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The Clausius theorem (1855), also known as the ''Clausius inequality'', states that for a
thermodynamic system A thermodynamic system is a body of matter and/or radiation, confined in space by walls, with defined permeabilities, which separate it from its surroundings. The surroundings may include other thermodynamic systems, or physical systems that are ...
(e.g.
heat engine In thermodynamics and engineering, a heat engine is a system that converts heat to mechanical energy, which can then be used to do mechanical work. It does this by bringing a working substance from a higher state temperature to a lower stat ...
or
heat pump A heat pump is a device that can heat a building (or part of a building) by transferring thermal energy from the outside using a refrigeration cycle. Many heat pumps can also operate in the opposite direction, cooling the building by removing ...
) exchanging heat with external thermal reservoirs and undergoing a
thermodynamic cycle A thermodynamic cycle consists of a linked sequence of thermodynamic processes that involve transfer of heat and work into and out of the system, while varying pressure, temperature, and other state variables within the system, and that eventu ...
, :-\oint dS_\text = \oint \frac \leq 0, where \oint dS_\text is the total
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
change in the external thermal reservoirs (surroundings), \delta Q is an infinitesimal amount of heat that is from each reservoir and absorbed by the system (\delta Q > 0 if heat from the reservoir is absorbed by the system, and \delta Q < 0 if heat is leaving from the system to the reservoir) and T_ is the
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
of the reservoir at a particular instant in time. The closed integral is carried out along a
thermodynamic process path Thermodynamic diagrams are diagrams used to represent the thermodynamic states of a material (typically fluid) and the consequences of manipulating this material. For instance, a temperature–entropy diagram ( T–s diagram) may be used to demon ...
from the initial/final state to the same initial/final state (thermodynamic cycle). In principle, the closed integral can start and end at an arbitrary point along the path. The Clausius theorem or inequality obviously implies \oint dS_\text \geq 0 per thermodynamic cycle, meaning that the entropy of the reservoirs increases or does not change, never decrease, per cycle. For multiple thermal reservoirs with different temperatures \left(T_1, T_2, \dots, T_N\right) interacting a thermodynamic system undergoing a thermodynamic cycle, the Clausius inequality can be written as the following for expression clarity: :-\oint dS_\text = \oint \left(\sum_^N\frac\right) \leq 0. where \delta Q_n is an infinitesimal heat from the reservoir n to the system. In the special case of a reversible process, the equality holds, and the reversible case is used to introduce the
state function In the thermodynamics of equilibrium, a state function, function of state, or point function for a thermodynamic system is a mathematical function relating several state variables or state quantities (that describe equilibrium states of a system ...
known as
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
. This is because in a cyclic process the variation of a state function is zero per cycle, so the fact that this integral is equal to zero per cycle in a reversible process implies that there is some function (entropy) whose infinitesimal change is \frac. The generalized "inequality of Clausius" :dS_ \geq \frac for dS_ as an infinitesimal change in entropy of a system (denoted by sys) under consideration applies not only to cyclic processes, but to any process that occurs in a closed system. The Clausius inequality is a consequence of applying the
second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unle ...
at each infinitesimal stage of heat transfer. The Clausius statement states that it is impossible to construct a device whose sole effect is the transfer of heat from a cool reservoir to a hot reservoir. Equivalently, heat spontaneously flows from a hot body to a cooler one, not the other way around.Giancoli, Douglas C. ''Physics: Principles with Applications''. 6th ed., Pearson/Prentice Hall, 2005.


