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 solve the problem of finding the mole ratio of benzene and toluene in the vapor phase, we will use Raoult's Law and some basic algebra. Here’s a step-by-step solution:

### Step 1: Write down the given data
- Vapor pressure of pure benzene, \( P^0_B = 150 \) torr
- Vapor pressure of pure toluene, \( P^0_T = 50 \) torr
- Total vapor pressure of the solution, \( P_T = 120 \) torr

### Step 2: Apply Raoult's Law
According to Raoult's Law, the total vapor pressure \( P_T \) is given by:
\[ P_T = P^0_B \cdot X_B + P^0_T \cdot X_T \]
where \( X_B \) is the mole fraction of benzene and \( X_T \) is the mole fraction of toluene.

### Step 3: Express \( X_T \) in terms of \( X_B \)
Since \( X_B + X_T = 1 \), we can express \( X_T \) as:
\[ X_T = 1 - X_B \]

### Step 4: Substitute \( X_T \) into the equation
Substituting \( X_T \) into the Raoult's Law equation gives:
\[ 120 = 150 \cdot X_B + 50 \cdot (1 - X_B) \]

### Step 5: Simplify the equation
Expanding and simplifying the equation:
\[ 120 = 150X_B + 50 - 50X_B \]
\[ 120 = 100X_B + 50 \]
\[ 120 - 50 = 100X_B \]
\[ 70 = 100X_B \]
\[ X_B = 0.7 \]

### Step 6: Calculate \( X_T \)
Now, we can find \( X_T \):
\[ X_T = 1 - X_B = 1 - 0.7 = 0.3 \]

### Step 7: Find the mole fractions in the vapor phase
Using the relationship between the vapor phase mole fractions and the liquid phase mole fractions:
\[ Y_B = \frac{P^0_B \cdot X_B}{P_T} \]
Substituting the values:
\[ Y_B = \frac{150 \cdot 0.7}{120} = \frac{105}{120} = 0.875 \]

For toluene:
\[ Y_T = 1 - Y_B = 1 - 0.875 = 0.125 \]

### Step 8: Calculate the mole ratio
The mole ratio of benzene to toluene in the vapor phase is:
\[ \text{Mole Ratio} = \frac{Y_B}{Y_T} = \frac{0.875}{0.125} = 7 \]
Thus, the mole ratio of benzene to toluene is \( 7:1 \).

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

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