# Calculate chemical potentials to find electrode Fermi level

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?