n moles of an ideal gas are initially at pressure P₁ and volume V₁. The gas undergoes the following two-step reversible process:
Step-1: Isothermal expansion at temperature T until its volume becomes kV₁ (k > 1).
Step-2: Adiabatic compression until its pressure once again becomes P₁.
(i) Derive an expression for the volume V₃ of the gas after the adiabatic compression in terms of V₁, k and γ.
(ii) Show that the work done by the gas in the adiabatic step can be written purely in terms of P₁, V₁, k and γ.
(iii) Hence obtain a simplified expression for the net work done by the gas over the complete process in terms of n, R, T, k and γ.