Question

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 at room temperature: $$ \mathrm{Cd}\left(\mathrm{NO}_{3}\right)_{2} \cdot 4 \mathrm{H}_{2} \mathrm{O}(s) \longrightarrow \mathrm{Cd}\left(\mathrm{NO}_{3}\right)_{2}(s)+4 \mathrm{H}_{2} \mathrm{O}(g) $$ This process is spontaneous and \(\Delta H^{\circ}\) is positive at room temperature. (a) What is the sign of \(\Delta S^{\circ}\) at room temperature? (b) If the hydrated compound is placed in a large, closed vessel that already contains a large amount of water vapor, does \(\Delta S^{\circ}\) change for this reaction at room temperature?

Short answer

(a) \(\Delta S^{\circ}\) is positive at room temperature. (b) No, placing the hydrated compound in a large, closed vessel already containing a large amount of water vapor does not change the intrinsic entropy change (\(\Delta S^{\circ}\)) of the reaction at room temperature.

Step-by-step solution

Calculate change in entropy \(\Delta S^{\circ}\) and its sign at room temperature

The relationship between Gibbs free energy (\(\Delta G^{\circ}\)), enthalpy (\(\Delta H^{\circ}\)), and entropy (\(\Delta S^{\circ}\)) is given by the equation: \[\Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ}\] Since the process is spontaneous, the change in Gibbs free energy (\(\Delta G^{\circ}\)) must be negative. We are given that the change in enthalpy (\(\Delta H^{\circ}\)) is positive for this process. The only way to satisfy a negative value for \(\Delta G^{\circ}\) is if the term \(T\Delta S^{\circ}\) is greater in magnitude than \(\Delta H^{\circ}\) and therefore has a positive value: \[\Delta H^{\circ} - T\Delta S^{\circ} < 0 \Rightarrow T\Delta S^{\circ} > \Delta H^{\circ}\] Since room temperature is a positive value, the change in entropy (\(\Delta S^{\circ}\)) must also be positive to satisfy the inequality. Thus, we can conclude that: (a) \(\Delta S^{\circ}\) is positive at room temperature.

Effect of additional water vapor on \(\Delta S^{\circ}\) at room temperature

Placing the hydrated compound in a large, closed vessel already containing a large amount of water vapor will not change the intrinsic change in entropy (\(\Delta S^{\circ}\)) as \(\Delta S^{\circ}\) depends on the reaction itself and not on the initial conditions of the system. However, it can potentially affect partial pressures of the water below and above the hydrate, which in turn can affect the spontaneity of the process. Nevertheless, the intrinsic change in entropy at room temperature for the reaction will remain the same. Therefore, we can conclude that: (b) No, placing the hydrated compound in a large, closed vessel already containing a large amount of water vapor does not change the intrinsic entropy change (\(\Delta S^{\circ}\)) of the reaction at room temperature.

Key concepts

Spontaneity

The spontaneity of a reaction refers to whether a process will occur naturally without any external influence. A spontaneous process is one that can proceed on its own. An important component to determine spontaneity is the Gibbs free energy change, represented as \( \Delta G^{\circ} \).
- If \( \Delta G^{\circ} \) is negative, the reaction is spontaneous.- If \( \Delta G^{\circ} \) is positive, the reaction is non-spontaneous.In the example of the crystalline hydrate losing water, the process is noted as spontaneous, meaning it releases water vapor without requiring additional energy or changes. This natural shift, despite having a positive enthalpy change, is still favorable because the entropy change \( \Delta S^{\circ} \) is significant enough to drive the reaction.forward.

Enthalpy

Enthalpy, represented as \( \Delta H^{\circ} \), is a measure of the total heat content of a system. Changes in enthalpy relate to the heat absorbed or released due to a reaction at constant pressure.
- A positive \( \Delta H^{\circ} \) means the reaction absorbs heat, also known as an endothermic reaction.- A negative \( \Delta H^{\circ} \) indicates the release of heat, known as an exothermic reaction.In the crystalline hydrate example, the enthalpy change is positive, signifying that the process of releasing water vapor absorbs heat. Despite this, because the process is spontaneous, another factor—like a significant increase in entropy—compensates for the positive \( \Delta H^{\circ} \). This dissociation highlights that enthalpy alone doesn't determine spontaneity, but must be considered along with entropy and temperature.

Gibbs Free Energy

Gibbs free energy is a crucial thermodynamic quantity, denoted as \( \Delta G^{\circ} \), that dictates the spontaneity of a reaction at constant temperature and pressure. It combines enthalpy, temperature, and entropy through the formula:\[ \Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ} \]
The equation shows how temperature and entropy impact the Gibbs free energy and ultimately the spontaneity:- Lower \( \Delta G^{\circ} \) (i.e., negative) suggests that a reaction will occur spontaneously.In the example of the crystalline hydrate, even with a positive \( \Delta H^{\circ} \), the process remains spontaneous. This indicates that the \( T\Delta S^{\circ} \) term is sufficiently large in positive value, making \( \Delta G^{\circ} \) negative. This interplay highlights how reactions that absorb heat can still proceed naturally if they result in increases in disorder or randomness (entropy). Thus, Gibbs free energy effectively predicts the feasibility of reactions in varying conditions.