Some water at `0^@C` is placed in a large insulated enclosure (vessel). The water vapour formed is poured out continuously. What fraction of the water will ultimately freeze, if the latent heat of vaporization is seven times the latent heat of fusion?

Options

A. `5/2`
B. `7/8`
C. `1/3`
D. `9/5`

Explanation

To solve the problem, we need to determine what fraction of the water will ultimately freeze when water vapor is continuously removed from an insulated vessel. Given that the latent heat of vaporization (L_V) is seven times the latent heat of fusion (L_F), we can set up the following steps:

### Step-by-Step Solution:

1. **Define Variables:**
- Let the total mass of water be \( M \).
- Let the fraction of water that freezes be \( F \).
- The mass of frozen water will then be \( M \cdot F \).
- The mass of water that remains unfrozen will be \( M \cdot (1 - F) \).

2. **Calculate Heat Lost by Frozen Water:**
- The heat lost by the water that freezes can be calculated using the latent heat of fusion:
\[ \text{Heat lost} = \text{mass of frozen water} \times L_F = M \cdot F \cdot L_F \]

3. **Calculate Heat Gained by Vapor:**
- The heat gained by the vapor formed from the unfrozen water can be calculated using the latent heat of vaporization:
\[ \text{Heat gained} = \text{mass of vapor} \times L_V = M \cdot (1 - F) \cdot L_V \]

4. **Set Heat Lost Equal to Heat Gained:**
- Since the vessel is insulated, the heat lost by the frozen water must equal the heat gained by the vapor:
\[ M \cdot F \cdot L_F = M \cdot (1 - F) \cdot L_V \]

5. **Cancel Mass \( M \):**
- Since \( M \) is common on both sides, we can cancel it out:
\[ F \cdot L_F = (1 - F) \cdot L_V \]

6. **Substitute \( L_V \):**
- Given that \( L_V = 7 \cdot L_F \), substitute this into the equation:
\[ F \cdot L_F = (1 - F) \cdot (7 \cdot L_F) \]

7. **Cancel \( L_F \):**
- We can also cancel \( L_F \) from both sides (assuming \( L_F \neq 0 \)):
\[ F = (1 - F) \cdot 7 \]

8. **Expand and Rearrange:**
- Expanding the right side gives:
\[ F = 7 - 7F \]
- Rearranging the equation:
\[ F + 7F = 7 \]
\[ 8F = 7 \]

9. **Solve for \( F \):**
- Dividing both sides by 8:
\[ F = \frac{7}{8} \]

### Conclusion:
The fraction of the water that will ultimately freeze is \( \frac{7}{8} \).