Butler–Volmer_equation

# Butler–Volmer equation The **Butler–Volmer equation** is the central rate law of [[Electrochemical_kinetics|electrode kinetics]]: it gives the net current density *j* crossing an electrode surface as the difference between two exponentials in the [[Overpotential|overpotential]] η, the amount by which the electrode is driven away from its equilibrium potential. In the form used here, with concentrations at the surface held equal to those in the bulk, it reads `j = j*·[exp(eta/b_a) − exp(−eta/b_c)]`, where `b_a = R·T/(alpha_O·F)` and `b_c = R·T/(alpha_R·F)` are the two Tafel slopes and the transfer coefficients satisfy `alpha_O + alpha_R = 1`.[^hav-bv] The prefactor *j\** is the [[Exchange_current_density|exchange current density]], the equal and opposite current that flows in both directions at equilibrium, where the net current is zero but the interface is far from idle. In the microsim below the reader slides η from −0.30 V to +0.30 V for a symmetric electrode, α = ½, for which the two exponentials collapse into a single hyperbolic sine, `j = 2·j*·sinh(eta/b)` with `b = 2·R·T/F`.[^hav-sinh] Two linked panels answer the same question in different coordinates: `j/j*` against η on linear axes, and `log10|j/j*|` against η as a Tafel plot. Shaded on both is the linear window `|eta| < R·T/F`, which is 25.7 mV at 298.15 K, and drawn across both are the Tafel asymptotes `eta = b_a·ln(j/j*)`, whose slope on the logarithmic panel is 118 mV per decade at that temperature.[^hav-tafel][^hav-linear] A readout gives the marker's percentage error against each approximation, so the reader can watch one exact curve fail two straight-line descriptions in opposite directions. On the [[Chemistry|Chemistry]] flagship's spine this page is the main article for Part IX — Redox, section *Electrode kinetics: Butler–Volmer and Tafel*. It sits between the thermodynamics that fixes where equilibrium is — [[Nernst_equation|the Nernst equation]] and the [[Standard_electrode_potential|standard electrode potential]] — and the engineering pages that spend overpotential as a cost: [[Alkaline_water_electrolysis|alkaline water electrolysis]], [[Fuel_cell|fuel cells]], [[Corrosion|corrosion]] and [[Lithium-ion_battery|lithium-ion batteries]]. Its nearest sibling is [[Tafel_equation|the Tafel equation]], which is one of the two limits treated below. ## Butler–Volmer equation The equation is the electrochemical analogue of an [[Arrhenius_equation|Arrhenius]] rate law, with the electrode potential doing the work that temperature does in ordinary [[Chemical_kinetics|chemical kinetics]]. Shifting an electrode's potential by η changes the free-energy barrier for the forward and reverse reactions by complementary fractions of *F*η, so one direction accelerates exponentially while the other is suppressed exponentially, and the measured current is the difference. At η = 0 the two are equal and cancel: the exchange current density *j\** is what remains flowing in each direction, invisibly. The sign convention matters and is easy to lose. An anodic (oxidizing) overpotential is positive and produces positive current; a cathodic overpotential and its current are both negative.[^hav-signs] The full concentration-dependent form of the equation, the one that carries surface concentrations explicitly, is treated in its own section below; this section's version assumes the surface sees bulk concentrations, which is true until the current is large enough to deplete the reactant faster than [[Diffusion|diffusion]] can replace it.[^hav-bv] #### Symbols, and how large the slopes are *R* is the molar gas constant and *F* the Faraday constant, about 96,485 coulombs per mole of electrons.[^hav-faraday] Their combination `R·T/F` is the natural voltage scale of the whole subject: at 298.15 K it is 25.7 mV, which is why almost every characteristic voltage in electrode kinetics is a small multiple of about 25 mV.