I'm running some simple simulations of the semiconductor-electrolyte interface and I'm having a hard time figuring out the equilibrium condition. Can someone please help me with my electrochemical potentials equation? Here's my thought flow so far:

I'm interested in finding out what is the Fermi level in the SC at equilibrium for OER: $4OH^- = O_2 + 2H_2O + 4e^-$. So I start by writing the condition for equilibrium:

$4\tilde \mu_{OH^-} = \tilde \mu_{O_2} + 4 \tilde \mu_{e^-} + 2 \tilde \mu_{H_2O}$

Since $O_2$ and $H_2O$ have no charge, their electrochemical potential is just their chemical potential ($\tilde \mu_{O_2} = \mu_{O_2}$ and $\tilde \mu_{H_2O} = \mu_{H_2O}$).

Moreover, the $\tilde \mu_{e^-}$ electrochemical potential of $e^-$ in my semiconductor electrode is just the Fermi level $E_F$ I'm looking for. Keeping in mind that $\tilde \mu_i = \mu_i^0 + RTln(a_i) + z_iF\phi$ and that the activity of water $a_{H_2O}=1$, I arranged the balance equation to:

$E_F = (\mu^0_{OH^-} - \frac{1}{4}\mu^0_{O_2} - \frac{1}{2}\mu^0_{H_2O}) + RTln\left(\frac{a_{OH^-}}{\left( \frac{p_{O_2}}{p^0}\right)}\right) -F\phi_{sol}$.

I can calculate the electrical potential $\phi_{sol}$ at the interface in the solution with the Poisson equation. I can calculate the activity coefficient $a_{OH^-}$ based on the $[OH^-]$ concentration at the interface and of course I know the partial pressure of $O_2$. My problem is how do I determine the chemical potentials at standard conditions from the first parantheses?

$(\mu^0_{OH^-} - \frac{1}{4}\mu^0_{O_2} - \frac{1}{2}\mu^0_{H_2O}) = ???$

Could I related them somehow to the SHE electrode which I know to be approx $4.5\ eV$ vs. vacuum?


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