$(20\%)$ Consider the case of binary VLE at low pressures.

$($a$)$ Determine an expression for the relative volatility ${}_{12}$ of species $1$ to species $2$ ; by definition,

${}_{12}-=\frac{K_1}{K_2}=\frac{\frac{y_1}{x_1}}{\frac{y_2}{x_2}}$

where the $K_i$ are the equilibrium $K-$values.

$($b$)$ Use the result of part $($a$)$ to develop a procedure for checking for azeotrope formation in a binary system.