Question

Liquids A and B form ideal solution over the entire range of composition. At temperature T, equimolar binary solution of liquids A and B has vapour pressure $45\text{torr}$. At the same temperature, a new solution of A and B having mole fractions $x_A$ and $x_B$, respectively, has vapour pressure of $22.5\text{torr}$. The value of $\frac{x_A}{x_B}$ in the new solution is .
(Given that the vapour pressure of pure liquid A is 20 Torr at temperature T)

Answer 1

By using the formula,
$p_{total}=p_A^o\times x_A+p_B^o{x}_B$
and for equimolar solutions
$x_A=x_B=\frac{1}{2}$
Given,
$p_{total}=45torr$ for equimolar solution
$p_A^o=20\text{torr}$
$\Rightarrow 45=p_A^o\times \frac{1}{2}+p_B^o\frac{1}{2}=\frac{1}{2}(p_A^o+p_B^o)$
$Rightarrow{p}_A^o+p_B^o=90\text{torr}$
But we know,
$p_A^o=20\text{torr}$
Now, for the new solution from the same formula.
Given,
$p_{total}=22.5\text{torr}$
So, $22.5=20x_A+70(1-x_A)$
(As $x_A+x_B=1)$
$22.5=70-50x_A$
$x_A=\frac{70-22.5}{50}=0.95$
Thus,
$x_B=1-0.95=0.05$
Hence, the ratio $\frac{x_A}{x_B}=\frac{0.95}{0.05}=19$