Abstract :
It has rigorously been shown that the average rate constant, derived from the mean lifetime, of solution reactions including first- and second-order ones can be expressed as 1/(kTST −1 + kf−1) under the condition of initial thermalization of reactants [H. Sumi; J. Phys. Chem. 95 (1991) 3334 and 100 (1996) 4831]. In this formula, kTST represents the rate constant expected from the transition state theory (TST). Since TST assumes that thermalization of reactants is always maintained in the course of reaction due to fast solvent fluctuations, kTST does not depend on how fast the thermalization is. On the other hand, kf (> 0) depends on the time τ for the thermalization, decreasing in many cases as a fractional (less-than-unity) power of τ−1 as τ increases. Usually, τ is proportional to the solvent viscosity η. TST is invalidated in the large τ region of kf ⪡ kTST where 1/(kTST−1 + kf−1) approaches kf. In this region, reaction becomes controlled by slow speeds of solvent fluctuations, and the fractional-power dependence becomes observed as that of the average rate constant. In fact, kf represents the rate constant with which solute-solvent rearrangements most favorable for reaction are realized as a result of solvent fluctuations. The fractional-power dependence of kf on τ−1 occurs when the rearrangements, described by a coordinate X, are not unique but distributed. To be more exact, when an intrinsic reactivity at each X is written as ki(X), it occurs when 〈[dki(X)−1/dX]2〉e/〈ki(X)−1〉e2 diverges for first-order reactions, where 〈…〉e represents the thermal average in X in the reactant state, with a similar condition for second-order reactions.
