Chapter 19

Consider a reaction \(\mathrm{A}_{2}(g)+\mathrm{B}_{2}(g) \rightleftharpoons 2 \mathrm{AB}(g)\), with atoms of A shown in red and atoms of B shown in blue. (a) If \(K_{c}=1\), which system is at equilibrium? (b) What is the sign of \(\Delta G\) for any process in which the contents of a reaction vessel move to equilibrium? (c) Rank the boxes in order of increasing magnitude of \(\Delta G\) for the reaction. [Sections \(19.5\) and \(19.7]\)

Which of the following processes are spontaneous, and which are nonspontaneous: (a) the ripening of a banana, (b) dissolution of sugar in a cup of hot coffee, (c) the reaction of nitrogen atoms to form \(\mathrm{N}_{2}\) molecules at \(25^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\), (d) lightning, (e) formation of \(\mathrm{CH}_{4}\) and \(\mathrm{O}_{2}\) molecules from \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\) at room temperature and 1 atm of pressure?

Which of the following processes are spontaneous: (a) the melting of ice cubes at \(10^{\circ} \mathrm{C}\) and 1 atm pressure; (b) separating a mixture of \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\) into two separate samples, one that is pure \(\mathrm{N}_{2}\) and one that is pure \(\mathrm{O}_{2}\); (c) alignment of iron filings in a magnetic field; (d) the reaction of sodium metal with chlorine gas to form sodium chloride; (e) the dissolution of \(\mathrm{HCl}(g)\) in water to form concentrated hydrochloric acid?

(a) Give two examples of endothermic processes that are spontaneous. (b) Give an example of a process that is spontaneous at one temperature but nonspontaneous at a different temperature.

The crystalline hydrate \(\mathrm{Cd}\left(\mathrm{NO}_{3}\right)_{2} \cdot 4 \mathrm{H}_{2} \mathrm{O}(s)\) loses water when placed in a large, closed, dry vessel: \(\mathrm{Cd}\left(\mathrm{NO}_{3}\right)_{2} \cdot 4 \mathrm{H}_{2} \mathrm{O}(s) \longrightarrow \mathrm{Cd}\left(\mathrm{NO}_{3}\right)_{2}(s)+4 \mathrm{H}_{2} \mathrm{O}(g)\) This process occurs even though it is endothermic; that is, \(\Delta H\) is positive. Is this process an exception to Bertholet's generalization? Explain.

Consider the vaporization of liquid water to steam at a pressure of 1 atm. (a) Is this process endothermic or exothermic? (b) In what temperature range is it a spontaneous process? (c) In what temperature range is it a nonspontaneous process? (d) At what temperature are the two phases in equilibrium?

The normal freezing point of 1 -propanol \(\left(\mathrm{C}_{3} \mathrm{H}_{3} \mathrm{O}\right)\) is \(-127{ }^{\circ} \mathrm{C}\). (a) Is the freezing of 1 -propanol an endothermic or exothermic process? (b) In what temperature range is the freezing of 1 -propanol a spontaneous process? (c) In what temperature range is it a nonspontaneous process? (d) Is there any temperature at which liquid and solid 1-propanol are in equilibrium? Explain.

(a) What is special about a reversible process? (b) Suppose a reversible process is reversed, restoring the system to its original state. What can be said about the surroundings after the process is reversed? (c) Under what circumstances will the vaporization of water to steam be a reversible process? (d) Are any of the processes that occur in the world around us reversible in nature? Explain.

(a) What is meant by calling a process irreversible? (b) After an irreversible process the system is restored to its original state. What can be said about the condition of the surroundings after the system is restored to its original state? (c) Under what conditions will the condensation of a liquid be an irreversible process?

Consider a process in which an ideal gas changes from state 1 to state 2 in such a way that its temperature changes from \(300 \mathrm{~K}\) to \(200 \mathrm{~K}\). (a) Describe how this change might be carried out while keeping the volume of the gas constant. (b) Describe how it might be carried out while keeping the pressure of the gas constant. (c) Does the change in \(\Delta E\) depend on the particular pathway taken to carry out this change of state? Explain.

