An ideal gas is to be compressed from an initial state (Pi,Vi) to a final state (Pf,Vf) at the same temperature T.
Two alternative methods are proposed:
Method I: a slow, reversible isothermal compression.
Method II: a sudden single-step compression against a constant external pressure Pf, after which the system is allowed to re-equilibrate.
Analyse the two methods with respect to mechanical work.
(i) Derive an expression for the work required in Method I and show that it can be written as W1=nRTln(ViVf).
(ii) Obtain an expression for the work required in Method II and express it in terms of Pf and Vi−Vf.
(iii) Prove that, for the same initial and final equilibrium states, ∣W2∣>∣W1∣; that is, the single-step compression requires more work input than the slow reversible compression.
(iv) Explain physically why the difference in work arises, referring to the P–V representation of the two processes.