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

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.

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.

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.

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 \( \Delta G = 0 \).

Step 5: Apply these criteria to each reaction (a)–(d) from Problem 9.132 to determine their spontaneity at different temperatures.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Spontaneity of Reactions

The spontaneity of a reaction refers to whether a reaction can occur without external intervention. A reaction is spontaneous if it leads to an increase in the entropy of the universe, which is often assessed using Gibbs free energy. If the change in Gibbs free energy (ΔG) is negative, the reaction is spontaneous; if positive, it is nonspontaneous.

Gibbs Free Energy

Gibbs free energy (G) is a thermodynamic potential that helps predict the direction of chemical reactions. It combines enthalpy (H) and entropy (S) into the equation ΔG = ΔH - TΔS, where T is the temperature in Kelvin. A negative ΔG indicates a spontaneous process, while a positive ΔG suggests nonspontaneity. The equilibrium temperature occurs when ΔG equals zero.

Equilibrium Temperature

The equilibrium temperature is the specific temperature at which a reaction shifts from being spontaneous to nonspontaneous, or vice versa. At this temperature, the Gibbs free energy change (ΔG) is zero, indicating that the forward and reverse reactions occur at the same rate. This concept is crucial for understanding how temperature influences reaction spontaneity.