$(15\%)$ The Gibbs free energy mixing curve of a binary regular solution system of Components A and $B$ at a given temperature, $T,$ and pressure, $P,$ is given by: $\frac{G^M}{R}T=x_{A}ln{x}_A+x_{B}ln{x}_B+x_A{x}_B,$ where $=\frac{}{R}T,$ is set to $16630J$ and $R$ is $8\mathrm{.}314J{K}^{-}{?}^{1}mo{l}^{-1}.$ For an ideal solution, $=0.$

$($a$)$ Determine the mathematical expression for the miscibility curve connecting the binodal points of a family of the Gibbs free energy mixing curves in the $T$ versus $x_B$ space. $($Hint: let del$\frac{G^M}{d}el{x}_B=0).$

$($b$)$ Plot out the miscibility curve by feeding $x_B=0,0\mathrm{.}05,0\mathrm{.}10,0\mathrm{.}15,0\mathrm{.}20,0\mathrm{.}25,0\mathrm{.}30,0\mathrm{.}35,0\mathrm{.}40,0\mathrm{.}45,0\mathrm{.}50,0\mathrm{.}55,0\mathrm{.}60,0\mathrm{.}65,0\mathrm{.}70,0\mathrm{.}75,0\mathrm{.}80,0\mathrm{.}85,0\mathrm{.}90,0\mathrm{.}95$ and $1\mathrm{.}0$ into the function of $T=f(x_B)$ to obtain $T$ in Part $($a$).$

$($c$)$ The binary system containing $30\%$ of Component B is raised to $800K.$ Will it experience phase separation based on the miscibility curve in Part $($b$)?$