A liquid with coefficient of volume expansion `gamma` is filled in a container of a material having coefficient of linear expansion `alpha` . If the liquid overflows on heating, then

Options

A. `gamma=3alpha`
B. `gammagt3alpha`
C. `gammalt3alpha`
D. `gamma=alpha^(3)`

Explanation

To solve the problem step by step, we need to analyze the relationship between the volume expansion of the liquid and the linear expansion of the container.

### Step-by-Step Solution:

1. **Understanding Coefficients of Expansion**:
- The coefficient of volume expansion of the liquid is denoted as \( \gamma \).
- The coefficient of linear expansion of the container material is denoted as \( \alpha \).
- The volume expansion of a liquid is related to its initial volume \( V \) and the change in temperature \( \Delta T \) by the formula:
\[ \Delta V_{\text{liquid}} = \gamma V \Delta T \]

2. **Volume Expansion of the Container**:
- For the container, which expands linearly, the change in volume \( \Delta V_{\text{container}} \) can be expressed in terms of its linear expansion coefficient \( \alpha \):
\[ \Delta V_{\text{container}} = 3 \alpha V \Delta T \]
- This factor of 3 comes from the fact that volume is three-dimensional, and thus the linear expansion coefficient must be multiplied by 3.

3. **Condition of Overflow**:
- The problem states that the liquid overflows when heated. This implies that the volume expansion of the liquid must exceed that of the container.
- Therefore, we set up the inequality:
\[ \Delta V_{\text{liquid}} > \Delta V_{\text{container}} \]
- Substituting the expressions we derived:
\[ \gamma V \Delta T > 3 \alpha V \Delta T \]

4. **Simplifying the Inequality**:
- Since \( V \Delta T \) is common on both sides and assuming \( V \Delta T \neq 0 \), we can cancel it out:
\[ \gamma > 3 \alpha \]

5. **Conclusion**:
- From the above inequality, we conclude that the coefficient of volume expansion \( \gamma \) of the liquid is greater than three times the coefficient of linear expansion \( \alpha \) of the container material.
- Thus, the final answer is:
\[ \gamma > 3 \alpha \]

### Final Answer:
The correct option is that \( \gamma \) is greater than \( 3\alpha \).

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