I want to write a quick derivation of the Nernst Equation, as it is widely used in the Li-S battery modelling community to describe the potentials of electrochemical reactions. It will be useful to understand what the equation assumes to be true and what approximations are often made in the literature. This will essentially be a consolidation of the material in Atkins’ Physical Chemistry textbook.
In part 1, I will give the state variables which describe the macroscopic variables of the battery. These state variables will then be related in a form more suitable for the next part.
In part 2, I will use the state variables and some extra assumptions to derive the Gibbs energy for a perfect gas. The form of the Gibbs energy for a perfect gas will be altered to allow non-perfect gases. This introduces an empirical function, the fugacity. Although not directly related to chemical potential, this is a helpful stepping stone. Once the idea of fugacity is understood, it is a small step to inserting the activity functions of (electro)chemical reactions.
In part 3, I will define the Gibbs reaction energy. This takes the form very close to Gibbs energy for a perfect gas and the introduction of activities parallels that of fugacity. This is all brought together in direct application to electrochemical reactions, yielding the Nernst Equation. Here, I will make a short comment on the assumptions used and how this equation is generally formulated in the Li-S modelling community.