Thermodynamic Analysis of the Hammett Equation, the Temperature Dependence of ρ, and the Isoequilibrium (Isokinetic) Relationship

Abstract

Exact thermodynamic analysis of the Hammett equation has led to four differential equations relating δΔH⁰, δΔS⁰, δΔC_p⁰, dρ/dT, and d²ρ/dT². Similar equations can be obtained in terms of activation parameters ΔH^≠, etc. For temperature independent δΔH⁰ and δΔS⁰ and therefore δΔC_p⁰ = 0, two of these differential equations lead to ρ = ρ_∞ [1–(β₁/T)] and the familiar isoequilibrium (isokinetic) equation δΔH⁰ = β₁δΔS⁰. The "isoequilibrium (isokinetic) temperature" represented here by β₁ is a temperature independent constant of integration. For constant non-zero δΔC_p⁰ we similarly obtain more complicated expressions for ρ and the "isoequilibrium (isokinetic) temperature." These findings are considered in relation to a model in which environmental contributions (due to solute–solvent interactions) to δΔH⁰ and δΔS⁰ are related by a parameter β_c. The relationship between β₁, and β_c is established, and it is shown that in general β₁ ≠ β_c.