Editing Carnot heat engine (section)

== Carnot's theorem ==
{{main|Carnot's theorem (thermodynamics)}}
[[File:Real vs Carnot.svg|upright=1.8|thumb|right|Real ideal engines (left) compared to the Carnot cycle (right). The entropy of a real material changes with temperature. This change is indicated by the curve on a [[temperature–entropy diagram|T–S diagram]]. For this figure, the curve indicates a vapor-liquid equilibrium (''See [[Rankine cycle]]''). Irreversible systems and losses of heat (for example, due to friction) prevent the ideal from taking place at every step.]]

'''Carnot's theorem''' is a formal statement of this fact: ''No engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between the same reservoirs.''

<math display="block">\eta_{I}=\frac{W}{Q_{\mathrm{H}}}=1-\frac{T_{\mathrm{C}}}{T_{\mathrm{H}}}</math>

===Explanation===
This maximum efficiency {{math|{{var|η}}{{sub|I}}}} is defined as above:
* {{mvar|W}} is the work done by the system (energy exiting the system as work),
* {{math|{{var|Q}}{{sub|H}}}} is the heat put into the system (heat energy entering the system),
* {{math|{{var|T}}{{sub|C}}}} is the [[absolute temperature]] of the cold reservoir, and
* {{math|{{var|T}}{{sub|H}}}} is the [[absolute temperature]] of the hot reservoir.

A corollary to Carnot's theorem states that: All reversible engines operating between the same heat reservoirs are equally efficient.

It is easily shown that the efficiency {{mvar|η}} is maximum when the entire cyclic process is a [[Reversible process (thermodynamics)|reversible process]]. This means the total [[entropy]] of system and surroundings (the entropies of the hot furnace, the "working fluid" of the heat engine, and the cold sink) remains constant when the "working fluid" completes one cycle and returns to its original state. (In the general and more realistic case of an irreversible process, the total entropy of this combined system would increase.)

Since the "working fluid" comes back to the same state after one cycle, and entropy of the system is a state function, the change in entropy of the "working fluid" system is 0. Thus, it implies that the total entropy change of the furnace and sink is zero, for the process to be reversible and the efficiency of the engine to be maximum. This derivation is carried out in the next section.

The [[coefficient of performance]] (COP) of the heat engine is the reciprocal of its efficiency.