Carnot's theorem, developed in 1824, also called Carnot's rule is a principle that specifies limits on the maximum efficiency any heat engine

Heat engine

In thermodynamics, a heat engine is a system that performs the conversion of heat or thermal energy to mechanical work. It does this by bringing a working substance from a high temperature state to a lower temperature state. A heat "source" generates thermal energy that brings the working substance...

 can obtain, which thus solely depends on the difference between the hot and cold temperature reservoirs.

Carnot's theorem states:

• All irreversible heat engines between two heat reservoirs are less efficient than a Carnot engine operating between the same reservoirs.
• All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.

The formula for this maximum efficiency is
where T_C is the absolute temperature of the cold reservoir, T_H is the absolute temperature of the hot reservoir, and the efficiency is the ratio of the work done by the engine to the heat drawn out of the hot reservoir.

Based on modern thermodynamics, Carnot's theorem is a result of the second law of thermodynamics

Second law of thermodynamics

The second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...

. Historically, however, it was based on contemporary caloric theory

Caloric theory

The caloric theory is an obsolete scientific theory that heat consists of a self-repellent fluid called caloric that flows from hotter bodies to colder bodies. Caloric was also thought of as a weightless gas that could pass in and out of pores in solids and liquids...

 and preceded the establishment of the second law.

Proof

The theorem may be proved in the following way for the cases of the irreversible and the reversible heat engines.

Irreversible engine

The derivation assumes an irreversible heat engine that has an efficiency , operating between reservoirs and . It is combined with a reversed Carnot engine that has efficiency as shown by the diagram to the right.

If , then the combined engine does nothing in a cycle, which contradicts the irreversibility.

If , then the net effect of the combined engine is draining heat
from the colder reservoir and releases the same amount to the hotter reservoir without affecting any other changes. This is clearly in violation of the Clausius statement of the second law.

The conclusion is that the only possibility remaining is .

Reversible engine

In the case of a reversible heat engine, the proof may be conducted similarly, in that .

The said engine is reversed and combined with a regular Carnot engine. In a similar manner the conclusion is obtained: .
Thus,.

Definition of thermodynamic temperature

The efficiency of the engine is the work divided by the heat introduced to the system or,

where w_cy is the work done per cycle. Thus, the efficiency depends only on q_C/q_H.

Because all reversible engines operating between the same heat reservoirs are equally efficient, any reversible heat engine operating between temperatures T₁ and T₂ must have the same efficiency, meaning, the efficiency is the function of the temperatures only:

In addition, a reversible heat engine operating between temperatures T₁ and T₃ must have the same efficiency as one consisting of two cycles, one between T₁ and another (intermediate) temperature T₂, and the second between T₂andT₃. This can only be the case if

Specializing to the case that is a fixed reference temperature: the temperature of the triple point of water. Then for anyT₂and T₃,


Therefore, if thermodynamic temperature is defined by


then the function f, viewed as a function of thermodynamic temperature, is
and the reference temperature T₁ has the value 273.16. (Of course any reference temperature and any positive numerical value could be used—the choice here corresponds to the Kelvin scale.)

It follows immediately that

Substituting Equation 3 back into Equation 1 gives a relationship for the efficiency in terms of temperature:

Applicability to fuel cells and batteries

Since fuel cell

Fuel cell

A fuel cell is a device that converts the chemical energy from a fuel into electricity through a chemical reaction with oxygen or another oxidizing agent. Hydrogen is the most common fuel, but hydrocarbons such as natural gas and alcohols like methanol are sometimes used...

s and batteries

Battery (electricity)

An electrical battery is one or more electrochemical cells that convert stored chemical energy into electrical energy. Since the invention of the first battery in 1800 by Alessandro Volta and especially since the technically improved Daniell cell in 1836, batteries have become a common power...

 can generate useful power when all components of the system are at the same temperature (), they are clearly not limited by Carnot's theorem, which states that no power can be generated when . 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

Second law of thermodynamics

The second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...

still provides restrictions on fuel cell and battery energy conversion.