a. What is the equilibrium constant for a reaction that is carried out at 25 °C (298 K) with ∆H° = 20 kcal/mol and ∆S° = 5.0 × 10^-2 kcal mol^-1 K^-1?
b. What is the equilibrium constant for the same reaction carried out at 125 °C?

Verified step by step guidance

1. Step 1: Recall the relationship between the Gibbs free energy change (ΔG°) and the equilibrium constant (K). The equation is ΔG° = -RT ln(K), where R is the gas constant (1.987 cal mol⁻¹ K⁻¹), T is the temperature in Kelvin, and K is the equilibrium constant.
2. Step 2: Use the thermodynamic relationship ΔG° = ΔH° - TΔS° to calculate ΔG° at each temperature. For part (a), substitute ΔH° = 20 kcal/mol, ΔS° = 5.0 × 10⁻² kcal mol⁻¹ K⁻¹, and T = 298 K into the equation. For part (b), use T = 125 °C, which must first be converted to Kelvin (T = 125 + 273 = 398 K).
3. Step 3: Convert ΔH° and ΔS° into consistent units if necessary. Since R is given in cal mol⁻¹ K⁻¹, convert ΔH° and ΔS° from kcal to cal by multiplying by 1000 (1 kcal = 1000 cal). This ensures all units are consistent for calculations.
4. Step 4: Calculate ΔG° for each temperature using the equation ΔG° = ΔH° - TΔS°. Substitute the values for ΔH°, T, and ΔS° into the equation for both temperatures (298 K and 398 K).
5. Step 5: Rearrange the equation ΔG° = -RT ln(K) to solve for K. Use the calculated ΔG° values for each temperature and substitute them into the equation ln(K) = -ΔG° / (RT). Finally, exponentiate both sides to solve for K: K = e^(-ΔG° / (RT)).

Key Concepts

Gibbs Free Energy

Gibbs Free Energy (G) is a thermodynamic potential that measures the maximum reversible work obtainable from a thermodynamic system at constant temperature and pressure. The change in Gibbs Free Energy (∆G) is related to the equilibrium constant (K) of a reaction through the equation ∆G = -RT ln(K), where R is the universal gas constant and T is the temperature in Kelvin. A negative ∆G indicates a spontaneous reaction, while a positive ∆G suggests non-spontaneity.

Equilibrium Constant (K)

The equilibrium constant (K) is a dimensionless value that expresses the ratio of the concentrations of products to reactants at equilibrium for a given reaction at a specific temperature. It is derived from the standard Gibbs Free Energy change (∆G°) of the reaction, and it can be calculated using the equation K = e^(-∆G°/RT). The value of K provides insight into the position of equilibrium, indicating whether products or reactants are favored.

Van 't Hoff Equation

The Van 't Hoff equation relates the change in the equilibrium constant (K) of a reaction to the change in temperature (T) and the enthalpy change (∆H°) of the reaction. It is expressed as ln(K2/K1) = -∆H°/R(1/T2 - 1/T1). This equation allows for the calculation of the equilibrium constant at different temperatures, providing a way to understand how temperature influences the position of equilibrium in a chemical reaction.