Question 16.9: CALCULATING AN EQUILIBRIUM CONSTANT FROM ΔG° FOR THE REACTIO...
CALCULATING AN EQUILIBRIUM CONSTANT FROM ΔG° FOR THE REACTION
Methanol (CH_{3}OH), an important alcohol used in the manufacture of adhesives, fibers, and plastics, is synthesized industrially by the reaction
CO(g) + 2 H_{2}(g) \rightleftharpoons CH_{3}OH(g)
Use the thermodynamic data in Appendix B to calculate the equilibrium constant for this reaction at 25 °C.
STRATEGY
First calculate ΔG° for the reaction from the tabulated values of for reactants and products. Then use the equation ΔG° = – RT ln K to find the value of the equilibrium constant.
ΔG° = ΔG°_{f}(CH_{3}OH) – [ΔG°_{f}(CO) + 2 ΔG°_{f}(H_{2})]
= (1 mol)(-162.3 kJ/mol) – [(1 mol)(-137.2 kJ/mol) + (2 mol)(0 kJ/mol)]
ΔG° = -25.1 kJ
Solving the equation ΔG° = – RT ln K
for ln K gives ln K = \frac{- ΔG°}{RT} = \frac{-(-25.1 × 10^{3}J/mol)}{[8.314 J/(K · mol)](298 K)} = 10.1
Therefore, K = K_{p} = antiln 10.1 = e^{10.1} = 2 × 10^{4}
The equilibrium constant obtained by this procedure is K_{p} because the reactants and products are gases and their standard states are defined in terms of pressure. If we want the value of K_{c}, we must calculate it from the relation K_{p} = K_{c}(RT)^{Δn} (Section 13.3), where R must be expressed in the proper units [R = 0.082 06 (L · atm)/(K · mol)].
BALLPARK CHECK
ΔG° is negative, so the equilibrium mixture should be mainly products (K > 1), in agreement with the solution.