$6\mathrm{.}67.$ A methanol$-$water feed stream is introduced to a vaporizer in which a molar fraction $f$ of the feed is vaporized. The feed has a methanol mole fraction of $x_F=0\mathrm{.}4,$ and the vaporizer operates at a pressure of $1$atm absolute and $80C.$ Vapor and liquid leaving the device are in equilibrium at the temperature and pressure of the system and have methanol mole fractions of $y$ and $x,$ respectively.

A Txy diagram for methanol$-$water mixtures at $1$atm absolute is shown below. The feed to the vaporizer and the liquid and vapor product streams are shown as points $B,A,$ and $C,$ respectively.

$($a$)$ Prove that $f$ can be determined from the equation

$f=\frac{\text{m}\text{o}\text{l}\text{e}\text{s}of\text{v}\text{a}\text{p}\text{o}\text{r}}{\text{m}\text{o}\text{l}\text{e}\text{s}of\text{l}\text{i}\text{q}\text{u}\text{i}\text{d}}=\frac{x_F-x}{y-x}$

Use this result to determine $f$ for the specific conditions cited above