ButlerVolmer

The Butler–Volmer equation is a foundational relation in electrochemistry that connects the current density at an electrode to the overpotential of a redox reaction. It describes how the forward (oxidation) and reverse (reduction) electron-transfer rates depend exponentially on the applied potential, assuming activation-controlled kinetics and surface species in quasi-equilibrium. The equation is widely applicable to simple, single-step transfers and serves as a standard model for interpreting polarization curves in batteries, fuel cells, and corrosion studies.

For a generic Ox + ne− ⇌ Red reaction, the current density i is given by

i = n F k0 [C_O* exp(−α n F η / RT) − C_R* exp((1 − α) n F η / RT)],

where η = Eapp − Eeq is the overpotential, k0 is the standard heterogeneous rate constant, α is the transfer coefficient (0 < α < 1), and C_O*, C_R* are the surface activities of Ox and Red. An equivalent formulation uses the exchange current density i0:

i = i0 [exp(α n F η / RT) − exp(−(1 − α) n F η / RT)],

with i0 = n F k0 (C_O*)^α (C_R*)^(1−α).

Limits and variants: at small overpotentials, the equation can be linearized as i ≈ (i0 n F / RT) η. At large overpotentials, the form reduces to the Tafel equation, linking η to ln i for each branch. The equation assumes isothermal conditions, a uniform active surface, and negligible mass-transport limitation; deviations arise when diffusion or multi-step kinetics become significant.

History: the equation is named for Max Volmer and J. Butler, who developed the kinetic description in the 1920s–1930s. It remains a central tool in electrochemical kinetics and process modeling.