Consider VLE of a binary system at $303\mathrm{.}15K.$ The liquid$-$phase mixture is a non$-$ideal

solution. The liquid$-$phase activity coefficients are given by:

$ln{}_1=x_2^2,$ and $,ln{}_2=x_1^2$

The saturation pressures of species $1$ and $2$ at $303\mathrm{.}15K$ are $P_1^{\text{s}\text{a}\text{t}}=24\mathrm{.}36$kPa and $P_2^{\text{s}\text{a}\text{t}}=$

$3\mathrm{.}77$kPa, respectively and $=1\mathrm{.}165$

$($a$)$ Determine the liquid and vapor phase composition when total pressure is $17\mathrm{.}9$kPa.

Show step$-$by$-$step development of the equation used, starting from the criterion of

phase equilibrium. A trial$-$and$-$error calculation may be required.

$($b$)$ Refer to conditions in part $($a$),$ i$.$e$.,$ when total pressure is $17\mathrm{.}9$kPa. Determine the

excess gibbs energy for the liquid$-$phase mixture.

Please solve it clearly with steps