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Carnot's theorem, also called Carnot's rule or Carnot's law, is a principle of
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
developed by
Nicolas Léonard Sadi Carnot Nicolas Léonard Sadi Carnot (; 1 June 1796 – 24 August 1832) was a French people, French military engineering, military engineer and physicist. A graduate of the École polytechnique, Carnot served as an officer in the Engineering Arm (''le ...
in 1824 that specifies limits on the maximum
efficiency Efficiency is the often measurable ability to avoid making mistakes or wasting materials, energy, efforts, money, and time while performing a task. In a more general sense, it is the ability to do things well, successfully, and without waste. ...
that any
heat engine A heat engine is a system that transfers thermal energy to do mechanical or electrical work. While originally conceived in the context of mechanical energy, the concept of the heat engine has been applied to various other kinds of energy, pa ...
can obtain. Carnot's theorem states that all heat engines operating between the same two thermal or heat reservoirs cannot have efficiencies greater than a reversible heat engine operating between the same reservoirs. A
corollary In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
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 A Carnot heat engine is a theoretical heat engine that operates on the Carnot cycle. The basic model for this engine was developed by Nicolas Léonard Sadi Carnot in 1824. The Carnot engine model was graphically expanded by Benoît Paul Émile ...
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 hot and cold reservoirs, denoted as and respectively, is the ratio of the temperature difference between the reservoirs to the hot reservoir temperature, expressed in the equation :\eta_ = \frac, where and are the
absolute temperature Thermodynamic temperature, also known as absolute temperature, is a physical quantity which measures temperature starting from absolute zero, the point at which particles have minimal thermal motion. Thermodynamic temperature is typically expres ...
s of the hot and cold reservoirs, respectively, and the efficiency is the ratio of the
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an ani ...
done by the engine (to the
surroundings Surroundings, or environs is an area around a given physical or geographical point or place. The exact definition depends on the field. Surroundings can also be used in geography (when it is more precisely known as vicinity, or vicinage) and ...
) 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 The second law of thermodynamics is a physical law based on Universal (metaphysics), universal empirical observation concerning heat and Energy transformation, energy interconversions. A simple statement of the law is that heat always flows spont ...
. Historically, it was based on contemporary
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 ...
, and preceded the establishment of the second law.


Proof

The proof of the Carnot theorem is a
proof by contradiction In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical pr ...
or
reductio ad absurdum In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical argument'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absur ...
(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 M with a greater efficiency \eta_ is driving a reversible heat engine L with a less efficiency \eta_, causing the latter to act as a
heat pump A heat pump is a device that uses electricity to transfer heat from a colder place to a warmer place. Specifically, the heat pump transfers thermal energy using a heat pump and refrigeration cycle, cooling the cool space and warming the warm s ...
. The requirement for the engine L to be reversible is necessary to explain work W and heat Q associated with it by using its known efficiency. However, since \eta_>\eta_, the net heat flow would be backwards, i.e., into the hot reservoir: :Q^\text_\text = Q < \fracQ=Q^\text_\text, where Q represents heat, \text denotes input to an object, \text for output from an object, and h for the hot thermal reservoir. If heat Q^\text_\text flows from the hot reservoir then it has the sign of + while if Q^\text_\text flows from the hot reservoir then it has the sign of -. This expression can be easily derived by using the definition of the
efficiency Efficiency is the often measurable ability to avoid making mistakes or wasting materials, energy, efforts, money, and time while performing a task. In a more general sense, it is the ability to do things well, successfully, and without waste. ...
of a heat engine, \eta=W/Q_\text^\text, 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 W, 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 The second law of thermodynamics is a physical law based on Universal (metaphysics), universal empirical observation concerning heat and Energy transformation, energy interconversions. A simple statement of the law is that heat always flows spont ...
. * It may seem odd that a hypothetical reversible heat pump with a low efficiency is used to violate the second law of thermodynamics, but the
figure of merit A figure of merit (FOM) is a performance metric that characterizes the performance of a device, system, or method, relative to its alternatives. Examples *Absolute alcohol content per currency unit in an alcoholic beverage *accurizing, Accuracy o ...
for refrigerator units is not the efficiency, W/Q_\text^\text, but the
coefficient of performance The coefficient of performance or COP (sometimes CP or CoP) of a heat pump, refrigerator or air conditioning system is a ratio of useful heating or cooling provided to work (energy) required. Higher COPs equate to higher efficiency, lower energy ( ...
(COP), which is Q_\text^\text/W where this W has the sign opposite to the above (+ for work done to the engine). Let's find the values of work Wand heat Q depicted in the right figure in which a reversible heat engine L with a less efficiency \eta_ is driven as a heat pump by a heat engine M with a more efficiency \eta_. The definition of the
efficiency Efficiency is the often measurable ability to avoid making mistakes or wasting materials, energy, efforts, money, and time while performing a task. In a more general sense, it is the ability to do things well, successfully, and without waste. ...
is \eta = W/Q_\text^\text for each engine and the following expressions can be made: :\eta_M= \frac = \frac=\eta_M, : \eta_L = \frac = \frac = \eta_L. The denominator of the second expression, Q^_\text = -\fracQ, is made to make the expression to be consistent, and it helps to fill the values of work and heat for the engine L. For each engine, the absolute value of the energy entering the engine, E_\text^\text , must be equal to the absolute value of the energy leaving from the engine, E_\text^\text . 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: :E_\text^\text = Q = (1-\eta_M)Q + \eta_M Q = E_\text^\text, :E_\text^\text = \eta_M Q + \eta_M Q \left(\frac- 1 \right ) = \fracQ = E_\text^\text. In the second expression, \left, Q^_\text \ = \left, - \fracQ \ is used to find the term \eta_M Q \left(\frac- 1 \right ) 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 T_1 and T_2 must have the same efficiency, assume that two reversible heat engines have different efficiencies, and let the relatively more efficient engine M drive the relatively less efficient engine L 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: :''All reversible heat engines that operate between the same two thermal'' (''heat'') ''reservoirs have the same efficiency.'' 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 Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
S is unique for all reversible processes: :\Delta S = \int_a^b \frac T as the entropy change, that is made during a transition from a
thermodynamic equilibrium Thermodynamic equilibrium is a notion of thermodynamics with axiomatic status referring to an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable ...
state a to a state b 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 A state variable is one of the set of Variable (mathematics), variables that are used to describe the mathematical "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behavi ...
.


