Equilibrium between phases in heterogeneous closed systems

3 Equilibrium between phases in heterogeneous closed systems 

If we combine the Clausius inequality given in Equation (113) with Equation (170), we obtain for both irreversible and reversible systems 

This inequality applies to all incremental changes towards the equilibrium state, and the equality holds at the equilibrium state where any change is reversible. It follows immediately that when S and V are constant, 

Gibbs was the first to prove that in a heterogeneous (multiphase) closed system, the chemical potential of every phase is equal to the chemical potential of the other phase, which is in equilibrium with it  

When we apply Euler s reciprocity relation to obtain the total differential of entropy by using Equation (176) for a constant interfacial flat boundary area, A, between them, we have 

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