The f.p. of `1%` solution of `Ca(NO_(3))_(2)` in water will be

Options

A. below `0^(@)C`
B. `0^(@)`
C. `1^(@)C`
D. `2^(@)C`

Answer

To find the freezing point of a 1% solution of calcium nitrate \((Ca(NO_3)_2)\) in water, we can follow these steps:

### Step 1: Understand the Freezing Point Depression Concept
When a solute is dissolved in a solvent, the freezing point of the solution is lower than that of the pure solvent. This phenomenon is known as freezing point depression.

### Step 2: Identify the Freezing Point Depression Formula
The freezing point depression can be calculated using the formula:
\[ \Delta T_f = i \cdot K_f \cdot m \]
where:
- \(\Delta T_f\) = freezing point depression
- \(i\) = van 't Hoff factor (number of particles the solute dissociates into)
- \(K_f\) = freezing point depression constant of the solvent (for water, \(K_f = 1.86 \, °C \cdot kg/mol\))
- \(m\) = molality of the solution

### Step 3: Calculate the Van 't Hoff Factor \(i\)
For calcium nitrate \((Ca(NO_3)_2)\):
- It dissociates into 3 ions: 1 calcium ion \((Ca^{2+})\) and 2 nitrate ions \((NO_3^-)\).
- Therefore, \(i = 3\).

### Step 4: Calculate the Molality of the Solution
To find the molality \(m\), we need to know the number of moles of solute in 1 kg of solvent (water).

1. **Calculate the molar mass of \(Ca(NO_3)_2\)**:
- Calcium (Ca) = 40.08 g/mol
- Nitrogen (N) = 14.01 g/mol (2 N in the formula)
- Oxygen (O) = 16.00 g/mol (6 O in the formula)
- Molar mass of \(Ca(NO_3)_2 = 40.08 + (2 \times 14.01) + (6 \times 16.00) = 164.10 \, g/mol\)

2. **Calculate the number of grams in 1% solution**:
- 1% solution means 1 g of \(Ca(NO_3)_2\) in 100 mL of solution.
- Since the density of water is approximately 1 g/mL, we can assume 100 mL of solution is roughly 100 g of water.

3. **Calculate moles of \(Ca(NO_3)_2\)**:
- Moles = mass (g) / molar mass (g/mol)
- Moles of \(Ca(NO_3)_2 = \frac{1 \, g}{164.10 \, g/mol} \approx 0.0061 \, mol\)

4. **Calculate molality \(m\)**:
- Molality \(m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} = \frac{0.0061 \, mol}{0.1 \, kg} = 0.061 \, mol/kg\)

### Step 5: Calculate the Freezing Point Depression
Now we can substitute the values into the freezing point depression formula:
\[ \Delta T_f = i \cdot K_f \cdot m = 3 \cdot 1.86 \, °C \cdot 0.061 \, mol/kg \]
\[ \Delta T_f \approx 0.34 \, °C \]

### Step 6: Determine the New Freezing Point
The normal freezing point of water is \(0 \, °C\). Therefore, the new freezing point will be:
\[ \text{New Freezing Point} = 0 \, °C - \Delta T_f = 0 \, °C - 0.34 \, °C \approx -0.34 \, °C \]

### Final Answer
The freezing point of a 1% solution of \(Ca(NO_3)_2\) in water will be approximately \(-0.34 \, °C\).

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