Concentration dependence of electron transfer

Question

Regarding a single electron transfer: $$O +e^- \rightarrow R$$ We find the current dependent on the overpotential as specified by the Butler Volmer equation: $$i = i_0 (exp(\frac{\eta}{b})-exp(\frac{-\eta}{c}))$$ With $\eta =E^{applied}-E^{eq}$ the overpotential, $i_0$ the exchange current density, b and c Tafel slopes for the anodic and cathodic reaction. $E^{eq}$ is defined by the Nernst equation: $$E^{eq}=E^0-\frac{RT}{F}ln(Q)$$ We find thus that the overpotential at the electrode surface is dependent on the concentration at the surface. The exchange current density itself however is said to be: $$i_0 = Fk_0C_O^{\alpha}C_R^{\beta}$$ In some books, pages online, and review articles I've seen these thrown around so much that it is not clear anymore how the current density is affected by the concentration. If you rework the first equation with the Nernst equation one can get the Q out of the exponential, but is this already taken into account in the exchange current density factor or not? Is the current density "doubly" dependent on the concentration as in the overpotential is dependent on the current density and there is some dependence of the exchange current density?

Answer 1

Yes, the current density is "doubly" dependent on the concentration, in the following sense that you have written:

1. $ C_O(0,t) $ and $ C_R(0,t) $ appear in $E^{eq}$ within the overpotential defind as $ \eta = E - E^{eq} $ where $ E $ is the potential difference between the electrode surface and the bulk solution, i.e., $ E = \phi_M - \phi_S $.
2. $ C_O(0,t) $ and $ C_R(0,t) $ appear also in the exchange current density, which formally, has the following form

$$ j_0 = F k^\Theta \bigg(\dfrac{C_O(0,t)}{C_O^\Theta}\bigg)^{1 - \alpha} \bigg(\dfrac{C_R(0,t)}{C_R^\Theta}\bigg)^\alpha $$ where $ \alpha $ is the symmetry factor, and $C_O^\Theta$ with $C_R^\Theta$ are the reference concentrations associated to the rate constant $ k^\Theta$.

You can see this "double" dependence in textbooks that derive B-V kinetics by using the Transition State Theory, or Eyring Theory, applied to charge transfer reactions.

Disclaimer: If you always talk about equilibrium, the concentrations at the electrode surface and the bulk are the same, so there is only one concentration to talk about. However, the kinetics are evaluated at the electrode surface, so it is a good practice to put $ (0,t) $, when it applies. When you write "$C_O$" or "$C_R$" in electrochemistry, my first question is, where is the $O$ and $R$ you are talking to me about? (my explanation does not take into account double-layer effects, where the evaluation is more subtle)