Butler–Volmer equation

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:

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

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

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

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

where ${\displaystyle a}$ and ${\displaystyle b}$ 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 ${\displaystyle b}$ can be defined as:

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

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

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.

The following derivation of the extended Butler–Volmer equation is adapted from that of Bard and Faulkner[3] and Newman and Thomas-Alyea.[5] For a simple unimolecular, one-step reaction of the form:

O+ne⁻ → 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:

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

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

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:[Note 2]

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

The figure above plots various Gibbs energy curves as a function of the reaction coordinate ξ. The reaction coordinate is 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 ${\displaystyle \Delta G^{*}(0)}$. Applying a potential E to the electrode will move the energy curve downward[Note 3] (to the red curve) by nFE and the intersection point will move to ${\displaystyle \Delta G^{*}(E)}$. ${\displaystyle \Delta ^{\ddagger }G_{c}}$ and ${\displaystyle \Delta ^{\ddagger }G_{a}}$ are the activation energies (energy barriers) to be overcome by the oxidized and reduced species respectively for a general E, while ${\displaystyle \Delta ^{\ddagger }G_{oc}}$ and ${\displaystyle \Delta ^{\ddagger }G_{oa}}$ are the activation energies for E=0. [Note 4]

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

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

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

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, 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:

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

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

It follows that:

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

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

For conciseness, define:

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

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

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

The rate constants can now be expressed as:

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

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

where the rate constants at zero potential are:

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

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

The current density j as a function of applied potential E may now be written:[5]^{: § 8.3}

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

At a certain voltage E_e, equilibrium will attain and the forward and backward rates (v_f and v_b) will be equal. This is represented by the green curve in the above figure. 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.

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

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

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

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

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

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:[5]^{: § 8.3}

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

Assuming that equilibrium holds in the bulk solution, with concentrations ${\displaystyle c_{o}^{*}}$ and ${\displaystyle c_{r}^{*}}$, it follows that ${\displaystyle c_{oe}=c_{o}^{*}}$ and ${\displaystyle c_{re}=c_{r}^{*}}$, 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.

For the simple reaction, the change in Gibbs energy is:[Note 5]

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

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:

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

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

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

Defining the formal potential:[3]^{: § 2.1.6}

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

the equilibrium potential is then:

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

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

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

which may also be written in terms of the standard rate constant k^o as:[3]^{: § 3.3.2}

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

The standard rate constant is an 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 ${\displaystyle k^{0}\rightarrow 0}$, 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 ${\displaystyle k^{0}\rightarrow \infty }$, the electrode becomes an ideal non-polarizable electrode and will behave as an electrical short. For a nearly ideal electrodes with large k^o, small changes in the overpotential will generate large changes in current.