A system goes from state 1 to state 2 and back to state 1 . (a) What is the relationship between the value of \(\Delta E\) for going from state 1 to state 2 to that for going from state 2 back to state \(1 ?\) (b) Without further information, can you conclude anything about the amount of heat transferred to the system as it goes from state 1 to state 2 as compared to that upon going from state 2 back to state \(1 ?\) (c) Suppose the changes in state are reversible processes. Can you conclude anything about the work done by the system upon going from state 1 to state 2 as compared to that upon going from state 2 back to state \(1 ?\)

Consider a system consisting of an ice cube. (a) Under what conditions can the ice cube melt reversibly? (b) If the ice cube melts reversibly, is \(\Delta E\) zero for the process? Explain.

Consider what happens when a sample of the explosive TNT (Section 8.8: "Chemistry Put to Work: Explosives and Alfred Nobel") is detonated. (a) Is the detonation a spontaneous process? (b) What is the sign of \(q\) for this process? (c) Can you determine whether \(w\) is positive, negative, or zero for the process? Explain. (d) Can you determine the sign of \(\Delta E\) for the process? Explain.

(a) How can we calculate \(\Delta S\) foran isothermal process? (b) Does \(\Delta S\) for a process depend on the path taken from the initial to the final state of the system? Explain.

Suppose we vaporize a mole of liquid water at \(25^{\circ} \mathrm{C}\) and another mole of water at \(100{ }^{\circ} \mathrm{C}\). (a) Assuming that the enthalpy of vaporization of water does not change much between \(25^{\circ} \mathrm{C}\) and \(100^{\circ} \mathrm{C}\), which process involves the larger change in entropy? (b) Does the entropy change in either process depend on whether we carry out the process reversibly or not? Explain.

Thenormal boiling point of methanol \(\left(\mathrm{CH}_{3} \mathrm{OH}\right)\) is \(64.7{ }^{\circ} \mathrm{C}\), and its molar enthalpy of vaporization is \(\Delta H_{\text {vap }}=\) \(71.8 \mathrm{~kJ} / \mathrm{mol} .\) (a) When \(\mathrm{CH}_{3} \mathrm{OH}(l)\) boils at its normal boiling point, does its entropy increase or decrease? (b) Calculate the value of \(\Delta S\) when \(1.00\) mol of \(\mathrm{CH}_{3} \mathrm{OH}(l)\) is vaporized at \(64.7{ }^{\circ} \mathrm{C}\).

The element cesium (Cs) freezes at \(28.4^{\circ} \mathrm{C}\), and its molar enthalpy of fusion is \(\Delta H_{\text {fus }}=2.09 \mathrm{~kJ} / \mathrm{mol}\). (a) When molten cesium solidifies to \(\mathrm{Cs}(s)\) at its normal melting point, is \(\Delta S\) positive or negative? (b) Calculate the value of \(\Delta S\) when \(15.0 \mathrm{~g}\) of \(\mathrm{Cs}(l)\) solidifies at \(28.4^{\circ} \mathrm{C}\).

(a) Express the second law of thermodynamics in words. (b) If the entropy of the system increases during a reversible process, what can you say about the entropy change of the surroundings? (c) In a certain spontaneous process the system undergoes an entropy change, \(\Delta S=42 \mathrm{~J} / \mathrm{K} .\) What can you conclude about \(\Delta S_{\text {surr }} ?\)

(a) Express the second law of thermodynamics as a mathematical equation. (b) In a particular spontaneous process the entropy of the system decreases. What can you conclude about the sign and magnitude of \(\Delta S_{\text {surr }} ?\) (c) During a certain reversible process, the surroundings undergo an entropy change, \(\Delta S_{\text {surr }}=-78 \mathrm{~J} / \mathrm{K}\). What is the entropy change of the system for this process?

The volume of \(0.100 \mathrm{~mol}\) of helium gas at \(27^{\circ} \mathrm{C}\) is increased isothermally from \(2.00 \mathrm{~L}\) to \(5.00 \mathrm{~L}\). Assuming the gas to be ideal, calculate the entropy change for the process.

The pressure on \(0.850\) mol of neon gas is increased from \(1.25\) atm to \(2.75\) atm at \(100^{\circ} \mathrm{C}\). Assuming the gas to be ideal, calculate \(\Delta S\) for this process.

