The Butler–Volmer Equation: Energy Barriers and Exchange Current
An electrode reaction has to climb an energy barrier, just like any chemical reaction. The overpotential tilts the energy profile: it lowers the barrier in one direction and raises it in the other, which is why the current depends exponentially on η. The exchange current density plays the part of the Arrhenius pre-exponential factor. Work through the three stages in order.
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Predict, then see: Zn ⇄ Zn2+ + 2e−,
Or explore freely:
0.00 V
Predict, then see:
Or explore freely:
0.000 V
Each moving dot is one zinc atom or ion crossing the barrier: orange for oxidation (Zn → Zn2+ + 2e−), blue for reduction. The close-up of the electrode shows the same events: ions leaving or joining the metal, and electrons moving in the electrode.
The electrons released by oxidation end up in the electrode, at its Fermi level. A positive η lowers the Fermi level, so Zn2+ + 2e− lies lower in energy and the oxidation barrier is smaller.
The faint curves are the other metals in the lecture table: compare how far each must be pushed before a current flows.
Use the button to find the overpotential needed to deposit each metal at 10 mA cm−2.
The orange half of the graph is oxidation (anodic, i > 0); the blue half is reduction (cathodic, i < 0).
Copper and silver cross at about −0.12 V. A larger i0 means a larger current near equilibrium, but n sets how steeply the current grows: with n = 2, copper's current rises twice as steeply as silver's.
Remember
Butler–Volmer equation: i = i0[eαnFη/RT − e−(1−α)nFη/RT]. The first term is oxidation (anodic, positive current); the second is reduction (cathodic, negative current).
An overpotential η changes the free energy of the products relative to the reactants by −nFη. With α = 0.5, the oxidation barrier falls by αnFη and the reduction barrier rises by (1 − α)nFη.
The barriers change linearly with η, so the rates change exponentially (Arrhenius): kox/kred = enFη/RT.
The exchange current density, i0, is the current in each direction at equilibrium. It is the electrochemical equivalent of the Arrhenius factor A: a small i0 means slow electron transfer, so a larger η is needed for the same current.
α, the transfer coefficient (symmetry factor), gives the position of the transition state. In this course α = 0.5.
Stage 1 uses an illustrative barrier of 50 kJ mol−1 at equilibrium. Exchange current densities are those in the lecture table. The value of n used in the exponent is the number of electrons in the half-reaction, as in the lecture examples.