Global Kinetic-Thermodynamic Responses: Derivation of a General Non-Linear Equation and Demonstrations on Chemical Reactions

How to accelerate a reaction has been a critical question in physical organic chemistry, eliciting multiple models describing the interplay between kinetics and thermodynamic driving forces. However, existing models often come with inevitable limitations: valid only within finite thermodynamic ranges, rely on heavy physical assumptions, or treat specific observed reactivity as isolated occurrences. We present the derivation of the non-linear relationship that generalises the relationship between activation energies with respect to reaction energies in chemical reactions, providing a broader and more overarching model than the classical Bell-Evans-Polanyi relationships, Leffler equation, and Marcus equation. This formulation is reached with minimal assumptions, stemming from the principle of microscopic reversibility. In contrast to the commonly observed Brønsted slopes and intrinsic barriers which represent the local approximation of the kinetic-thermodynamic response of a family of reactions only within a small range of reaction energies, we introduce a new set of thermodynamic-independent parameters (Emin, Eeq and θ) that capture the global kinetic-thermodynamic relationship across the entire thermodynamic range. We showcase the applicability of this model and the insights it provides from a set of 1,2-hydride shifts and Beckmann rearrangements, identifying the origin of differences in activation barriers across substrates and enabling the rational evaluation of reactivity trends. The findings demonstrate the chemical significance of the three parameters in the non-linear model inherently affording information about the reaction unattainable through simple data fitting. The equation governing the non-linear relationship provides new paths for the rational modulation of energy barriers, new perspectives towards developing reaction methodology. Our global non-linear relationship can still be applied to a linear approximation within finite thermodynamic ranges, while also enabling the examination and improvement of its reliability, providing straightforward practical guidance for reaction analysis and optimisation.