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.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 where are the absolute temperatures of the hot and cold reservoirs, respectively, and the efficiency is greater than zero if and only if there is a temperature difference between the two thermal reservoirs.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] 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 net heat flow would be backwards, i.e., into the hot reservoir: whereThis expression can be easily derived by using the definition of the efficiency of a heat engine,, 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).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.depicted in the right figure in which a reversible heat engine, is made to make the expression to be consistent, and it helps to fill the values of work and heat for the engine, must be equal to the absolute value of the energy leaving from the engine,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.[3] To see that every reversible engine operating between reservoirs at temperaturesAs 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.This conclusion is an important result because it helps establish the Clausius theorem, which implies that the change in entropyis unique for all reversible processes:[4] as the entropy change, that is made during a transition from a thermodynamic equilibrium statein 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.is the heat to the engine from the hot reservoir, per cycle.[6] Because all reversible heat engines operating between temperaturesmust 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(Of course any reference temperature and any positive numerical value could be used — the choice here corresponds to the Kelvin scale.)), they are clearly not limited by Carnot's theorem, which states that no power can be generated whenThis is because Carnot's theorem applies to engines converting thermal energy to work, whereas fuel cells instead convert chemical energy to work.[7] Nevertheless, the second law of thermodynamics still provides restrictions on fuel cell energy conversion.[8] A Carnot battery is a type of energy storage system that stores electricity in thermal energy storage and converts the stored heat back to electricity through thermodynamic cycles.