Carnot's theorem (thermodynamics)

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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:

“ No engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between the same reservoirs. ”

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}},


Figure 1: Carnot heat engine diagram The first law of thermodynamics requires The second law of thermodynamics requires The third law of thermodynamics requires the use of an absolute temperature scale. Together they imply a maximum heat-to-work efficiency of:
Figure 1: Carnot heat engine diagram

The first law of thermodynamics requires Q_H = Q_C + W\,\!

The second law of thermodynamics requires \frac{Q_H}{T_H} = \frac{Q_C}{T_C}

The third law of thermodynamics requires the use of an absolute temperature scale.

Together they imply a maximum heat-to-work efficiency of: \eta \equiv \frac{W_{out}}{Q_{in}} = \frac{\Delta T} {T_H}

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

\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),
QH is the heat put into the system (heat energy entering the system),
TC is the absolute temperature of the cold reservoir, and
TH 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.

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