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

Options

A. `3.11`
B. `0.381`
C. `2.05`
D. `2.61`

Explanation

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

### Step 1: Write down the given data
- Freezing point depression constant for water (Kf) = 1.86 °C kg/mol
- Mass of Na₂SO₄ (solute) = 5.00 g
- Mass of water (solvent) = 45.0 g
- Change in freezing point (ΔTf) = -3.82 °C

### Step 2: Calculate the molality (m) of the solution
Molality (m) is defined as the number of moles of solute per kilogram of solvent.

1. First, calculate the number of moles of Na₂SO₄:
- Molar mass of Na₂SO₄ = (2 × 23) + (32) + (4 × 16) = 46 + 32 + 64 = 142 g/mol
- Moles of Na₂SO₄ = mass / molar mass = 5.00 g / 142 g/mol = 0.0352 mol

2. Convert the mass of water to kilograms:
- Mass of water = 45.0 g = 45.0 / 1000 = 0.045 kg

3. Calculate molality (m):
- m = moles of solute / kg of solvent = 0.0352 mol / 0.045 kg = 0.7822 mol/kg

### Step 3: Use the freezing point depression formula
The freezing point depression is given by the formula:
\[ \Delta T_f = i \cdot K_f \cdot m \]

Where:
- ΔTf = change in freezing point
- i = van't Hoff factor
- Kf = freezing point depression constant
- m = molality

### Step 4: Rearrange the formula to solve for i
Rearranging the formula gives:
\[ i = \frac{\Delta T_f}{K_f \cdot m} \]

### Step 5: Substitute the values into the equation
Substituting the known values:
\[ i = \frac{3.82 °C}{1.86 °C \cdot 0.7822 \, \text{mol/kg}} \]

### Step 6: Calculate the van't Hoff factor (i)
1. Calculate the denominator:
- \( K_f \cdot m = 1.86 °C \cdot 0.7822 \, \text{mol/kg} = 1.454 \, °C \)

2. Now calculate i:
- \( i = \frac{3.82}{1.454} \approx 2.62 \)

### Final Answer
The van't Hoff factor (i) for Na₂SO₄ is approximately **2.62**.

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