$11\mathrm{.}4.$ The Wilson model for $G^{Ex}$ is particularly useful in predicting non$-$idealities in both polar

and nonpolar mixtures and it can easily be extended to multicomponent systems using

only binary parameters $({}_{ij}$ or ${}_{ij}).$

$(1)$ For an equimolar mixture of two components ${}_1=1\mathrm{.}2$ and ${}_2=1\mathrm{.}5$ at a particular

temperature and pressure. Determine Wilson parameters and ${}_{21})$ and plot

In ${}_1$ and $ln{}_2$ over the range of compositions from $x_1=0$ to $1\mathrm{.}0.$

$(2)$ Develop an expression to calculate the activity coefficients at infinite dilution $({}_i^n)$

and evaluate ${}_1^r$ and ${}_2^2.$

$(3)$ Determine values of Wilson parameters that give extrema is variations of $ln{}_1$ as a

function of $x_1$ at constant $T$ and $P.$