When 0.9 g of a non-volatile solute was dissovled in 45 g of benezene, the elevation of boiling point is `0.88^@C`. If `K_b` for benezene is `2.53 K kg "mol"^(-1)` , Calculate the molar mass of solute.

Answer

To calculate the molar mass of the non-volatile solute, we can follow these steps:

### Step 1: Understand the formula for boiling point elevation
The elevation of boiling point (\( \Delta T_b \)) is given by the formula:
\[ \Delta T_b = K_b \cdot m \]
where:
- \( \Delta T_b \) = elevation in boiling point (in °C)
- \( K_b \) = ebullioscopic constant of the solvent (in K kg/mol)
- \( m \) = molality of the solution (in mol/kg)

### Step 2: Identify the given values
From the question, we have:
- Mass of solute (\( m_{solute} \)) = 0.9 g
- Mass of solvent (benzene) (\( m_{solvent} \)) = 45 g = 0.045 kg (since we need it in kg for molality)
- Elevation of boiling point (\( \Delta T_b \)) = 0.88 °C
- \( K_b \) for benzene = 2.53 K kg/mol

### Step 3: Calculate molality
Molality (\( m \)) is defined as the number of moles of solute per kilogram of solvent. We need to find the number of moles of solute first:
\[ \text{Number of moles of solute} = \frac{m_{solute}}{M} \]
where \( M \) is the molar mass of the solute.

Now, molality can be expressed as:
\[ m = \frac{\text{Number of moles of solute}}{m_{solvent}} = \frac{\frac{m_{solute}}{M}}{m_{solvent}} \]
Substituting the values:
\[ m = \frac{\frac{0.9 \, \text{g}}{M}}{0.045 \, \text{kg}} = \frac{0.9}{0.045M} \]

### Step 4: Substitute into the boiling point elevation formula
Now we can substitute this expression for molality into the boiling point elevation formula:
\[ \Delta T_b = K_b \cdot m \]
Substituting the values:
\[ 0.88 = 2.53 \cdot \frac{0.9}{0.045M} \]

### Step 5: Solve for molar mass \( M \)
Rearranging the equation to solve for \( M \):
\[ M = \frac{2.53 \cdot 0.9}{0.88 \cdot 0.045} \]

Calculating the right side:
\[ M = \frac{2.277}{0.0396} \approx 57.5 \, \text{g/mol} \]

### Conclusion
The molar mass of the solute is approximately \( 57.5 \, \text{g/mol} \).

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