Elevation in boiling point of an aqueous solution of urea is `0.52 ( k_(b) "for water"=0.52K"molality"^(-1))`. The mole fraction of urea in this solution is :

Options

A. 0.982
B. 0.0567
C. 0.943
D. 0.018

Answer

To find the mole fraction of urea in the solution given the elevation in boiling point, we can follow these steps:

### Step 1: Understand the relationship between boiling point elevation, molality, and the ebullioscopic constant.
The formula for boiling point elevation is given by:
\[ \Delta T_b = K_b \cdot m \]
where:
- \(\Delta T_b\) = elevation in boiling point (in °C or K)
- \(K_b\) = ebullioscopic constant (for water, \(K_b = 0.52 \, \text{K kg/mol}\))
- \(m\) = molality of the solution (in mol/kg)

### Step 2: Substitute the given values into the equation.
Given:
- \(\Delta T_b = 0.52 \, \text{K}\)
- \(K_b = 0.52 \, \text{K kg/mol}\)

Substituting these values into the equation:
\[ 0.52 = 0.52 \cdot m \]

### Step 3: Solve for molality (m).
To find molality \(m\), we can rearrange the equation:
\[ m = \frac{0.52}{0.52} = 1 \, \text{mol/kg} \]
This means there is 1 mole of urea in 1 kg (1000 g) of water.

### Step 4: Calculate the number of moles of urea and water.
- Number of moles of urea = 1 mole (as per the definition of 1 molal solution).
- To find the number of moles of water, we use the formula:
\[ \text{Number of moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \]
The molar mass of water (H₂O) is \(2 \times 1 + 16 = 18 \, \text{g/mol}\).

Calculating the number of moles of water:
\[ \text{Number of moles of water} = \frac{1000 \, \text{g}}{18 \, \text{g/mol}} \approx 55.56 \, \text{moles} \]

### Step 5: Calculate the total number of moles in the solution.
Total moles = moles of urea + moles of water:
\[ \text{Total moles} = 1 + 55.56 = 56.56 \, \text{moles} \]

### Step 6: Calculate the mole fraction of urea.
The mole fraction \(X\) of urea is given by:
\[ X_{\text{urea}} = \frac{\text{moles of urea}}{\text{total moles}} = \frac{1}{56.56} \]
Calculating this gives:
\[ X_{\text{urea}} \approx 0.0177 \]

### Step 7: Round the value.
Rounding off the mole fraction:
\[ X_{\text{urea}} \approx 0.018 \]

### Conclusion
The mole fraction of urea in the solution is approximately \(0.018\).

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