How would each of the following changes affect the number of microstates available to a system: (a) increase in temperature, (b) decrease in volume, (c) change of state from liquid to gas?

(a) What do you expect for the sign of \(\Delta S\) in a chemical reaction in which two moles of gaseous reactants are converted to three moles of gaseous products? (b) For which of the processes in Exercise \(19.9\) does the entropy of the system increase?

How does the entropy of the system change when (a) a solid melts, (b) a gas liquefies, (c) a solid sublimes?

How does the entropy of the system change when (a) the temperature of the system increases, (b) the volume of a gas increases, (c) equal volumes of ethanol and water are mixed to form a solution.

(a) State the third law of thermodynamics. (b) Distinguish between translational motion, vibrational motion, and rotational motion of a molecule. (c) Illustrate these three kinds of motion with sketches for the HCl molecule.

(a) The energy of a gas is increased by heating it. Using \(\mathrm{CO}_{2}\) as an example, illustrate the different ways in which additional energy can be distributed among the molecules of the gas. (b) You are told that the number of microstates for a system increases. What does this tell you about the entropy of the system?

Propanol \(\left(\mathrm{C}_{3} \mathrm{H}_{7} \mathrm{OH}\right)\) melts at \(-126.5^{\circ} \mathrm{C}\) and boils at \(97.4^{\circ} \mathrm{C}\). Draw a qualitative sketch of how the entropy changes as propanol vapor at \(150^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) is cooled to solid propanol at \(-150^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\).

For each of the following pairs, choose the substance with the higher entropy per mole at a given temperature: (a) \(\operatorname{Ar}(l)\) or \(\operatorname{Ar}(g)\), (b) \(\mathrm{He}(g)\) at 3 atm pressure or \(\mathrm{He}(\mathrm{g})\) at \(1.5\) atm pressure, (c) \(1 \mathrm{~mol}\) of \(\mathrm{Ne}(\mathrm{g})\) in \(15.0 \mathrm{~L}\) or 1 mol of \(\mathrm{Ne}(\mathrm{g})\) in \(1.50 \mathrm{~L},(\mathrm{~d}) \mathrm{CO}_{2}(g)\) or \(\mathrm{CO}_{2}(s)\)

Foreach of the following pairs, indicate which substance possesses the larger standard entropy: (a) 1 mol of \(\mathrm{P}_{4}(g)\) at \(300{ }^{\circ} \mathrm{C}, 0.01 \mathrm{~atm}\), or \(1 \mathrm{~mol}\) of \(\mathrm{As}_{4}(g)\) at \(300^{\circ} \mathrm{C}, 0.01 \mathrm{~atm} ;\) (b) \(1 \mathrm{~mol}\) of \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) at \(100^{\circ} \mathrm{C}, 1 \mathrm{~atm}\), or \(1 \mathrm{~mol}\) of \(\mathrm{H}_{2} \mathrm{O}(l)\) at \(100^{\circ} \mathrm{C}, 1 \mathrm{~atm} ;\) (c) \(0.5 \mathrm{~mol}\) of \(\mathrm{N}_{2}(\mathrm{~g})\) at \(298 \mathrm{~K}, 20\) - \(\mathrm{L}\) volume, or \(0.5 \mathrm{~mol} \mathrm{CH}_{4}(g)\) at \(298 \mathrm{~K}, 20\) -L volume; (d) \(100 \mathrm{~g}\), \(\mathrm{Na}_{2} \mathrm{SO}_{4}(s)\) at \(30^{\circ} \mathrm{C}\) or \(100 \mathrm{~g} \mathrm{Na}_{2} \mathrm{SO}_{4}(a q)\) at \(30^{\circ} \mathrm{C}\).

