Calculate the mass of urea (`NH_2CONH_2`) required in making 4.5kg of 0.25 molal aqueous solution.

Answer

To calculate the mass of urea (NH₂CONH₂) required to make a 4.5 kg of a 0.25 molal aqueous solution, we can follow these steps:

### Step 1: Understand the Definition of Molality
Molality (m) is defined as the number of moles of solute per kilogram of solvent. In this case, we have a 0.25 molal solution, meaning there are 0.25 moles of urea in 1 kg of water (solvent).

### Step 2: Calculate the Moles of Urea Required for 4.5 kg of Solution
Since the solution is 0.25 molal, we need to determine how many kilograms of solvent (water) are in 4.5 kg of the solution. The mass of the solution is the sum of the mass of the solute (urea) and the mass of the solvent (water).

Let:
- \( m_{solution} = 4.5 \, \text{kg} \)
- \( m_{urea} = x \, \text{kg} \)
- \( m_{water} = 4.5 - x \, \text{kg} \)

Using the definition of molality:
\[ 0.25 = \frac{n_{urea}}{m_{water}} = \frac{n_{urea}}{4.5 - x} \]

### Step 3: Calculate the Molar Mass of Urea
The molar mass of urea (NH₂CONH₂) is calculated as follows:
- Nitrogen (N) = 14 g/mol × 2 = 28 g/mol
- Carbon (C) = 12 g/mol × 1 = 12 g/mol
- Oxygen (O) = 16 g/mol × 1 = 16 g/mol
- Hydrogen (H) = 1 g/mol × 4 = 4 g/mol

Adding these together:
\[ \text{Molar mass of urea} = 28 + 12 + 16 + 4 = 60 \, \text{g/mol} \]

### Step 4: Calculate the Mass of Urea Required
From the molality, we know that for every 1 kg of water, we need 0.25 moles of urea. Therefore, if we have 1 kg of water, the mass of urea required is:
\[ \text{Mass of urea} = n_{urea} \times \text{Molar mass of urea} = 0.25 \, \text{moles} \times 60 \, \text{g/mol} = 15 \, \text{g} \]

### Step 5: Calculate the Total Mass of the Solution
The total mass of the solution when 15 g of urea is added to 1 kg of water is:
\[ \text{Total mass of solution} = \text{mass of water} + \text{mass of urea} = 1000 \, \text{g} + 15 \, \text{g} = 1015 \, \text{g} = 1.015 \, \text{kg} \]

### Step 6: Calculate the Mass of Urea in 4.5 kg of Solution
Now, we need to find out how much urea is required in 4.5 kg of solution. Using the ratio:
\[ \text{Mass of urea in 4.5 kg of solution} = \frac{15 \, \text{g}}{1.015 \, \text{kg}} \times 4.5 \, \text{kg} \]
Calculating this gives:
\[ \text{Mass of urea} = \frac{15}{1.015} \times 4.5 \approx 66.5 \, \text{g} \]

### Final Answer
The mass of urea required to make 4.5 kg of a 0.25 molal aqueous solution is approximately **66.5 grams**.

---