Carbon monoxide and hydrogen react to form methanol according to
CO(g) + 2 H₂(g) ⇌ CH₃OH(g).
Thermodynamic data for the reaction are:
ΔH° = –90.7 kJ mol⁻¹ ΔS° = –0.236 kJ mol⁻¹ K⁻¹.
A sealed 2.00 L reactor at temperature T initially contains 0.500 mol of CO, 1.20 mol of H₂, and 0.020 mol of CH₃OH, all behaving ideally. Assume that ΔH° and ΔS° are temperature-independent over the range considered. (R = 8.314 J mol⁻¹ K⁻¹.)
(1) Derive an expression for the Gibbs free energy change, ΔG, for the reaction in terms of T and the initial partial pressures of the three gases in the reactor.
(2) Using the initial composition given, obtain an explicit inequality that T must satisfy for the forward reaction to be thermodynamically favored in the reactor.
(3) Calculate the maximum temperature (to the nearest kelvin) at which the forward reaction will be spontaneous under the stated initial conditions.
(4) Discuss qualitatively how increasing the initial amount of CH₃OH while keeping the amounts of CO and H₂ constant would affect the maximum temperature calculated in part (c), and justify your reasoning.