[^hav-linear] Because *b* is proportional to *T*, every one of those scales widens as the cell heats up. For the symmetric case α = ½, the anodic Tafel slope `b_a = R·T/(alpha_O·F)` is twice `R·T/F`, which the source rounds to about 0.05 V at ambient temperature and which is 51.4 mV at 298.15 K.[^hav-tafel] The quantity an engineer quotes is not *b* but the voltage cost of a tenfold increase in current, `b_a·ln 10`: the book gives about 120 mV per decade, and at 298.15 K the exact figure is 118 mV.[^hav-tafel] That arithmetic is the most common slip in the field — a slope quoted "per decade" is *b* multiplied by ln 10, so converting between the two without the factor 2.303 changes the answer by more than a factor of two.[^hav-signs] #### The symmetric electrode and the microsim Setting α_O = α_R = ½ makes the two Tafel slopes equal, and the difference of exponentials becomes `j = 2·j*·sinh(eta/b)` with `b = 2·R·T/F` — the only case of the Butler–Volmer equation that can be inverted in closed form, as `eta = b·asinh(j/(2·j*))`.[^hav-sinh] That single expression is what the microsim evaluates. There is no bake: the curve is about 400 exponential evaluations, recomputed on each slider change in well under a tenth of a millisecond, so the sim is in the live tier. The reader's one control is η, from −0.30 V to +0.30 V in 1 mV steps, default +0.05 V. What moves is a marker on both panels at once, and the two error readouts beneath them. Near the origin the sinh curve is indistinguishable from the straight line of a resistor, and the Tafel asymptotes are hopelessly wrong — at η = 0 the Tafel form predicts zero current where the truth is zero current for a completely different reason, and the logarithmic panel's asymptote heads to minus infinity. A few tens of millivolts out, the picture reverses: the linear approximation is already low by a large factor while the Tafel line has become an excellent fit. The window in between, roughly `|eta| < 25.7 mV` on one side and `j ≳ 2·j*` on the other, is where neither description is comfortable and only the exact expression will do.[^hav-tafel][^hav-linear] The aha the sim is built to deliver is that one curve carries two straight lines. Near equilibrium an electrode behaves like an ohmic resistor; a few *b* away it behaves like a diode. Both are the same equation, and the transition between them happens across a range of voltages a reader can sweep with a finger. ### Limiting cases Because the Butler–Volmer equation cannot generally be inverted analytically, working practice is built almost entirely on its two limits.[^hav-invert] Each is a one-line formula, each is exact in its own regime, and each fails badly outside it. The two limits also answer different questions: one tells you how stiff an electrode is against a small disturbance, the other how much voltage a large current will cost. The microsim exists to show where the boundary between them falls, and how wide the band is in which neither is usable. #### Low overpotential: the charge-transfer resistance For `|eta|` small compared with `R·T/F`, both exponentials can be expanded to first order and the difference becomes linear: `eta = (R·T/(F·j*))·j`, valid for η of order 25 mV or less.[^hav-linear] The coefficient has the units of an area-specific resistance, and it behaves like one — an electrode near equilibrium is, to a good approximation, a resistor whose value is set entirely by *j\**. A sluggish reaction with a small exchange current density is a large resistance; a fast one is a small resistance. This is why the exchange current density, not the equilibrium potential, is the number that decides whether an electrode material is worth using, and why [[Platinum|platinum]] remains the reference catalyst for hydrogen evolution. The practical consequence runs the other way too: because the linear coefficient contains `1/j*`, measuring the small-signal resistance of a cell is a direct measurement of how fast its slowest interface reaction is, without ever driving the cell far from equilibrium. It is the least destructive diagnostic in [[Electrochemical_engineering|electrochemical engineering]], and the reason a battery's internal resistance is quoted at all. #### High overpotential: the Tafel regime When the overpotential is large enough in the anodic direction, the reverse exponential is negligible and the equation reduces to `eta_a = b_a·ln(j/j*)`, a straight line on a semi-logarithmic plot.