$2\mathrm{.}24.$ Activity coefficients over the composition range from infinite$-$dilution values.

For ethanol$(1)-$isooctane$(2)$ mixtures at $50C,$ the infinitedilution, liquid$-$phase activity coefficients are ${}_1^{}=21\mathrm{.}17$ and ${}_2^{}=9\mathrm{.}84.$

$($a$)$ Calculate the constants $A_{12}$ and $A_{21}$ in the van Laar equations.

$($b$)$ Calculate the constants ${}_{12}$ and ${}_{21}$ in the Wilson equations.

$($c$)$ Using the constants from $($a$)$ and $($b$),$ calculate ${}_1$ and ${}_2$ over the composition range and plot the points as $log$ versus $x_1.$

$($d$)$ How well do the van Laar and Wilson predictions agree with the azeotropic point $x_1=0\mathrm{.}5941,{}_1=1\mathrm{.}44,$ and ${}_2=2\mathrm{.}18?$

$($e$)$ Show that the van Laar equation erroneously predicts two liquid phases over a portion of the composition range by calculating and plotting a $y-x$ diagram like Figure $2\mathrm{.}16.$