$(4)$ For acetone $(1)/$methanol $(2)$ system a vapor mixture for which $Z_1=0\mathrm{.}25$ and $Z_2=0\mathrm{.}75$ is cooled to temperature $T$ in the two$-$phase region and flows into a separation chamber at ar pressure of $100$kPa. If the composition of the liquid product is to be $x_1=0\mathrm{.}20,$ determine the following:

$($a$)$ The required value of $T$

$($b$)$ The mole fraction of methanol in the vapor.

For the liquid mixture of this system, the activity coefficients are given by the following equations:

$ln{}_1=0\mathrm{.}62x_2^2,ln{}_2=0\mathrm{.}62x_1^2$

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