A hyperbolic modification of linear free energy relationships

Abstract

In a one-step reaction the activation energies in either forward or reverse reactions cannot be negative. This limitation specifically excludes very wide-range linear dependences of E_a upon ΔH. Analogously, all rate constants have upper limits (in addition to the diffusion limit), and thus linear free energy relationships such as the Hammett or Brønsted relationships, which when extrapolated lead to absurdly high rate constants, are also impossible. The problem of extrapolation is attacked by using a hyperbolic relationship between the logarithms of rate constants and equilibrium constants. In addition to satisfactory extrapolation, the treatment contains within itself the reactivity–selectivity principle and implies an inverse relationship between the curvature of a free energy plot and reactivity. The hyperbola may be related to a naive pictorial representation of the reaction co-ordinate, from which two measures of product-like character in the transition state ensue; one uses E_a and ΔH, the other compares substituent effects on rate constants and equilibrium constants. The hyperbolic relationship is compared with the equations used by Marcus. The treatments provide a norm for rate–equilibrium correlations of single-step reactions with transition states described as hybrids of reagent and product-like structures; marked deviation from this norm is evidence of contributions of special structures in the transition state.