History

The Clausius theorem is a mathematical representation of the
second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unle ...
. It was developed by
Rudolf Clausius Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist and mathematician and is considered one of the central founding fathers of the science of thermodynamics. By his restatement of Sadi Carnot's princip ...
who intended to explain the relationship between the heat flow in a system and the entropy of the system and its surroundings. Clausius developed this in his efforts to explain entropy and define it quantitatively. In more direct terms, the theorem gives us a way to determine if a cyclical process is reversible or irreversible. The Clausius theorem provides a quantitative formula for understanding the second law. Clausius was one of the first to work on the idea of entropy and is even responsible for giving it that name. What is now known as the Clausius theorem was first published in 1862 in Clausius' sixth memoir, "On the Application of the Theorem of the Equivalence of Transformations to Interior Work". Clausius sought to show a proportional relationship between entropy and the energy flow by heating (δ''Q'') into a system. In a system, this heat energy can be transformed into work, and work can be transformed into heat through a cyclical process. Clausius writes that "The algebraic sum of all the transformations occurring in a cyclical process can only be less than zero, or, as an extreme case, equal to nothing." In other words, the equation :\oint \frac = 0 with 𝛿''Q'' being energy flow into the system due to heating and ''T'' being absolute temperature of the body when that energy is absorbed, is found to be true for any process that is cyclical and reversible. Clausius then took this a step further and determined that the following relation must be found true for any cyclical process that is possible, reversible or not. This relation is the "Clausius inequality", :\oint \frac \leq 0 where \delta Q is an infinitesimal amount of heat that is from the thermal reservoir interacting with the system and absorbed by the system (\delta Q > 0 if heat from the reservoir is absorbed by the system, and \delta Q < 0 if heat is leaving from the system to the reservoir) and T_ is the
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
of the reservoir at a particular instant in time. Now that this is known, there must be a relation developed between the Clausius inequality and entropy. The amount of entropy ''S'' added to the system during the cycle is defined as :\Delta S \oint \frac It has been determined, as stated in the
second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unle ...
, that the entropy is a state function: It depends only upon the state that the system is in, and not what path the system took to get there. This is in contrast to the amount of energy added as heat (𝛿''Q'') and as work (𝛿''W''), which may vary depending on the path. In a cyclic process, therefore, the entropy of the system at the beginning of the cycle must equal to the entropy at the end of the cycle (because the entropy is a state function), \Delta S=0, regardless of whether the process is reversible or irreversible. In irreversible cases, the net entropy is added to the system reservoirs (\Delta S_\text>0) per thermodynamic cycle while in reversible cases, no entropy is created or added to the reservoirs. If the amount of energy added by heating can be measured during the process, and the temperature can be measured during the process, then the Clausius inequality can be used to determine whether the process is reversible or irreversible by carrying out the integration in the Clausius inequality. If integral result is equal to zero then it is a reversible process, while if greater than zero then an irreversible process (less than zero cannot be possible).


Proof

The temperature that enters in the denominator of the integrand in the Clausius inequality is the temperature of the external thermal reservoir with which the system exchanges heat. At each instant of the process, the system is in contact with an external reservoir. ''Because of the Second Law of Thermodynamics, in each infinitesimal heat exchange process between the system and the reservoirs, the net change in entropy of the "universe", so to say, is dS_\text = dS_\text + dS_\text \geq 0 , where Sys and Res stand for System and Reservoir, respectively.'' In the proof of the Clausius theorem or inequality, a sign convention of heat is used; in the perspective of an object under consideration, when heat is absorbed by the object then the heat is positive, while when heat leaves from the object then the heat is negative. When the system takes heat from a hotter (hot)
reservoir A reservoir (; from French ''réservoir'' ) is an enlarged lake behind a dam. Such a dam may be either artificial, built to store fresh water or it may be a natural formation. Reservoirs can be created in a number of ways, including contr ...
by an infinitesimal amount \delta Q_(\geq 0), for the net change in entropy dS_ to be positive or zero (i.e., non-negative) in this step (called the step 1 here) to fulfill the Second Law of Thermodynamics, the temperature of the hot
reservoir A reservoir (; from French ''réservoir'' ) is an enlarged lake behind a dam. Such a dam may be either artificial, built to store fresh water or it may be a natural formation. Reservoirs can be created in a number of ways, including contr ...
T_\text needs to be equal to or greater than the temperature of the system at that instant; if the temperature of the system is given by T_ at that instant, then dS_ = \frac as the entropy change in the system at the instant, and T_\text\geq T_ forces us to have: : -dS_ =\frac\leq \frac = dS_ This means the magnitude of the entropy "loss" from the hot reservoir, \left, dS_\ = \frac is equal to or less than the magnitude of the entropy "gain" dS_ = \frac(\geq 0) by the system, so the net entropy change dS_ is zero or positive. Similarly, when the system at temperature T_ expels heat in magnitude \left , \delta Q_ \right , = -\delta Q_ (\delta Q_\leq 0) into a colder (cold)
reservoir A reservoir (; from French ''réservoir'' ) is an enlarged lake behind a dam. Such a dam may be either artificial, built to store fresh water or it may be a natural formation. Reservoirs can be created in a number of ways, including contr ...
(at temperature T_\text\leq T_) in an infinitesimal step (called the step 2), then again, for the Second Law of Thermodynamics to hold, one would have, in a very similar manner: : -dS_=\frac\leq \frac= dS_ Here, the amount of heat 'absorbed' by the system is given by \delta Q_ \leq 0, signifying that heat is actually transferring (leaving) from the system to the cold reservoir, with dS_\leq 0. The magnitude of the entropy gained by the cold reservoir dS_ = - \frac is equal to or greater than the magnitude of the entropy loss of the system \left, dS_\, so the net entropy change dS_ is also zero or positive in this case. Because the total change in entropy for the system is zero in a thermodynamic cyclic process where all state functions of the system are reset or returned to initial values (values at the process starts) upon the completion of each cycle, if one adds all the infinitesimal steps of heat intake from and heat expulsion to the reservoirs, signified by the previous two equations, with the temperature of each reservoir at each instant given by T_\text, one gets : -\oint dS_\text= \oint \frac\leq \oint dS_\text = 0. In particular, :\oint \frac\leq 0, which was to be proven (and is now proven). In summary, (the inequality in the third statement below, being obviously guaranteed by the
second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unle ...
, which is the basis of our calculation), :\oint dS_\text\geq 0 :\oint dS_\text=0 (as a cyclic process) :\oint dS_\text=\oint dS_\text+\oint dS_\text\geq 0 For a reversible
cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in so ...
process, there is no generation of entropy in each of the infinitesimal heat transfer processes since there is practically no temperature difference between the system and the thermal reservoirs (I.e., the system entropy change and the reservoirs entropy change is equal in magnitude and opposite in sign at any instant.), so the following equality holds, :\oint \frac=0 :\oint dS_\text = 0 :\oint dS_\text=0 (as a cyclic process) :\oint dS_\text=\oint dS_\text+\oint dS_\text = 0 The Clausius inequality is a consequence of applying the
second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unle ...
at each infinitesimal stage of heat transfer, and is thus in a sense a weaker condition than the Second Law itself.


