difference between the result in (a) and \(4.2 \ \mathrm{J} / \mathrm{K} \cdot \mathrm{mol}\). [Hint: Assume that each CO molecule has two choices for orientation and use Equation (17.1) to calculate the residual entropy.]

Comment on the correctness of the analogy sometimes used to relate a student's dormitory room becoming untidy to an increase in entropy.

The standard enthalpy of formation and the standard entropy of gaseous benzene are 82.93 kJ/mol and \(269.2 \ \mathrm{J} / \mathrm{K} \cdot \mathrm{mol}\), respectively. Calculate \(\Delta H^{\circ}\), \(\Delta S^{\circ}\), and \(\Delta G^{\circ}\) for the process at \(25^{\circ} \mathrm{C}\). \(\mathrm{C}_{6} \mathrm{H}_{6}(l) \longrightarrow \mathrm{C}_{6} \mathrm{H}_{6}(g)\) Comment on your answers.

In chemistry, the standard state for a solution is 1 \(M\) (see Table 17.2). This means that each solute concentration expressed in molarity is divided by 1 \(M\). In biological systems, however, we define the standard state for the \(\mathrm{H}^{+}\) ions to be \(1 \times 10^{-7} M\) because the physiological pH is about 7. Consequently, the change in the standard Gibbs free energy according to these two conventions will be different involving uptake or release of \(\mathrm{H}^{+}\) ions, depending on which convention is used. We will therefore replace \(\Delta G^{\circ}\) with \(\Delta G^{\circ \prime}\), where the prime denotes that it is the standard Gibbs free-energy change for a biological process. (a) Consider the reaction \(\mathrm{A}+\mathrm{B} \longrightarrow \mathrm{C}+x \mathrm{H}^{+}\) where \(x\) is a stoichiometric coefficient. Use Equation (17.13) to derive a relation between \(\Delta G^{\circ}\) and \(\Delta G^{\circ \prime}\), keeping in mind that \(\Delta G\) is the same for a process regardless of which convention is used. Repeat the derivation for the reverse process: \(\mathrm{C}+x \mathrm{H}^{+} \longrightarrow \mathrm{A}+\mathrm{B}\) (b) \(\mathrm{NAD}^{+}\) and NADH are the oxidized and reduced forms of nicotinamide adenine dinucleotide, two key compounds in the metabolic pathways. For the oxidation of NADH: \(\mathrm{NADH}+\mathrm{H}^{+} \longrightarrow \mathrm{NAD}^{+}+\mathrm{H}_{2}\) \(\Delta G^{\circ}\) is - 21.8 kJ/mol at 298 \(K\). Calculate \(\Delta G^{\circ \prime}\). Also calculate \(\Delta G\) using both the chemical and biological conventions when \([\mathrm{NADH}]=1.5 \times 10^{-2} M\), \(\left[\mathrm{H}^{+}\right]=3.0 \times 10^{-5} M\), \([\mathrm{NAD}]=4.6 \times 10^{-3} M\), and \(P_{\mathrm{H}_{2}}=0.010\) atm.

The following diagram shows the variation of the equilibrium constant with temperature for the reaction \(\mathrm{I}_2(\mathrm{~g}) \rightleftharpoons 2 \mathrm{I}(\mathrm{g})\) Calculate \(\Delta G^{\circ}, \Delta H^{\circ}\), and \(\Delta S^{\circ}]\) for the reaction at 872 K. (Hint: See Problem 17.51.)

Consider the gas-phase reaction between \(\mathrm{A}_{2}\) (green) and \(\mathrm{B}_{2}\) (red) to form AB at 298 \(K\): \(\mathrm{A}_{2}(g)+\mathrm{B}_{2}(g) \rightleftharpoons 2 \mathrm{AB}(g) \quad \Delta G^{\circ}=-3.4 \ \mathrm{kJ} / \mathrm{mol}\) (1) Which of the following reaction mixtures is at equilibrium? (2) Which of the following reaction mixtures has a negative \(\Delta G\) value? (3) Which of the following reaction mixtures has a positive \(\Delta G\) value? The partial pressures of the gases in each frame are equal to the number of \(\mathrm{A}_{2}\), \(\mathrm{B}_{2}\), and AB molecules times 0.10 atm. Round your answers to two significant figures.

The \(K_{P}\) for the reaction \(\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}\) is \(2.4 \times 10^{-3}\) at \(720^{\circ} \mathrm{C}\). What is the minimum partial pressure of \(\mathrm{N}_{2}\), required for the reaction to be spontaneous in the forward direction if the partial pressures of \(\mathrm{H}_{2}\), and \(\mathrm{NH}_{3}\); are 1.52 atm and \(2.1 \times 10^{-2}\) atm, respectively?

The table shown here lists the ion-product constant \(\left(K_{\mathrm{w}}\right)\) of water at several temperatures. Determine graphically the \(\Delta H^{\circ}\) for the ionization of water. \(\begin{array}{cccccc} K_{\mathrm{w}} & 0.113 \times & 0.292 \times & 1.008 \times & 2.917 \times & 5.474 \times \\ & 10^{-14} & 10^{-14} & 10^{-14} & 10^{-14} & 10^{-14} \\ t\left({ }^{\circ} \mathrm{C}\right) & 0 & 10 & 25 & 40 & 50 \end{array}\) (Hint: See Problem 14.118.)

The reaction \(\mathrm{NH}_{3}(g)+\mathrm{HCl}(g) \longrightarrow \mathrm{NH}_{4} \mathrm{Cl}(s)\) is spontaneous at room temperature (see Figure 5.20). Estimate the temperature at which the reaction is no longer spontaneous under standard conditions.

The boiling point of diethyl ether is \(34.6^{\circ} \mathrm{C}\). Estimate (a) its molar heat of vaporization and (b) its vapor pressure at \(20^{\circ} \mathrm{C}\). (Hint: See Problems 17.48 and 17.51.)

Nicotine is the compound in tobacco responsible for addiction to smoking. While most of the nicotine in tobacco exists in the neutral form, roughly 90 percent of the nicotine in the bloodstream is protonated, as represented in the following chemical equation. Estimate \(\Delta G^{\circ}\) for the reaction.

Estimate \(\Delta S\) for the process depicted in Figure 17.1(a) if the apparatus contained 20 molecules in the flask on the left in the initial distribution, and each flask contained 10 molecules in the final distribution. Useful information: The number of ways to distribute \(n\) objects between two bins such that \(r\) particles are in one bin is called the number of combinations (C) and is given by the equation \(C(n, r)=\frac{n !}{r !(n-r) !}\) where n! ("n factorial") = \(1 \times 2 \times 3 \times \cdots \times n\), and 0! is defined to be 1.

At what point in the series \(\mathrm{H}-\mathrm{O}_{n}-\mathrm{H}(g)\) (n = 1, 2, 3,...) does formation of the compound from the elements \(\mathrm{H}_{2}(g)\) and \(\mathrm{O}_{2}(g)\) become nonspontaneous?