Vapour pressure of pure benzene is 119 torr and that of toluene is 37.0 torr at the same temperature. Mole fraction of toluene in vapour phase which is in equilibrium with a solution of benzene and toluene having a mole fraction of toluene 0.50, will be :

Given the vapour pressure of pure benzene is 119 torr and that of pure toluene is 37.0 torr at the same temperature. The solution has a mole fraction of toluene ($$X_{\text{toluene}}$$) equal to 0.50. We need to find the mole fraction of toluene in the vapour phase when the solution is in equilibrium.

First, since the mole fraction of toluene in the liquid phase is 0.50, the mole fraction of benzene ($$X_{\text{benzene}}$$) is:

$$X_{$$ benzene $$} = 1 - X_{$$ toluene $$} = 1 - 0.50 = 0.50$$

According to Raoult's law, the partial vapour pressure of benzene ($$P_{\text{benzene}}$$) is:

$$P_{$$ benzene $$} = X_{$$ benzene $$} \times P^{\circ}_{$$ benzene $$} = 0.50 \times 119$$

Calculating that:

$$P_{$$ benzene $$} = 0.50 \times 119 = 59.5$$ torr

Similarly, the partial vapour pressure of toluene ($$P_{\text{toluene}}$$) is:

$$P_{$$ toluene $$} = X_{$$ toluene $$} \times P^{\circ}_{$$ toluene $$} = 0.50 \times 37.0$$

Calculating that:

$$P_{$$ toluene $$} = 0.50 \times 37.0 = 18.5$$ torr

The total vapour pressure ($$P_{\text{total}}$$) is the sum of the partial pressures:

$$P_{$$ total $$} = P_{$$ benzene $$} + P_{$$ toluene $$} = 59.5 + 18.5$$

Calculating that:

$$P_{$$ total $$} = 59.5 + 18.5 = 78.0$$ torr

In the vapour phase, the mole fraction of toluene ($$Y_{\text{toluene}}$$) is given by the ratio of its partial pressure to the total pressure:

$$Y_{$$ toluene $$} = \frac{P_{$$ toluene $$}}{P_{$$ total $$}} = \frac{18.5}{78.0}$$

To compute this, we can simplify the fraction:

$$\frac{18.5}{78.0} = \frac{185}{780} \quad$$ (multiplying numerator and denominator by 10)

Now, divide both numerator and denominator by 5:

$$\frac{185 \div 5}{780 \div 5} = \frac{37}{156}$$

Converting to decimal:

$$37 \div 156 \approx 0.237179$$

Rounding to three decimal places, we get approximately 0.237.

Comparing with the options:

A. 0.137

B. 0.237

C. 0.435

D. 0.205

Hence, the correct answer is Option B.