A vessel is partly filled with a liquid. Coefficients of cubical expansion of material of the vessel and liquid are `gamma_v` 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 the above

Explanation

To solve the problem, we need to analyze the situation where a vessel is partly filled with a liquid, and we are given the coefficients of cubical expansion for both the vessel and the liquid. Let's denote:

- \( \gamma_v \): Coefficient of cubical expansion of the vessel material.
- \( \gamma_l \): Coefficient of cubical expansion of the liquid.

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
- Let the initial volume of the vessel be \( V \).
- Let the initial volume of the liquid be \( V_l \).
- Since the vessel is partly filled, we have \( V_l < V \).

2. **Determine the Change in Volume**:
- When the temperature changes by \( \Delta T \), the change in volume for the vessel can be expressed as:
\[ \Delta V_v = V \cdot \gamma_v \cdot \Delta T \]
- Similarly, the change in volume for the liquid is:
\[ \Delta V_l = V_l \cdot \gamma_l \cdot \Delta T \]

3. **Calculate the New Volumes**:
- The new volume of the vessel after expansion will be:
\[ V' = V + \Delta V_v = V + V \cdot \gamma_v \cdot \Delta T = V(1 + \gamma_v \cdot \Delta T) \]
- The new volume of the liquid will be:
\[ V'_l = V_l + \Delta V_l = V_l + V_l \cdot \gamma_l \cdot \Delta T = V_l(1 + \gamma_l \cdot \Delta T) \]

4. **Analyze the Relationship Between the Volumes**:
- Since the vessel is partly filled, we need to compare the new volume of the liquid \( V'_l \) with the new volume of the vessel \( V' \).
- The unoccupied volume in the vessel after expansion can be expressed as:
\[ V_{unoccupied} = V' - V'_l \]

5. **Consider the Case When \( \gamma_v = \gamma_l \)**:
- If \( \gamma_v = \gamma_l \), then:
\[ V' = V(1 + \gamma \cdot \Delta T) \]
\[ V'_l = V_l(1 + \gamma \cdot \Delta T) \]
- Since both the vessel and the liquid expand at the same rate, the unoccupied volume will increase because the vessel's volume increases more than the liquid's volume.

6. **Conclusion**:
- Therefore, if \( \gamma_v = \gamma_l \), the volume unoccupied by the liquid will increase.

### Final Answer:
The volume unoccupied by the liquid will increase if the coefficients of cubical expansion of the vessel and the liquid are equal.