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When a liquid is taken in a long cylindr...

When a liquid is taken in a long cylindrical vessel of material, with linear coeffcient of expansion `alpha`, is heated, the level of liquid did not change. The volume coefficient of expansion of liquid is

A

`3alpha`

B

`2alpha`

C

`alpha`

D

`4alpha`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the situation where a liquid is contained in a long cylindrical vessel that expands when heated, but the level of the liquid remains unchanged. This indicates a relationship between the expansion of the vessel and the expansion of the liquid. ### Step-by-Step Solution: 1. **Understand the scenario**: When the vessel is heated, both the vessel and the liquid expand. However, since the level of the liquid does not change, the volume expansion of the liquid must be equal to the volume expansion of the vessel. 2. **Define the coefficients of expansion**: - Let the linear coefficient of expansion of the vessel be denoted as \( \alpha \). - The volume coefficient of expansion of the vessel can be expressed in terms of the linear coefficient as: \[ \gamma_v = 3\alpha \] - Let the volume coefficient of expansion of the liquid be denoted as \( \gamma_l \). 3. **Set up the relationship**: Since the level of the liquid does not change, the volume expansion of the vessel must equal the volume expansion of the liquid: \[ \gamma_v = \gamma_l \] 4. **Substitute the expression for the vessel's volume expansion**: \[ 3\alpha = \gamma_l \] 5. **Conclusion**: Therefore, the volume coefficient of expansion of the liquid is: \[ \gamma_l = 3\alpha \] ### Final Answer: The volume coefficient of expansion of the liquid is \( 3\alpha \). ---
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