The Mathematics of Specific Catalysis

Let s start with an obvious example of specific-add catalysis. We use water for demonstration purposes, but the kinetic development would be the same for other solvents. Eq. 9.11 shows a schematic mechanism in which hydronium protonates a reactant or substrate (R) prior to a reaction that is rate-determining. The slow step could be first order as shown in Eq. 9.11, or it commonly involves addition of a nucleophile that would then become part of the kinetic expression. Water is a common nucleophile. We keep the scenario simple here, because having additional reactants involved in the rate-determining step does not affect the conclusion we are leading to regarding the kinetic behavior found for the acid catalyst. After the reaction has proceeded, the proton is lost back to the solution, giving the product (P). 

We start the kinetic analysis with Eq. 9.12, and substitute for RH using Eq. 9.13. Eq. 9.14 gives the kinetic expression for the mechanism of Eq. 9.11, assuming that the equilibrium between R and RH is completely established. The kinetic expression contains [HsO ], as the definition of specific-acid catalysis implies. Hence, the reaction rate depends upon the pH. The expression also contains the acid dissociation constant (KaRHO of RH , which is an important factor that we will return to below. Note that k, [HsO ], and KaRH are constants during the reaction. Hence, we create a new rate constant, showing that the reaction appears 

Let s now examine the same reaction but under conditions for which it is not quite so obvious whether the reaction is catalyzed by the specific acid. If we add an acid such as acetic acid to water, small amounts of hydronium ion are produced, but the add in highest concentration is acetic acid. Eq. 9.16 shows a mechanism in which the added add protonates the reactant in an equilibrium prior to the rate-determining step. We designate the acid as HA with the implication that it could be acetic acid performing the protonation. If it were H3O performing the protonation, we would simply have the same scenario as presented in Eqs. 9.14 and 9.15. 

This derivation teaches an important lesson. If the acid catalyst is involved in an equilibrium prior to the rate-determining step, and it is not involved in the rate-determining step, then the kinetics of the reaction will depend solely upon the concentration of the specific acid. This is true even if an added acid (such as acetic acid) is involved in protonating the reactant. The reason for this is that when a prior equilibrium is established, the concentration of RH determines the rate of the reaction (Eqs. 9.12 and 9.17). The concentration of RH depends solely upon the pH and the pfC of RH+, and does not depend upon the concentration of the acid HA that was added to solution. 

A similar kinetic expression can be derived for the use of a catalytic base, B. When B is involved in an equilibrium with a reactant (RH) prior to a rate-determining step, Eqs. 9.22, 9.23, and 9.24 describe the situation (see Exercise 2). Now K, and [HsO ] trade places in the numerator and denominator relative to the acid-catalyzed scenario (Eq. 9.21). The same expression will be derived if either the specific base or an added base is used in the equilibrium. Once again, because [R ] controls the rate of the reaction, it is the pH of the solution and the pK of RH that are important, not the amount of B present in the solution. Finally, recognizing that kKaRn/fHsO ] is a constant during the reaction gives a kinetic expression that is first order in [RH] only (Eq. 9.25, where fcobs = [HsO ]). 

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