Hydrolysis of a hypothetical ester X−CO2CH3 in aqueous acidic solution is believed to occur by the following pathway:
Step 1 H3O++X−CO2CH3⇌HX−CO2CH3++H2O (fast)
Step 2 HX−CO2CH3+→X−CO2H+CH3+ (slow)
Step 3 CH3++H2O→CH4+H3O+ (fast)
Laboratory studies give an observed rate law of
rate=k[X−CO2CH3][H3O+].
Answer the following questions about the kinetic behavior of this reaction.
(a) Derive the rate law predicted by the mechanism in terms of only X−CO2CH3 and H3O+, showing the logical steps (no algebraic equilibrium expressions are required—state proportionalities at each stage).
(b) Explain why CH3+, although formed during the reaction, does not appear in the experimentally observed rate law.
(c) Suppose an inert gas is added to the sealed reaction vessel at constant total pressure, thereby increasing the total volume. Predict qualitatively whether the reaction rate increases, decreases, or remains unchanged, and support your answer with mechanistic reasoning involving concentration changes of the relevant species.
(d) Discuss how a very large equilibrium constant for Step 1 (favoring the right-hand side) would alter the apparent reaction order with respect to H3O+. Provide an explanation rooted in the elementary steps.