Heat engine efficiency

In the heat engine model with two thermal reservoirs (hot and cold reservoirs), the limit of the efficiency of any heat engine \eta =\frac, where W and Q_1 are work done by the heat engine and heat transferred from the hot thermal reservoir to the engine, respectively, can be derived by the first law of thermodynamics (i.e., the law of conservation of energy) and the Clausius theorem or inequality. In respecting the abovementioned sign convention of heat, :+=W\to \eta =\frac=1+\frac, where Q_2 is heat transferred from the engine to the cold reservoir. The Clausius inequality \frac+\frac\le 0 can be expressed as \frac\le -\frac. By substituting this inequality to the above equation results in, :\eta =\frac\le 1-\frac. This is the limit of heat engine efficiencies, and the equality of this expression is what is called the
Carnot efficiency A Carnot cycle is an ideal thermodynamic cycle proposed by French physicist Sadi Carnot in 1824 and expanded upon by others in the 1830s and 1840s. By Carnot's theorem, it provides an upper limit on the efficiency of any classical thermodynam ...
, that is the efficiency of all reversible heat engines and the maximum efficiency of all heat engines.


See also

* Kelvin-Planck statement * Carnot's theorem (thermodynamics) *
Carnot heat engine A Carnot heat engine is a 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 Clapeyron in 1 ...
* Introduction to entropy


References


Further reading

*Morton, A. S., and P.J. Beckett. ''Basic Thermodynamics''. New York: Philosophical Library Inc., 1969. Print. *Saad, Michel A. ''Thermodynamics for Engineers''. Englewood Cliffs: Prentice-Hall, 1966. Print. *Hsieh, Jui Sheng. ''Principles of Thermodynamics''. Washington, D.C.: Scripta Book Company, 1975. Print. *Zemansky, Mark W. ''Heat and Thermodynamics''. 4th ed. New York: McGwaw-Hill Book Company, 1957. Print. *Clausius, Rudolf. ''The Mechanical Theory of Heat''. London: Taylor and Francis, 1867. eBook


External links

* * * {{DEFAULTSORT:Clausius Theorem Laws of thermodynamics Physics theorems