At $1$atm pressure, the ethanol$-$water has a composition of $50$mol$\%$ water in a gas phase, at a temperature of $78\mathrm{.}15C.$ At this temperature the pure species vapor pressures are water $0\mathrm{.}434$atm and ethanol $0\mathrm{.}993$atm. The activity coefficient of liquid phase can be obtained by the following van Laar equation $($A: $0\mathrm{.}7292,$ B:$0\mathrm{.}4104).$ Calculate the composition of liquid phase. Assume that the Lewis$-$Randall fugacity rule holds for the gas phase, with ${}_{\text{e}\text{t}\text{h}\text{a}\text{n}\text{o}\text{l}}=0\mathrm{.}8,$ and ${}_{\text{w}\text{a}\text{t}\text{e}\text{r}}=0\mathrm{.}73$

$log{}_{\text{e}\text{t}\text{h}\text{a}\text{n}\text{o}\text{l}}=\frac{A{x}_{\text{w}\text{a}\text{t}\text{e}\text{r}}^2}{(\frac{A}{B}{x}_{\text{e}\text{t}\text{h}\text{a}\text{n}\text{o}\text{l}}+x_{\text{w}\text{a}\text{t}\text{e}\text{r}}{)}^2},log{}_{\text{w}\text{a}\text{t}\text{e}\text{r}}=\frac{A{x}_{\text{e}\text{t}\text{h}\text{a}\text{n}\text{o}\text{l}}^2}{(x_{\text{e}\text{t}\text{h}\text{a}\text{n}\text{o}\text{l}}+\frac{A}{B}{x}_{\text{w}\text{a}\text{t}\text{e}\text{r}}{)}^2}$