A vessel is partly filled with liquid. When the vessel is cooled to a lower temperature, the space in the vessel unoccupied by the liquid remains constant. Then the volume of the liquid `(V_L)` volume of the vessel `(V_V)` the coefficient of cubical expansion of the material of the vessel `(gamma_v)` and of the solid `(gamma_L)` are related as

Options

A. `gamma_(L)gtgamma_(v)`
B. `gamma_(L)ltgamma_(V)`
C. `(gamma_(V))/(gamma_(L))=(V_(V))/(V_(L))`
D. `(gamma_(V))/(gamma_(L))=(V_(L))/(V_(V))`

Explanation

To solve the problem, we need to establish the relationship between the volumes of the liquid and the vessel, as well as their respective coefficients of cubical expansion when the vessel is cooled.

### Step-by-Step Solution:

1. **Understand the Problem**: We have a vessel that is partly filled with a liquid. When the temperature decreases, we want the unoccupied space in the vessel to remain constant. This means that any change in volume of the liquid must be balanced by a change in the volume of the vessel.

2. **Define Variables**:
- Let \( V_L \) be the volume of the liquid.
- Let \( V_V \) be the volume of the vessel.
- Let \( \gamma_L \) be the coefficient of cubical expansion of the liquid.
- Let \( \gamma_V \) be the coefficient of cubical expansion of the vessel.

3. **Volume Change Due to Temperature Change**:
- The change in volume of the liquid when cooled can be expressed as:
\[ \Delta V_L = \gamma_L V_L \Delta T \]
- The change in volume of the vessel can be expressed as:
\[ \Delta V_V = \gamma_V V_V \Delta T \]

4. **Condition for Constant Unoccupied Space**:
- For the unoccupied space to remain constant, the increase in volume of the liquid must equal the increase in volume of the vessel:
\[ \Delta V_L = \Delta V_V \]
- Substituting the expressions for volume change:
\[ \gamma_L V_L \Delta T = \gamma_V V_V \Delta T \]

5. **Canceling Out \(\Delta T\)**:
- Since \(\Delta T\) is common on both sides and is not zero, we can cancel it out:
\[ \gamma_L V_L = \gamma_V V_V \]

6. **Rearranging the Equation**:
- We can rearrange this equation to express the relationship between the coefficients of cubical expansion and the volumes:
\[ \frac{\gamma_V}{\gamma_L} = \frac{V_L}{V_V} \]

### Conclusion:
The relationship between the coefficient of cubical expansion of the vessel and the liquid, along with their volumes, is given by:
\[ \frac{\gamma_V}{\gamma_L} = \frac{V_L}{V_V} \]