Understanding the mathematics of voltammetry: a step-by-step guide

Although voltammetry is one of the most widely used electrochemical techniques, the mathematical foundations of its characteristic current responses are rarely available in a didactic and self-contained form. As a consequence, students and researchers often make use of inadequate equations and approaches without a clear comprehension of the conditions under which they are valid. This work addresses this educational gap by providing a thorough, step-by-step, illustrated derivation of voltammetric current profiles. Starting from first principles and physical intuition, the main assumptions necessary to model a voltammetric experiment are presented, along with the key mathematical tools that are employed throughout the text. The paradigm system studied is a reversible electrochemical reaction on a macroelectrode in a semi-infinite domain, in which mass transport occurs only through diffusion. The derivation follows a six-step procedure, progressing from the partial differential equations that describe mass transport to the voltammetric current expression. Each of the six steps can be adapted and applied to solve different physical problems in electrochemistry. After deriving the current profiles, they are thoroughly examined to yield peak current (Randles–Ševčík) and peak potential equations, which are discussed in detail using intuitive terms. Finally, the six-step method is applied to chronoamperometry and finite kinetics electrochemical reactions, showcasing its wide applicability. By making explicit the logical structure behind electrochemical models, this work aims to provide students and researchers with tools needed to critically interpret voltammograms and adapt theoretical models to new electrochemical systems. Graphical Abstract