# Butler–Volmer equation

The **Butler–Volmer equation** is the standard empirical relation of electrochemical kinetics, describing how the current density at an electrode depends on the overpotential, the deviation of the electrode potential from its equilibrium value.

| Key fact | Detail |
|---|---|
| What it describes | Net current density as the difference of anodic and cathodic exponential terms in the overpotential η = E − E<sub>eq</sub><sup>[1](https://www.sciencedirect.com/science/article/pii/S1572665720303283)</sup><sup> • </sup><sup>[2](https://researchmgt.monash.edu/ws/portalfiles/portal/127423998/92022330_oa.pdf)</sup>  |
| Central parameters | Exchange current density i<sub>0</sub> (A/m²), transfer coefficient α (dimensionless), heterogeneous rate constant k<sub>0</sub> (m/s)<sup>[3](https://doc.comsol.com/6.3/doc/com.comsol.help.fce/fce_ug_electrochem.07.121.html)</sup> |
| Coefficient relation | For a one-electron reduction, α<sub>a</sub> + α<sub>c</sub> = 1; most systems have α between 0.3 and 0.7<sup>[4](https://www.mdpi.com/2227-9040/9/10/277)</sup><sup> • </sup><sup>[5](https://chem.libretexts.org/Bookshelves/Analytical_Chemistry/Supplemental_Modules_%28Analytical_Chemistry%29/Analytical_Sciences_Digital_Library/Courseware/Analytical_Electrochemistry%3A_The_Basic_Concepts/03_Fundamentals_of_Electrochemistry/B%3A_The_Electrode_Process/02_Reversibility__Chemical_vs._Electrochemical)</sup>  |
| High-overpotential limit | The Tafel equations, valid for |η| above roughly 100 mV, where one exponential term dominates<sup>[6](https://stax.strath.ac.uk/downloads/r494vk50j?locale=en)</sup> |
| Low-overpotential limit | A linear relation with charge-transfer resistance Rct = RT/(F i0), valid for η of order RT/F, about 25 mV at room temperature<sup>[7](https://etd.ohiolink.edu/acprod/odb_etd/ws/send_file/send?accession=osu1354652448&disposition=attachment)</sup><sup> • </sup><sup>[8](https://doc.comsol.com/6.1/doc/com.comsol.help.echem/ElectrochemistryModuleUsersGuide.pdf)</sup><sup> • </sup><sup>[19](https://doi.org/10.1149/1945-7111/ad4399)</sup> |
| Origin | Butler's three works (two from 1924, one from 1932) and the 1930 paper by Erdey-Grúz and Volmer on hydrogen overpotential<sup>[1](https://www.sciencedirect.com/science/article/pii/S1572665720303283)</sup> |

## Form and parameters

All forms of the equation share one structure: the current density is the sum of an anodic and a cathodic term, each exponential in a measure of electrode polarization and weighted by unitless transfer coefficients.<sup>[1](https://www.sciencedirect.com/science/article/pii/S1572665720303283)</sup> In the common exchange-current-density form,

\[ j = j_{0} \left[ \exp\left( \frac{\alpha_{a} F \eta}{RT} \right) - \exp\left( -\frac{\alpha_{c} F \eta}{RT} \right) \right], \]

where j is the current density, j<sub>0</sub> the exchange current density, η = E − E<sub>r</sub> the overpotential, F the [Faraday constant](https://www.edgepedia.org/faraday-constant), R the gas constant, and T the temperature.<sup>[2](https://researchmgt.monash.edu/ws/portalfiles/portal/127423998/92022330_oa.pdf)</sup> The overpotential is the extra voltage beyond the equilibrium potential that drives the Faradaic current.<sup>[9](https://ocw.mit.edu/courses/10-626-electrochemical-energy-systems-spring-2014/56cfa6e0f28bc8fc1a647cbe679384d1_MIT10_626S14_S11lec13.pdf)</sup>