Predict the sign of the entropy change of the system for each of the following reactions: (a) \(2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{SO}_{3}(g)\) (b) \(\mathrm{Ba}(\mathrm{OH})_{2}(s) \stackrel{\mathrm{L}}{\longrightarrow} \mathrm{BaO}(s)+\mathrm{H}_{2} \mathrm{O}(g)\) (c) \(\mathrm{CO}(\mathrm{g})+2 \mathrm{H}_{2}(\mathrm{~g}) \longrightarrow \mathrm{CH}_{3} \mathrm{OH}(l)\) (d) \(\mathrm{FeCl}_{2}(s)+\mathrm{H}_{2}(g) \longrightarrow \mathrm{Fe}(s)+2 \mathrm{HCl}(g)\)

Predict the sign of \(\Delta S_{\text {sys }}\) for each of the following processes: (a) Gaseous Ar is liquefied at \(80 \mathrm{~K}\). (b) Gaseous \(\mathrm{N}_{2} \mathrm{O}_{4}\) dissociates to form gaseous \(\mathrm{NO}_{2}\). (c) Solid potassium reacts with gaseous \(\mathrm{O}_{2}\) to form solid potassium superoxide, \(\mathrm{KO}_{2}\). (d) Lead bromide precipitates upon mixing \(\mathrm{Pb}\left(\mathrm{NO}_{3}\right)_{2}(a q)\) and \(\mathrm{KBr}(a q)\)

In each of the following pairs, which compound would you expect to have the higher standard molar entropy: (a) \(\mathrm{C}_{2} \mathrm{H}_{2}(g)\) or \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\) (b) \(\mathrm{CO}_{2}(g)\) or \(\mathrm{CO}(g) ?\) Explain.

Use Appendix \(C\) to compare the standard entropies at \(25^{\circ} \mathrm{C}\) for the following pairs of substances: (a) \(\mathrm{Sc}(s)\) and \(\mathrm{Sc}(g) ;\) (b) \(\mathrm{NH}_{3}(g)\) and \(\mathrm{NH}_{3}(a q) ;\) (c) \(1 \mathrm{~mol} \mathrm{P}_{4}(g)\) and \(2 \mathrm{~mol}\) \(\mathrm{P}_{2}(\mathrm{~g}) ;\) (d) C(graphite) and \(\mathrm{C}\) (diamond). In each case explain the difference in the entropy values.

Using Appendix \(C\), compare the standard entropies at \(25^{\circ} \mathrm{C}\) for the following pairs of substances: (a) \(\mathrm{CuO}(s)\) and \(\mathrm{Cu}_{2} \mathrm{O}(\mathrm{s}) ;\) (b) \(1 \mathrm{~mol} \mathrm{~N}_{2} \mathrm{O}_{4}(g)\) and \(2 \mathrm{~mol} \mathrm{NO}_{2}(g) ;\) (c) \(\mathrm{SiO}_{2}(\mathrm{~s})\) and \(\mathrm{CO}_{2}(\mathrm{~g}) ;\) (d) \(\mathrm{CO}(\mathrm{g})\) and \(\mathrm{CO}_{2}(\mathrm{~g})\). Fur each pair, explain the difference in the entropy values.

The standard entropies at \(298 \mathrm{~K}\) for certain of the group \(4 \mathrm{~A}\) elements are as follows: \(\mathrm{C}(s\), diamond \()=2.43 \mathrm{~J} / \mathrm{mol}-\mathrm{K} ; \quad \mathrm{Si}(s)=18.81 \mathrm{~J} / \mathrm{mol}-\mathrm{K} ;\) \(\mathrm{Ge}(s)=31.09 \mathrm{~J} / \mathrm{mol}-\mathrm{K} ; \quad\) and \(\quad \mathrm{Sn}(s)=51.18 \mathrm{~J} / \mathrm{mol}-\mathrm{K}\) All but Sn have the diamond structure. How do you account for the trend in the \(S^{\circ}\) values?

Using \(S^{\circ}\) values from Appendix \(C\), calculate \(\Delta S^{\circ}\) values for the following reactions. In each case account for the sign of \(\Delta S^{n}\). (a) \(\mathrm{C}_{2} \mathrm{H}_{4}(g)+\mathrm{H}_{2}(g) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{6}(g)\) (b) \(\mathrm{N}_{2} \mathrm{O}_{4}(g) \longrightarrow 2 \mathrm{NO}_{2}(g)\) (c) \(\mathrm{Be}(\mathrm{OH})_{2}(s) \longrightarrow \mathrm{BeO}(s)+\mathrm{H}_{2} \mathrm{O}(g)\) (d) \(2 \mathrm{CH}_{3} \mathrm{OH}(g)+3 \mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{CO}_{2}(g)+4 \mathrm{H}_{2} \mathrm{O}(g)\)

