18.2 g of urea is dissolved in 100g of water at `50^(@)C`. The lowering of vapour pressure produced is 5 mm Hg. Calculate the molecular mass of urea. The vapour pressure of water at `50^(@)C` is 92 mm Hg.

Answer

To calculate the molecular mass of urea from the given data, we can follow these steps:

### Step 1: Identify the given values
- Mass of urea (solute) = 18.2 g
- Mass of water (solvent) = 100 g
- Lowering of vapor pressure (ΔP) = 5 mm Hg
- Vapor pressure of pure water at 50°C (P₀) = 92 mm Hg

### Step 2: Calculate the vapor pressure of the solution
The vapor pressure of the solution (P) can be calculated using the formula:
\[ P = P₀ - ΔP \]
Substituting the values:
\[ P = 92 \, \text{mm Hg} - 5 \, \text{mm Hg} = 87 \, \text{mm Hg} \]

### Step 3: Calculate the mole fraction of the solute
The relative lowering of vapor pressure is given by:
\[ \frac{ΔP}{P₀} = \frac{n_{solute}}{n_{solute} + n_{solvent}} \]
Where:
- \( n_{solute} \) = number of moles of urea
- \( n_{solvent} \) = number of moles of water

### Step 4: Calculate the number of moles of water
The molecular mass of water (H₂O) is approximately 18 g/mol. Therefore, the number of moles of water (n₁) can be calculated as:
\[ n_{solvent} = \frac{100 \, \text{g}}{18 \, \text{g/mol}} \approx 5.56 \, \text{mol} \]

### Step 5: Set up the equation for the mole fraction
We can express the mole fraction of the solute as:
\[ \frac{ΔP}{P₀} = \frac{n_{solute}}{n_{solute} + n_{solvent}} \]
Substituting the known values:
\[ \frac{5}{92} = \frac{n_{solute}}{n_{solute} + 5.56} \]

### Step 6: Solve for the number of moles of urea
Let \( n_{solute} = \frac{18.2 \, \text{g}}{M} \), where M is the molecular mass of urea. Substituting this into the equation:
\[ \frac{5}{92} = \frac{\frac{18.2}{M}}{\frac{18.2}{M} + 5.56} \]

Cross-multiplying gives:
\[ 5 \left( \frac{18.2}{M} + 5.56 \right) = 92 \cdot \frac{18.2}{M} \]

### Step 7: Simplify and solve for M
Expanding and rearranging:
\[ 5 \cdot 5.56 = 92 \cdot \frac{18.2}{M} - 5 \cdot \frac{18.2}{M} \]
\[ 27.8 = \frac{(92 - 5) \cdot 18.2}{M} \]
\[ 27.8M = 87 \cdot 18.2 \]
\[ M = \frac{87 \cdot 18.2}{27.8} \]

Calculating this gives:
\[ M \approx 57.05 \, \text{g/mol} \]

### Conclusion
The molecular mass of urea is approximately 57.05 g/mol.

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