In a research laboratory, compound A reacts with B to form product P. The uncatalyzed and catalyzed pathways occur simultaneously under identical conditions.
Uncatalyzed pathway (single step): A + B → P (rate constant ku)
Catalyzed pathway (involving catalyst C):
Step 1 C + A → AC k1 (slow)
Step 2 AC + B → C + P k2 (fast)
No other steps contribute measurably to the formation of P. At the start of each experiment, [A]0≫[C]0 and [B]0 is comparable to [A]0.
(a) Derive an algebraic expression for the initial rate v0 that accounts for both the uncatalyzed and catalyzed pathways. Express v0 in terms of ku, k1, k2, and the initial concentrations [A]0, [B]0, and [C]0.
(b) Under the laboratory conditions k1≪k2 and k1[A]0≫ku. Simplify your expression from part (a) to a form suitable for direct comparison with experimental initial-rate data.
(c) Using the simplified rate law from part (b), predict how a ten-fold increase in the initial catalyst concentration would affect the observed initial rate, assuming all other initial concentrations and rate constants remain unchanged.
(d) The chemist wishes to determine ku experimentally. Describe a strategy, based on your work in parts (a)–(c), for designing an experiment that yields ku with minimal interference from the catalyzed pathway.