Carnot theorem implies an absolute temperature scale

Consider a reversible heat engine operating between two reservoirs, “hot” and “cold,” with reservoir temperatures in the sense of thermodynamic temperature .

Statement

Assume Carnot's theorem : all reversible engines between the same two reservoirs have the same efficiency, and no engine can exceed it.

Then there exists a positive temperature function $T$ (unique up to an overall multiplicative constant) such that for every reversible engine between the two reservoirs,

where $Q_H$ is the heat absorbed from the hot reservoir and $Q_C$ is the heat rejected to the cold reservoir (both taken as positive magnitudes).

Equivalently, the efficiency of a reversible (Carnot) engine is

i.e. the Carnot efficiency formula .

Key hypotheses

• The reservoirs remain at fixed temperatures during each cycle.
• The engine cycle is reversible (no entropy production).
• Carnot’s theorem holds (as a consequence of the second law ).

Key conclusions

• For reversible engines, the heat ratio $Q_H/Q_C$ depends only on the reservoir temperatures.
• This defines an absolute temperature scale (Kelvin scale) up to the choice of units.

Proof idea / significance

Compare reversible engines between different pairs of reservoirs and compose cycles (running one engine in reverse as a heat pump). Carnot universality forces a multiplicative functional equation for reversible heat ratios, implying the existence of a function $T$ with $Q_H/Q_C = T_H/T_C$. Choosing temperature units fixes the overall multiplicative constant. This is the thermodynamic origin of “absolute” temperature.