MATHEMATICAL Model beyond the Master Electrochemical mechanism (ECrevEC’) under conditions of Protein-Film Voltammetry-Surface ECrevEC’ mechanism under Butler-Volmer formalism

The surface-confined two-step ECrevEC′ mechanistic framework linked with two different chemical reactions provides a rigorous mathematical basis for interpreting complex redox transformations in protein-film cyclic voltammetry. In this model, all electroactive species are strongly adsorbed on the electrode surface, and their temporal evolution is governed by a coupled system of nonlinear differential equations derived from Butler–Volmer kinetics for two sequential electron-transfer steps and first-order chemical rate laws for the intermediate transformations. The inclusion of a reversible chemical step (Crev) between intermediates and a regenerative catalytic step (C′) introduces dynamic coupling between thermodynamic equilibria and kinetic pathways, leading to rich and highly diagnostic voltammetric responses.
From a mathematical standpoint, the model captures the interplay between electron-transfer kinetics, chemical reversibility, and catalytic regeneration through dimensionless parameters that scale with the experimental time window, enabling systematic mechanistic identification. In protein-film cyclic voltammetry, this framework is particularly important because many redox enzymes—especially those containing metal centers such as Mo—undergo multi-step, surface-confined electron transfer coupled to substrate conversion. The ECrevEC′ model therefore serves as a unified platform for extracting mechanistic, kinetic, and thermodynamic information from experimentally accessible features such as peak splitting, peak symmetry, and the appearance of multiple redox waves. Ultimately, this approach advances the quantitative understanding of enzymatic electrochemistry and supports the rational design of bioelectrocatalytic systems and electrochemical biosensors. Entire MATHCAD Protocol designed to simulate this mechanism under conditions of cyclic voltammetry is available for free on the platform of the repository of Goce Delcev University, Stip, Macedonia in 2026.