Q$1.$ At $270K,$ a binary liquid mixture of species $1$ and $2$ phase separates into two liquid phases $$ and $$

with $x_1^{}=0\mathrm{.}22$ and $x_1^{}=0\mathrm{.}78.$ At $340K,$ the same mixture phase separates into the two liquid phases

with $x_1^{}=0\mathrm{.}34$ and $x_1^{}=0\mathrm{.}66.$ For the mixture, $\frac{g^E}{RT}=A(T)x_1{x}_2$ where $A(T)=a_0+a_{1}T.$

$($a$)$ Determine $A(T).$

$[10$ points$]$

$($b$)$ Determine the temperature at which the mixture is completely miscible.

$[5$ points$]$