**Exchange current density.** At the equilibrium potential the forward and backward reaction rates are equal, so the net current is zero; the common magnitude of these opposing currents is the exchange current density.<sup>[10](https://www.sciencedirect.com/topics/engineering/butler-volmer-kinetics)</sup> It is the key measure of intrinsic catalyst activity: the higher the exchange current density, the smaller the potential change produced by a given current.<sup>[10](https://www.sciencedirect.com/topics/engineering/butler-volmer-kinetics)</sup><sup> • </sup><sup>[11](https://pmc.ncbi.nlm.nih.gov/articles/PMC3264446/)</sup>  In electroanalysis, kinetics are instead expressed through the heterogeneous rate constant k<sub>0</sub> in m/s, related to the exchange current density by i<sub>0</sub> = k<sub>0</sub>Fc for equal concentrations c of the redox couple.<sup>[3](https://doc.comsol.com/6.3/doc/com.comsol.help.fce/fce_ug_electrochem.07.121.html)</sup>

**Transfer coefficient.** The dimensionless transfer coefficient α characterizes the symmetry of the energy barrier in the transition state for electron transfer, partitioning the electrical activation energy between reduction and oxidation, with 0 ≤ α ≤ 1.<sup>[12](https://pmc.ncbi.nlm.nih.gov/articles/PMC12371579/)</sup> For a one-electron reduction the anodic and cathodic coefficients satisfy α<sub>a</sub> + α<sub>c</sub> = 1.<sup>[4](https://www.mdpi.com/2227-9040/9/10/277)</sup> Most electrochemical systems fall between 0.3 and 0.7.<sup>[5](https://chem.libretexts.org/Bookshelves/Analytical_Chemistry/Supplemental_Modules_%28Analytical_Chemistry%29/Analytical_Sciences_Digital_Library/Courseware/Analytical_Electrochemistry%3A_The_Basic_Concepts/03_Fundamentals_of_Electrochemistry/B%3A_The_Electrode_Process/02_Reversibility__Chemical_vs._Electrochemical)</sup>

## Limiting cases

The equation's usefulness comes from its two asymptotic forms, which connect it to the older empirical laws of electrochemistry.

**Tafel behavior.** At high overpotential, when one exponential term dominates, the Butler–Volmer equation reduces to the Tafel equations: for η above about 100 mV the anodic form log(i) = log(i<sub>0</sub>) + α<sub>a</sub>nFη/(RT), and for η below about −100 mV the cathodic form.<sup>[6](https://stax.strath.ac.uk/downloads/r494vk50j?locale=en)</sup> A semilogarithmic Tafel plot of log(current) against overpotential is then linear, with a slope set by the transfer coefficient and an intercept giving the exchange current.<sup>[9](https://ocw.mit.edu/courses/10-626-electrochemical-energy-systems-spring-2014/56cfa6e0f28bc8fc1a647cbe679384d1_MIT10_626S14_S11lec13.pdf)</sup> The Tafel law assumes an irreversible reaction; if the reverse reaction can occur in practice, Tafel kinetics are not correct.<sup>[8](https://doc.comsol.com/6.1/doc/com.comsol.help.echem/ElectrochemistryModuleUsersGuide.pdf)</sup>

**Linear response.** For small overpotentials, of order RT/F (about 25 mV at room temperature), the exponentials can be linearized, giving a current proportional to η.<sup>[8](https://doc.comsol.com/6.1/doc/com.comsol.help.echem/ElectrochemistryModuleUsersGuide.pdf)</sup> The ratio −η/i has units of resistance and is called the charge-transfer resistance, and linearizing the Butler–Volmer equation for the one-electron reaction gives Rct = RT/(F i0), a quantity extracted from electrochemical impedance spectroscopy because that technique inherently uses a small signal.<sup>[7](https://etd.ohiolink.edu/acprod/odb_etd/ws/send_file/send?accession=osu1354652448&disposition=attachment)</sup><sup> • </sup><sup>[19](https://doi.org/10.1149/1945-7111/ad4399)</sup> Near equilibrium the interface therefore behaves like a resistor whose value is inversely proportional to i<sub>0</sub>.<sup>[13](https://yusukehashimotolab.github.io/AI-Knowledge-Notes/knowledge/en/MS/electrochemistry-introduction/chapter-3.html)</sup>

**Equilibrium limit.** Setting the current to zero in the electroanalytical form of the equation recovers the thermodynamic [Nernst equation](https://www.edgepedia.org/nernst-equation) relating the equilibrium concentrations of the reacting species.<sup>[3](https://doc.comsol.com/6.3/doc/com.comsol.help.fce/fce_ug_electrochem.07.121.html)</sup>

