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

The reaction rate constant is expressed as Z exp(-G_a/RT). Z is the binary collision frequency. G_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_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_a on ΔG°, 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_a = αΔG° + G_a°M(α). Here α is the Brønsted slope, α = -∂ ln k(T)/∂(ΔG°/RT), G_a° is independent of ΔG° and M(α), the Legendre transform of G_a, is a function only of α. For small changes in ΔG°, the general result reduces to the familiar "linear" free energy relation δG_a = αδΔG°. It is concluded from general considerations that M(α) is a symmetric, convex function of α and hence that α is a monotonically increasing function of ΔG°. Experimental data appear to conform well to the form α = 1/[1 + exp(-ΔG°/G_a°)]. A simple interpretation of the ΔG° dependence of G_a, based on an interpolation of the free energy from that of the reagents to that of the products, is offered.