What is the mole ratio of benzene `(P_(B)^(@)=150 t o r r)` and toluence `(P_(tau)^(@)=50 t o rr)` in vapour phase if the given solution has a vapour phase if the given solution has a vapour pressure of `120` torr ?

Options

A. `7:1`
B. `7:3`
C. `8:1`
D. `7:8`

Explanation

To find the mole ratio of benzene and toluene in the vapor phase, we can use Raoult's Law and the given information about the vapor pressures. Here’s a step-by-step solution:

### Step 1: Understand the Given Data
- Vapor pressure of pure benzene \( P_B^0 = 150 \, \text{torr} \)
- Vapor pressure of pure toluene \( P_T^0 = 50 \, \text{torr} \)
- Total vapor pressure of the solution \( P_T = 120 \, \text{torr} \)

### Step 2: Calculate the Mole Fraction of Benzene in the Vapor Phase
Using Raoult's Law, we can express the mole fraction of benzene in the vapor phase \( Y_B \) as:
\[ Y_B = \frac{P_B^0 \cdot X_B}{P_T} \]
Where:
- \( X_B \) is the mole fraction of benzene in the liquid phase.

### Step 3: Calculate the Mole Fraction of Benzene in the Liquid Phase
Using the total vapor pressure equation:
\[ P_T = P_B^0 \cdot X_B + P_T^0 \cdot X_T \]
Since \( X_T = 1 - X_B \), we can rewrite the equation as:
\[ P_T = P_B^0 \cdot X_B + P_T^0 \cdot (1 - X_B) \]
Substituting the values:
\[ 120 = 150 \cdot X_B + 50 \cdot (1 - X_B) \]
Expanding and rearranging gives:
\[ 120 = 150X_B + 50 - 50X_B \]
\[ 120 = 100X_B + 50 \]
\[ 70 = 100X_B \]
\[ X_B = 0.7 \]

### Step 4: Calculate the Mole Fraction of Benzene in the Vapor Phase
Now, we can substitute \( X_B \) back into the equation for \( Y_B \):
\[ Y_B = \frac{P_B^0 \cdot X_B}{P_T} = \frac{150 \cdot 0.7}{120} \]
Calculating this gives:
\[ Y_B = \frac{105}{120} = 0.875 \]

### Step 5: Calculate the Mole Fraction of Toluene in the Vapor Phase
Since the total mole fraction must equal 1:
\[ Y_T = 1 - Y_B = 1 - 0.875 = 0.125 \]

### Step 6: Calculate the Mole Ratio of Benzene to Toluene in the Vapor Phase
The mole ratio of benzene to toluene in the vapor phase is:
\[ \text{Mole Ratio} = \frac{Y_B}{Y_T} = \frac{0.875}{0.125} = 7 \]

### Final Answer
The mole ratio of benzene to toluene in the vapor phase is **7:1**.

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