If x molal solution of a compound is benzene has mole fraction of solute equal to 0.4. Then the value of x is

Options

A. 4.2
B. 8.5
C. 3.2
D. 5.1

Answer

To solve the problem, we need to find the molality \( x \) of a solution where the mole fraction of the solute is given as 0.4. The solvent is benzene.

### Step-by-Step Solution:

1. **Understanding Mole Fraction**:
The mole fraction of the solute (\( \chi_{solute} \)) is given by the formula:
\[ \chi_{solute} = \frac{n_{solute}}{n_{solute} + n_{solvent}} \]
where \( n_{solute} \) is the number of moles of solute and \( n_{solvent} \) is the number of moles of solvent.

2. **Setting Up the Equation**:
Given \( \chi_{solute} = 0.4 \), we can express this as:
\[ 0.4 = \frac{n_{solute}}{n_{solute} + n_{solvent}} \]
Rearranging gives:
\[ n_{solute} = 0.4(n_{solute} + n_{solvent}) \]

3. **Solving for Moles**:
Let's assume the total moles of the solution is 1 (for simplicity). Thus:
\[ n_{solute} + n_{solvent} = 1 \]
From the mole fraction equation:
\[ n_{solute} = 0.4 \quad \text{and} \quad n_{solvent} = 1 - n_{solute} = 1 - 0.4 = 0.6 \]

4. **Calculating the Mass of Solvent**:
The number of moles of solvent (benzene) is 0.6. The molar mass of benzene (\( C_6H_6 \)) is calculated as:
\[ \text{Molar mass of benzene} = 12 \times 6 + 1 \times 6 = 78 \, \text{g/mol} \]
Therefore, the mass of the solvent is:
\[ \text{Mass of solvent} = n_{solvent} \times \text{Molar mass} = 0.6 \, \text{mol} \times 78 \, \text{g/mol} = 46.8 \, \text{g} \]

5. **Converting Mass to Kilograms**:
To find molality, we need the mass of the solvent in kilograms:
\[ \text{Mass of solvent in kg} = \frac{46.8 \, \text{g}}{1000} = 0.0468 \, \text{kg} \]

6. **Calculating Molality**:
The formula for molality \( x \) is:
\[ x = \frac{n_{solute}}{\text{Mass of solvent in kg}} = \frac{0.4 \, \text{mol}}{0.0468 \, \text{kg}} \]
Performing the calculation:
\[ x = \frac{0.4}{0.0468} \approx 8.5 \, \text{mol/kg} \]

### Final Answer:
The value of \( x \) is **8.5**.