The more general form of the Butler–Volmer equation, applicable to the mass transfer-influenced conditions, can be written as:
where:
* j is the current density, A/m,
* c and c refer to the concentration of the species to be oxidized and to be reduced, respectively,
* c(0,t) is the time-dependent concentration at the distance zero from the surface of the electrode.
The above form simplifies to the conventional one (shown at the top of the article) when the concentration of the electroactive species at the surface is equal to that in the bulk.
There are two rates which determine the current-voltage relationship for an electrode. First is the rate of the chemical reaction at the electrode, which consumes reactants and produces products. This is known as the charge transfer rate. The second is the rate at which reactants are provided, and products removed, from the electrode region by various processes including diffusion, migration, and convection. The latter is known as the mass-transfer rate
. These two rates determine the concentrations of the reactants and products at the electrode, which are in turn determined by them. The slowest of these rates will determine the overall rate of the process.
The simple Butler–Volmer equation assumes that the concentrations at the electrode are practically equal to the concentrations in the bulk electrolyte, allowing the current to be expressed as a function of potential only. In other words, it assumes that the mass transfer rate is much greater than the reaction rate, and that the reaction is dominated by the slower chemical reaction rate. Despite this limitation, the utility of the Butler–Volmer equation in electrochemistry is wide, and it is often considered to be "central in the phenomenological electrode kinetics".
The extended Butler–Volmer equation does not make this assumption, but rather takes the concentrations at the electrode as given, yielding a relationship in which the current is expressed as a function not only of potential, but of the given concentrations as well. The mass-transfer rate may be relatively small, but its only effect on the chemical reaction is through the altered (given) concentrations. In effect, the concentrations are a function of the potential as well. A full treatment, which yields the current as a function of potential only, will be expressed by the extended Butler–Volmer equation, but will require explicit inclusion of mass transfer effects in order to express the concentrations as functions of the potential.