Carnot's theorem, also called Carnot's rule is a principle which sets a limit on the maximum amount of efficiency any possible engine can obtain, which thus solely depends on the difference between the hot and cold temperature reservoirs. Carnot's theorem states:

The rule was an essential stepping stone towards the formulation of the second law of thermodynamics. When transforming thermal energy into mechanical energy, the thermal efficiency of a heat engine is the percentage of energy that is transformed into work. Thermal efficiency is defined as

$eta\_\{th\}\; equiv\; frac\{W\_\{out\}\}\{Q\_\{in\}\}$,

Carnot showed that the maximum efficiency possible by any sort of engine has a limit defined by the following efficiency $eta$:

$eta=frac\{W\}\{Q\_H\}=frac\{T\_H\}\{T\_H\}-frac\{T\_C\}\{T\_H\}\; =\; frac\{Delta\; T\}\; \{T\_H\}$

where:

$W$ is the work done by the system (energy exiting the system as work),

$Q\_H$ is the heat put into the system (heat energy entering the system),

$T\_C$ is the absolute temperature of the cold reservoir, and

$T\_H$ is the absolute temperature of the hot reservoir.

Carnot's theorem sets essential limitations on the yield of a cyclic heat engine such as steam engines or internal combustion engines, which operate on the Carnot Cycle. They can extract only a certain proportion of mechanical energy from the heat of the working fluid, and this maximal amount is realized by the ideal Carnot heat engine.