Calculate the molarity and mole fraction of the solute in aqueous solution containing `3.0 g` of urea per `250gm` of water (Mol. Wt. of urea = 60)

Options

A. ` 0.2m ,0.00357 `
B. ` 0.4m, 0.00357 `
C. ` 0.5m, 0.00357 `
D. ` 0.7m, 0.00357 `

Explanation

To solve the problem of calculating the molarity and mole fraction of urea in an aqueous solution, we will follow these steps:

### Step 1: Calculate the moles of urea
To find the moles of urea, we use the formula:

\[ \text{Moles of solute} = \frac{\text{mass of solute (g)}}{\text{molar mass of solute (g/mol)}} \]

Given:
- Mass of urea = 3.0 g
- Molar mass of urea = 60 g/mol

Calculating the moles of urea:

\[ \text{Moles of urea} = \frac{3.0 \, \text{g}}{60 \, \text{g/mol}} = 0.05 \, \text{mol} \]

### Step 2: Calculate the mass of water in kg
The mass of water is given as 250 g. To convert this to kg:

\[ \text{Mass of water (kg)} = \frac{250 \, \text{g}}{1000} = 0.25 \, \text{kg} \]

### Step 3: Calculate the molality of the solution
Molality (m) is defined as the number of moles of solute per kilogram of solvent:

\[ \text{Molality} = \frac{\text{moles of solute}}{\text{mass of solvent (kg)}} \]

Substituting the values:

\[ \text{Molality} = \frac{0.05 \, \text{mol}}{0.25 \, \text{kg}} = 0.2 \, \text{mol/kg} \]

### Step 4: Calculate the moles of water
To find the moles of water, we use the formula:

\[ \text{Moles of water} = \frac{\text{mass of water (g)}}{\text{molar mass of water (g/mol)}} \]

Given:
- Molar mass of water = 18 g/mol

Calculating the moles of water:

\[ \text{Moles of water} = \frac{250 \, \text{g}}{18 \, \text{g/mol}} \approx 13.89 \, \text{mol} \]

### Step 5: Calculate the mole fraction of urea
The mole fraction of a component is calculated using the formula:

\[ \text{Mole fraction of solute} = \frac{\text{moles of solute}}{\text{moles of solute} + \text{moles of solvent}} \]

Substituting the values:

\[ \text{Mole fraction of urea} = \frac{0.05 \, \text{mol}}{0.05 \, \text{mol} + 13.89 \, \text{mol}} \approx \frac{0.05}{13.94} \approx 0.0036 \]

### Final Results
- **Molarity of the solution**: 0.2 mol/kg
- **Mole fraction of urea**: 0.0036