A liquid (coefficient of cubical expansion `gamma_(1)`) is contained in a glass vessel of volume `V_(0)` (coefficient of cubical expansion `gamma_(g)`) at a temperature. The volume of liquid at this temperature is `V_(l)`. Now the system is heated and it is found that at all temperatures, the volume of vessel, unoccupied by liquid remains always same, then

Options

A. `(V_(g))/(V_(l))=(gamma_(l))/(gamma_(g))`
B. `(V_(g))/(V_(l))=(gamma_(g))/(gamma_(l))`
C. `V_(g)-V_(l)=gamma_(g)-gamma_(l)`
D. `V_(g)+V_(l)=gamma_(g)+gamma_(l)`

Answer

To solve the problem, we need to analyze the relationship between the volumes of the liquid and the glass vessel as they undergo thermal expansion.

### Step-by-Step Solution:

1. **Understand the Variables**:
- Let \( V_0 \) be the initial volume of the glass vessel.
- Let \( V_L \) be the volume of the liquid at the initial temperature.
- The coefficient of cubical expansion for the liquid is \( \gamma_1 \).
- The coefficient of cubical expansion for the glass is \( \gamma_g \).

2. **Volume Change Due to Temperature**:
- When the temperature changes by \( \Delta T \), the change in volume of the liquid can be expressed as:
\[ \Delta V_L = \gamma_1 V_L \Delta T \]
- The change in volume of the glass vessel can be expressed as:
\[ \Delta V_G = \gamma_g V_0 \Delta T \]

3. **Total Volume Consideration**:
- The problem states that the volume of the vessel unoccupied by the liquid remains constant as the system is heated. This means that the volume of the glass vessel increases, but the volume of liquid that occupies the vessel also increases in such a way that the unoccupied volume does not change.

4. **Setting Up the Equation**:
- Since the unoccupied volume remains constant, we can write:
\[ V_0 + \Delta V_G - (V_L + \Delta V_L) = \text{constant} \]
- This simplifies to:
\[ \Delta V_G - \Delta V_L = 0 \]
- Substituting the expressions for \( \Delta V_G \) and \( \Delta V_L \):
\[ \gamma_g V_0 \Delta T - \gamma_1 V_L \Delta T = 0 \]

5. **Canceling \( \Delta T \)**:
- Since \( \Delta T \) is common on both sides and not equal to zero, we can cancel it:
\[ \gamma_g V_0 = \gamma_1 V_L \]

6. **Final Relationship**:
- Rearranging gives us the relationship between the coefficients of cubical expansion and the volumes:
\[ \frac{\gamma_1}{\gamma_g} = \frac{V_0}{V_L} \]