Question

Which of the following must be independent of temperature when properly applying the van't Hoff equation? a. \(K_{e q}\) b. \(\Delta S^{\circ}\) c. \(1 / T\) d. \(\mathrm{pH}\)

Short answer

\(\Delta S^{\circ}\) is independent of temperature in the van't Hoff equation.

Step-by-step solution

Understand the van't Hoff equation

The van't Hoff equation is given by \(\ln K_{eq} = -\frac{\Delta H^{\circ}}{R} \left(\frac{1}{T}\right) + \frac{\Delta S^{\circ}}{R}\). This equation relates the equilibrium constant \(K_{eq}\) to the standard enthalpy change \(\Delta H^{\circ}\), the standard entropy change \(\Delta S^{\circ}\), the gas constant \(R\), and the temperature \(T\) in Kelvin.

Identify what the van't Hoff equation depends on

The expression \(\ln K_{eq}\) depends on \(T\), \(\Delta H^{\circ}\), and \(\Delta S^{\circ}\). Specifically, \(K_{eq}\) changes with temperature, as it affects the \(1/T\) term, and \(\Delta S^{\circ}\) represents a constant in the equation, independent of temperature.

Analyze each option

- \(K_{eq}\) must change with temperature to maintain equilibrium.- \(\Delta S^{\circ}\) is a parameter representing a constant for a given reaction, and does not change with temperature.- \(1/T\) is a variable that directly depends on temperature.- \(\mathrm{pH}\) can depend on temperature because it changes with the equilibrium position of ions, but it's not directly related to the van't Hoff equation itself.

Determine the correct answer

Only \(\Delta S^{\circ}\) remains unchanged with temperature variation as per the van't Hoff equation. This makes \(\Delta S^{\circ}\) independent of temperature when properly applying this equation.

Key concepts

Equilibrium constant

The equilibrium constant, denoted as \( K_{eq} \), plays a crucial role in chemical reactions and their thermodynamics. It reflects the ratio of product concentrations to reactant concentrations at equilibrium, raised to the power of their stoichiometric coefficients. The value of \( K_{eq} \) helps predict the direction of a reaction under equilibrium conditions. If \( K_{eq} > 1 \), the products are favored, whereas if \( K_{eq} < 1 \), the reactants are favored.
\( K_{eq} \) is influenced by temperature, as described by the van’t Hoff equation:

• \( \ln K_{eq} = -\frac{\Delta H^{\circ}}{R} \left(\frac{1}{T}\right) + \frac{\Delta S^{\circ}}{R} \)

Here, \( T \) represents temperature in Kelvin, \( R \) is the gas constant, \( \Delta H^{\circ} \) is the standard enthalpy change, and \( \Delta S^{\circ} \) the standard entropy change. Notice that as temperature changes, \( K_{eq} \) will adjust in response to maintain equilibrium. This makes \( K_{eq} \) temperature dependent, distinguishing it from other constants like \( \Delta S^{\circ} \) which remain unaffected by temperature changes.

Standard enthalpy change

Standard enthalpy change, denoted \( \Delta H^{\circ} \), is the heat absorbed or released during a reaction at standard conditions (1 atm pressure and 298 K). It tells us whether a reaction is endothermic (absorbs heat, \( \Delta H^{\circ} > 0 \)) or exothermic (releases heat, \( \Delta H^{\circ} < 0 \)).
In the van’t Hoff equation:

• \( \ln K_{eq} = -\frac{\Delta H^{\circ}}{R} \left(\frac{1}{T}\right) + \frac{\Delta S^{\circ}}{R} \)

The term \(-\frac{\Delta H^{\circ}}{R} \left(\frac{1}{T}\right)\) indicates how changes in temperature affect the equilibrium position. This equation reveals that the equilibrium constant is temperature-dependent because the value \( \Delta H^{\circ} \) helps determine how \( K_{eq} \) shifts with temperature variation. Although temperature affects \( K_{eq} \), \( \Delta H^{\circ} \) remains constant for a given reaction under standard conditions. Understanding \( \Delta H^{\circ} \) is key to predicting how the reaction shifts to restore equilibrium when the temperature changes.

Standard entropy change

Standard entropy change, symbolized by \( \Delta S^{\circ} \), measures the change in disorder or randomness resulting from a chemical reaction under standard conditions. Entropy gives insight into the spontaneity of a reaction, with a positive \( \Delta S^{\circ} \) suggesting increased randomness, making a reaction more likely to occur spontaneously.
In the van’t Hoff equation, the term \( \frac{\Delta S^{\circ}}{R} \) acts as a constant for any given reaction. Unlike \( K_{eq} \) and temperature, \( \Delta S^{\circ} \) is not influenced by changes in temperature. This implies that the intrinsic distribution of energy levels (or configuration) of the reactants and products remains unchanged as long as the reaction conditions remain consistent.
\( \Delta S^{\circ} \) helps to identify whether a reaction is favorable in terms of entropy. When combined with \( \Delta H^{\circ} \) in determining \( G \), the Gibbs free energy, it aids in predicting the overall spontaneity of a process. Hence, in the context of the van’t Hoff equation, \( \Delta S^{\circ} \)'s constancy provides critical insight into the equilibrium constant's temperature dependence.