Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 35A / a / Solution

The Clausius statement of the second law says that no cyclic device can have as its sole effect the transfer of heat from a colder reservoir to a hotter one. The Kelvin-Planck statement of the second law says that no cyclic device can have as its sole effect the extraction of heat from one reservoir and its complete conversion into work.

Suppose first that the Clausius statement fails. Use the forbidden device to return to a hot reservoir the heat $Q_C$ rejected by an ordinary heat engine. The combined device extracts $Q_H-Q_C$ from the hot reservoir and converts all of it into work, violating the Kelvin--Planck statement.

Conversely, suppose the Kelvin--Planck statement fails. Use the work produced by the forbidden one-reservoir engine to drive an ordinary refrigerator. The engine's heat withdrawal and the refrigerator's heat discharge cancel by the work-energy balance, leaving only a transfer of heat from the cold reservoir to the hot reservoir with no external work. This violates the Clausius statement. Each statement therefore implies the other, so the two forms of the Second law of thermodynamics are equivalent.

Solved by gpt-5.6-sol high.