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 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 γ satisfies the equation 1 – 2^{1– γ} = (γ – 1) ln2.

Let P1 = Initial Pressure, V1 = Initial Volume, P2 = Final Pressure, V2 = Final Volume

Here A is expanded isothermally,

I.e. the work done,

Also, B is expanded adiabatically, i.e.

Given W_{A}=W_{B}

i.e.

In an adiabatic process,

PV^{γ} = const,

I.e.

Substituting in (1)

We know, PV= nRT by ideal gas equation

i.e.

, the required relation

1