Extended electron transfer and the Frumkin correction

It is now well established that the rate constant for electron transfer bet ween a metal electrode and a given redox moiety (at distance x from the ele ctrode) decays according to exp[ - betax]; values of beta are of the order of 10(8) cm(-1) and depend primarily upon the nature of the intervening med ium. The present work demonstrates how this 'extended' electron transfer ca n modify the Frumkin correction for diffuse layer effects in the measuremen t of heterogeneous rate constants. We quantify the effect of this 'extended ' electron transfer by comparing the electron transfer rate constants deduc ed with (k'(eff)) and without (k(eff)) extended electron transfer. Butler-V olmer kinetics and a single electron transfer are assumed. The ratio k'(eff )/k(eff) is deduced as a function of beta/kappa, where kappa is the recipro cal Debye length; (F/RT)phi (PCA) where phi (PCA) is the diffuse layer pote ntial at the plane of closest approach; and z(ox) - alpha, where z(ox) is t he charge of the oxidized redox species and alpha is the Butler-Volmer tran sfer coefficient. The deviation of k'(eff)/k(eff) from unity is greatest wh en the diffuse layer potential, phi (x), decays over distances of a few ang stroms from the plane of closest approach, as is the case with higher conce ntrations of supporting electrolyte and/or large values of the diffuse laye r potential. When /Z(ox)/ less than or equal to /z(se)/ and /Z(red)/ less t han or equal to /Z(se)/ the classical Frumkin expression is applicable as l ong as -ln[1 + beta/kappa] less than or equal to F/RT phi (PCA)(z(ox) - alpha) les s than or equal to ln[1 + beta/kappa] When /Z(ox)/ > /Z(se)/ or /Z(red)/ > /Z(se)/ and (F/RT)phi (PCA)(Z(ox) - al pha)< 0 redox ions can become a significant component of the diffuse layer and the classical Gouy-Chapman theory is no longer applicable. To avoid thi s complication and apply the classical Frumkin correction the condition to be met is 0 <less than or equal to> (F/RT)phi (PCA)(z(ox) - alpha) less than or equal to ln[1 + beta/kappa] thus ensuring that the redox species is not disproportionately accumulated in the diffuse layer. We also show that when beta/kappa less than or equal to 0.5, phi (PCA)Z(ox) greater than or equal to 0 and phi (PCA)z(red) great er than or equal to 0 then virtually all ET occurs outside the diffuse laye r and there is no Frumkin effect. However, this condition is not easily ach ieved experimentally. (C) 2000 Elsevier Science B.V. All rights reserved.