The global kinetic–thermodynamic relationship derived from first principles

Abstract

What governs the relationship between the reaction rate and thermodynamic driving force? Despite decades of rate theory, no general physically grounded equation exists to relate rate and driving force across all regimes. Classical models, such as the Marcus equation and Leffler equations, either rely on under-realistic assumptions or only capture the local behaviour, failing outside narrow regimes. We derive a general, non-linear equation from microscopic reversibility, arriving at three physically meaningful parameters: a minimum preorganisational barrier (E_min), a reaction symmetry offset (E_eq), and a kinetic curvature factor (θ). This model captures global behaviour, recovers known limits, and explains why classical models like Leffler equation exhibit the rate–driving force responsiveness (the Brønsted slope) as they do, by revealing their physical origin, not by fitting them. The model enables a causal reinterpretation of experimentally observed curved rate–driving force plots, such as in hydrogen atom transfer to Fe(IV)O. Importantly, this model does not require replacing existing models, it explains their physical foundations, enabling chemists to continue using them while understanding when and why they apply, and where they break down. Beyond case studies including hydride shifts, rearrangements, and cyclisations, the framework's strength lies in its deductive foundation enabling the physically grounded design of reactions with desired kinetics across diverse chemical systems. By revealing the global structure of the rate–driving force relationship, this framework enables chemists to recognise, predict, and design reactivity that would otherwise appear anomalous or inaccessible, better clarifying the unknowns. Examples include highly exergonic regimes near E_min, where further increases in exergonicity offer little rate improvement, and control shifts to structural factors; even when the rate–driving force plot appears linear, the model uncovers hidden curvature and deeper physical meaning.