$(20$ points$)$ For a regular solution of a binary system A and $B,($a$)$ Please prove $$ function ${}_i=\frac{ln{}_i}{(1-x_1{)}^2}$prop$\frac{1}{(T)}.$ Hint: Using G$-$H eqn and $\frac{}{b}ar(G{)}_i=RTln{a}_i.(5$ points$)($b$)$ Use $($a$)$ to explain the pressure changes at rising temperature in the positive and negative deviations. $(5$ points$)($c$)$ Given $H^M=*x_A{x}_B,$ please explain the physical meaning of $$ and its statistic relations with the enthalpies $\frac{}{b}ar(H{)}_A$ and $\frac{}{b}ar(H{)}_B$ in the aspect of the binding probabilities $x_A$ and $x_B.(5$ points$)($d$)$ What does RT ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{}_{A}d{x}_A$ calculate for? Must derive and explain the equation. $(5$ points$)$; $H^M$ is the enthalpy after mixing. Hint for $($c$)($d$)$: How many A and $B$ break their bonds and bind to $B$ and $A.$