A binary system of species $1$ and $2$ consists of vapor and liquid phases in equilib$-$

rium at temperature $T,$ for which $ln{}_1=1\mathrm{.}8x_2^2,ln{}_2=1\mathrm{.}8x_1^2,P_1^{\text{s}\text{a}\text{t}}=\frac{1\mathrm{.}24}{b}ar,$ and

$P_2^{\text{s}\text{a}\text{t}}=40\mathrm{.}50$kPa. Assuming the validity of Eq$.(13\mathrm{.}19),$

$($a$)$ For what range of values of the overall mole fraction $z_1$ can this two$-$phase system

exist with a liquid mole fraction $x_1=0\mathrm{.}65?$

$($b$)$ What are the pressure $P$ and vapor mole fraction $y_1$ within this range?

$($c$)$ What are the pressure and composition of the azeotrope at temperature $T?$