Question

Liquid A and B form an ideal solution. The vapour pressure of pure liquids A and B are 350 and 750 mm Hg respectively at the same temperature. If $ x_A $ and $ x_B $ are the mole fraction of A and B in solution while $ y_A $ and $ y_B $ are the mole fraction of A and B in vapour phase then :

• A. \( \frac{x_A}{x_B}<\frac{y_A}{y_B} \)
• B. \( \frac{x_A}{x_B} = \frac{y_A}{y_B} \)
• C. \( \frac{x_A}{x_B}>\frac{y_A}{y_B} \)
• D. \( (x_A - y_A)<(x_B - y_B) \)

Hint:

In an ideal solution, the vapour phase is richer in the more volatile component (the one with the higher vapour pressure). Here, B has a higher vapour pressure than A, so the vapour phase will have a higher mole fraction of B compared to the solution. This implies that the ratio of mole fractions of A to B will be smaller in the vapour phase than in the solution.

Answer 1

The Correct Option is C

Step 1: Apply Raoult's Law for an ideal solution.
For an ideal solution, the partial pressure of each component in the vapour phase is equal to the mole fraction of that component in the solution multiplied by the vapour pressure of the pure component.
\( p_A = x_A P_A^\circ \) \( p_B = x_B P_B^\circ \) where \( P_A^\circ = 350 \) mm Hg and \( P_B^\circ = 750 \) mm Hg are the vapour pressures of pure liquids A and B, respectively.
Step 2: Use Dalton's Law of partial pressures.
The total vapour pressure of the solution is \( P_{total} = p_A + p_B = x_A P_A^\circ + x_B P_B^\circ \). We also know that \( x_A + x_B = 1 \), so \( x_B = 1 - x_A \). \( P_{total} = x_A P_A^\circ + (1 - x_A) P_B^\circ = x_A P_A^\circ + P_B^\circ - x_A P_B^\circ = P_B^\circ + x_A (P_A^\circ - P_B^\circ) \) Since \( P_A^\circ<P_B^\circ \), \( (P_A^\circ - P_B^\circ) \) is negative, so \( P_{total} \) will be between \( P_A^\circ \) and \( P_B^\circ \).
Step 3: Relate the mole fractions in the vapour phase to the partial pressures.
The mole fraction of each component in the vapour phase is given by the ratio of its partial pressure to the total pressure: \( y_A = \frac{p_A}{P_{total}} = \frac{x_A P_A^\circ}{x_A P_A^\circ + x_B P_B^\circ} \) \( y_B = \frac{p_B}{P_{total}} = \frac{x_B P_B^\circ}{x_A P_A^\circ + x_B P_B^\circ} \)
Step 4: Consider the ratio of mole fractions in the vapour phase and the solution.
We want to compare \( \frac{x_A}{x_B} \) with \( \frac{y_A}{y_B} \). \[ \frac{y_A}{y_B} = \frac{\frac{x_A P_A^\circ}{x_A P_A^\circ + x_B P_B^\circ}}{\frac{x_B P_B^\circ}{x_A P_A^\circ + x_B P_B^\circ}} = \frac{x_A P_A^\circ}{x_B P_B^\circ} = \frac{x_A}{x_B} \cdot \frac{P_A^\circ}{P_B^\circ} \] Given \( P_A^\circ = 350 \) mm Hg and \( P_B^\circ = 750 \) mm Hg, we have \( \frac{P_A^\circ}{P_B^\circ} = \frac{350}{750} = \frac{7}{15} \). So, \( \frac{y_A}{y_B} = \frac{x_A}{x_B} \cdot \frac{7}{15} \). Since \( \frac{7}{15}<1 \), we can conclude that \( \frac{y_A}{y_B}<\frac{x_A}{x_B} \). Alternatively, \( \frac{x_A}{x_B}>\frac{y_A}{y_B} \).
Step 5: Verify with a specific example.
Let \( x_A = 0.5 \) and \( x_B = 0.5 \). Then \( y_A = \frac{0.5 \times 350}{0.5 \times 350 + 0.5 \times 750} = \frac{175}{175 + 375} = \frac{175}{550} = \frac{7}{22} \) \( y_B = \frac{0.5 \times 750}{0.5 \times 350 + 0.5 \times 750} = \frac{375}{550} = \frac{15}{22} \) \( \frac{x_A}{x_B} = \frac{0.5}{0.5} = 1 \) \( \frac{y_A}{y_B} = \frac{7/22}{15/22} = \frac{7}{15} \) Here, \( 1>\frac{7}{15} \), so \( \frac{x_A}{x_B}>\frac{y_A}{y_B} \).