Question

If \[{\text{P}}^\circ {\text{a}}\] and \[{\text{P}}^\circ {\text{b}}\] are 108 and 36 torr respectively. What will be the mole fraction of A in vapour phase if B has a mole fraction in solution \[0.5\] :
A.\[0.25\]
B.\[0.75\]
C.\[0.60\]
D.\[0.35\]

Answer 1

Hint: First of all total vapour pressure will be calculated by using mole fraction and pure vapour pressure. Then using Dalton's formula we will calculate the mole fraction.
Formula used:
\[{{\text{V}}_{\text{p}}} = {\chi _{\text{A}}}{\text{P}}^\circ {\text{a}} + {\chi _{\text{B}}}{\text{P}}^\circ {\text{b}}\]
Here \[{{\text{V}}_{\text{p}}}\] is the total vapour pressure, \[\chi \] represents mole fraction and \[{\text{P}}^\circ \] represents pressure of pure solvent.

Complete step by step solution:
Vapour pressure of a solution containing both solute and solvent of volatile nature is equal to the sum of partial vapour pressure of each component in the solution. On the other hand vapour pressure of a solution containing non volatile solute and volatile solvent, for such solution relative lowering of vapour pressure is equal to mole fraction of non volatile solute.
According to question, both A and B are volatile as they both have vapour pressure. Vapour pressure of pure liquid A is 108 torr and vapour pressure of pure liquid B is 36 torr. As it is given that mole fraction of B in solution is \[0.5\] which means rest mole fraction of \[0.5\] will be of A in solution. The mole fraction of all components present in a mixture is equal to 1. Vapour pressure of solution is equal to sum of partial vapour pressure of A and B.
\[{{\text{V}}_{\text{p}}} = {\text{ }}0.5{\text{ }} \times {\text{ }}108{\text{ }} + 0.5{\text{ }} \times {\text{ }}36\]
\[ \Rightarrow {{\text{V}}_{\text{p}}} = {\text{ }}54{\text{ }} + {\text{ }}18 = {\text{ }}72{\text{ torr}}\].
This means vapour pressure of solution is 72 torr which is contributed by both A and B in their vapour form. The mole fraction of A in vapour phase (which can be different from solution's mole fraction) can be calculated as:
We already knew vapour pressure of solution (72 torr) and partial vapour pressure of A (54 torr):
\[{\chi _a}{\text{ or mole fraction of A in vapour phase}} = {\text{ }}\dfrac{{54}}{{72}} = {\text{ }}0.75\]

Thus, the correct option is B.

Note:
Vapour pressure is defined as the pressure exerted on the surface of liquid by the vapour in a closed surface at equilibrium conditions. It is a surface phenomenon. The surface molecules have lesser force of attraction causing them to easily form vapour and also they have more kinetic energy. Vapour pressures of non volatile substances are taken as zero. A more volatile substance like alcohol has more vapour pressure than one with low volatility like water. The vapour pressure of a solution depends on the nature of its solute and solvent.