Question about the Butler-Volmer equation in electrochemistry

[Dishsoap]

Homework Statement

Not an actual homework question, but as a physicist learning about electrochemistry I feel it is about that level.

My question is, how do I calculate the equilibrium potential $E_{eq}$ of an electrode in a fluid with no initial concentration of products?

Relevant Equations

(also pasting in main text in case LaTex doesn't work here)

$i\propto \frac{C_R(0,t)}{C_R^*}e^{(1-\alpha)f \eta}$
$\eta = E-E_{eq}$
$E_{eq}=E^{0'}+\frac{RT}{F}ln(\frac{C_O^*}{C_R^*})$

I am reading Bard & Faulkner, who states that the current given by the Butler-Volmer equation is related to the overpotential $\eta$, which is the $\eta=E_{app}-E^0-E_{eq}$. The equilibrium potential depends strongly on the initial bulk concentration of products and reactants, $E_{eq}=E^{0'}+\frac{RT}{F}ln(\frac{C_O^*}{C_R^*})$.Eapp.

This makes sense for e.g. the water electrolysis reaction, and says that the potential needed for the reaction to proceed is around 0.8V at neutral pH instead of the formal potential of 1.3V due to the presence of $H^+$ in the solution.

However, what about reactions where there is no product in the fluid initially? For instance, the chlorine redox reaction $2Cl^- -> 2e^- + Cl_2(g)$. If there is no chlorine in the fluid to begin with, how does one write down the equilibrium potential?

My first thought is that in the logarithm for $E_{eq}$, either $C_O^*$ or $C_R^*$ would be zero depending on which direction the reaction would proceed. However, the logarithm of zero is undefined. How does one proceed?

I just want to figure out at what electrode potential I should expect to begin performing the redox reaction on Cl- ions in the fluid.

Thank you everyone!

 

[Borek]

When there is no products from one system some other system jumps in (plenty of these) and limits the potential. If there is no Cl₂ O₂ is everywhere.