Carnot theorem

Statement

Consider a cyclic heat engine that exchanges heat only with two thermal reservoirs, a hot reservoir and a cold reservoir, at fixed temperatures $T_H>T_C$. Let $Q_H>0$ be the heat absorbed from the hot reservoir per cycle, $Q_C>0$ the heat rejected to the cold reservoir per cycle, and $W=Q_H-Q_C$ the work output per cycle. The efficiency is

Carnot theorem asserts:

1. Maximality of reversible efficiency. For any (possibly irreversible) engine operating between the same two reservoirs,

   $$\eta \le \eta_{\mathrm{rev}}(T_H,T_C),$$

   where $\eta_{\mathrm{rev}}(T_H,T_C)$ is the efficiency of a reversible engine between those reservoirs.

2. Universality among reversible engines. Any two reversible engines operating between the same two reservoirs have the same efficiency. Equivalently, $\eta_{\mathrm{rev}}$ depends only on the reservoir temperatures (and not on the working substance or details of the cycle).

Key hypotheses and conclusions

Hypotheses

• A cyclic device (engine) interacting with exactly two reservoirs at fixed temperatures (hot and cold).
• The engine’s working body is always in (or can be idealized as passing through) thermodynamic equilibrium states.
• The second law of thermodynamics holds (in any standard formulation; see equivalence below).

Conclusions

• No engine between $(T_H,T_C)$ can exceed the efficiency of a reversible one.
• Reversible efficiency is a function only of $(T_H,T_C)$; this underlies the absolute temperature scale and leads to the explicit Carnot efficiency formula .

Cross-links to definitions

• Reservoir temperatures use the thermodynamic notion of temperature .
• The directionality constraint is encoded by the second law , and the equivalence of standard formulations is summarized in Kelvin–Planck–Clausius equivalence .
• The entropy viewpoint connects via Clausius’ theorem (entropy) and the Clausius inequality .
• The induced absolute scale is captured in Carnot’s corollary on absolute temperature .

Proof idea / significance

Idea (standard contradiction argument). Suppose there exists an engine $\mathcal{E}$ between $T_H$ and $T_C$ more efficient than a reversible engine $\mathcal{R}$ between the same reservoirs. Run $\mathcal{R}$ in reverse as a refrigerator/heat pump and couple it to $\mathcal{E}$ so that the net heat exchange with one reservoir cancels. The remaining net effect is either:

• extraction of heat from a single reservoir and complete conversion to work (violating the Kelvin–Planck statement), or
• transfer of heat from cold to hot with no net work input (violating the Clausius statement).

By equivalence , either outcome contradicts the second law, so the assumption was impossible.

Significance. Carnot’s theorem isolates a universal performance bound for heat engines, independent of microscopic details, and is the starting point for defining absolute temperature and entropy in macroscopic thermodynamics.