Consider the vapor$-$liquid equilibrium of a binary mixture of two high boiling point

species. At $300$K$,$ species $(1)$ has vapor pressure of $415$kPa and species$-2$ has vapor

pressure of $820$kPa.

The gibbs excess energy $(g^E)$ of the liquid mixture can be described by the following

equation:

$\frac{g^E}{R}T=x_1{x}_2(A+B{x}_2)$

where, $A=0\mathrm{.}25$ and $B=0\mathrm{.}35$

a$)$ Derive the expression for activity coefficients for each species, i$.$e$.$ derive the

expressions for $ln({}_1)$ and $ln({}_2).$ Step$-$by$-$step derivation is expected for

securing full marks.

b$)$ Assuming the vapor$-$phase is an ideal gas, calculate the vapor composition and

total pressure of the vapor$-$liquid system with a liquid phase composition of $x_1=0\mathrm{.}7$

at $300K.$ State all assumptions made.

c$)$ If species$-1$ and species$-2,$ both at $300K,$ are mixed in equimolar ratio to obtain a

liquid mixture at $300K,$ calculate the total change in gibbs energy of mixing at $300K.$

Please solve it clearly with steps