A thermal insulated vessel contains some water at `0^(@)C`. The vessel is connected to a vaccum pump to pum out water vapour. This results in some water getting frozen. It is given latent heat of vaporization of water at `0^(@)C = 21 xx 10^(5) J//kg` and latent heat of freezing of water `=3.36 xx 10^(5) J//kg`. the maximum percentage amount of water vapour that will be solidified in this manner will be:

Options

A. `86.2%`
B. `33.6%`
C. `21%`
D. `24.36%`

Answer

To solve the problem, we need to apply the principle of conservation of energy, specifically in the context of calorimetry. The key idea is that the heat lost by the water vapor during its condensation and subsequent freezing will equal the heat gained by the ice formed.

### Step-by-step Solution:

1. **Identify Variables:**
- Let \( M \) be the total mass of water in the vessel (in kg).
- Let \( x \) be the mass of water that gets frozen (in kg).
- The mass of water vapor remaining will then be \( M - x \).

2. **Latent Heats:**
- Latent heat of freezing of water, \( L_f = 3.36 \times 10^5 \, \text{J/kg} \).
- Latent heat of vaporization of water, \( L_v = 21 \times 10^5 \, \text{J/kg} \).

3. **Energy Balance Equation:**
- The heat gained by the ice (frozen water) is given by:
\[ Q_{\text{gained}} = x \cdot L_f \]
- The heat lost by the water vapor is given by:
\[ Q_{\text{lost}} = (M - x) \cdot L_v \]

4. **Set Up the Equation:**
- According to the conservation of energy:
\[ Q_{\text{gained}} = Q_{\text{lost}} \]
- Thus, we have:
\[ x \cdot L_f = (M - x) \cdot L_v \]

5. **Substituting Values:**
- Substitute \( L_f \) and \( L_v \):
\[ x \cdot (3.36 \times 10^5) = (M - x) \cdot (21 \times 10^5) \]

6. **Rearranging the Equation:**
- Expanding the equation:
\[ 3.36 \times 10^5 x = 21 \times 10^5 M - 21 \times 10^5 x \]
- Combine like terms:
\[ 3.36 x + 21 x = 21 M \]
\[ 24.36 x = 21 M \]
- Solving for \( x \):
\[ x = \frac{21 M}{24.36} \]

7. **Finding the Fraction of Water that Freezes:**
- The fraction of the total mass \( M \) that freezes is:
\[ \frac{x}{M} = \frac{21}{24.36} \]

8. **Calculating the Percentage:**
- To find the percentage:
\[ \text{Percentage} = \left( \frac{21}{24.36} \right) \times 100 \]
- Performing the calculation:
\[ \text{Percentage} \approx 86.2\% \]

### Final Answer:
The maximum percentage amount of water vapor that will be solidified is approximately **86.2%**.