We have shown that, for pure components, equilibrium is given when:

$dG = 0$

or

$g_L = g_v$

For a mixture of multiple components (and at least two phases) we can start at the same place:

$dG = 0 = dG_L + dG_v = \left [-SdT + VdP + \sum \bar G_i dn_i \right ]_L + \left [-SdT + VdP + \sum \bar G_i dn_i \right ]_v$

Since we are again assuming mechanical and thermal equilibrium, we have

$T_L = T_v$ and $P_L=P_v$ so that

$dT_L = dT_v = 0$ and $dP_L=dP_v=0$

this simplifies our $dG$ expression to

$dG = 0 = \left [\sum \bar G_i dn_i \right ]_L + \left [\sum \bar G_i dn_i \right ]_v$

At the microscopic scale, the rate of change of material
*from* one phase must be balanced by the rate of change of
material *to* that phase *for each component*, so
that $dn_{i_v} =-dn_{i_L}$ and

$dG = 0 = \sum \left [ \bar G_{i_L} - \bar G_{i_v} \right ]dn_{i_L}$

This is true when the partial molar Gibbs free energy of each component is equal across the phases. That is

$\bar G_{i_L}=\bar G_{i_v}$

Differences in the partial molar Gibbs free energy create a
*potential* for the material to change phase. Thus, thie
quantity is often called the **chemical potential** or
species, $i$, and denoted as $\mu_i$.

$\mu_i = \bar G_i$

so we can write our phase equilibrium condition (for mixtures) as

$\mu_{i_L}=\mu_{i_v}$

Explain why the chemical potential is the relevant property for determining solution equilibrium