## Origins

Contemporary textbooks point to three publications by [John Alfred Valentine Butler](https://www.edgepedia.org/john-alfred-valentine-butler) (two from 1924, one from 1932) and a single 1930 article coauthored by Tibor Erdey-Grúz and [Max Volmer](https://www.edgepedia.org/max-volmer); because Erdey-Grúz was the lesser-known coauthor, his name has often been dropped from the equation over time.<sup>[1](https://www.sciencedirect.com/science/article/pii/S1572665720303283)</sup> Butler's 1924 papers did not construct a theory of heterogeneous electron-transfer rates; they gave a kinetic account of how the Nernst equilibrium potential arises through small degrees of oxidation or reduction shifting the electrode's [Fermi level](https://www.edgepedia.org/fermi-level), invoking a transition state of partially transferred charge.<sup>[1](https://www.sciencedirect.com/science/article/pii/S1572665720303283)</sup> The Erdey-Grúz and Volmer paper addressed hydrogen evolution overpotential: its innovation was to treat the rate-determining step as one to which both anodic and cathodic directions contribute, with equal rates at the equilibrium potential.<sup>[1](https://www.sciencedirect.com/science/article/pii/S1572665720303283)</sup> The equation is a product of the contributions of Butler, Volmer, and Erdey-Grúz to theoretical electrochemistry.<sup>[14](https://knowledge.electrochem.org/encycl/art-p05-pillars-of-ec.htm)</sup>

## Scope and limitations

The equation is essentially an empirical relation inspired by transition-state arguments, and its coefficients can only be determined electrochemically rather than predicted independently.<sup>[1](https://www.sciencedirect.com/science/article/pii/S1572665720303283)</sup> It was originally derived for single-electron transfer reactions, implying reactions that do not involve breaking or creating a chemical bond.<sup>[8](https://doc.comsol.com/6.1/doc/com.comsol.help.echem/ElectrochemistryModuleUsersGuide.pdf)</sup> It is accurate for outer-sphere reactions, in which the only process is electron tunneling between electrode and analyte, with no adsorption or intermediate steps; the Ag/AgCl reaction, for example, does not fall into this category.<sup>[11](https://pmc.ncbi.nlm.nih.gov/articles/PMC3264446/)</sup> Even for the hydrogen evolution/oxidation reaction, one of the simplest electrochemical reactions, the equation is applicable only in limited cases.<sup>[15](https://pmc.ncbi.nlm.nih.gov/articles/PMC4642571/)</sup>

**Mass transport.** The equation does not include mass-transfer effects on the overpotential, so it is a good approximation of the full current–overpotential equation only when the current density is smaller than 10% of the smaller of the anodic or cathodic limiting current.<sup>[16](https://www.sciencedirect.com/topics/engineering/butler-volmer-equation)</sup> Beyond that range, depletion of reactants at the interface limits the current regardless of overpotential.<sup>[11](https://pmc.ncbi.nlm.nih.gov/articles/PMC3264446/)</sup>

**Multistep processes.** Applying textbook formulations to multistep electrocatalytic processes such as the hydrogen or oxygen reactions is a common source of error; for multistep reactions the transfer coefficients generally satisfy α<sub>Rd</sub> + α<sub>Ox</sub> = n/ν, where n is the number of electrons in the overall reaction and ν the stoichiometric number.<sup>[1](https://www.sciencedirect.com/science/article/pii/S1572665720303283)</sup><sup> • </sup><sup>[10](https://www.sciencedirect.com/topics/engineering/butler-volmer-kinetics)</sup>  The equation is also quoted in a wide variety of mathematical forms without uniformity in the meaning of variables such as overpotential, exchange current density, and transfer coefficient, with a particular disparity between electroanalytical chemistry and electrochemical energy conversion and storage science (batteries, fuel cells, and electrolysers).<sup>[1](https://www.sciencedirect.com/science/article/pii/S1572665720303283)</sup>

