We have shown that the criteria for equilibrium is:

$dG \le 0$

For a mixture of two phases we can write:

$G = n_lg_l + n_vg_v$

so that

$dG = n_ldg_l + g_ldn_l + n_vdg_v + g_vdn_v$

Since we are assuming thermal equilibrium (constant $T$) and mechanical equilibrium (constant $P$), the state postulate says that $g$ is fixed so that $dg_l = dg_v = 0$, so

$dG = g_ldn_l + g_vdn_v$

At equilibrium the rate of change of material *from* one
phase must be balanced by the rate of change of material
*to* that phase, so that $dn_v = -dn_l$ and

$dG = (g_l-g_v)dn_v \le 0$

Once we reach equilibrium the inequality takes on the value of zero, so

$g_l=g_v$

Write down the conditions for equilibrium for: a pure single phase system, a pure multi-phase system, and a multi-phase mixture