Shapes of curves of pH-dependence of reactions

A simple case is considered in which the rate of a two-step reaction depends on pH because the intermediate formed in the first step has to gain (or lose) a proton before it can react in the second step, and in which the rate-determining step therefore changes with pH. The curves of reaction rate against pH are shown to be symmetrical, and the sharpest peak possible has a width at half its height of 1.53pH units, i.e. of 2log(3+2 radical2). Any particular curve for this situation proves to be identical with a curve that could be generated for the pH-dependence of a single-step reaction in which the rate is proportional to the concentration of a particular ionic form of a reactant. Curves for the latter situation, however, can have forms impossible for the former case in which the rate-determining step changes, but only if the protonations that activate and deactivate the reactant are co-operative. The peak can then become even sharper, and its width at half its height can fall to 1.14pH units, i.e. to 2log(2+ radical3).