[^hav-tafel] The source gives the condition for its validity as roughly `j ≳ 2·j*`, and adds an upper bound that the equation itself cannot see: the Tafel form holds only until concentration gradients form at the surface, after which transport rather than charge transfer limits the current.[^hav-tafel] Extrapolating the Tafel line back to η = 0 is the standard experimental route to *j\**, and the slope is the standard route to α — which is why a Tafel plot is drawn for almost every new electrocatalyst. A worked cathodic example in the source, with `j* = 0.01 mA/cm²`, α_R = 0.65 and `j = −20 mA/cm²`, has no printed answer; evaluating it at 298 K gives η_c ≈ −0.30 V, a **derived** figure rather than a book result, and the reason the microsim's slider stops there.[^hav-ex11] ## Extended Butler–Volmer equation The form used above deliberately holds the surface concentrations fixed. The extended equation restores them, multiplying each exponential by the ratio of the surface concentration of the relevant species to its equilibrium value, so that a depleted surface reduces the current the same way a smaller rate constant would. Once that is done, the exchange current density is no longer an independent constant: it is itself built from the rate constants and the equilibrium concentrations, `j* = n·F·(k_O·c_R,eq)^alpha_R·(k_R·c_O,eq)^alpha_O`, a form recovered from a source whose exponents were split across lines in extraction and flagged for checking against the printed page.[^hav-eq127] Two length scales govern when the extension is needed, and confusing them is a standing error. The electrical double layer at the interface is of order a nanometre thick; the [[Boundary_layer|diffusion layer]] over which concentrations relax to their bulk values is 10 to 100 micrometres, four to five orders of magnitude larger.[^hav-tafel] Charge transfer happens across the first; the concentration corrections in the extended equation come from the second. As the current rises, the surface concentration of the reactant falls toward zero, the extra term grows without bound, and the electrode approaches a limiting current that no further overpotential can exceed. In a real cell, the electrode's overpotential is only one of several voltage losses, and the extended treatment is usually written at cell level. For an [[Electrolysis_of_water|alkaline water electrolyser]] the source assembles the whole polarization curve as `|V| = 1.23 + b_c·ln(j/j*_c) + b_a·ln(j/j*_a) + j·(L/kappa + AR)`: an equilibrium voltage, two Tafel terms, and a linear ohmic term for the [[Electrolyte|electrolyte]] gap and the electronic resistance.[^hav-polar] With the source's own parameters — `b_a ≈ b_c ≈ 50 mV`, `j*_c = 10⁻² A/cm²`, `j*_a = 10⁻⁴ A/cm²`, 0.5 Ω·cm² of electrolyte and 0.01 Ω·cm² of electronics — that curve gives 1.742 V at 0.1 A/cm² and 2.431 V at 1.0 A/cm² (**derived** from the book's parameters), against a printed figure spanning roughly 1.23 to 2.4 V over the same range.[^hav-polar] At low current the two activation terms dominate and the slow oxygen electrode, whose exchange current density is a hundred times smaller, costs by far the most; at high current the ohmic gap takes over. The logarithmic form has a defect the source repairs in a footnote. Below `j = j*` the term `ln(j/j*)` turns negative, so the model predicts an overpotential of the wrong sign — at 1 mA/cm² the cathodic term comes out at −0.115 V, which is unphysical. Replacing `ln(x)` with `asinh(x/2)` fixes it without changing the high-current behaviour, and it moves the predicted cell voltage at 10⁻³ A/cm² from 1.231 V to 1.349 V (**derived**).