Problem 75 - Chemistry and Chemical Reactivity

Question

Decide whether each of the following statements is true or false. If false, rewrite it to make it true. (a) The entropy of a substance increases on going from the liquid to the vapor state at any temperature. (b) An exothermic reaction will always be spontaneous. (c) Reactions with a positive \(\Delta H_{\mathrm{rxn}}^{\circ}\) and a positive \(\Delta S_{\mathrm{rxm}}^{\circ}\) can never be product-favored. (d) If \(\Delta G_{\text {ran }}^{\circ}\) for a reaction is negative, the reaction will have an equilibrium constant greater than 1

Step-by-Step Solution

Answer

(a) True, (b) False, (c) False, (d) True. Corrected versions for (b) and (c) provided.

Step 1: Analyze Statement (a)

The statement claims that the entropy of a substance increases when transitioning from the liquid to vapor state, which reflects an increase in disorder. As substances vaporize, molecules move more freely, increasing randomness (entropy). Thus, the statement is true.

Step 2: Analyze Statement (b)

The statement suggests that an exothermic reaction (releases heat) is always spontaneous. While exothermic reactions frequently tend to be spontaneous, they are not necessarily always so, especially if the entropy change is unfavorable or if the temperature is very low. The statement is false. Corrected statement: An exothermic reaction is often spontaneous but not always, depending on the entropy change and temperature.

Step 3: Analyze Statement (c)

The statement asserts that reactions with both positive \(\Delta H^{\circ}\) (enthalpy change) and positive \(\Delta S^{\circ}\) (entropy change) can never be product-favored. However, such reactions can be favorable at high temperatures where the entropy term \(T\Delta S^{\circ}\) dominates and \(\Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ}\) becomes negative. The statement is false. Corrected statement: Reactions with positive \(\Delta H^{\circ}\) and positive \(\Delta S^{\circ}\) can be product-favored at high temperatures.

Step 4: Analyze Statement (d)

The statement claims that if \(\Delta G^{\circ}\) for a reaction is negative, the equilibrium constant \(K\) will be greater than 1. Since \(\Delta G^{\circ} = -RT\ln K\), a negative \(\Delta G^{\circ}\) implies \(\ln K > 0\), meaning \(K > 1\). Thus, the statement is true.

Key Concepts

Entropy

Entropy is a measure of how disordered or random a system is. When a substance transitions from a liquid to a vapor, its entropy increases significantly. This is because the vapor state allows molecules to move more freely and occupy more space compared to the structured arrangement in a liquid state.
This change to a higher entropy state is a natural tendency as systems evolve towards maximum randomness. This is a fundamental principle of thermodynamics, where spontaneous processes often move towards increasing entropy. For example:

• In gases, the molecules are distributed widely and randomly, contributing to a high entropy state.
• The melting of ice to form water is another case where entropy increases as solid structures break to form more disordered liquids.

Understanding entropy is crucial when evaluating chemical reactions and their spontaneity, as this concept helps predict whether a reaction will take place under given conditions.

Exothermic Reactions

Exothermic reactions are processes that release heat into their surroundings. Commonly, these reactions are assumed to be spontaneous because they release energy, often resulting in a temperature increase.
However, spontaneity in chemistry is not only determined by the energy release. It also depends on the change in entropy and the temperature at which the reaction occurs. While exothermic reactions favor spontaneity, it is crucial to note:

• Reactions may not be spontaneous if they decrease entropy significantly, even if they are exothermic.
• The temperature can also play a role — some reactions might not be spontaneous at very low temperatures despite being exothermic.

This highlights the importance of considering all thermodynamic parameters when predicting the spontaneity of a reaction, not just whether it is exothermic or endothermic.

Product-Favored Reactions

A reaction being product-favored means that products are predominantly formed when the reaction reaches equilibrium. This is often interpreted through the sign of Gibbs free energy change ( \( \Delta G^{\circ} \)) and the value of the equilibrium constant ( \( K \)). A reaction with a positive enthalpy change ( \( \Delta H^{\circ} \)) and positive entropy change ( \( \Delta S^{\circ} \)) can still be product-favored at high temperatures.
Consider the equation for Gibbs free energy: \( \Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ} \). At higher temperatures, the entropy term (\( T\Delta S^{\circ} \)) can outweigh the positive enthalpy change, resulting in a negative \( \Delta G^{\circ} \), thus favoring product formation.
Therefore, a positive enthalpy change does not preclude a reaction from being product-favored, especially if the system's entropy significantly increases and the reaction occurs at a sufficiently high temperature.

Gibbs Free Energy

Gibbs free energy (\( \Delta G \)) is a thermodynamic potential that helps predict whether a process will proceed spontaneously at constant temperature and pressure. A negative \( \Delta G \) indicates that a reaction is spontaneous and will go forward towards forming products.One of the significant relationships involving Gibbs free energy is its connection with the equilibrium constant ( \( K \)):

• \( \Delta G^{\circ} = -RT\ln K \) , where \( R \) is the universal gas constant and \( T \) is the temperature in Kelvin.

For a negative \( \Delta G^{\circ} \), the logarithmic term results in a positive value for \( K \), meaning \( K > 1 \). This implies that at equilibrium, the concentration of products is greater than that of reactants. Thus, the reaction favors product formation.
Understanding this fundamental relationship helps students predict the behavior of chemical reactions under different conditions, emphasizing the profound importance of Gibbs free energy in chemical thermodynamics.