Extended electron transfer and the Frumkin correction

It is now well established that the rate constant for electron transfer between a metal electrode and a given redox moiety (at distance x from the electrode) decays according to exp[−βx]; values of β are of the order of 108 cm−1 and depend primarily upon the nature of the intervening medium. The present work demonstrates how this ‘extended’ electron transfer can modify the Frumkin correction for diffuse layer effects in the measurement of heterogeneous rate constants. We quantify the effect of this ‘extended’ electron transfer by comparing the electron transfer rate constants deduced with (k′eff) and without (keff) extended electron transfer. Butler–Volmer kinetics and a single electron transfer are assumed. The ratio k′eff/keff is deduced as a function of β/κ, where κ is the reciprocal Debye length; (F/RT)φPCA where φPCA is the diffuse layer potential at the plane of closest approach; and zox−α, where zox is the charge of the oxidized redox species and α is the Butler–Volmer transfer coefficient. The deviation of k′eff/keff from unity is greatest when the diffuse layer potential, φx, decays over distances of a few angstroms from the plane of closest approach, as is the case with higher concentrations of supporting electrolyte and/or large values of the diffuse layer potential. When |zox|≤|zse| and |zred|≤|zse| the classical Frumkin expression is applicable as long as−ln1+βκ≤FRT φPCAzox−α≤ln1+βκWhen |zox|>|zse| or |zred|>|zse| and (F/RT)φPCA(zox−α)<0 redox ions can become a significant component of the diffuse layer and the classical Gouy–Chapman theory is no longer applicable. To avoid this complication and apply the classical Frumkin correction the condition to be met is0≤(F/RT)φPCA(zox−α)≤ln[1+β/κ]thus ensuring that the redox species is not disproportionately accumulated in the diffuse layer. We also show that when β/κ≤0.5, φPCAzox≥0 and φPCAzred≥0 then virtually all ET occurs outside the diffuse layer and there is no Frumkin effect. However, this condition is not easily achieved experimentally.