[^hav-asinh] The repair is the extended Butler–Volmer equation reasserting itself: `asinh` is exactly the inverse of the symmetric sinh form, so the fix amounts to using the full two-exponential expression instead of one of its limits. ### Derivation The derivation starts from [[Chemical_thermodynamics|thermodynamics]] and adds one kinetic assumption. Thermodynamically, the free energy available from a cell reaction fixes the equilibrium voltage through `dG = −n·F·V_eq`, which for the splitting of water gives 237 kJ/mol over two electrons, or 1.23 V.[^hav-faraday] How that equilibrium potential shifts with composition is the Nernst relation, `E_eq = E0' + (R·T/(n·F))·ln(c_O/c_R)`.[^hav-nernst] None of this says anything about rate: a cell held exactly at its equilibrium voltage passes no net current, however fast or slow the underlying reaction is. The kinetic assumption is the transfer coefficient. When the interfacial potential is shifted by η, the [[Transition_state|activated complex]] sits part-way across the interface, so only a fraction α_O of the energy *F*η lowers the barrier to oxidation, and the remaining fraction α_R = 1 − α_O raises the barrier to reduction.[^hav-bv] Writing each direction as an Arrhenius rate with its modified barrier and subtracting gives the two-exponential form directly, with `b_a = R·T/(alpha_O·F)` and `b_c = R·T/(alpha_R·F)`. Eliminating the equilibrium potential between the two rate expressions, using α_O + α_R = 1, yields the composite exchange current density quoted above.[^hav-eq127] Three consequences follow immediately and are worth stating as such. The symmetric case α = ½ gives equal slopes and the invertible sinh form. The equation has no closed-form inverse otherwise, which is why polarization curves are solved numerically and why the electrolyser curve above must be inverted by bisection to find the current at a given voltage.[^hav-invert] And because every slope carries a factor *T*, an electrode that is barely usable at room temperature can become acceptable hot — the argument behind high-temperature [[Electrolysis|electrolysis]] and behind the temperature dependence that [[Electrochemical_engineering|electrochemical engineering]] designs around. ## See also - [[Tafel_equation]] - [[Overpotential]] - [[Exchange_current_density]] - [[Electrochemistry]] - [[Electrochemical_kinetics]] - [[Nernst_equation]] - [[Alkaline_water_electrolysis]] - [[Corrosion]] ## Notes Sign conventions on this page follow the source: anodic overpotential and current are positive, cathodic ones negative, and the electrolyser figures are quoted as magnitudes because an electrolytic cell has `V_cell < V_eq < 0` on the source's convention. Tafel slopes quoted "per decade" are the slope *b* multiplied by ln 10, and the two are never interchangeable. The nanometre-scale double layer and the 10–100 µm diffusion layer are different objects. Values marked **derived** were computed from the source's own inputs and formulas because the printed result was lost in text extraction; they are candidates for checking against the original page, not book results. The supporting citations for all of these are given under References. ## References [^hav-bv]: Haverkort, J. W. (2024). *Electrolysers, Fuel Cells and Batteries: Analytical Modelling*. Chapter 1, Electrochemistry, p. 30 and pp. 33–34 (the concentration-independent Butler–Volmer equation `j = j*[exp(η/b_a) − exp(−η/b_c)]`; `b_a = RT/(α_O F)`, `b_c = RT/(α_R F)`, `α_O + α_R = 1`). Portal Book 053. https://open.umn.edu/opentextbooks/textbooks/electrolysers-fuel-cells-and-batteries-analytical-modelling [^hav-sinh]: Haverkort, J. W. (2024). *Electrolysers, Fuel Cells and Batteries: Analytical Modelling*. Chapter 1, pp. 34–35 (for α = ½ the equation becomes `j = 2 j* sinh(η/b)` with `b = 2RT/F`, invertible as `η = b·asinh(j/(2 j*))`; the sinh form requires α = ½). Portal Book 053. [^hav-tafel]: Haverkort, J. W. (2024). *Electrolysers, Fuel Cells and Batteries: Analytical Modelling*. Chapter 1, p. 33 (anodic Tafel form `η_a = b_a ln(j/j*)`, valid for j ≳ 2j* and only before concentration gradients form; `b_a ≈ 0.05 V` at ambient temperature for α = ½; one decade of current costs `b_a ln 10 ≈ 120 mV`; the double layer of order 1 nm against the 10–100 µm diffusion layer). The exact 