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Question

An ideal gas has a molar heat capacity at constant volume CV. Find the molar heat capacity of this gas as a function of its volume V, if it undergoes a process T=ToeαV2, where To and α are constants.

A
CV+RαV
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B
CV+αRmV
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C
CV+RαV2
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D
CV+R2αV2
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Solution

The correct option is D CV+R2αV2
Given,
T=ToeαV2
Differentiating on both sides with respect to 'T',
d(T)dT=ToeαV2×2αVdVdT
1=T(2αVdVdT) .......(1)
By using first law of thermodynamics, we can say that,
ΔQ=ΔU+ΔW
or nCdT=nCVdT+PdV
or C=CV+PndVdT .....(2)
Using ideal gas equation in (1), we get,
1=(PVnR)2αVdVdT
1=2αV2R(PndVdT)
(PndVdT)=R2αV2 .......(3)
Substituting (3) in (2),
C=CV+R2αV2
Thus, option (d) is the correct answer.

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