Carnot's rule

Carnot's theorem, developed in 1824 by Nicolas Léonard Sadi Carnot, also called Carnot's rule, is a principle that specifies limits on the maximum efficiency any heat engine can obtain. The efficiency of a Carnot engine depends solely on the difference between the hot and cold temperature reservoirs.

Carnot's theorem states:

The formula for this maximum efficiency is

where T_C is the absolute temperature of the cold reservoir, T_H is the absolute temperature of the hot reservoir, and the efficiency ${\displaystyle \eta }$ is the ratio of the work done by the engine to the heat drawn out of the hot reservoir.

Based on modern thermodynamics, Carnot's theorem is a result of the second law of thermodynamics. Historically, however, it was based on contemporary caloric theory and preceded the establishment of the second law.

The proof of the Carnot theorem is a proof by contradiction, or reductio ad absurdum, as illustrated by the figure showing two heat engines operating between two reservoirs of different temperature. The heat engine with more efficiency (${\displaystyle \eta _{M}}$) is driving a heat engine with less efficiency (${\displaystyle \eta _{L}}$), causing the latter to act as a heat pump. This pair of engines receives no outside energy, and operates solely on the energy released when heat is transferred from the hot and into the cold reservoir. However, if ${\displaystyle \eta _{M}>\eta _{L}}$, then the net heat flow would be backwards, i.e., into the hot reservoir:

...
Wikipedia