The activity coefficient of component $A$ in a binary regular solution is known to be

${}_A=$exp$[\frac{C}{T}(1-x_A{)}^2]$

where $C$ is a constant, $x_A$ is the mole fraction of A$,T$ is the temperature.

$($a$)$ Derive the partial molar free energy of component B$.$

$($b$)$ Derive the molar Gibbs free energy change of mixing.

$($c$)$ Derive the molar entropy change of mixing.

$($d$)$ Derive the molar enthalpy change of mixing.

The answer must be expressed in terms of $R,T,C,x_A,G_A^0,G_B^0.$