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Minimum work for a two-step isothermal compression of an ideal gas from 2 bar to 8 bar

A gas-tight cylinder of 0.5 mol of ideal gas stands in a 600 K thermostat bath. Press start and the bench does the rest: weights land on the piston in two steps, the gas is squeezed, the work appears as a shaded area on the indicator diagram, and the experiment repeats this for many first-step pressures, plots the total work against P, finds the lowest point and reveals the answer. No prediction stage - input, run, watch, result.

Sub-topics covered

• Work done on a gas against a constant external pressure, w = -Pext dV
• Irreversible isothermal compression in steps
• Isothermal ideal gas: V = nRT/P at each stopping point
• Optimising the intermediate pressure: P = sqrt(P1 P2)
• Reversible work as the lower bound, one-step work as the upper
• Indicator (P-V) diagrams and work as area
• First law for an isothermal ideal gas: dU = 0, q = -w

What it teaches

• Against a constant external pressure the work done on the gas is Pext times the volume swept, whatever the gas does inside the cylinder; on a P-V diagram it is the rectangle under the external-pressure line
• Each step stops when the gas pressure equals the external pressure, and the bath holds the temperature, so each step ends on the isotherm at V = nRT/P
• W(P) = nRT(P/P1 + P2/P - 2): a small first-step P makes step 1 cheap but step 2 heavy, a large P the reverse, so the total has a minimum in between
• dW/dP = 0 at P = sqrt(P1 P2) - the geometric mean - and there the two steps do exactly equal work, nRT(sqrt(P2/P1) - 1) each
• W is symmetric in ln P about ln sqrt(P1 P2), which is why a sweep of P evenly spaced in ln P brackets the minimum at its centre
• W_min = 2nRT(sqrt(P2/P1) - 1) = 2 x 0.5 x 600 x (2 - 1) R = 600R J
• The reversible isothermal work nRT ln(P2/P1) = 415.9R is a lower bound that no finite number of steps can reach; more equal-ratio steps approach it (1, 2, 4, 100 steps: 900, 600, 497, 419R)
• For an isothermal ideal gas dU = 0, so all the work done on the gas leaves as heat to the bath

Simulation ID

• ADV-2026-P1-CHE-Q01 · revision 1

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