Carnot theorem implies an absolute temperature scale

Carnot's theorem yields a temperature function with for reversible engines, defining absolute temperature up to units.

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,

\frac{Q_H}{Q_C}=\frac{T_H}{T_C},

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

\eta_{\mathrm{rev}} = 1-\frac{T_C}{T_H},

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.