Carnot's theorem (thermodynamics)

Thermodynamics
The classical Carnot heat engine
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Carnot's theorem, also called Carnot's rule, is a principle of thermodynamics developed by Nicolas Léonard Sadi Carnot in 1824 that specifies limits on the maximum efficiency that any heat engine can obtain.

Carnot's theorem states that all heat engines operating between the same two thermal or heat reservoirs cannot have efficiencies greater than a reversible heat engine operating between the same reservoirs. A corollary of this theorem is that every reversible heat engine operating between a pair of heat reservoirs is equally efficient, regardless of the working substance employed or the operation details. Since a Carnot heat engine is also a reversible engine, the efficiency of all the reversible heat engines is determined as the efficiency of the Carnot heat engine that depends solely on the temperatures of its hot and cold reservoirs.

The maximum efficiency (i.e., the Carnot heat engine efficiency) of a heat engine operating between hot and cold reservoirs, denoted as H and C respectively, is the ratio of the temperature difference between the reservoirs to the hot reservoir temperature, expressed in the equation

${\displaystyle \eta _{\text{max}}={\frac {T_{\mathrm {H} }-T_{\mathrm {C} }}{T_{\mathrm {H} }}},}$

where ⁠${\displaystyle T_{\mathrm {H} }}$⁠ and ⁠${\displaystyle T_{\mathrm {C} }}$⁠ are the absolute temperatures of the hot and cold reservoirs, respectively, and the efficiency ⁠${\displaystyle \eta }$⁠ is the ratio of the work done by the engine (to the surroundings) to the heat drawn out of the hot reservoir (to the engine).

⁠${\displaystyle \eta _{\text{max}}}$⁠ is greater than zero if and only if there is a temperature difference between the two thermal reservoirs. Since ⁠${\displaystyle \eta _{\text{max}}}$⁠ is the upper limit of all reversible and irreversible heat engine efficiencies, it is concluded that work from a heat engine can be produced if and only if there is a temperature difference between two thermal reservoirs connecting to the engine.

Carnot's theorem is a consequence of the second law of thermodynamics. Historically, it was based on contemporary caloric theory, and preceded the establishment of the second law.^[1]