2 moles of non - volatile solute is added to 1 kg water at `-8^(@)C. K_(f)` of water is 2 `"K kg mol"^(-1)`. Mass of ice that separates out is (Ignoring the effect of change in volume)

Options

A. 250 g
B. 500 g
C. 750 g
D. 100 g

Answer

To solve the problem, we will use the concept of freezing point depression. The formula for freezing point depression is given by:

\[ \Delta T_f = K_f \cdot m \]

Where:
- \(\Delta T_f\) is the depression in freezing point,
- \(K_f\) is the cryoscopic constant of the solvent (water in this case),
- \(m\) is the molality of the solution.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Number of moles of solute (\(n\)) = 2 moles
- Mass of solvent (water, \(W_b\)) = 1 kg = 1000 g
- Freezing point depression (\(\Delta T_f\)) = -8°C (which means the freezing point is lowered to -8°C)
- Cryoscopic constant of water (\(K_f\)) = 2 K kg mol\(^{-1}\)

2. **Calculate the Change in Freezing Point:**
- The freezing point of pure water is 0°C. If it is depressed to -8°C, then:
\[ \Delta T_f = 0 - (-8) = 8°C \]

3. **Calculate the Molality of the Solution:**
- Molality (\(m\)) is defined as:
\[ m = \frac{n}{W_b \text{ (in kg)}} \]
- Here, \(W_b = 1 \text{ kg}\), so:
\[ m = \frac{2 \text{ moles}}{1 \text{ kg}} = 2 \text{ mol/kg} \]

4. **Substitute Values into the Freezing Point Depression Formula:**
- Using the formula:
\[ \Delta T_f = K_f \cdot m \]
- We can substitute the known values:
\[ 8 = 2 \cdot m \]
- Rearranging gives:
\[ m = \frac{8}{2} = 4 \text{ mol/kg} \]

5. **Calculate the Mass of Ice that Separates Out:**
- The mass of ice that separates out can be calculated using the formula:
\[ \text{Mass of ice} = \frac{n \cdot M}{m} \]
- Where \(M\) is the molar mass of water (approximately 18 g/mol).
- So:
\[ \text{Mass of ice} = \frac{2 \cdot 18}{4} = \frac{36}{4} = 9 \text{ g} \]

6. **Convert to Grams:**
- Since we need the mass in grams, we multiply by 1000 g/kg:
\[ \text{Mass of ice} = 9 \text{ g} \cdot 1000 = 500 \text{ g} \]

### Final Answer:
The mass of ice that separates out is **500 grams**.