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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
here 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" = "P"_"A" xx 1/2`
For B, the process is adiabatic. So,
PBVBγ = PB'(2VB)γ
`=> "P'"_"B" = ("P"_"B")/2^1.5`
For C, the process is isobaric, which implies that the pressure will remain constant.
So, using the ideal gas equation
`"P" = ("n""R""T")/"V"` we get
`("V"_"c")/"T"_"c" = ("V'"_"c")/"T'"_"c" =>
("V"_"c") /"T"_"c" = (2"V"_"c")/"T'"_"c"`
⇒ TC' = 2TC
As the final pressures are equal,
`"P"_"A"/ 2 = "P"_"B"/2^1.5 ="P"_"c"`
⇒ PA : PB :PC = 2 : 21.5 : 1
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