# Why is the electrochemical potential $\mu$ constant in the phase transition?

I was studying thermodynamics and my book shows that the electrochemical potentials have to be equal in a phase transition and I can't understand why. (the book I'm using is Callen).

• Because the Gibbs free energy is constant. Jan 9, 2019 at 1:19
• @JonCuster And what does that imply? Jan 9, 2019 at 2:00
• That thermochemical properties are constant when the two phases are in equilibrium. Jan 9, 2019 at 2:03
• @JonCuster We assume that it is in equilibrium when it passes from one state to another? Jan 9, 2019 at 2:36
• In a standard thermodynamics context (like a textbook) the two phases are indeed in equilibrium. That is how you can relate, say, the enthalpy of fusion to the entropy change. There are, certainly, non-equilibrium phase transitions (supercooling, etc.). Here you can still use thermodynamics but it is slightly trickier. Jan 9, 2019 at 13:46

## 1 Answer

This answer considers the case in which electrostatics are ignored (and thus the chemical potential is relevant rather than the electrochemical potential) because this is what I am most familiar with. I believe that the explanation could be generalized by replacing "$$T$$ and $$P$$" with "$$T$$, $$P$$, and $$q$$ (charge)" and "chemical potential" with "electrochemical potential.

Let's imagine a closed system at constant $$T$$ and $$P$$. It can be shown (and I will take it as a given) that, under these conditions, a process is spontaneous if and only if it leads to a reduction in Gibbs energy (chemical potential).

For a $$T$$ and $$P$$ at which the system exists as a single phase, that phase has a lower chemical potential than all other phases would at the same $$T$$ and $$P$$. Transitions from any other phase to the most stable phase are spontaneous, while transitions from the stable phase to any other phase are impossible. We therefore see only the one phase (at equilibrium, at least).

In order to be able to observe two phases at equilibrium given a constant $$T$$ and $$P$$, both phases must be equally stable at that $$T$$ and $$P$$. This means that they must both have the same chemical potential, and that transitions from one phase to the other are reversible at that $$T$$ and $$P$$.