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Butler-Volmer equation

Equation characterising electrochemical kinetics

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Source updated August 22, 2026

In electrochemistry, the Butler-Volmer equation (named after John Alfred Valentine Butler and Max Volmer), also known as Erdey-Grúz-Volmer equation (after Tibor Erdey-Grúz), is one of the most fundamental relationships in electrochemical kinetics. It describes how the electrical current through an electrode depends on the voltage difference between the electrode and the bulk electrolyte for a simple, unimolecular redox reaction, considering that both a cathodic and an anodic reaction occur on the same electrode:

01Butler-Volmer equation

The Butler-Volmer equation is:

$j=j_{0}\cdot \left\{\exp \left[{\frac {\alpha _{\rm {a}}zF}{RT}}(E-E_{\rm {eq}})\right]-\exp \left[-{\frac {\alpha _{\rm {c}}zF}{RT}}(E-E_{\rm {eq}})\right]\right\}$

or in a more compact form:

$j=j_{0}\cdot \left\{\exp \left[{\frac {\alpha _{\rm {a}}zF\eta }{RT}}\right]-\exp \left[-{\frac {\alpha _{\rm {c}}zF\eta }{RT}}\right]\right\}$

where:

$j$ : electrode current density, A/m² (defined as $j=I/S$ )
$j_{0}$ : exchange current density, A/m²
$E$ : electrode potential, V
$E_{\rm {eq}}$ : equilibrium potential, V
$T$ : absolute temperature, K
$z$ : number of electrons involved in the electrode reaction
$F$ : Faraday constant
$R$ : universal gas constant
$\alpha _{\rm {c}}$ : so-called cathodic charge transfer coefficient, dimensionless
$\alpha _{\rm {a}}$ : so-called anodic charge transfer coefficient, dimensionless
$\eta$ : activation overpotential (defined as $\eta =E-E_{\rm {eq}}$ ).

The right hand figure shows plots valid for $\alpha _{\rm {a}}=1-\alpha _{\rm {c}}$ .

Limiting cases

There are two limiting cases of the Butler-Volmer equation:

the low overpotential region (called "polarization resistance", i.e., when E ≈ E_eq), where the Butler-Volmer equation simplifies to:

$j=j_{0}{\frac {zF}{RT}}(E-E_{\rm {eq}})$ ;

the high overpotential region, where the Butler-Volmer equation simplifies to the Tafel equation. When $(E-E_{\rm {eq}})>>0$ , the first term dominates, and when $(E-E_{\rm {eq}})<<0$ , the second term dominates.

$E-E_{\rm {eq}}=a_{\rm {c}}-b_{\rm {c}}\log j$ for a cathodic reaction, when E << E_eq, or

$E-E_{\rm {eq}}=a_{\rm {a}}+b_{\rm {a}}\log j$ for an anodic reaction, when E >> E_eq

where $a$ and $b$ are constants (for a given reaction and temperature) and are called the Tafel equation constants. The theoretical values of the Tafel equation constants are different for the cathodic and anodic processes. However, the Tafel slope $b$ can be defined as:

$b=\left({\frac {\partial E}{\partial \ln |I_{\rm {F}}|}}\right)_{c_{i},T,p}$

where $I_{\rm {F}}$ is the faradaic current, expressed as $I_{\rm {F}}=I_{\rm {c}}+I_{\rm {a}}$ , being $I_{\rm {c}}$ and $I_{\rm {a}}$ the cathodic and anodic partial currents, respectively.

02Extended Butler-Volmer equation

The more general form of the Butler-Volmer equation, applicable to the mass transfer-influenced conditions, can be written as:

$j=j_{0}\left\{{\frac {c_{\rm {o}}(0,t)}{c_{\rm {o}}^{*}}}\exp \left[{\frac {\alpha _{\rm {a}}zF\eta }{RT}}\right]-{\frac {c_{\rm {r}}(0,t)}{c_{\rm {r}}^{*}}}\exp \left[-{\frac {\alpha _{\rm {c}}zF\eta }{RT}}\right]\right\}$

where:

j is the current density, A/m²,
c_o and c_r refer to the concentration of the species to be oxidized and to be reduced, respectively,
c(0,t) is the time-dependent concentration at a distance of zero from the surface of the electrode.

