$(50$p$)100$kmol mixture containing $35\%$ ethanol $($A$)$ and $65\%$ water $($B$)$ by mole; It is heated and evaporated under differential conditions at a pressure of $101\mathrm{.}32$kPa until $25$ kmol is distilled.

a$)(20$p$)$ Find the composition of the liquid remaining at the end of distillation,

b$)(10$p$)$ Calculate the average composition of distilled steam $($distillate$),$

c$)(10$p$)$ Find the compositions of the bottom and top product when equilibrium distillation $($flash$)$ is applied to the same mixture to obtain $25$ kmol of steam $($distillate$).$

d$)(10p)$ Compare the product recoxery and product purity values obtained by using the differential and flash distillation operations.

Table $2.$ Vapor$-$liquid equilibrium data for ethanol$-$water system

$\backslash$table$[[$xA ${?}_A,0,0\mathrm{.}02,0\mathrm{.}072,0\mathrm{.}12,0\mathrm{.}23,0\mathrm{.}40,0\mathrm{.}51,0\mathrm{.}57,0\mathrm{.}68,0\mathrm{.}75,0\mathrm{.}89],[$XA$,0,0\mathrm{.}17,0\mathrm{.}39,0\mathrm{.}47,0\mathrm{.}54,0\mathrm{.}61,0\mathrm{.}66,0\mathrm{.}68,0\mathrm{.}74,0\mathrm{.}78,0\mathrm{.}89]]$