Irreversible engines

Consider two engines, M and L, which are irreversible and reversible respectively. We construct the machine shown in the right figure, with M driving L as a heat pump. Then if M is more efficient than L, the machine will violate the second law of thermodynamics. Since a Carnot heat engine is a reversible heat engine, and all reversible heat engines operate with the same efficiency between the same reservoirs, we have the first part of Carnot's theorem: :''No irreversible heat engine is more efficient than a Carnot heat engine operating between the same two thermal reservoirs.''


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 w_\text is the work done by the engine, q_C is the heat to the cold reservoir from the engine, and q_H is the heat to the engine from the hot reservoir, per cycle. Thus, the efficiency depends only on \frac. The sign of ''qC'' > 0 for the waste heat lost by the system violates the sign convention of
heat In thermodynamics, heat is energy in transfer between a thermodynamic system and its surroundings by such mechanisms as thermal conduction, electromagnetic radiation, and friction, which are microscopic in nature, involving sub-atomic, ato ...
.
Because all reversible heat engines operating between temperatures T_1 and T_2 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 T_1 and T_3 must have the same efficiency as one consisting of two cycles, one between T_1 and another (intermediate) temperature T_2, and the second between T_2 and T_3 (T_1 < T_2 < T_3). This can only be the case if Specializing to the case that T_1 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 The kelvin (symbol: K) is the base unit for temperature in the International System of Units (SI). The Kelvin scale is an absolute temperature scale that starts at the lowest possible temperature (absolute zero), taken to be 0 K. By de ...
scale.) Then for any T_2 and T_3, :f(T_2,T_3) = \frac = \frac. Therefore, if thermodynamic temperature is defined by :T' = 273.16 \cdot f(T_1,T), then the function viewed as a function of thermodynamic temperature, is :f(T_2,T_3) = \frac. It follows immediately that Substituting this equation back into the above equation \frac = f(T_H,T_C) gives a relationship for the efficiency in terms of thermodynamic temperatures:


Applicability to fuel cells

Since
fuel cell A fuel cell is an electrochemical cell that converts the chemical energy of a fuel (often hydrogen fuel, hydrogen) and an oxidizing agent (often oxygen) into electricity through a pair of redox reactions. Fuel cells are different from most bat ...
s can generate useful power when all components of the system are at the same temperature (T=T_H=T_C), they are clearly not limited by Carnot's theorem, which states that no power can be generated when T_H=T_C. This is because Carnot's theorem applies to engines converting thermal energy to work, whereas fuel cells instead convert
chemical energy Chemical energy is the energy of chemical substances that is released when the substances undergo a chemical reaction and transform into other substances. Some examples of storage media of chemical energy include batteries, Schmidt-Rohr, K. (20 ...
to work. Nevertheless, the
second law of thermodynamics The second law of thermodynamics is a physical law based on Universal (metaphysics), universal empirical observation concerning heat and Energy transformation, energy interconversions. A simple statement of the law is that heat always flows spont ...
still provides restrictions on fuel cell energy conversion. 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.


See also

* Chambadal–Novikov efficiency * Heating and cooling efficiency bounds


References

{{DEFAULTSORT:Carnot's Theorem (Thermodynamics) Eponymous theorems of physics Laws of thermodynamics Thought experiments in physics