How much work can you get by removing weights from a piston?

Abstract

In thermodynamics, reversible adiabatic expansion can be understood as a limit of stepwise irreversible processes. We make this idea concrete by studying an ideal gas in an insulating cylinder with a frictionless piston supporting a load divided into $N$ blocks. If blocks are removed one at a time, the resulting expansion is a stepwise, irreversible process, but we prove that for a fixed total load, the work done by the gas approaches the reversible limit as the largest block mass tends to zero. On the way to this limit, an interesting work optimization question arises at finite $N$: how does the work done by the gas depend on the order and sizes of the removed blocks? Guided by numerical experiments accessible to advanced undergraduates, we motivate and then prove general answers to these questions. Our main finite-$N$ result proves that for fixed $N$, the optimal stepwise expansion corresponds to a geometric progression of equilibrium pressures, settling a conjecture previously made by Andresen, Berry, Nitzan, and Salamon.