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Question

The figure shows an insulated cylinder divided into three parts A, B and C. Piston I and II are connected by a rigid rod and can move without friction inside the cylinder. Piston I is perfectly conducting while piston II is perfectly insulating. The initial state of gas (γ=1.5) present in each compartment A, B and C is as shown. Now, compartment A is slowly given heat through a heater H such that the final volume of C becomes 4V09. Assume the gas to be ideal and find the heat supplied by the heater.
135585_07b70e7c0eb747b4ac4e7b1a3772402c.png

A
18PoVo
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B
12PoVo
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C
9PoVo
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D
25PoVo
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Solution

The correct option is A 18PoVo
heat supplied ΔQ=ΔU+ΔW
ΔU=ΔUA+ΔUB+0 [no heat change in case of compartment C]
ΔW=ΔWA+0+ΔWC [no change in volume in case of compartment B hence work done = 0]

initial conditions : Po Pressure Vo Volume To Temperature
final conditions :

compartment C

PVγ=PVγ=P×(4V09)1.5=PoV1.5o
P=27Po8
PoVoTo=27Po8×4Vo9T
T=3To2

compartment A

P=27Po8 compartment A and C has to have same pressure for pistons to come at rest.
PoVoTo=27Po8×(Vo+5Vo9)T
T=21To4

compartment B

T=21To4 temperature of compartment A and B should be same at equilibrium
PoVoTo=P×Vo21To4
P=21Po4

γ=CpCv=f+2f
f=4
Cv=fR2=2R


ΔUA=ΔUB
ΔUA=nCvΔT

ΔUA=P0VoRTo×Cv(21To4To)=PoVoRTo×2R×17To4=17PoVo2

ΔWA=ΔWC as the gas in chamber A is working on chamber C
ΔQC=0 as its a Adibatic process hence
ΔQC=ΔUC+ΔWC
ΔUC=ΔWC=nCvΔT=PoVoRTo×2R×(3To2To)=PoVo

ΔU=ΔUA+ΔUB+0 [no heat change in case of compartment C]
ΔW=ΔWA+0+ΔWC [no change in volume in case of compartment B hence work done = 0]

head supplied by the heater = heat supplied to compartment A + heat flown through piston I
ΔQ=ΔUA+ΔUB+ΔWAB=ΔUA+ΔUB+ΔWA=2ΔUA+ΔWA
ΔQ=18PoVo

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