The mole fraction of toluene in the vapour phase which is in equilibrium with a solution of benzene `(P_(B)^(@)=120 "torr")` and toluene `(P_(T)^(@)=80 "torr")` having `2.0 mol` of each, is

Options

A. `0.50`
B. `0.25`
C. `0.60`
D. `0.40`

Explanation

To find the mole fraction of toluene in the vapor phase that is in equilibrium with a solution of benzene and toluene, we can follow these steps:

### Step 1: Determine the total number of moles in the solution
Given that there are 2 moles of benzene (B) and 2 moles of toluene (T):
\[ N_B = 2 \, \text{mol} \quad \text{and} \quad N_T = 2 \, \text{mol} \]
Total moles in the solution:
\[ N_{total} = N_B + N_T = 2 + 2 = 4 \, \text{mol} \]

### Step 2: Calculate the mole fractions of benzene and toluene in the solution
The mole fraction of benzene (\(x_B\)) is:
\[ x_B = \frac{N_B}{N_{total}} = \frac{2}{4} = \frac{1}{2} \]
The mole fraction of toluene (\(x_T\)) is:
\[ x_T = \frac{N_T}{N_{total}} = \frac{2}{4} = \frac{1}{2} \]

### Step 3: Calculate the partial pressures of benzene and toluene in the vapor phase
Using Raoult's Law:
\[ P_B = P_B^0 \cdot x_B \quad \text{and} \quad P_T = P_T^0 \cdot x_T \]
Where:
- \(P_B^0 = 120 \, \text{torr}\) (vapor pressure of pure benzene)
- \(P_T^0 = 80 \, \text{torr}\) (vapor pressure of pure toluene)

Calculating the partial pressure of benzene (\(P_B\)):
\[ P_B = 120 \, \text{torr} \cdot \frac{1}{2} = 60 \, \text{torr} \]

Calculating the partial pressure of toluene (\(P_T\)):
\[ P_T = 80 \, \text{torr} \cdot \frac{1}{2} = 40 \, \text{torr} \]

### Step 4: Calculate the total pressure in the vapor phase
Total pressure (\(P_{total}\)) is the sum of the partial pressures:
\[ P_{total} = P_B + P_T = 60 \, \text{torr} + 40 \, \text{torr} = 100 \, \text{torr} \]

### Step 5: Calculate the mole fraction of toluene in the vapor phase
The mole fraction of toluene in the vapor phase (\(y_T\)) is given by:
\[ y_T = \frac{P_T}{P_{total}} = \frac{40 \, \text{torr}}{100 \, \text{torr}} = 0.4 \]

### Final Answer
The mole fraction of toluene in the vapor phase is \(0.4\).

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