Carnot efficiency formula

Statement

Let a reversible cyclic heat engine operate between a hot reservoir at temperature $T_H$ and a cold reservoir at temperature $T_C$, with $T_H>T_C>0$. If $Q_H>0$ is the heat absorbed from the hot reservoir per cycle and $Q_C>0$ is the heat rejected to the cold reservoir per cycle, then the efficiency is

On the absolute thermodynamic temperature scale, a reversible engine satisfies

Key hypotheses and conclusions

Hypotheses

• The engine is reversible (no entropy production; it can be run backward as a refrigerator without additional dissipation).
• Heat exchange occurs only with two reservoirs at fixed temperatures $T_H$ and $T_C$.
• Temperatures are interpreted as thermodynamic temperatures on an absolute scale (see below).

Conclusions

• The reversible efficiency depends only on the temperature ratio $T_C/T_H$.
• By Carnot’s theorem , every engine between the same reservoirs satisfies $\eta\le 1-T_C/T_H$.

Cross-links to definitions

• Reservoir temperatures: temperature .
• Second-law input: second law of thermodynamics .
• Universality/maximality: Carnot theorem .
• Entropy formulation: entropy and Clausius’ theorem .
• Absolute scale consequence: Carnot absolute temperature corollary .

Proof idea / significance

Entropy-based derivation. For a reversible cycle exchanging heats $Q_H$ at $T_H$ and $Q_C$ at $T_C$ (and otherwise adiabatic), Clausius’ theorem implies the cyclic integral of $\delta Q_{\mathrm{rev}}/T$ vanishes. With the sign convention using magnitudes $Q_H,Q_C>0$, this gives

so $Q_C/Q_H=T_C/T_H$. Substituting into $\eta_{\mathrm{rev}}=1-Q_C/Q_H$ yields $\eta_{\mathrm{rev}}=1-T_C/T_H$.

Significance. This is the canonical “thermodynamic limit” on efficiency: it quantifies why making $T_C$ small and $T_H$ large is the only way (even ideally) to improve efficiency, and it pins down the operational meaning of the absolute temperature scale.