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Question

Following figure shows an adiabatic cylindrical container of volume Vo divided by an adiabatic smooth piston (area of cross section = A) in two equal parts. An ideal gas (CPCV=γ) is at pressure P1 and temperature T1 in left part and gas at pressure P2 and temperature T2 in right part. The piston is slowly displaced and released at a position where it can stay in equilibrium. The final pressure of the two parts can be (Suppose x is displacement of the piston)

A
P2
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B
P1
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C
P1(Vo2)γ(Vo2+Ax)γ
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D
P2(Vo2)γ(Vo2+Ax)γ
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Solution

The correct option is D P2(Vo2)γ(Vo2+Ax)γ
Let the initial volume of each container be Vo2. Let Pf be the final pressure on both sides(in equilibrium).
It is not specified which of the two pressures are bigger initially. So, we have to consider both the possibilities.

Case 1: P1>P2
If the displacement of the piston is x, then the final volumes will be different.
Final volume of left part, Vfl=Vo2+Ax
Final volume of right part, Vfr=Vo2Ax

The given process is adiabatic as there is no exchange of heat.
PiVγi=PfVγf
Applying this to the left part we get,
P1(Vo2)γ=Pf(Vo2+Ax)γ
Pf=P1(Vo2)γ(Vo2+Ax)γ
This is option (c)

Case 2:P1<P2
Final volume of left part, Vfl=Vo2Ax
Final volume of right part, Vfr=Vo2+Ax

PiVγi=PfVγf
Applying this to the right part we get,
P2(Vo2)γ=Pf(Vo2+Ax)γ
Pf=P2(Vo2)γ(Vo2+Ax)γ
This is option (d)

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