A vessel is partly filled with a liquid. Coefficients of cubical expansion of material of the vessel and liquid are `gamma_v` and `gamma_L` , then the volume unoccupied by the liquid will necessarily

Options

A. Remain unchanged if `gamma_(V)=gamma_(L)`
B. Increase if `gamma_(V)-gamma_(L)`
C. Decrease if `gamma_(V)-gamma_(L)`
D. None of above

Explanation

To solve the problem, we need to analyze how the volumes of the liquid and the vessel change with temperature due to their respective coefficients of cubical expansion.

### Step-by-Step Solution:

1. **Understanding the Coefficients of Cubical Expansion**:
- The coefficient of cubical expansion for the vessel is denoted as \( \gamma_v \).
- The coefficient of cubical expansion for the liquid is denoted as \( \gamma_L \).
- These coefficients indicate how much the volume of the material (vessel or liquid) changes with a change in temperature.

2. **Initial Volume Definitions**:
- Let \( V_0 \) be the initial volume of the liquid in the vessel.
- Let \( V_g \) be the initial volume of the vessel.

3. **Volume Change with Temperature**:
- When the temperature changes by \( \Delta T \), the change in volume of the liquid can be expressed as:
\[ \Delta V_L = V_0 \cdot \gamma_L \cdot \Delta T \]
- Similarly, the change in volume of the vessel can be expressed as:
\[ \Delta V_g = V_g \cdot \gamma_v \cdot \Delta T \]

4. **New Volumes After Temperature Change**:
- The new volume of the liquid after the temperature change will be:
\[ V_L = V_0 + \Delta V_L = V_0 + V_0 \cdot \gamma_L \cdot \Delta T = V_0 (1 + \gamma_L \cdot \Delta T) \]
- The new volume of the vessel after the temperature change will be:
\[ V_g = V_g + \Delta V_g = V_g + V_g \cdot \gamma_v \cdot \Delta T = V_g (1 + \gamma_v \cdot \Delta T) \]

5. **Volume Unoccupied by the Liquid**:
- The volume unoccupied by the liquid in the vessel can be expressed as:
\[ V_{unoccupied} = V_g - V_L \]
- Substituting the new volumes:
\[ V_{unoccupied} = V_g (1 + \gamma_v \cdot \Delta T) - V_0 (1 + \gamma_L \cdot \Delta T) \]

6. **Analyzing the Change in Unoccupied Volume**:
- If \( \gamma_v = \gamma_L \), then the terms involving \( \Delta T \) will cancel out, and the volume unoccupied by the liquid remains unchanged.
- If \( \gamma_v \neq \gamma_L \), the change in unoccupied volume will depend on the difference \( \gamma_v - \gamma_L \).

7. **Conclusion**:
- The volume unoccupied by the liquid will remain unchanged if the coefficients of cubical expansion of the vessel and the liquid are equal. If they are not equal, the unoccupied volume will change based on the difference between the two coefficients.

### Final Answer:
The volume unoccupied by the liquid will necessarily remain unchanged if \( \gamma_v = \gamma_L \).