Tafel Equation for Overpotential

In 1905 Julius Tafel, working on the electrochemical reduction of organic compounds, noticed that the extra potential needed to drive an electrode reaction rose with the logarithm of the current. He published it as an empirical rule. Twenty–five years later the Butler–Volmer equation explained it, and the Tafel form turned out to be its high-overpotential limit — the case where the reverse reaction has become negligible and only the forward one is left. The rule outlived its own derivation, which is usually a sign that it was seeing something real.

The logarithm is the whole content. A Tafel slope of 120 mV per decade means the overpotential must rise 120 mV for every tenfold increase in current density, so going from 1 to 1000 A/m² costs only 360 mV. This is why a corroding surface holds a nearly constant potential while the current on it swings over orders of magnitude, and why measuring the potential of a structure tells you far less about its corrosion rate than intuition suggests. Potential is a logarithmic instrument pointed at an exponential process.

The slope carries physical meaning. It emerges from the derivation as $b = 2.303 RT / (\alpha n F)$, where $\alpha$ is the charge transfer coefficient and $n$ the electrons in the rate-determining step. At room temperature with $\alpha$ near 0.5 and one electron, that gives about 120 mV per decade, which is why so many measured slopes cluster there. A slope near 60 mV suggests two electrons in the rate-determining step; a slope far outside 30–200 mV usually means the reaction is not under charge-transfer control at all, and the extrapolation you were about to do is invalid.

Which brings up the limits, because they are where this equation gets misused. The straight line exists only in a window. Below roughly 50 mV of overpotential the reverse reaction still matters and the relation curves toward the origin. Above some current the supply of reactant to the surface becomes limiting — oxygen diffusing through quiet water is the classic case — and the potential runs away while the current refuses to rise, giving a vertical branch that is not Tafel behaviour and has no slope worth reading. Tafel extrapolation, the technique of running the straight sections of both branches back to their intersection to find the corrosion current, is only as good as the straightness of what was extrapolated, and drawing a confident line through a curve is the most common way it fails.

One bookkeeping point that trips people constantly: the intercept $a$ is the overpotential where the current density equals one unit, so its value depends entirely on whether that unit is A/m², A/cm² or µA/cm². Quoting $a$ without its current-density units says nothing. The slope $b$ has no such problem — it is a ratio of the change over a decade, and a decade is a decade in any unit — which is why the slope travels between papers and the intercept does not.