Carnot's theorem (thermodynamics)

In thermodynamics, Carnot's theorem, developed in 1824 by Nicolas Léonard Sadi Carnot, also called Carnot's rule, is a principle 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 can't 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 cold and hot reservoirs, denoted as and respectively, is the ratio of the temperature difference between the reservoirs to the hot reservoir temperature, expressed in the equation 
where and are the absolute temperatures of the hot and cold reservoirs, respectively, and the efficiency 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).
 is greater than zero if and only if there is a temperature difference between the two thermal reservoirs. Since 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.
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 formula_2 with a greater efficiency formula_3 is driving a reversible heat engine formula_4 with a less efficiency formula_5, causing the latter to act as a heat pump. The requirement for the engine formula_4 to be reversible is necessary to explain work formula_7 and heat formula_8 associated with it by using its known efficiency. However, since formula_9, the net heat flow would be backwards, i.e., into the hot reservoir:
where formula_8 represents heat, formula_12 for input to an object denoted by the subscript, formula_13 for output from an object denoted by the subscript, and formula_14 for the hot thermal reservoir. If heat formula_15 flows from the hot reservoir then it has the sign of + while if formula_16 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, formula_17, 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 formula_7, with which the sign of + for work done by an engine to its surroundings, is employed.
The above expression means that heat into the hot reservoir from the engine pair (can be considered as a single engine) is greater than heat into the engine pair from the hot reservoir (i.e., the hot reservoir continuously gets energy). A reversible heat engine with a low efficiency delivers more heat (energy) to the hot reservoir for a given amount of work (energy) to this engine when it is being driven as a heat pump. All these mean that heat can transfer from cold to hot places without external work, and such a heat transfer is impossible by the second law of thermodynamics. 
Let's find the values of work formula_7and heat formula_8 depicted in the right figure in which a reversible heat engine formula_4 with a less efficiency formula_5 is driven as a heat pump by a heat engine formula_2 with a more efficiency formula_3.
The definition of the efficiency is formula_28 for each engine and the following expressions can be made:
The denominator of the second expression, formula_31, is made to make the expression to be consistent, and it helps to fill the values of work and heat for the engine formula_4.
For each engine, the absolute value of the energy entering the engine, formula_33, must be equal to the absolute value of the energy leaving from the engine, formula_34. Otherwise, energy is continuously accumulated in an engine or the conservation of energy is violated by taking more energy from an engine than input energy to the engine:
In the second expression, formula_37 is used to find the term formula_38 describing the amount of heat taken from the cold reservoir, completing the absolute value expressions of work and heat in the right figure.
Having established that the right figure values are correct, Carnot's theorem may be proven for irreversible and the reversible heat engines as shown below.
Reversible engines.
To see that every reversible engine operating between reservoirs at temperatures formula_39 and formula_40 must have the same efficiency, assume that two reversible heat engines have different efficiencies, and let the relatively more efficient engine formula_2 drive the relatively less efficient engine formula_4 as a heat pump. As the right figure shows, this will cause heat to flow from the cold to the hot reservoir without external work, which violates the second law of thermodynamics. Therefore, both (reversible) heat engines have the same efficiency, and we conclude that:
The reversible heat engine efficiency can be determined by analyzing a Carnot heat engine as one of reversible heat engine.
This conclusion is an important result because it helps establish the Clausius theorem, which implies that the change in entropy formula_43 is unique for all reversible processes:
as the entropy change, that is made during a transition from a thermodynamic equilibrium state formula_45 to a state formula_46 in a "V-T" (Volume-Temperature) space, is the same over all reversible process paths between these two states. If this integral were not path independent, then entropy would not be a state variable.
Irreversible engines.
Let's think two engines, one is formula_2 that is relatively more efficient irreversible engine while the other is formula_4 that is relatively less efficient reversible engine, and we construct a machine described in the right figure (formula_2 drives formula_4 as a heat pump). Then this machine violates the second law of thermodynamics. Since a Carnot heat engine is a reversible heat engine, with the conclusion in the discussion about two reversible heat engines above, we have the first part of Carnot's theorem:
Definition of thermodynamic temperature.
The efficiency of a heat engine is the work done by the engine divided by the heat introduced to the engine per engine cycle or
where formula_51 is the work done by the engine, formula_52 is the heat to the cold reservoir from the engine, and formula_53 is the heat to the engine from the hot reservoir, per cycle. Thus, the efficiency depends only on formula_54.
Because all reversible heat engines operating between temperatures formula_39 and formula_40 must have the same efficiency, the efficiency of a reversible heat engine is a function of only the two reservoir temperatures:
In addition, a reversible heat engine operating between temperatures formula_39 and formula_58 must have the same efficiency as one consisting of two cycles, one between formula_39 and another (intermediate) temperature formula_40, and the second between formula_40 and formula_58 (formula_63). This can only be the case if
Specializing to the case that formula_39 is a fixed reference temperature: the temperature of the triple point of water as 273.16. (Of course any reference temperature and any positive numerical value could be used — the choice here corresponds to the Kelvin scale.) Then for any formula_40 and formula_58,
Therefore, if thermodynamic temperature is defined by
then the function viewed as a function of thermodynamic temperature, is
It follows immediately that
Substituting this equation back into the above equation formula_70 gives a relationship for the efficiency in terms of thermodynamic temperatures:
Applicability to fuel cells and batteries.
Since fuel cells and batteries can generate useful power when all components of the system are at the same temperature (formula_71), they are clearly not limited by Carnot's theorem, which states that no power can be generated when formula_72. This is because Carnot's theorem applies to engines converting thermal energy to work, whereas fuel cells and batteries instead convert chemical energy to work. Nevertheless, the second law of thermodynamics still provides restrictions on fuel cell and battery energy conversion.
A Carnot battery is a type of energy storage system that stores electricity in heat storage and converts the stored heat back to electricity through thermodynamic cycles.