Short Answer 
To find the mole fraction of component B in the vapor phase, first calculate the mole fraction of B in the liquid phase (0.6), then use Raoult’s Law to determine the partial pressures (PA = 30 mm Hg, PB = 15 mm Hg). Finally, calculate the total vapor pressure (45 mm Hg) and the mole fraction of B in the vapor phase (0.333).
Step 1: Calculate the Mole Fraction of B in the Liquid Phase
First, determine the mole fraction of B in the liquid phase, which can be calculated from the mole fraction of A. Since the total mole fraction must equal 1, use the formula:
- XB = 1 – XA = 1 – 0.4 = 0.6
Step 2: Calculate Partial Pressures of A and B
Next, use Raoult’s Law to calculate the partial pressures of each component in the vapor phase. Multiply the mole fractions by their respective vapor pressures:
- PA = XA * vapor pressure of A = 0.4 * 75 mm Hg = 30 mm Hg
- PB = XB * vapor pressure of B = 0.6 * 25 mm Hg = 15 mm Hg
Step 3: Calculate Total Vapor Pressure and Mole Fraction of B in Vapor Phase
Now, find the total vapor pressure of the mixture by adding the partial pressures, then calculate the mole fraction of B in the vapor phase using its partial pressure:
- Total Vapor Pressure = PA + PB = 30 mm Hg + 15 mm Hg = 45 mm Hg
- Mole Fraction of B in Vapor = PB / Total Vapor Pressure = 15 mm Hg / 45 mm Hg = 0.333