$(19)*$ A distiller working in a Montreal Microdistillery is trying to get an improved understanding of his distillation process. He obtained the VLE diagram for an Ethanol$-$Water mixture $($hereunder$)$ at $P=1\mathrm{.}01325{10}^{5}Pa.$ His distillation is apparatus allows for only $1$VLE step to occur. The conditions for the use of the modified Raoult's law apply.

Data taken from Dortmund Data Bank

Vanor.l iauid Fauilihrium

a$.(3)$ The distiller is distilling a mixture of ethanol and water containing $10\%($molar$)$ ethanol. At what temperature will the mixture boil?

b$.(3)$ He measures the boiling point of another mixture at $358K,$ what is the composition of the vapor?

c$.(3)$ If the distiller condenses this vapor $($the one from b$.,$ boiling at $358K)$ and distils it again, what will be the boiling point of the liquid, what will be the composition of the new vapor?

d$.(6)$ Extract the equilibrium liquid and vapor compositions data at $356K$ from the graph. From this single data point, create a model for the activity coefficient using the Margules model, but assuming that