Calculate \(\Delta S^{\circ}\) values for the following reactions by using tabulated \(S^{\circ}\) values from Appendix \(C\). In each case explain the sign of \(\Delta S^{\circ}\). (a) \(\mathrm{N}_{2} \mathrm{H}_{4}(g)+\mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{NH}_{3}(g)\) (b) \(\mathrm{K}(s)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{KO}_{2}(s)\) (c) \(\mathrm{Mg}(\mathrm{OH})_{2}(s)+2 \mathrm{HCl}(g) \longrightarrow \mathrm{MgCl}_{2}(s)+2 \mathrm{H}_{2} \mathrm{O}(l)\) (d) \(\mathrm{CO}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{CH}_{3} \mathrm{OH}(g)\)

(a) For a process that occurs at constant temperature, express the change in Gibbs free energy in terms of changes in the enthalpy and entropy of the system. (b) For a certain process that occurs at constant \(T\) and \(P\), the value of \(\Delta G\) is positive. What can you conclude? (c) What is the relationship between \(\Delta G\) for a process and the rate at which it occurs?

(a) What is the meaning of the standard free-energy change, \(\Delta G^{\circ}\), as compared with \(\Delta G\) ? (b) For any process that occurs at constant temperature and pressure, what is the significance of \(\Delta G=0 ?(c)\) For a certain process, \(\Delta G\) is large and negative. Does this mean that the process necessarily occurs rapidly?

For a certain chemical reaction, \(\Delta H^{\circ}=-35.4 \mathrm{~kJ}\) and \(\Delta S^{\circ}=-85.5 \mathrm{~J} / \mathrm{K} .(\mathrm{a}) \mathrm{ls}\) the reaction exothermic or en- dothermic? (b) Does the reaction lead to an increase or decrease in the randomness or disorder of the system? (c) Calculate \(\Delta G^{\circ}\) for the reaction at \(298 \mathrm{~K}\). (d) Is the reaction spontaneous at \(298 \mathrm{~K}\) under standard conditions?

A certain reaction has \(\Delta H^{\circ}=-19.5 \mathrm{~kJ}\) and \(\Delta S^{\circ}=\) \(+42.7 \mathrm{~J} / \mathrm{K} .\) (a) Is the reaction exothermic or endothermic? (b) Does the reaction lead to an increase or decrease in the randomness or disorder of the system? (c) Calculate \(\Delta G^{\circ}\) for the reaction at \(298 \mathrm{~K}\). (d) Is the reaction spontaneous at \(298 \mathrm{~K}\) under standard conditions?

Using data in Appendix \(C\), calculate \(\Delta H^{\circ}, \Delta S^{\circ}\), and \(\Delta G^{\circ}\) at \(298 \mathrm{~K}\) for each of the following reactions. In each case show that \(\Delta G^{\circ}=\Delta H^{\circ}-T \Delta S^{\circ}\). (a) \(\mathrm{H}_{2}(g)+\mathrm{F}_{2}(g) \longrightarrow 2 \mathrm{HF}(g)\) (b) \(\mathrm{C}(\mathrm{s}\), graphite \()+2 \mathrm{Cl}_{2}(\mathrm{~g}) \longrightarrow \mathrm{CCl}_{4}(\mathrm{~g})\) (c) \(2 \mathrm{PCl}_{3}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{POCl}_{3}(\mathrm{~g})\) (d) \(2 \mathrm{CH}_{3} \mathrm{OH}(g)+\mathrm{H}_{2}(g) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{6}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)\)

Use data in Appendix \(\mathrm{C}\) to calculate \(\Delta H^{\circ}, \Delta S^{\circ}\), and \(\Delta G^{\circ}\) at \(25^{\circ} \mathrm{C}\) for each of the following reactions. In each case show that \(\Delta G^{\circ}=\Delta H^{\circ}-T \Delta S^{\circ}\). (a) \(2 \mathrm{Cr}(s)+3 \mathrm{Br}_{2}(g) \longrightarrow 2 \mathrm{CrBr}_{3}(s)\) (b) \(\mathrm{BaCO}_{3}(s) \longrightarrow \mathrm{BaO}(s)+\mathrm{CO}_{2}(g)\) (c) \(2 \mathrm{P}(s)+10 \mathrm{HF}(g) \longrightarrow 2 \mathrm{PF}_{5}(g)+5 \mathrm{H}_{2}(g)\) (d) \(\mathrm{K}(s)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{KO}_{2}(s)\)

