Using data from Appendix C, write the equilibrium-constant expression and calculate the value of the equilibrium constant and the free-energy change for these reactions at 298 K: (b) 2 HBr(g) + Cl2(g) ⇌ 2 HCl(g) + Br2(g) (c) 2 SO2(g) + O2(g) ⇌ 2 SO3(g)

Verified step by step guidance

1. Step 1: Write the equilibrium-constant expression for each reaction. For a general reaction aA + bB ⇌ cC + dD, the equilibrium constant expression Kc is given by \( K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b} \). Apply this to each reaction.
2. Step 2: For reaction (b) 2 HBr(g) + Cl2(g) ⇌ 2 HCl(g) + Br2(g), the equilibrium-constant expression is \( K_c = \frac{[HCl]^2[Br_2]}{[HBr]^2[Cl_2]} \).
3. Step 3: For reaction (c) 2 SO2(g) + O2(g) ⇌ 2 SO3(g), the equilibrium-constant expression is \( K_c = \frac{[SO_3]^2}{[SO_2]^2[O_2]} \).
4. Step 4: Use the standard Gibbs free energy change equation \( \Delta G^\circ = -RT \ln K \) to relate the equilibrium constant to the free energy change. Here, R is the gas constant (8.314 J/mol·K) and T is the temperature in Kelvin (298 K).
5. Step 5: Calculate \( \Delta G^\circ \) for each reaction using \( \Delta G^\circ = \sum \Delta G^\circ_{\text{products}} - \sum \Delta G^\circ_{\text{reactants}} \) with values from Appendix C, then solve for K using \( K = e^{-\Delta G^\circ / RT} \).

Key Concepts

Equilibrium Constant (K)

The equilibrium constant (K) is a numerical 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 balanced chemical equation and is calculated using the formula K = [products]^[coefficients] / [reactants]^[coefficients]. A large K value indicates that products are favored at equilibrium, while a small K value suggests that reactants are favored.

Gibbs Free Energy (ΔG)

Gibbs free energy (ΔG) is a thermodynamic quantity that indicates the spontaneity of a reaction at constant temperature and pressure. It is calculated using the equation ΔG = ΔG° + RT ln(Q), where ΔG° is the standard free energy change, R is the universal gas constant, T is the temperature in Kelvin, and Q is the reaction quotient. A negative ΔG indicates a spontaneous reaction, while a positive ΔG suggests non-spontaneity.

Reaction Quotient (Q)

The reaction quotient (Q) is a measure of the relative concentrations of products and reactants at any point in a reaction, not necessarily at equilibrium. It is calculated similarly to the equilibrium constant but uses the current concentrations instead of equilibrium concentrations. Comparing Q to K helps predict the direction in which a reaction will proceed to reach equilibrium: if Q < K, the reaction will shift to the right (toward products), and if Q > K, it will shift to the left (toward reactants).