Kelvin–Planck–Clausius equivalence

Statement

Two standard formulations of the second law of thermodynamics are:

• Kelvin–Planck statement. No cyclic device can extract heat from a single thermal reservoir and convert it entirely into work (i.e., no 100% efficient single-reservoir heat engine).

• Clausius statement. No cyclic device can transfer heat from a colder reservoir to a hotter reservoir without net work input (i.e., no “spontaneous” pumping of heat uphill for free).

Equivalence theorem. The Kelvin–Planck statement holds if and only if the Clausius statement holds.

Key hypotheses and conclusions

Hypotheses

• Standard macroscopic bookkeeping consistent with the first law of thermodynamics (energy conservation over cycles).
• Reservoirs are idealized as bodies at fixed temperature that can supply/absorb heat without changing temperature.

Conclusions

• Either statement may be taken as the second law without loss of generality.
• Carnot-type bounds and entropy inequalities can be derived from either formulation:
  • Carnot theorem
  • Clausius theorem (entropy)
  • Clausius inequality

Cross-links to definitions

• Second law: second law of thermodynamics .
• Energy conservation over cycles: first law .
• Temperature and reservoirs: temperature .

Significance. This equivalence justifies treating “the second law” as a robust principle independent of which operational impossibility statement one starts from, and it underpins both Carnot’s theorem and entropy-based formulations.