Kelvin–Planck statement

A heat engine is a device that executes a cycle while exchanging heat with one or more thermal reservoirs and exchanging work with a work reservoir . The Kelvin–Planck statement of the second law says:

It is impossible to construct a device that operates in a cycle whose sole net effect is to absorb heat from a single thermal reservoir and deliver an equivalent amount of work.

Equivalently, no heat engine can have 100% efficiency when it interacts with only one reservoir; if net work is produced, some heat must be rejected to additional surroundings at a lower temperature.

Physical interpretation

The statement rules out a “perpetual motion machine of the second kind”: energy conservation (the first law ) alone would allow a cyclic device with $\Delta U=0$ to turn all absorbed heat into work, but the second law forbids this complete conversion. A temperature difference is a necessary resource for extracting sustained work from heat.

Key relations

• Energy balance over a cycle. For a cyclic engine, $\Delta U=0$, so net work output equals net heat absorbed. If the engine absorbs heat $Q_h>0$ from a hot reservoir and rejects heat $Q_c>0$ to a colder reservoir (here $Q_h,Q_c$ are magnitudes), then

  $$W_{\text{out}} = Q_h - Q_c,$$

  and Kelvin–Planck requires $Q_c>0$ whenever $W_{\text{out}}>0$.

• Equivalent formulations. Violating Kelvin–Planck would allow a cyclic device that, when combined appropriately, violates the Clausius statement (moving heat from cold to hot with no work input), and vice versa. Both are captured quantitatively by the Clausius inequality .

• Temperature dependence in the reversible limit. For an ideal reversible engine operating between two reservoirs, the limiting performance depends only on their absolute temperatures , reflecting that the second law ties work extraction to temperature differences rather than to detailed material properties.