The above form simplifies to the conventional expression (shown at the top of the article) when the concentration of the electroactive species at the surface is equal to that in the bulk. The concentration at the surface of the electrode can differ from the bulk concentration because of the formation of the electrical double layer (EDL) and mass transportation limitations during the reaction.

The effect of the EDL was first formalized by Alexander N. Frumkin and is incorporated in the Frumkin-Butler-Volmer theory. The variation of the electrochemical potential in the vicinity of the electrode surface modifies the local concentration of ionic species, thereby affecting electrochemical reaction involving them.

Mass transportation also influences the reaction rate. 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 supplied to, and products are removed from, the electrode region via processes such as 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 they in turn determine. 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 approximately equal to those 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 slower chemical rate dominates the reaction. 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; instead, it treats the concentrations at the electrode as given, yielding a relationship in which the current is expressed as a function not only of potential but also of the given concentrations. The mass-transfer rate may be relatively low, but its only effect on the chemical reaction is through the altered (given) concentrations. In effect, the concentrations are also a function of the potential. 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 to express the concentrations as functions of the potential.

Derivation

General expression

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 redox reaction of the form:

Oxidant + n e⁻ → Reductant

O + n e⁻ → R

The forward and backward reaction rates (v_f and v_b) and, from Faraday's laws of electrolysis, the associated electrical current densities (j), may be written as:

$v_{f}=k_{f}c_{o}=j_{f}/nF$

$v_{b}=k_{b}c_{r}=j_{b}/nF$

where k_f and k_b are the reaction rate constants, with units of frequency (1/time) and c_o and c_r are the surface concentrations (mol/area) of the oxidized and reduced molecules, respectively (written as c_o(0,t) and c_r(0,t) in the previous section). The net rate of reaction v and net current density j are then:

$v=v_{b}-v_{f}={\frac {j_{b}-j_{f}}{nF}}={\frac {j}{nF}}$

The figure above plots various Gibbs energy curves as a function of the reaction coordinate ξ. The reaction coordinate is approximately a measure of distance, with the electrode body on the left and the bulk solution on the right. The blue energy curve shows the increase in Gibbs energy of an oxidized molecule as it moves toward the electrode surface in the absence of an applied potential. 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 $\Delta G^{*}(0)$ . 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 $\Delta G^{*}(E)$ . $\Delta ^{\ddagger }G_{c}$ and $\Delta ^{\ddagger }G_{a}$ are the activation energies (energy barriers) to be overcome by the oxidized and reduced species, respectively, for a general E. In contrast, $\Delta ^{\ddagger }G_{oc}$ and $\Delta ^{\ddagger }G_{oa}$ are the activation energies for E = 0.

Assume that the rate constants are well approximated by an Arrhenius equation,

$k_{f}=A_{f}\exp[-\Delta ^{\ddagger }G_{c}/RT]$

$k_{b}=A_{b}\exp[-\Delta ^{\ddagger }G_{a}/RT]$

where the A_f and A_b are constants such that A_f c_o = A_b c_r is the "correctly oriented" O-R collision frequency. 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:

$\Delta G=S_{c}\xi +K_{c}$	(blue curve)
$\Delta G=S_{c}\xi +K_{c}-nFE$	(red curve)
$\Delta G=-S_{a}\xi +K_{a}$	(black curve)

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:

$\alpha ={\frac {S_{c}}{S_{a}+S_{c}}}$

It follows that:

$\Delta ^{\ddagger }G_{c}=\Delta ^{\ddagger }G_{oc}+\alpha nFE$

$\Delta ^{\ddagger }G_{a}=\Delta ^{\ddagger }G_{oa}-(1-\alpha )nFE$

For conciseness, define:

$f_{\alpha }=\alpha nF/RT$

$f_{\beta }=(1-\alpha )nF/RT$

$f=f_{\alpha }+f_{\beta }=nF/RT$

The rate constants can now be expressed as:

$k_{f}=k_{fo}e^{-f_{\alpha }E}$

$k_{b}=k_{bo}e^{f_{\beta }E}$

where the rate constants at zero potential are:

$k_{fo}=A_{f}e^{-\Delta ^{\ddagger }G_{oc}/RT}$

$k_{bo}=A_{b}e^{-\Delta ^{\ddagger }G_{oa}/RT}$

The current density j as a function of applied potential E may now be written:

$j=nF(c_{r}k_{bo}e^{f_{\beta }E}-c_{o}k_{fo}e^{-f_{\alpha }E})$

Expression in terms of the equilibrium potential

At a certain voltage E_e, equilibrium will be attained and the forward and backward rates (v_f and v_b) will be equal. The green curve in the above figure represents this. The equilibrium rate constants will be written as k_fe and k_be, and the equilibrium concentrations will be written c_oe and c_re. The equilibrium currents (j_ce and j_ae) will be equal and are written as j_o, which is known as the exchange current density.

