$($a$)$ The volume for liquid mixtures of benzene $(1)$ and cyclohexane $(2)$ at $25C$ is given by

$V=109\mathrm{.}4-16\mathrm{.}8x_1-2\mathrm{.}64x_1^2.$ Calculate the volume change of mixing $(V)$ and excess

volume $(V^E)$ at $x_1=0\mathrm{.}4.$

$($b$)$ The solution behaviour of a certain class of substance is described by the following

expression $G={\displaystyle \underset{?}{\overset{?}{}}}{x}_i\frac{O}{{?}_i}+RT{\displaystyle \underset{?}{\overset{?}{}}}{x}_{i}ln(x_{i}P),(\frac{O}{{?}_i}$ is a function of T only$).$

Obtain the expression of volume of pure liquid $(V_l),$ partial molar Gibbs free energy $(\frac{?}{\text{b}\text{a}\text{r}}(G))$

and partial molar volume of component $i(\frac{?}{\text{b}\text{a}\text{r}}(V)).$

$($c$)$ For a binary system, the Gibbs free energy of a binary liquid mixture can be

represented by $\frac{G^B}{RT}=0\mathrm{.}5x_1{x}_2.$ Derive the expression of activity coefficient of

component $1.$

$10+10+5$