Matching isothermal processes with work expressions:
A. Reversible expansion: In a reversible isothermal expansion from Vi to Vf (where Vf>Vi), the gas does work against external pressure equal to the surroundings. The work is w=nRTlnViVf, matching II.
B. Free expansion: The gas expands into a vacuum with no external pressure opposition. No work is done: w=0, matching I.
C. Irreversible expansion: Isothermal expansion from Vi to Vf against constant external pressure pex. Work done is w=−pex(Vf−Vi), matching III.
D. Irreversible compression: Isothermal compression from Vi to Vf (where Vf<Vi) against constant external pressure. Work is w=−pex(Vi−Vf), matching IV.
The correct pairings are A-II, B-I, C-III, D-IV.