An X molal solution of a compound in benzene has mole fraction of solute `=0.2.` The value of X is

Options

A. ` 14.0 `
B. ` 3.2 `
C. ` 1.4 `
D. ` 2.0 `

Explanation

To find the value of \( X \) in an \( X \) molal solution of a compound in benzene with a mole fraction of solute equal to 0.2, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- Mole fraction of solute \( (X_{solute}) = 0.2 \)
- Therefore, mole fraction of solvent \( (X_{solvent}) = 1 - X_{solute} = 1 - 0.2 = 0.8 \)

2. **Define Moles of Solute and Solvent**:
- Let \( n_2 \) be the moles of solute.
- Let \( n_1 \) be the moles of solvent (benzene).
- The mole fraction of solute can be expressed as:
\[ X_{solute} = \frac{n_2}{n_1 + n_2} = 0.2 \]
- The mole fraction of solvent can be expressed as:
\[ X_{solvent} = \frac{n_1}{n_1 + n_2} = 0.8 \]

3. **Set Up the Ratio of Moles**:
- From the mole fraction equations, we can set up the following ratio:
\[ \frac{n_2}{n_1} = \frac{0.2}{0.8} = \frac{1}{4} \]
- This implies that \( n_2 = \frac{1}{4} n_1 \).

4. **Calculate Moles of Solvent**:
- Assume we have 1 kg (1000 grams) of benzene (solvent).
- The molecular mass of benzene \( (C_6H_6) \) is calculated as:
\[ \text{Molecular mass of benzene} = 12 \times 6 + 1 \times 6 = 78 \, \text{g/mol} \]
- The number of moles of solvent \( n_1 \) can be calculated as:
\[ n_1 = \frac{\text{mass of solvent}}{\text{molecular mass}} = \frac{1000 \, \text{g}}{78 \, \text{g/mol}} \approx 12.82 \, \text{moles} \]

5. **Calculate Moles of Solute**:
- Using the ratio \( n_2 = \frac{1}{4} n_1 \):
\[ n_2 = \frac{1}{4} \times 12.82 \approx 3.205 \, \text{moles} \]

6. **Determine the Molality**:
- Molality \( (m) \) is defined as the number of moles of solute per kilogram of solvent:
\[ m = \frac{n_2}{\text{mass of solvent in kg}} = \frac{3.205 \, \text{moles}}{1 \, \text{kg}} = 3.205 \, \text{molal} \]

7. **Final Answer**:
- The value of \( X \) is approximately \( 3.2 \, \text{molal} \).

### Conclusion:
The value of \( X \) is \( 3.2 \, \text{molal} \).