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Question

Three samples A, B and C of the same gas (γ = 1.5) have equal volumes and temperatures. The volume of each sample is doubled, the process being isothermal for A, adiabatic for B and isobaric for C. If the final pressures are equal for the three samples, find the ratio of the initial pressures.

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Solution

There are three gases A, B and C.
It is given that initially,
VA = VB = VC and TA = TB = TC .

For A, the process is isothermal and for an isothermal process, PV = constant.
PAVA = P'A2VA
P'A=PA×12

For B, the process is adiabatic. So,
PBVBγ = PB'(2VB)γ
PB'=PB21.5

For C, the process is isobaric, which implies that the pressure will remain constant.
So, using the ideal gas equation
P=nRTV, we get
VCTC=V'CT'CVCTC=2VCT'C
⇒ TC' = 2TC

As the final pressures are equal,
PA2=PB21.5=PCPA : PB : PC=2 : 21.5 : 1

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