The Gibbs energy can be used to predict phase separation. We derived the Gibbs energy of a mixture as

$g={\displaystyle \underset{?}{\overset{?}{}}}{x}_i{g}_i+RT{\displaystyle \underset{?}{\overset{?}{}}}{x}_{i}ln{x}_i+g^E.$ Use a reference state of the pure species at system $T$ and $P$ so the first

term is zero. This is then equivalent to $g_{\text{m}\text{i}\text{x}}(g_{\text{m}\text{i}\text{x}}=RT{\displaystyle \underset{?}{\overset{?}{}}}{x}_{i}ln{x}_i+g^E).$ For the following mixtures,

assume the three$-$suffix Margules equation accurately models $g^E.$

a$)$ Given a mixture of species a and $b,{}_a^{}=6$ and ${}_b^{}=32.$ Plot $\frac{g_{\text{m}\text{i}\text{x}}}{R}T$ vs $x_a$ and determine the

number of phases, the composition of each phase, and overall $\frac{g_{\text{m}\text{i}\text{x}}}{R}T$ for the solution at

$($i$)x_a=0\mathrm{.}1,($ii$)x_a=0\mathrm{.}5,$ and $($iii$)x_a=0\mathrm{.}8.$

b$)$ Is a miscible, partially miscible, or immiscible in $b?$

c$)$ Given a mixture of species a and $c,{}_a^{}=0\mathrm{.}8$ and ${}_c^{}=0\mathrm{.}2.$ Plot $\frac{g_{\text{m}\text{i}\text{x}}}{R}T$ vs $x_a$ and determine the

number of phases, the composition of each phase, and overall $\frac{g_{\text{m}\text{i}\text{x}}}{R}T$ for the solution at

$($i$)x_a=0\mathrm{.}1,($ii$)x_a=0\mathrm{.}5,$ and $($iii$)x_a=0\mathrm{.}8.$

d$)$ Is a miscible, partially miscible, or immiscible in $c?$