Which of the reactions (a)–(d) in Problem 9.132 are spontaneous at all temperatures, which are nonspontaneous at all temperatures, and which have an equilibrium temperature?

Verified step by step guidance

1. Step 1: Recall the Gibbs free energy equation, \( \Delta G = \Delta H - T\Delta S \), where \( \Delta G \) is the change in Gibbs free energy, \( \Delta H \) is the change in enthalpy, \( T \) is the temperature in Kelvin, and \( \Delta S \) is the change in entropy.
2. Step 2: Understand that a reaction is spontaneous at all temperatures if \( \Delta H < 0 \) and \( \Delta S > 0 \), because \( \Delta G \) will be negative regardless of the temperature.
3. Step 3: Recognize that a reaction is nonspontaneous at all temperatures if \( \Delta H > 0 \) and \( \Delta S < 0 \), because \( \Delta G \) will be positive regardless of the temperature.
4. Step 4: Identify that a reaction has an equilibrium temperature if \( \Delta H \) and \( \Delta S \) have the same sign. Calculate the equilibrium temperature using \( T = \frac{\Delta H}{\Delta S} \) when \( \Delta G = 0 \).
5. Step 5: Apply these criteria to each reaction (a)–(d) from Problem 9.132 to determine their spontaneity at different temperatures.