text: The following derivation of the extended Butler–Volmer equation is adapted from that of Bard and Faulkner and Newman and Thomas-Alyea. For a simple unimolecular, one-step reaction of the form:
  :O+ne → R
  The forward and backward reaction rates (v and v) and, from Faradays laws of electrolysis, the associated electrical current densities (j'),  may be written as:
  where k and k are the reaction rate constants, with units of frequency (1/time) and c and c are the surface concentrations (mol/area) of the oxidized and reduced molecules, respectively (written as c(0,t) and c(0,t) in the previous section). The net rate of reaction v and net current density j are then:
  The figure above plots various Gibbs energy curves as a function of the reaction coordinate &xi;. The reaction coordinate is roughly a measure of distance, with the body of the electrode being on the left, the bulk solution being on the right. The blue energy curve shows the increase in Gibbs energy for an oxidized molecule as it moves closer to the surface of the electrode when no potential is applied. The black energy curve shows the increase in Gibbs energy as a reduced molecule moves closer to the electrode. The two energy curves intersect at . Applying a potential E to the electrode will move the energy curve downward (to the red curve) by nFE and the intersection point will move to .  and  are the activation energies (energy barriers) to be overcome by the oxidized and reduced species respectively for a general E, while  and  are the activation energies for E=0. 
  Assume that the rate constants are well approximated by an Arrhenius equation,
  where the A and A are constants such that A c = A c is the "correctly oriented" O-R collision frequency, and the exponential term (Boltzmann factor) is the fraction of those collisions with sufficient energy to overcome the barrier and react.
  Assuming that the energy curves are practically linear in the transition region, they may be represented there by:
  The charge transfer coefficient for this simple case is equivalent to the symmetry factor, and can be expressed in terms of the slopes of the energy curves:
  It follows that:
  For conciseness, define:
  The rate constants can now be expressed as:
  where the rate constants at zero potential are:
  The current density j as a function of applied potential E may now be written: