On Contemporary Tafel Analysis

For nearly a century, Tafel slope analysis has anchored mechanistic interpretation across electrochemistry. We demonstrate that the contemporary framework relating Tafel slopes to rate-determining step identity and position is mathematically incomplete: its derivation omits a chain-rule term that vanishes only for an effectively empty surface. We derive one master expression for the stationary Tafel slope of any linear reaction sequence, valid across adsorption models, and a closed matrix form for arbitrary reaction networks. At its center stands the surface stiffness: the layer of adsorbed intermediates behaves as a spring that stiffens as it fills, and any Tafel slope above its electron-transfer-only value records the additional overpotential spent against this stiffness. The contemporary relation is recovered as the low-coverage case; for bound adsorption, the electrons preceding the rate-determining step must be counted from the step consuming the most abundant surface species. The same network solution yields Campbell's degree of rate control in closed form: small Tafel slopes place rate control on a late irreversible step behind reversible predecessors, while high slopes move it upstream or into charge withheld by the surface. Applied to the hydrogen evolution reaction, this picture holds across paths; the corrected framework captures our measured 140-190 mV/dec on platinum in alkaline media, values the classical relations cannot produce, and removes the need for heuristically defined Tafel regions.