An aqueous solution of urea containing 18 g urea in 1500 `cm^(3)` of solution has density equal to 1.052. If the molecular weight of urea is 60, find the molality of the solution.

Explanation

To find the molality of the urea solution, we can follow these steps:

### Step 1: Calculate the mass of the solution
Given:
- Volume of the solution = 1500 cm³
- Density of the solution = 1.052 g/cm³

Using the formula for mass:
\[ \text{Mass of solution} = \text{Density} \times \text{Volume} \]
\[ \text{Mass of solution} = 1.052 \, \text{g/cm}^3 \times 1500 \, \text{cm}^3 = 1578 \, \text{g} \]

### Step 2: Calculate the mass of the solvent
Given:
- Mass of solute (urea) = 18 g

Using the formula:
\[ \text{Mass of solvent} = \text{Mass of solution} - \text{Mass of solute} \]
\[ \text{Mass of solvent} = 1578 \, \text{g} - 18 \, \text{g} = 1560 \, \text{g} \]

### Step 3: Convert the mass of the solvent to kilograms
Since molality is defined in terms of kilograms of solvent:
\[ \text{Mass of solvent in kg} = \frac{1560 \, \text{g}}{1000} = 1.56 \, \text{kg} \]

### Step 4: Calculate the number of moles of urea
Given:
- Molecular weight of urea = 60 g/mol

Using the formula for moles:
\[ \text{Number of moles of urea} = \frac{\text{Mass of urea}}{\text{Molecular weight of urea}} \]
\[ \text{Number of moles of urea} = \frac{18 \, \text{g}}{60 \, \text{g/mol}} = 0.3 \, \text{mol} \]

### Step 5: Calculate the molality of the solution
Using the formula for molality:
\[ \text{Molality (m)} = \frac{\text{Number of moles of solute}}{\text{Mass of solvent in kg}} \]
\[ \text{Molality} = \frac{0.3 \, \text{mol}}{1.56 \, \text{kg}} \approx 0.192 \, \text{mol/kg} \]

### Final Answer
The molality of the solution is approximately **0.192 mol/kg**.

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