The freezing point depression constant for water is `1.86^(@)Cm^(-1)`. If `5.00g NaSO_(4)` is dissolved in `45.0 g H_(2)O` the freezing point is changed by `-3.82^(@)C`. Calculate the van't Hoff factor for `Na_(2)SO_(4)`.

Options

A. `2.63`
B. `3.11`
C. `0.381`
D. `2.05`

Explanation

To calculate the van't Hoff factor (i) for sodium sulfate (Na₂SO₄) based on the given information, we can follow these steps:

### Step 1: Understand the formula for freezing point depression
The freezing point depression (ΔTf) can be calculated using the formula:
\[ \Delta T_f = i \cdot K_f \cdot m \]
Where:
- \( \Delta T_f \) = change in freezing point
- \( i \) = van't Hoff factor
- \( K_f \) = freezing point depression constant
- \( m \) = molality of the solution

### Step 2: Calculate the molality (m)
Molality (m) is defined as the number of moles of solute per kilogram of solvent. First, we need to calculate the number of moles of Na₂SO₄.

1. **Calculate the molar mass of Na₂SO₄**:
- Sodium (Na): 22.99 g/mol × 2 = 45.98 g/mol
- Sulfur (S): 32.07 g/mol × 1 = 32.07 g/mol
- Oxygen (O): 16.00 g/mol × 4 = 64.00 g/mol
- Total molar mass of Na₂SO₄ = 45.98 + 32.07 + 64.00 = 142.05 g/mol (approximately 142 g/mol)

2. **Calculate the number of moles of Na₂SO₄**:
\[ \text{Moles of Na₂SO₄} = \frac{\text{mass}}{\text{molar mass}} = \frac{5.00 \text{ g}}{142 \text{ g/mol}} \approx 0.0352 \text{ moles} \]

3. **Calculate the mass of the solvent (water) in kg**:
\[ \text{Mass of water} = 45.0 \text{ g} = 0.045 \text{ kg} \]

4. **Calculate molality (m)**:
\[ m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} = \frac{0.0352 \text{ moles}}{0.045 \text{ kg}} \approx 0.7822 \text{ mol/kg} \]

### Step 3: Substitute the values into the freezing point depression formula
Now we can substitute the values into the freezing point depression formula:
- \( \Delta T_f = -3.82 \, ^\circ C \)
- \( K_f = 1.86 \, ^\circ C \cdot \text{kg/mol} \)
- \( m \approx 0.7822 \, \text{mol/kg} \)

Rearranging the formula to solve for \( i \):
\[ i = \frac{\Delta T_f}{K_f \cdot m} \]

### Step 4: Calculate the van't Hoff factor (i)
Substituting the values:
\[ i = \frac{-3.82}{1.86 \cdot 0.7822} \]
Calculating the denominator:
\[ 1.86 \cdot 0.7822 \approx 1.454 \]
Now substituting back:
\[ i = \frac{-3.82}{1.454} \approx 2.63 \]

### Conclusion
The van't Hoff factor \( i \) for sodium sulfate (Na₂SO₄) is approximately **2.63**.

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