Proof of equality of Gibbs free energy at a phase coexistence line

If the system is in equilibrium at a fixed pressure and temperature, the Gibbs free energy must be a minimum. That means that $G$ will be unchanged under a further small shift of mass from one phase to the other.

Now the total Gibbs free energy is the sum of the gibbs free energy of each phase. If we introduce the ``specific Gibbs free energy'', the Gibbs free energy per unit mass, and if the mass of each phase is $m_1$ and $m_2$, we can write

$$G=m_1 g_1+m_2 g_2$$

At equilibrium a small transfer of mass ${\rm d}m$ from one phase to the other won't change the total $G$:

$${\rm d}G={\rm d}m_1 g_1+{\rm d}m_2 g_2={\rm d}m g_1-{\rm d}m g_2=0$$

This can only be true if $g_1= g_2$.

So the condition for phase coexistence is that the Gibbs free energy per unit mass in each phase is the same.