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. (a)remain unchanged if `gamma_v=gamma_L`
B. (b)increase if `gamma_v=gamma_L`
C. (c)decrease if `gamma_v=gamma_L`
D. (d)none of the above

Explanation

To solve the problem, we need to analyze the situation of a vessel that is partially filled with a liquid, taking into account the coefficients of cubical expansion of both the vessel and the liquid.

### Step-by-Step Solution:

1. **Understanding the Coefficients of Cubical Expansion**:
- The coefficient of cubical expansion (γ) measures how much a substance expands per degree change in temperature. For the vessel, we denote it as γ_v (for the vessel) and for the liquid as γ_l (for the liquid).

2. **Initial Volume of the Vessel and Liquid**:
- Let the initial volume of the liquid be V_l and the initial volume of the vessel be V_v. The volume of the vessel that is unoccupied by the liquid is V_u = V_v - V_l.

3. **Change in Volume Due to Temperature Change**:
- When the temperature changes by ΔT, the change in volume of the liquid (ΔV_l) can be expressed as:
\[ \Delta V_l = V_l \cdot \gamma_l \cdot \Delta T \]
- Similarly, the change in volume of the vessel (ΔV_v) is given by:
\[ \Delta V_v = V_v \cdot \gamma_v \cdot \Delta T \]

4. **Volume Relationships**:
- The volume of the liquid expands as it heats up, and the volume of the vessel also expands. The total volume of the vessel remains constant, which means the volume that is unoccupied will change based on the expansions of both the liquid and the vessel.

5. **Setting Up the Equation**:
- For the volume that remains unoccupied (V_u), we can write:
\[ V_u + \Delta V_l = V_v + \Delta V_v \]
- Rearranging gives us:
\[ V_u = V_v + \Delta V_v - \Delta V_l \]

6. **Equating Changes**:
- Since the unoccupied volume must remain constant, we can set the changes equal:
\[ \Delta V_l = \Delta V_v \]
- This leads us to conclude that:
\[ V_l \cdot \gamma_l \cdot \Delta T = V_v \cdot \gamma_v \cdot \Delta T \]
- Simplifying this, we find:
\[ \gamma_l = \gamma_v \]

7. **Conclusion**:
- Therefore, the coefficient of cubical expansion of the liquid must equal that of the vessel for the unoccupied volume to remain unchanged. Thus, the correct option is **A**.