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 liquid `(gamma_L)` are related as

Options

A. `gamma_L gt gamma_V`
B. `gamma_L lt gamma_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 analyze the relationship between the volumes of the liquid and the vessel, as well as their respective coefficients of cubical expansion when the temperature changes.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a vessel that is partially filled with a liquid. When the temperature of the vessel and the liquid is lowered, the volume of the liquid does not change, meaning the space in the vessel that is unoccupied by the liquid remains constant.

2. **Defining Variables**:
- Let:
- \( V_L \) = Volume of the liquid
- \( V_V \) = Volume of the vessel
- \( \gamma_V \) = Coefficient of cubical expansion of the vessel material
- \( \gamma_L \) = Coefficient of cubical expansion of the liquid

3. **Applying the Concept of Cubical Expansion**:
- The change in volume due to temperature change for the vessel and the liquid can be expressed as:
- Change in volume of the vessel:
\[ \Delta V_V = \gamma_V \cdot V_V \cdot \Delta T \]
- Change in volume of the liquid:
\[ \Delta V_L = \gamma_L \cdot V_L \cdot \Delta T \]

4. **Setting Up the Equation**:
- Since the volume of the unoccupied space remains constant, the change in volume of the vessel must equal the change in volume of the liquid:
\[ \Delta V_V = \Delta V_L \]
- Substituting the expressions for the changes in volume:
\[ \gamma_V \cdot V_V \cdot \Delta T = \gamma_L \cdot V_L \cdot \Delta T \]

5. **Simplifying the Equation**:
- Since \(\Delta T\) is the same for both the vessel and the liquid, we can cancel it out:
\[ \gamma_V \cdot V_V = \gamma_L \cdot V_L \]

6. **Rearranging the Equation**:
- Rearranging gives us the relationship between the coefficients of cubical expansion and the volumes:
\[ \frac{\gamma_V}{\gamma_L} = \frac{V_L}{V_V} \]

7. **Conclusion**:
- This relationship indicates that the ratio of the coefficients of cubical expansion is equal to the ratio of the volumes of the liquid and the vessel.

### Final Answer:
\[ \frac{\gamma_V}{\gamma_L} = \frac{V_L}{V_V} \]