## Theoretical status

The equation's phenomenological character has drawn criticism, but recent theory has strengthened its footing. Coupled ion–electron transfer (CIET) theory provides a rigorous quantum-mechanical derivation of Butler–Volmer kinetics, including Tafel's limiting law, clarifying its range of applicability and underlying physics.<sup>[17](https://pubs.rsc.org/en/content/articlehtml/2023/fd/d3fd00108c)</sup> For moderate overpotentials, that theory reduces to the Butler–Volmer equation and explains its relevance in both the large electron-transfer (solvent reorganization) energy limit and the opposite limit of large ion-transfer energy.<sup>[18](https://arxiv.org/abs/2007.12980)</sup> In practice, exchange current density and charge-transfer coefficient remain the two most important Butler–Volmer parameters for determining the intrinsic activity of electrocatalysts in fuel cells, electrolysers, and batteries.<sup>[10](https://www.sciencedirect.com/topics/engineering/butler-volmer-kinetics)</sup>

## References

1. [The Butler-Volmer equation in electrochemical theory: Origins, value, and practical application, Current Opinion in Electrochemistry](https://www.sciencedirect.com/science/article/pii/S1572665720303283)
2. [Electrochemistry of and in Ionic Liquids, Monash University repository](https://researchmgt.monash.edu/ws/portalfiles/portal/127423998/92022330_oa.pdf)
3. [COMSOL 6.3: The Electroanalytical Butler–Volmer Equation](https://doc.comsol.com/6.3/doc/com.comsol.help.fce/fce_ug_electrochem.07.121.html)
4. [Electrochemical Performance of Lithographically-Defined Micro-Electrodes, ChemEngineering (MDPI)](https://www.mdpi.com/2227-9040/9/10/277)
5. [Reversibility: Chemical vs. Electrochemical, Chemistry LibreTexts](https://chem.libretexts.org/Bookshelves/Analytical_Chemistry/Supplemental_Modules_%28Analytical_Chemistry%29/Analytical_Sciences_Digital_Library/Courseware/Analytical_Electrochemistry%3A_The_Basic_Concepts/03_Fundamentals_of_Electrochemistry/B%3A_The_Electrode_Process/02_Reversibility__Chemical_vs._Electrochemical)
6. [Department of Pure and Applied Chemistry thesis, University of Strathclyde](https://stax.strath.ac.uk/downloads/r494vk50j?locale=en)
7. [Low-Field Kinetics, OhioLINK thesis](https://etd.ohiolink.edu/acprod/odb_etd/ws/send_file/send?accession=osu1354652448&disposition=attachment)
8. [COMSOL 6.1 Electrochemistry Module User's Guide](https://doc.comsol.com/6.1/doc/com.comsol.help.echem/ElectrochemistryModuleUsersGuide.pdf)
9. [MIT 10.626 Lecture 13: Butler-Volmer equation (Bazant)](https://ocw.mit.edu/courses/10-626-electrochemical-energy-systems-spring-2014/56cfa6e0f28bc8fc1a647cbe679384d1_MIT10_626S14_S11lec13.pdf)
10. [Butler-Volmer Kinetics: an overview, ScienceDirect Topics](https://www.sciencedirect.com/topics/engineering/butler-volmer-kinetics)
11. [Microfabricated Reference Electrodes and their Biosensing Applications, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC3264446/)
12. [Taffit: An Excel Tool for Fitting Tafel Data, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC12371579/)
13. [Chapter 3: Kinetics — Overpotential and Tafel Analysis, AI Terakoya](https://yusukehashimotolab.github.io/AI-Knowledge-Notes/knowledge/en/MS/electrochemistry-introduction/chapter-3.html)
14. [Pillars of modern electrochemistry, Electrochemistry Encyclopedia](https://knowledge.electrochem.org/encycl/art-p05-pillars-of-ec.htm)
15. [Insight on Tafel slopes from a microkinetic analysis of aqueous electrocatalysis, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC4642571/)
16. [Butler-Volmer Equation: an overview, ScienceDirect Topics](https://www.sciencedirect.com/topics/engineering/butler-volmer-equation)
17. [Unified quantum theory of electrochemical kinetics by coupled ion–electron transfer, Faraday Discussions (RSC)](https://pubs.rsc.org/en/content/articlehtml/2023/fd/d3fd00108c)
18. [Theory of coupled ion-electron transfer kinetics, arXiv](https://arxiv.org/abs/2007.12980)
19. [doi.org](https://doi.org/10.1149/1945-7111/ad4399)

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*Topic: Encyclopedia › Physical world and mathematics › Chemistry › Chemical principles and methods › Analytical chemistry › Electroanalysis and electrochemistry › Electrode kinetics and electron transfer*

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