The Tafel Equation: When the Reverse Reaction Can Be Ignored
The Butler–Volmer equation contains two terms: one for oxidation and one for reduction. Close to equilibrium both matter, but a little way from it the reverse reaction becomes so slow that it can be ignored. The equation then simplifies to the Tafel equation, and a plot of log |i| against η becomes a straight line from which the exchange current density can be read. Work through the three stages in order.
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Predict, then see:
Or explore freely:
0.000 V
Predict, then see:
Or explore freely:
−0.200 V
The dashed curves are the two terms of the Butler–Volmer equation: orange for oxidation, blue for reduction. The solid curve is their sum, the net current.
The bar under the graph compares the reverse reaction with the forward one. Once it is below 1 %, the reverse term can be ignored: this is the Tafel region.
The dashed lines are the Tafel equation. Far from η = 0 they lie on top of the Butler–Volmer curve; close to η = 0 the curve bends away, because the reverse reaction is no longer negligible.
Compare silver (n = 1) with copper (n = 2): copper's line rises twice as steeply, which is why the two Butler–Volmer curves cross in the previous lesson.
Remember
Butler–Volmer: i = i0[eαnFη/RT − e−(1−α)nFη/RT]. With α = 0.5 the reverse term is a fraction e−nF|η|/RT of the forward term.
The reverse reaction falls below 1 % of the forward reaction when |η| > 0.118 V/n: 0.12 V for n = 1 and 0.06 V for n = 2.
Beyond this point (the Tafel region) only one term remains. For reduction, |i| = i0e(1−α)nF|η|/RT; taking logarithms gives the Tafel equation, |η| = b log(|i|/i0).
Tafel slope: b = 2.303RT/(αnF), the extra overpotential needed for each tenfold increase in current. With α = 0.5 at 298 K: 118 mV per decade for n = 1, 59 mV per decade for n = 2.
On a Tafel plot (log |i| against η), the straight line extrapolated back to η = 0 meets the axis at log i0.
Exchange current densities are those in the Lecture 5 and 6 tables. The value of n in the exponent is the number of electrons in the half-reaction, as in the lecture examples, and α = 0.5 throughout.