Question

The vapour pressure of pure M and N are$700$mm of Hg and $450$mm of Hg respectively. Which of the following options is correct?
Given: ${X_N},{X_M}$- mole fraction of N and M in liquid phase
${Y_N},{Y_M}$= mole fraction of N and M in vapour phase.
A. ${X_M} - {X_N} > {Y_M} - {Y_N}$
B. $\dfrac{{{X_M}}}{{{X_N}}} > \dfrac{{{Y_M}}}{{{Y_N}}}$
C. $\dfrac{{{X_M}}}{{{X_N}}} < \dfrac{{{Y_M}}}{{{Y_N}}}$
D. $\dfrac{{{X_M}}}{{{X_N}}} = \dfrac{{{Y_M}}}{{{Y_N}}}$

Answer 1

To solve this question, you must recall the Raoult’s law and Dalton’s law of partial pressure. Raoult’s Law states that in a solution, the partial vapour pressure of each volatile compound is directly proportional to its mole fraction.

Formula used: ${{\text{P}}_{\text{M}}}{\text{ = P}}_{\text{M}}^{\text{0}}{{\text{X}}_{\text{M}}}$ and ${{\text{P}}_{\text{N}}}{\text{ = P}}_{\text{N}}^{\text{0}}{{\text{X}}_{\text{N}}}$
Where, ${{\text{P}}_{\text{M}}}$is partial vapour pressure of M
${{\text{P}}_{\text{N}}}$ is the partial vapour pressure of N
And ${X_N},{X_M}$ are mole fractions of N and M in the liquid phase respectively.
From Dalton’s law, we have ${{\text{P}}_{\text{M}}}{\text{ = P}}_{\text{M}}^{\text{0}}{{\text{Y}}_{\text{M}}}$ and ${{\text{P}}_{\text{N}}}{\text{ = P}}_{\text{N}}^{\text{0}}{{\text{Y}}_{\text{N}}}$.
Where, ${Y_N},{Y_M}$are mole fraction of N and M in the vapour phase.

Complete step by step solution:
We are given in the question, ${\text{P}}_{\text{M}}^{\text{0}} = 700{\text{mmHg}}$ and ${\text{P}}_{\text{N}}^{\text{0}} = 450{\text{mmHg}}$.
From the values of vapour pressure of the pure liquids, we can conclude that M is more volatile than N, since the vapour pressure of M is greater.
This implies that the mole fraction of M in vapour phase will be more than the mole fraction of N in vapour phase, that is, ${{\text{Y}}_{\text{M}}}{\text{ > }}{{\text{Y}}_{\text{N}}}$.
And thus, it is evident that the mole fraction of M in liquid phase will be less than the mole fraction of N in the liquid phase, that is, ${{\text{X}}_{\text{M}}}{\text{ < }}{{\text{X}}_{\text{N}}}$.
Using Dalton’s law and Raoult’s law, we can write,
$\dfrac{{{{\text{P}}_{\text{M}}}}}{{{{\text{P}}_{\text{N}}}}}{\text{ = }}\dfrac{{{\text{P}}_{\text{M}}^{\text{0}}{{\text{X}}_{\text{M}}}}}{{{\text{P}}_{\text{N}}^{\text{0}}{{\text{X}}_{\text{N}}}}}{\text{ = }}\dfrac{{{{\text{P}}_{\text{T}}}{{\text{Y}}_{\text{M}}}}}{{{{\text{P}}_{\text{T}}}{{\text{Y}}_{\text{N}}}}}$
Substituting the values of vapour pressure of pure liquids,
$\Rightarrow \dfrac{{700 \times {X_M}}}{{450 \times {X_N}}} = \dfrac{{{Y_M}}}{{{Y_N}}}$
From this equation we can say that $\dfrac{{{X_M}}}{{{X_N}}} < \dfrac{{{Y_M}}}{{{Y_N}}}$.

Thus, the correct option is C.

Note: Dalton’s law of partial pressure states that the total pressure of a mixture of a number of non-reacting gases is equal to the sum of pressures exerted by individual gases.
The partial pressure of gas is given by the product of total pressure and mole fraction of the gas in the mixture.