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]

Proof

The proof of the Carnot theorem is a proof by contradiction or reductio ad absurdum (a method to prove a statement by assuming its falsity and logically deriving a false or contradictory statement from this assumption), based on a situation like the right figure where two heat engines with different efficiencies are operating between two thermal reservoirs at different temperature. The relatively hotter reservoir is called the hot reservoir and the other reservoir is called the cold reservoir. A (not necessarily reversible) heat engine ${\displaystyle M}$ with a greater efficiency ${\displaystyle \eta _{_{M}}}$ is driving a reversible heat engine ${\displaystyle L}$ with a less efficiency ${\displaystyle \eta _{_{L}}}$, causing the latter to act as a heat pump. The requirement for the engine ${\displaystyle L}$ to be reversible is necessary to explain work ${\displaystyle W}$ and heat ${\displaystyle Q}$ associated with it by using its known efficiency. However, since ${\displaystyle \eta _{_{M}}>\eta _{_{L}}}$, the net heat flow would be backwards, i.e., into the hot reservoir:

${\displaystyle Q_{\text{h}}^{\text{out}}=Q<{\frac {\eta _{_{M}}}{\eta _{_{L}}}}Q=Q_{\text{h}}^{\text{in}},}$

where ${\displaystyle Q}$ represents heat, ${\displaystyle {\text{in}}}$ denotes input to an object, ${\displaystyle {\text{out}}}$ for output from an object, and ${\displaystyle h}$ for the hot thermal reservoir. If heat ${\displaystyle Q_{\text{h}}^{\text{out}}}$ flows from the hot reservoir then it has the sign of + while if ${\displaystyle Q_{\text{h}}^{\text{in}}}$ flows from the hot reservoir then it has the sign of -. This expression can be easily derived by using the definition of the efficiency of a heat engine, ${\displaystyle \eta =W/Q_{h}^{in}}$, where work and heat in this expression are net quantities per engine cycle, and the conservation of energy for each engine as shown below. The sign convention of work ${\displaystyle W}$Zdroj:https://en.wikipedia.org?pojem=Carnot's_theorem_(thermodynamics)
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