When water is heated from `0^(@)C` to `4^(@)C` and `C_(P)` and `C_(V)` are its specific heats at constant pressure and constant volume respectively , then :-

Options

A. `C_(P)gt C_(V)`
B. `C_(P)lt C_(V)`
C. `C_(P) = C_(V)`
D. `C_(P) - C_(V) = R`

Explanation

To solve the problem of determining the relationship between the specific heats \( C_P \) and \( C_V \) of water when it is heated from \( 0^\circ C \) to \( 4^\circ C \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Specific Heats**:
- \( C_P \) is the specific heat at constant pressure.
- \( C_V \) is the specific heat at constant volume.
- We need to analyze the behavior of water as it is heated from \( 0^\circ C \) to \( 4^\circ C \).

2. **Applying the First Law of Thermodynamics**:
- The first law of thermodynamics states:
\[ \Delta Q = \Delta U + W \]
- Where \( \Delta Q \) is the heat added, \( \Delta U \) is the change in internal energy, and \( W \) is the work done.

3. **Expressing Heat Added**:
- The heat added when heating the water can be expressed as:
\[ \Delta Q = n C_P \Delta T \]
- Here, \( n \) is the number of moles and \( \Delta T \) is the change in temperature.

4. **Change in Internal Energy**:
- The change in internal energy can be expressed as:
\[ \Delta U = n C_V \Delta T \]

5. **Work Done**:
- The work done on or by the system can be expressed as:
\[ W = P \Delta V \]

6. **Combining the Equations**:
- Substituting these expressions into the first law gives:
\[ n C_P \Delta T = n C_V \Delta T + P \Delta V \]

7. **Analyzing the Volume Change**:
- When water is heated from \( 0^\circ C \) to \( 4^\circ C \), it actually contracts before expanding. At \( 4^\circ C \), water reaches its maximum density.
- This means that as water is heated, the volume decreases, leading to a negative work done (\( P \Delta V < 0 \)).

8. **Conclusion**:
- Rearranging the equation gives:
\[ n C_P \Delta T < n C_V \Delta T \]
- This implies:
\[ C_P < C_V \]

### Final Answer:
Thus, the relationship between the specific heats is \( C_P < C_V \).