A catalytic cycle for converting substrate S to product U is shown schematically below.
• Uncatalyzed one-step pathway: activation energy 135 kJ mol⁻¹.
• Catalyzed pathway: two steps of equal activation energy 70 kJ mol⁻¹ each, with an intermediate at +15 kJ mol⁻¹ relative to S.
An inhibitor I competes with S for the catalytic surface. When I is present, the activation energy for each catalytic step is raised by 10 kJ mol⁻¹ for the fraction of catalytic sites occupied by I.
At 310 K the catalyst is 60 % free and 40 % inhibited. Assume identical pre-exponential factors for all pathways and that the inhibited and uninhibited catalytic pathways operate in parallel.
(a) Derive an expression for the effective overall rate constant keff in terms of the individual rate constants for the uninhibited catalytic path (kcat), the inhibited catalytic path (kinh), and the uncatalyzed path (k0), given the fractions of each pathway operating independently.
(b) Using the Arrhenius relationship and the data provided, calculate the ratio keff/k0 at 310 K. Use R=8.314J mol−1K−1.
(c) The inhibitor concentration is doubled, changing the fraction of inhibited catalytic sites to 70 % while the fraction of free catalytic sites decreases to 30 %, with all other conditions unchanged. Recalculate keff/k0 and discuss qualitatively how the shape of the composite reaction–coordinate diagram would reflect this change.
(d) Propose one practical strategy (other than lowering inhibitor concentration) that a chemist could employ to regain the original rate enhancement and explain mechanistically how it would counteract the effect of the inhibitor.