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Question

Two samples A  and B of the same gas have equal volumes and pressuresa. THe gas in sample A is expanded isotherm ally to duble ist volume and the gas in B is expanded adiabat ically to double its volume. If the work done by the gas is the same for the two cases, show that  γ  satisfies the equation  121γ(γ1) In2


Solution

 P1  = Initial pressure,

 V1  = Initial volume

P2  = Final pressure,

V2  = Final volume

Given,    V2=2V1

Isothermal work done

nRT1InV2V1

Adiabatic work done

P1V1P2V2γ1

Given that work done at both cases are same.

Hence,

 nRT1In=V2V1

P1V1P2V2γ1                ....(1)

An adiabatic process,

P2=P1(V2V1)γ=P1(12)γ

From the equation  (1)

nRT1In2=P1V1(112γ.2)γ1

and     nRT1=P1V1

So   In 2 =  112γ.2γ1

or   (γ1)In2=121γ

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