$v_{fe}=k_{fe}c_{oe}=j_{o}/nF$

$v_{be}=k_{be}c_{re}=j_{o}/nF$

Note that the net current density at equilibrium will be zero. The equilibrium rate constants are then:

$k_{fe}=k_{fo}e^{-f_{\alpha }E_{e}}$

$k_{be}=k_{bo}e^{f_{\beta }E_{e}}$

Solving the above for k_fo and k_bo in terms of the equilibrium concentrations c_oe and c_re and the exchange current density j_o, the current density j as a function of applied potential E may now be written:

$j=j_{o}\left({\frac {c_{r}}{c_{re}}}e^{f_{\beta }(E-E_{e})}-{\frac {c_{o}}{c_{oe}}}e^{-f_{\alpha }(E-E_{e})}\right)$

Assuming that equilibrium holds in the bulk solution, with concentrations $c_{o}^{*}$ and $c_{r}^{*}$ , it follows that $c_{oe}=c_{o}^{*}$ and $c_{re}=c_{r}^{*}$ , and the above expression for the current density j is then the Butler-Volmer equation. Note that E-E_e is also known as η, the activation overpotential.

Expression in terms of the formal potential

For the simple reaction, the change in Gibbs energy is:

$\Delta G=\Delta G_{o}-\Delta G_{r}=(\Delta G_{o}^{o}-\Delta G_{r}^{o})+RT\ln \left({\frac {a_{oe}}{a_{re}}}\right)$

where a_oe and a_re are the activities at equilibrium. The activities a are related to the concentrations c by a = γc, where γ is the activity coefficient. The equilibrium potential is given by the Nernst equation:

$E_{e}=-{\frac {\Delta G}{nF}}=E^{o}-{\frac {RT}{nF}}\ln \left({\frac {a_{oe}}{a_{re}}}\right)$

where $E^{o}$ is the standard potential

$E^{o}=-(\Delta G_{o}^{o}-\Delta G_{r}^{o})/nF$

Defining the formal potential:

$E^{o'}=E^{o}-{\frac {RT}{nF}}\ln \left({\frac {\gamma _{oe}}{\gamma _{re}}}\right)$

The equilibrium potential is then:

$E_{e}=E^{o'}-{\frac {RT}{nF}}\ln \left({\frac {c_{oe}}{c_{re}}}\right)$

Substituting this equilibrium potential into the Butler-Volmer equation yields:

$j={\frac {j_{o}}{c_{oe}^{1-\alpha }c_{re}^{\alpha }}}\left(c_{r}e^{f_{\beta }(E-E^{o'})}-c_{o}e^{-f_{\alpha }(E-E^{o'})}\right)$

which may also be written in terms of the standard rate constant k^o as:

$j=nFk^{o}\left(c_{r}e^{f_{\beta }(E-E^{o'})}-c_{o}e^{-f_{\alpha }(E-E^{o'})}\right)$

The standard rate constant is an essential 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 $k^{0}\rightarrow 0$ , the electrode becomes an ideal polarizable electrode and will behave electrically as an open circuit (neglecting capacitance). For nearly ideal electrodes with small k^o, large changes in the overpotential are required to generate a significant current. In the limit as $k^{0}\rightarrow \infty$ , the electrode becomes an ideal non-polarizable electrode and will behave as an electrical short. For nearly ideal electrodes with large k^o, small changes in the overpotential will generate large changes in current.

A plot of various Gibbs energies as a function of reaction coordinate. The reaction will proceed towards the lower energy, reducing for the blue curve, oxidizing for the red curve. The green curve illustrates equilibrium.

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Sources and credits

This article is adapted from the Wikipedia article “Butler-Volmer equation”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

Current density versus potential according to butler volmer equation.svg by Toshiyouri, CC BY-SA 4.0
ButlerVolmerGibbsPlot.png by PAR, CC0

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