Phase Separation in Solutions

In this section, we first develop the general property of the G vs. X relation at the extreme compositions, which has important consequences in the problem of the stability of a solution. For a binary solution 3.1$$ \Delta {G^{{mix}}} = RT\left( {{X_1}\ln {a_1} + {X_2}\ln {a_2}} \right) $$ As X₂→0, we have, according to laws of dilute solution (Chap. l.III), $$ {\tilde{X}_2} $$, and $$ {a_1} = {\tilde{X}_1} $$, where $$ {\tilde{X}_i} $$ is an appropriate compositional parameter with the property that at X₂ = 0, $$ {\tilde{X}_2} = {X_2} $$ and $$ {\tilde{X}_1} = {X_1} $$ [see Eq. (1.9) and discussion]. For the sake of simplicity, let us assume that $$ {\tilde{X}_2} = {X_2} $$ and $$ {\tilde{X}_1} = {X_1} $$. Then, differentiating Eq. (3.1) with respect to X₂, and substituting the dilute solution properties of a₁ and a₂, we find that as X₂→0, 3.2$$ \frac{{\partial \Delta {G^{{mix}}}}}{{\partial {X_2}}} = RT\left( {\ln \frac{{{X_2}}}{{{X_1}}} + \ln {{K'}_H}} \right) $$ where $$ {K'_H} $$ is a constant [see Eq. (1.36)].