Using data from Appendix \(C\), calculate \(\Delta G^{\circ}\) for the following reactions. Indicate whether each reaction is spontaneous under standard conditions. (a) \(2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{SO}_{3}(g)\) (b) \(\mathrm{NO}_{2}(g)+\mathrm{N}_{2} \mathrm{O}(g) \longrightarrow 3 \mathrm{NO}(g)\) (c) \(6 \mathrm{Cl}_{2}(g)+2 \mathrm{Fe}_{2} \mathrm{O}_{3}(s) \longrightarrow 4 \mathrm{FeCl}_{3}(s)+3 \mathrm{O}_{2}(g)\) (d) \(\mathrm{SO}_{2}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{S}(s)+2 \mathrm{H}_{2} \mathrm{O}(g)\)

Using data from Appendix \(C\), calculate the change in Gibbs free energy for each of the following reactions. In each case indicate whether the reaction is spontaneous under standard conditions. (a) \(\mathrm{H}_{2}(\mathrm{~g})+\mathrm{Cl}_{2}(g) \longrightarrow 2 \mathrm{HCl}(g)\) (b) \(\mathrm{MgCl}_{2}(s)+\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{MgO}(s)+2 \mathrm{HCl}(g)\) (c) \(2 \mathrm{NH}_{3}(g) \longrightarrow \mathrm{N}_{2} \mathrm{H}_{4}(g)+\mathrm{H}_{2}(g)\) (d) \(2 \mathrm{NOCl}(g) \longrightarrow 2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g)\)

Cyclohexane \(\left(\mathrm{C}_{6} \mathrm{H}_{12}\right)\) is a liquid hydrocarbon at room temperature. (a) Write a balanced equation for the combustion of \(\mathrm{C}_{6} \mathrm{H}_{12}(l)\) to form \(\mathrm{CO}_{2}(\mathrm{~g})\) and \(\mathrm{H}_{2} \mathrm{O}(l)\). (b) Without using thermochemical data, predict whether \(\Delta G^{\circ}\) for this reaction is more negative or less negative than \(\Delta H^{\circ}\)

Sulfur dioxide reacts with strontium oxide as follows: $$ \mathrm{SO}_{2}(g)+\mathrm{SrO}(s) \longrightarrow \operatorname{SrSO}_{3}(s) $$ (a) Without using thermochemical data, predict whether \(\Delta G^{\circ}\) for this reaction is more negative or less negative than \(\Delta H^{\circ} .\) (b) If you had only standard enthalpy data for this reaction, how would you go about making a rough estimate of the value of \(\Delta G^{\circ}\) at \(298 \mathrm{~K}\), using data from Appendix \(\mathrm{C}\) on other substances?

From the values given for \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\), calculate \(\Delta G^{\circ}\) for each of the following reactions at \(298 \mathrm{~K}\). If the reaction is not spontaneous under standard conditions at \(298 \mathrm{~K}\), at what temperature (if any) would the reaction become spontaneous? (a) \(2 \mathrm{PbS}(s)+3 \mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{PbO}(s)+2 \mathrm{SO}_{2}(g)\) \(\Delta H^{\circ}=-844 \mathrm{~kJ} ; \Delta S^{\circ}=-165 \mathrm{~J} / \mathrm{K}\) (b) \(2 \mathrm{POCl}_{3}(g) \longrightarrow 2 \mathrm{PCl}_{3}(g)+\mathrm{O}_{2}(g)\) $$ \Delta H^{\circ}=572 \mathrm{~kJ} ; \Delta S^{\circ}=179 \mathrm{~J} / \mathrm{K} $$

A particular reaction is spontaneous at \(450 \mathrm{~K}\). The enthalpy change for the reaction is \(+34.5 \mathrm{~kJ}\). What can you conclude about the sign and magnitude of \(\Delta S\) for the reaction?