The Question is from neet-physics thermal-properties-of-matter thermal-expansion-of-liquids, and it was created by the author VENKATESWARARAO

Question

The coefficient of real expansion of a liquid is $\gamma_r$ and the coefficient of apparent expansion of the liquid in a vessel is $\gamma_a$. The coefficient of volume expansion of the vessel is:

VENKATESWARARAO · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
	**Primary Concept**: Thermal Expansion of Liquids and Solids
	**Related Topics**: Coefficient of Real Expansion, Coefficient of Apparent Expansion, Coefficient of Volume Expansion
	**Prerequisites**: Understanding of thermal expansion and its coefficients.
## SOLUTION PATHWAY
**Given Information**:
	$\gamma_r$: Coefficient of real expansion of the liquid.
	$\gamma_a$: Coefficient of apparent expansion of the liquid in the vessel.
**Method & Steps**:
1. **Understand the definitions**
	Reasoning: The coefficient of real expansion ($\gamma_r$) represents the actual change in volume of the liquid per unit volume per degree Celsius. The coefficient of apparent expansion ($\gamma_a$) represents the observed change in volume of the liquid in a vessel per unit volume per degree Celsius. The coefficient of volume expansion of the vessel ($\gamma_v$) represents the change in volume of the vessel per unit volume per degree Celsius.
2. **Establish the relationship**
	Working: The apparent expansion of the liquid is the difference between the real expansion of the liquid and the expansion of the vessel.  Mathematically, $\gamma_a = \gamma_r - \gamma_v$.
	Key Point: The apparent expansion is less than the real expansion because the vessel also expands, accommodating some of the increased liquid volume.
3. **Solve for $\gamma_v$**
	Result: Rearranging the equation, we get $\gamma_v = \gamma_r - \gamma_a$.
	Verification: This result aligns with the understanding that if the vessel expands at the same rate as the liquid ($\gamma_v = \gamma_r$), there would be no apparent expansion ($\gamma_a = 0$).
## OPTION ANALYSIS
[A] :x: $\gamma_r + \gamma_a$: This is incorrect.  Adding the real and apparent expansions does not give the expansion of the vessel. It would imply the vessel is shrinking, which is not the case in this context.
[B] ✓: $\gamma_r - \gamma_a$: This is the correct option, as derived in the solution pathway.
[C] :x: $\gamma_a - \gamma_r$: This would result in a negative value for the vessel's expansion coefficient, implying the vessel contracts upon heating, which is generally not true.
[D] :x: $\gamma_r * \gamma_a$: Multiplying the coefficients is not physically meaningful in this context. The relationship between the coefficients is additive/subtractive, not multiplicative.
## LEARNING AIDS
	**Common Mistakes**:
		1. Confusing real and apparent expansion: Remember that apparent expansion is what we observe, while real expansion is the actual change in the liquid's volume.
		2. Incorrectly applying the formula: Ensure the correct signs are used in the relationship $\gamma_a = \gamma_r - \gamma_v$.
	**Quick Tips**:
		1. Visualize the scenario: Imagine the liquid and vessel expanding. This helps understand why the apparent expansion is less than the real expansion.
		2. Remember the relationship: $\gamma_a = \gamma_r - \gamma_v$ is the key equation for this concept.

VENKATESWARARAO · Answer

γr + γa

VENKATESWARARAO · Answer

The correct Option is "γr - γa"

VENKATESWARARAO · Answer

γa - γr

VENKATESWARARAO · Answer

γr * γa

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