The oxidative cleavage of compound D by molecular oxygen is greatly accelerated by the presence of catalyst E. A detailed mechanism is proposed:
Step 1 E + D ⇌ ED (k1, k−1) (fast equilibrium)
Step 2 ED + O2 → E(O2)D k2 (fast)
Step 3 E(O2)D → E + products k3 (slow)
The reaction products do not react further with E or O2. In a series of experiments, the following observations are made at constant temperature:
• When [O2] is doubled at fixed [D]0 and [E]0, the initial rate doubles.
• When [D]0 is doubled at fixed [O2] and [E]0, the initial rate increases by less than a factor of 2.
• When [E]0 is doubled at fixed [D]0 and [O2], the initial rate nearly doubles.
All concentrations quoted are much smaller than the corresponding gas-phase solubility limits.
(a) Derive a general expression for the initial rate v0 in terms of k1, k−1, k2, k3, and the initial concentrations of D, O2, and E. Your final expression must not contain intermediate concentrations.
(b) Using your expression from part (a), obtain an approximate rate law that is consistent with all three experimental observations. State clearly any mathematical assumptions you invoke.
(c) Under the assumptions of part (b), determine the overall reaction order with respect to each reactant (D, O2) and the catalyst E.
(d) Propose a quantitative experiment (including the type of plot you would generate) that could be used to evaluate the numerical value of k3 independently of k1, k−1, and k2. Explain how the slope or intercept of the plot relates to k3. (Do not perform the actual data analysis.)