Two samples `A` and `B` of the same gas have equal volumes and pressures . The gas in sample `A` is expanded isothermally to double its volume and the the gas in `B` is expanded adiabatically to double its volume . If the work done by the gas is the same for the two cases, show that gamma satisfies the equation`(1-2 ^(1-gamma))= (gamma - 1 ) 1n2`.

Explanation

`(P_0,V_0)underset(A)overset("isothermal")to((P_0)/(2),2V_0)`
`(P_0,V_0)underset(B)overset("adiabatic")to(P',2V_0)`
For adiabatic process
`P_0V_0^(gamma)=P'(2V_0)^(gamma)impliesP'=(P_0)/(2^(gamma))`
`DeltaW_A=nRTln((2V_0)/(V_0))=P_0V_0ln2`
`DeltaW_B=(P_0V_0-P'.2V_0)/(gamma-1)=(P_0V_0-(P_0)/(2^(gamma)).2V_0)/(gamma-1)`
`=(P_0V_0(1-2^(1-gamma)))/(gamma-1)`
`DeltaW_A=DeltaW_B`
`P_0V_0ln=2=P_0V_0((1-2^(1-gamma))/(gamma-1))`
`(gamma-1)ln2=(1-2^(1-gamma))`