For the simple reaction, the change in Gibbs energy is:
where a and a are the activities at equilibrium. The activities a are related to the concentrations c by a=&gamma;c where &gamma; is the activity coefficient. The equilibrium potential is given by the Nernst equation:
where  is the standard potential
Defining the formal potential:
the equilibrium potential is then:
Substituting this equilibrium potential into the Butler–Volmer equation yields:
which may also be written in terms of the standard rate constant k as:
The standard rate constant is an important descriptor of electrode behavior, independent of concentrations. It is a measure of the rate at which the system will approach equilibrium. In the limit as , the electrode becomes an ideal polarizable electrode and will behave electrically as an open circuit (neglecting capacitance). For nearly ideal electrodes with small k, large changes in the overpotential are required to generate a significant current. In the limit as , the electrode becomes an ideal non-polarizable electrode and will behave as an electrical short. For a nearly ideal electrodes with large k, small changes in the overpotential will generate large changes in current.