Free energy of activation. Definition, properties, and dependent variables with special reference to "linear" free energy relations

Abstract

The reaction rate constant is expressed as Z exp(-G/sub a//RT). Z is the binary collision frequency. G/sub a/, the free energy of activation, is shown to be the difference between the free energy of the reactive reactants and the free energy of all reactants. The results are derived from both a statistical mechanical and a collision theoretic point of view. While the later is more suitable for an ab-initio computation of the reaction rate, it is the former that lends itself to the search of systematics and of correlations and to compaction of data. Different thermodynamic-like routes to the characterization of G/sub a/ are thus explored. The two most promising ones appear to be the use of thermodynamic type cycles and the changes of dependent variables using the Legendre transform technique. The dependence of G/sub a/ on ..delta..G/sup 0/, the standard free energy change in the reaction, is examined from the later point of view. It is shown that one can rigorously express this dependence as G/sub a/ = ..cap alpha delta..G/sup 0/ + G/sub a//sup 0/M(..cap alpha..). Here ..cap alpha.. is the Bronsted slope, ..cap alpha.. = -par. delta ln k(T)/par. delta(..delta..G/sup 0//RT), G/sub a//sup 0/ is independent ofmore » ..delta..G/sup 0/ and M(..cap alpha..), the Legendre transform of G/sub a/, is a function only of ..cap alpha... For small changes in ..delta..G/sup 0/, the general result reduces to the familiar ''linear'' free energy relation delta G/sub a/ = ..cap alpha.. delta ..delta..G/sup 0/. It is concluded from general considerations that M(..cap alpha..) is a symmetric, convex function of ..cap alpha.. and hence that ..cap alpha.. is a monotonically increasing function of ..delta..G/sup 0/. Experimental data appear to conform well to the form ..cap alpha.. = 1/(1 + exp(-..delta..G/sup 0//G/sub s//sup 0/)). A simple interpretation of the ..delta..G/sup 0/ dependence of G/sub a/, based on an interpolation of the free energy from that of the reagents to that of the products, is offered. 4 figures, 69 references.« less