A mole of gas undergoes an expansion through two different processes. The gas satisfies the equation of state pV = NR T where R is a constant, N the number of moles, p the pressure, T the temperature and V the volume of the gas. The initial and final temperatures are T₀. The walls of the gas container are diathermal. However, if a process takes place extremely fast, the walls can be considered as adiabatic. The initial pressure of the gas is p₁, the final pressure is p₂. Express the work performed on the gas in terms of p₁, p₂ and T₀ for the following processes:

a) A reversible isothermal process,

b) An extremely fast pressure variation, such that the external pressure on the gas is p2 during the expansion, then an isochoric process during which the temperature again reaches the equilibrium temperature T₀,