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Question

A polytropic process for an ideal gas is represented by equation PVα=constant. If γ is the ratio of specific heats CpCv , then the value of α for which molar heat capacity of the process is negative, is given by

A
γ>α
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B
γ>α>1
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C
α>γ
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D
none of these
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Solution

The correct option is B γ>α>1
In a polytropic process the molar heat capacity of the process doesnot vary.
Given PVα=constant
We need to find of values of α for which the molar heat capacity is negative.
By the definition of molar specific heat capacity, we get
ΔQ=nCΔT
C=1nΔQΔT=1ndQdT (For instantaneous value)
Using first law of thermodynamics, dQ=dU+dW in the above equation,
C=1ndU+dWdT
Substituting, dU=nCv dT and dW=PdV
C=1n(nCv dTdT+PdVdT)
=Cv+PdVndT....(i)

Using the ideal gas equation, PV=nRT
Differentiaing, PdV+VdPR=ndT
Substituting this in equation (i) we have,
C=Cv+PdVPdV+VdPR
C=Cv+R1+VdPPdV....(ii)

From the given equation,PVα=kP=kvα
Differenciating, dPdV=αkVα+1
Substituting this in equation (ii) we have,
C=Cv+R1+VdPPdV=Cv+11+Vk(k)αVα+1Vα
In general Cv=Rγ1
C=Rγ1+R1α
We want this value to be negative,
C=Rγ1+R1α<0
Simplyfing the above equation, (αγ)(γ1)(α1)<0
γ1 is positive.
(αγ)(α1)<0
Case 1: (αγ)<0 and (α1)>0
α<γ and α>1γ>α>1

Case 2:(αγ)>0 and (α1)<0
α>γ and α<1
This possibility is not possible as γ cannot be less than 1

γ>α>1 is correct.

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