Consider a mixture of $n-$hexane $(1)$ and n$-$heptane $(2)$ that is $35\%$ mole $n-$heptane in the liquid phase and at a pressure of $101\mathrm{.}325$kPa.

Determine the system temperature and the vapor phase mole fractions. In case of iteration, use the initial value of $T$ as $T=x_1{T}_1^{\text{s}\text{a}\text{t}}+x_2{T}_2^{\text{s}\text{a}\text{t}}.$ Show complete solution for the $1^{st}$ iteration, and for the final values of vapor phase mole fractions. State all your assumptions and the justifications of your assumptions. $(20$ points$)$

Consider a low$-$vapor pressure binary system $($consisting of species $1$ and species $2)$ in vapor$-$liquid equilibrium. The liquid phase is an equimolar mixture. If activity coefficients are given by $ln{}_1=A{x}_2^2$ and $ln{}_2=A{x}_1^2$ and the modified Raoult's Law is valid, prove that the mole fraction of species $1$ in the vapor phase is given by

$y_2=1-\frac{P_1^{\text{s}\text{a}\text{t}}}{P_1^{\text{s}\text{a}\text{t}}+P_2^{\text{s}\text{a}\text{t}}}$

where ${}_i=$ activity coefficients,

$P_i^{\text{s}\text{a}\text{t}}=$ pure species vapor pressure $of$ component $i$

$x_i=$ mole fraction $in$ liquid phase

$y_i=$ mole fraction $in$ vapor phase

Show your complete solution. Start solution from the Modified Raoult's Law. $(10$ points$)$