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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 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 Torr. The value of $x_{A} / x_{B}$ in the new solution is _____.
(given that the vapour pressure of pure liquid A is 20 Torr at temperature T)

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answer is 19.

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Detailed Solution

for equimolar binary solution

$P_{T o t a l} = P_{A}^{o} (\frac{1}{2}) + P_{B}^{o} (\frac{1}{2}) = 45$

$P_{A}^{o} + P_{B}^{o} = 90 . . . . . . . . (1)$

$P_{T o t a l} = P_{A}^{o} x_{A} + P_{B}^{o} x_{B} = 22.5$

if $P_{A}^{o} = 20 \Rightarrow P_{B}^{o} = 70$

$20 x_{A} + 70 x_{B} = 22.5$

$20 x_{A} + 70 (1 - x_{A}) = 22.5$

$70 - 22.5 = 50 x_{A}$

$x_{A} = \frac{47.50}{50}$

$x_{B} = \frac{2.5}{50}$

$\frac{x_{A}}{x_{B}} = \frac{47.5}{2.5} = \frac{475}{25} = 19$

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