A binary system exhibits $2$ liquid phases at a certain pressure, with compositions $x_1^{}=$

$0\mathrm{.}2$ and $x_1^{}=0\mathrm{.}9.$ The vapor pressures are $P_1$ sat $=2$ atm; $P_2$ sat $=7$atm.

a$)$ Estimate the activity coefficients ${}_1^{}$ and ${}_2^{}.$ Explain any assumptions.

b$)$ Evaluate the $3-$phase pressure and all the mole fractions $($in the liquids and

Vapor phase$)$ at the three$-$phase pressure.

c$)$ Sketch the $P-x_1-y_1$ diagram.