The vapour pressures of two liquids \(A\) and \(B\) in their pure states are in the ratio of \(1: 2\). A binary solution of \(A\) and \(B\) contains \(A\) and \(B\) in the mole proportion of \(1: 2\). The mole fraction of \(A\) in the vapour phase of the solution will be

Given, \(\mathrm{p}_{\mathrm{A}}: \mathrm{p}_{\mathrm{B}}=1: 2\)
\(\begin{array}{ll}\therefore & \mathrm{p}_{\mathrm{B}}=2 \mathrm{p}_{\mathrm{A}} \\ \text { Similarly } & \frac{\mathrm{n}_{\mathrm{A}}}{\mathrm{n}_{\mathrm{B}}}=\frac{1}{2}\end{array}\)
\(\therefore \mathrm{x}_{\mathrm{A}}=\frac{1}{1+2}=\frac{1}{3}\) and \(\mathrm{x}_{\mathrm{B}}=\frac{2}{1+2}=\frac{2}{3}\)
Total pressure, \(\mathrm{p}_{\mathrm{T}}=\mathrm{p}_{\mathrm{A}} \mathrm{x}_{\mathrm{A}}+\mathrm{p}_{\mathrm{B}} \mathrm{x}_{\mathrm{B}}\)
\[
\begin{aligned}
&=p_{A} \times \frac{1}{3}+2 p_{A} \times \frac{2}{3} \\
&=\frac{1}{3} p_{A}+\frac{4}{3} p_{A} \\
&=\frac{5}{3} p_{A}
\end{aligned}
\]