298.15 K values, `b_a = 51.4 mV` and 118 mV per decade, are derived from `RT/F` and are not printed in the book. Portal Book 053. [^hav-linear]: Haverkort, J. W. (2024). *Electrolysers, Fuel Cells and Batteries: Analytical Modelling*. Chapter 1, p. 35 (the linear regime `η = (RT/(F j*))·j`, valid for η of order `RT/F ≈ 25 mV`). The value `RT/F = 25.7 mV` at 298.15 K is derived. Portal Book 053. [^hav-signs]: Haverkort, J. W. (2024). *Electrolysers, Fuel Cells and Batteries: Analytical Modelling*. Chapter 1, pp. 28 and 34, with p. 25 and p. 33 (cathodic current and overpotential are negative; an electrolytic cell has `V_cell < V_eq < 0`; Tafel slopes quoted per decade must be divided by ln 10). Portal Book 053. [^hav-faraday]: Haverkort, J. W. (2024). *Electrolysers, Fuel Cells and Batteries: Analytical Modelling*. Chapter 1, pp. 25–26 (`ΔG = −n F V_eq` with F ≈ 96,485 C per mole of electrons; water splitting, 237 kJ/mol over two electrons, gives `V_eq ≈ 1.23 V`; the thermoneutral voltage is 1.48 V). Portal Book 053. [^hav-nernst]: Haverkort, J. W. (2024). *Electrolysers, Fuel Cells and Batteries: Analytical Modelling*. Chapter 1, p. 32 (single-electron Nernst form `E_eq = E0' + (RT/(nF)) ln(c_O/c_R)`). Portal Book 053. [^hav-eq127]: Haverkort, J. W. (2024). *Electrolysers, Fuel Cells and Batteries: Analytical Modelling*. Chapter 1, Eq. 1.27, p. 32 (the concentration-dependent Butler–Volmer equation and the composite exchange current density). The exponents of `j* = n F (k_O c_R,eq)^{α_R} (k_R c_O,eq)^{α_O}` were split across lines in text extraction; the standard form is supplied here and is flagged in the Wikitube sub-manual for verification against the printed page. Portal Book 053. [^hav-invert]: Haverkort, J. W. (2024). *Electrolysers, Fuel Cells and Batteries: Analytical Modelling*. Chapter 1, p. 32 (the Butler–Volmer equation cannot be inverted analytically except in its limits). Portal Book 053. [^hav-ex11]: Haverkort, J. W. (2024). *Electrolysers, Fuel Cells and Batteries: Analytical Modelling*. Chapter 1, Exercise 1.1, p. 40 (`j* = 0.01 mA/cm²`, α_R = 0.65, `j = −20 mA/cm²`). The exercise has no printed answer; the cathodic Tafel result η_c ≈ −0.30 V at 298 K is derived. Portal Book 053. [^hav-polar]: Haverkort, J. W. (2024). *Electrolysers, Fuel Cells and Batteries: Analytical Modelling*. Chapter 1, Eq. 1.36 and Fig. 1.8, pp. 36–38 (the alkaline electrolyser polarization curve as equilibrium voltage plus two Tafel terms plus a linear ohmic term; parameters `b_a ≈ b_c ≈ 50 mV`, `j*_c = 10⁻² A/cm²`, `j*_a = 10⁻⁴ A/cm²`, 0.5 Ω·cm² electrolyte and 0.01 Ω·cm² electronic; printed curve spanning about 1.23–2.4 V over 0–1 A/cm²). The point values 1.742 V at 0.1 A/cm² and 2.431 V at 1.0 A/cm² are derived from those parameters. Portal Book 053. [^hav-asinh]: Haverkort, J. W. (2024). *Electrolysers, Fuel Cells and Batteries: Analytical Modelling*. Chapter 1, footnote 19, pp. 36–37 (replacing `ln(x)` with `asinh(x/2)` so that no overpotential goes negative below `j*`). The unphysical −0.115 V at 1 mA/cm², and the shift from 1.231 V to 1.349 V at 10⁻³ A/cm², are derived from the book's parameters. Portal Book 053. ## External links - *Electrolysers, Fuel Cells and Batteries: Analytical Modelling* (2024), Portal Book 053 — Open Textbook Library record: https://open.umn.edu/opentextbooks/textbooks/electrolysers-fuel-cells-and-batteries-analytical-modelling - The Wikipedia pair's External links section lists the pair's own links.  **Microsim — three.js (Wikitube framework), pending deploy:** *Butler–Volmer equation* will play here once `https://wikitube-3d-microsims.netlify.app/matter/Butler–Volmer_equation.html` is live.   ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Butler–Volmer_equation) : [Wikitube](https://en.wikitube.io/wiki/Butler–Volmer_equation) · pinned revision [1370651490](https://en.wikipedia.org/w/index.php?oldid=1370651490) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Chemistry]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Chemistry row K45 · sim pending (matter/Butler–Volmer_equation).*