Kelvin-Planck Statement

Kelvin-Planck statement on limits of cyclic heat engines. Theory, formulas, worked examples and interactive graphs.

Conceptual context

The Kelvin-Planck statement is the heat-engine form of the second law. It says that no device operating in a $ciclo$ can have as its only effect the extraction of heat from one reservoir and the complete conversion of that heat into useful $W$. In direct terms, it explains why a perfect engine fed by a single thermal reservoir cannot exist.

The key point is not energy conservation. The first law gives the balance, but it does not decide which conversions are physically possible. Kelvin-Planck adds the directional restriction: part of the absorbed heat must be rejected as $Q_C$ to a cold reservoir, or the device cannot close a physical cycle.

🟢 Essential level

A heat engine receives heat from a hot reservoir, produces work, and rejects part of the heat to a cold reservoir. If someone proposes a cyclic engine that receives $Q_H$ and delivers the same amount as $W$, they are proposing a machine forbidden by the second law.

The intuitive reason is that heat is disordered energy, while work is organized energy. Converting all disordered energy into organized energy again and again, with no other effect on the surroundings, would erase irreversibility without compensation. The second law states that this does not happen in macroscopic systems.

The star magnitude of this leaf is $\eta$, the thermal efficiency. If $\eta$ were equal to one, all absorbed heat would become work. Kelvin-Planck requires every real cyclic engine to remain below total conversion.

[!NOTE] The statement does not say that heat cannot produce work. It says that a cyclic engine cannot convert all heat from a single reservoir into work as its only result.

🔵 Formal level

Basic formulas

The energy balance of a cyclic heat engine is:

Heat engine energy balance

Thermal efficiency is defined by:

Heat engine efficiency

The central Kelvin-Planck prohibition is summarized by:

Prohibition of total cyclic conversion

For a reversible ideal engine operating between two reservoirs, the efficiency ceiling is:

Carnot limit for a heat engine

The general entropic reading of the same limit is:

Entropic criterion for the engine cycle

Real student questions

Q: If energy is conserved, why not convert all heat into work?

A: Because conserving energy is not sufficient. The second law restricts energy quality and requires the cycle not to reduce total entropy.

Q: Is rejected heat only a technical loss?

A: No. Some rejection can be reduced by better design, but a cyclic engine between finite reservoirs always needs some $Q_C$.

Q: Do Kelvin-Planck and Clausius say different things?

A: They are equivalent formulations. If one could be violated, a device could be built that violates the other.

Final synthesis

Kelvin-Planck says that a cyclic engine cannot convert all heat from a single reservoir into useful work. Energy is conserved, but the second law imposes heat rejection or entropy production compatible with the surroundings. Therefore $\eta$ has a physical ceiling and a perfect engine of the second kind does not exist.