The Preequilibrium Assumption

The rate of reaction, expressed as the rate of product formation, is 

The problem can be greatly simplified if ki and k i are much larger than k2, for then it will be reasonable to assume that the reaction of A and B to form C is 

consider the nature of equilibrium from the kinetic viewpoint. If the reaction 

Because the equilibrium constant K is given by CqIcaCb, we obtain 

The terms fast and slow are relative. (The hydrolysis step is followed by another fast proton transfer, but this has no effect on the rate of hydrolysis.) 

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Figure 3-9. Companson of exact solutions and the preequilibrium assumption for Seheme XIV. For all of the calculations the values ki = 10 s", A i = 5 s were used. The values are given in the figure. The exact solutions were obtained with Eq. (3-135) and the approximate solution (corresponding to = 0) with Eq. (3-136).

Figure 3-9 shows plots of Eqs. (3-135) and (3-136) for some hypothetical systems. Obviously the equilibrium approximation is poor in the early stages of the reaction, but in the later stages the assumption can be quite good. The preequilibrium assumption, applied to Scheme XIV, amounts to the statement that 2 is negligible relative to ki and i. 

The time scale of a phenomenon whose characteristic quantity has the units seconds is given by the reciprocal of that quantity. In the present context, therefore, we can state that the preequilibrium assumption is valid if the A B reaction is. very fast on the time scale (1/ 2) of the B — C reaction. 

Further discussion of the preequilibrium assumption is given in the next subsection. 

In the preceding subsection we described the preequilibrium assumption. Let us now see how that assumption is related to the steady-state approximation. Scheme XIV will serve for the discussion. The equilibrium and steady-state expressions for the intermediate concentration are... 

In the first step the hydrated ion and ligand form a solvent-separated complex this step is believed to be relatively fast. The second, slow, step involves the readjustment of the hydration sphere about the complex. The measured rate constants can be approximately related to the constants in Scheme IX by applying the fast preequilibrium assumption the result is k = Koko and k = k Q. However, the situation can be more complicated than this. - °... 

The appearance of Cd in the denominator means that D is coupled to a reversible step prior to the rds. If k, and k- were so large that the fast preequilibrium assumption is valid, then the Cd term in the denominator would drop out, and we would have v = A 2 CaObCd, giving the composition of the rds transition state. If k2 is very much larger than it, and k i, Eq. (5-59) becomes v = A CaOb the first step is now the rds, and the rate equation gives the transition state composition. 

TScc is also the stage at which the enantiofacial selectivity of the reaction is determined [80]. This conflicts with the conventional assumption that the face selectivity is established in the initial Ti-complexation [40a], which is now shown to represent a preequilibrium state preceding TScc. The calculated activation energy taking the solvation of the lithium atoms into account shows reasonable agreement with recently determined experimental data [75]. 

In this chapter we first consider a mathematically tractable model mechanism and demonstrate that, depending upon the relative magnitudes of the rate constants, there are two chemical approximations that may be appropriate for simplifying analysis the preequilibrium and the steady-state assumptions. We then demonstrate how hypotheses based upon these simplifications are used to interpret rate law data and to develop chemically reasonable mechanistic descriptions for gas- and solution-phase reactions. Finally we consider the problem of catalysis, i.c., how addition of trace amounts of an intermediate permits a sluggish or kinetically forbidden reaction to become rapid if a new mechanistic pathway can be created. 

Another term used to indicate a rapid equilibrium assumption in a kinetic process. The most prominent example in biochemistry is the assumption that enzyme and substrate rapidly form a preequilibrium enzyme-substrate complex in the Michaelis-Menten treatment. 

It is interesting to note that the stepwise association model or stepwise aggregation model for micelle formation, as shown in Scheme 3.2, leads to the expression = (C, - [S])/(C , - [S]), where represents the number-average micellar aggregation number, C, is the total analytical concentration of surfactant (i.e., C, = [Surflj), C i is the total concentration of osmotically active particles (i.e., monomers + micelles or CMC + [D ]), and [S] (= CMC) is the monomer concentration." Although the stepwise association model is not exactly similar to the phase-separation model, the basis of preequilibrium kinetic model of micelle (Scheme 3.1), the expression = (Ct - [S])/(Cni - [S]) is exactly similar to assumption (5) in the PEK model. The mass action model or multiequilibrium model. Equation 3.1, which is equivalent to Equation 1.20 in Chapter 1... 

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