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# Butler-Volmer equation

The Butler-Volmer equation is one of the most fundamental relationships in electrochemistry. It describes how the electrical current on an electrode depends on the electrode potential, considering that both a cathodic and an anodic reaction occur on the same electrode:

:$I = A cdot i_0 cdot left\left\{ exp left \left[ frac \left\{ \left(1 - alpha\right) cdot n cdot F \right\} \left\{ R cdot T \right\} cdot \left( E - E_\left\{eq\right\} \right) ight\right] - exp left \left[ - \left\{ frac \left\{ alpha cdot n cdot F \right\} \left\{ R cdot T \right\} \right\} cdot \left( E - E_\left\{eq\right\} \right) ight\right] ight\right\}$

where::$I$: electrode current, A:$i_o$: exchange current density, A/m2:$E$: electrode potential, V:$E_\left\{eq\right\}$: equilibrium potential, V:$A$: electrode active surface area, m2:$T$: absolute temperature, K:$n$: number of electrons involved in the electrode reaction:$F$: Faraday constant:$R$: universal gas constant:$alpha$: so-called symmetry factor, dimensionless

The equation is named after chemists John Alfred Valentine Butler and Max Volmer.

Limitations

The equation is valid when the electrode reaction is controlled by electrical charge transfer at the electrode (and not by the mass transfer to or from the electrode surface from or to the bulk electrolyte). Nevertheless, the utility of the Butler-Volmer equation in electrochemistry is wide, and it is often considered to be "central in the phenomenological electrode kinetics". [J. O'M. Bockris, A.K.N.Reddy, and M. Gamboa-Aldeco, "Modern Electrochemistry 2A. Fundamentals of Electrodics.", Second Edition, Kluwer Academic/Plenum Publishers, p.1083, 2000.]

The limiting cases

There are two limiting cases of the Butler-Volmer equations:
* the low overpotential region (called "polarization resistance"), where the Butler-Volmer equation simplifies to:: $i=i_0 frac \left\{nF\right\} \left\{RT\right\} \left(E-E_\left\{eq\right\}\right)$
* the high overpotential region, where the Butler-Volmer equation simplifies to the Tafel equation::$E-E_\left\{eq\right\} = a - b log\left(i\right)$ for a cathodic reaction, or::$E-E_\left\{eq\right\} = a + b log\left(i\right)$ for an anodic reaction,

where a and b are constants (for a given reaction and temperature) and are called the Tafel equation constants.

* Nernst equation
* Tafel equation

References

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