# How to derive phase equillibrium criteria for more than 2 phases mathematically?

I know the derivation of phase equilibrium criteria derivation for 2 phases. I am not getting how to extend the derivation to more than 2 phases. Please help me out on that. Thank you

• Can you post what you do know? Posting what you have for two phases (along with any other useful information) will help others to provide you with answers.
– hazzey
Feb 5, 2019 at 17:16
• Are you talking about liquid vs. solid vs. gas? If so, I'm pretty sure there is no way to predict Triple Points for any element or chemical. Are you looking for some set of chemical reaction stability points? Feb 5, 2019 at 19:15
• @CarlWitthoft i am asking whether we can prove the chemical potential equality for equilibrium for more than 2 phases with more than 1 component. Feb 10, 2019 at 9:29
• @hazzey i have put up the derivation which i know Feb 10, 2019 at 9:30

The starting point at constant $$T, p$$ is the Gibb's energy in any phase $$\Pi$$ with components $$j$$.

$$dG^\Pi = \sum_j \mu_j^\Pi\ dz_j^\Pi$$

At phase equilibrium, the total Gibb's energy change is zero.

$$dG = 0 = \sum_\Pi \sum_j \mu_j^\Pi\ dz_j^\Pi$$

In any given phase, we have the mass balance equation.

$$\sum_j dz_j^\Pi = 0$$

We have as many of these as we have phases.

The remaining analysis is an exercise in applying the proper mathematics to combine the phase equilibrium equation with the $$\Pi$$ mass balance equations.