Editing Second law of thermodynamics (section)

== Various statements of the law ==
The second law of thermodynamics may be expressed in many specific ways,<ref name=MIT>{{cite web|title=Concept and Statements of the Second Law|url=http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node37.html|access-date=2010-10-07 |publisher=web.mit.edu}}</ref> the most prominent classical statements{{sfnp|Lieb|Yngvason|1999}} being the statement by [[Rudolf Clausius]] (1854), the statement by [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] (1851), and the statement in [[axiomatic]] thermodynamics by [[Constantin Carathéodory]] (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.{{sfnp|Rao|2004|p=213}}

=== Carnot's principle ===
The historical origin<ref>[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986).</ref> of the second law of thermodynamics was in [[Nicolas Léonard Sadi Carnot|Sadi Carnot]]'s theoretical analysis of the flow of heat in steam engines (1824). The centerpiece of that analysis, now known as a [[Carnot engine]], is an ideal [[heat engine]] fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of [[thermodynamic equilibrium]]. It represents the theoretical maximum efficiency of a heat engine operating between any two given thermal or heat reservoirs at different temperatures. Carnot's principle was recognized by Carnot at a time when the [[caloric theory]] represented the dominant understanding of the nature of heat, before the recognition of the [[first law of thermodynamics]], and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, Carnot's analysis is physically equivalent to the second law of thermodynamics, and remains valid today. Some samples from his book are:
:: ...''wherever there exists a difference of temperature, motive power can be produced.''<ref>Carnot, S. (1824/1986), p. 51.</ref>
:: The production of motive power is then due in [[steam engines]] not to an actual consumption of caloric, but ''to its transportation from a warm body to a cold body ...''<ref>Carnot, S. (1824/1986), p. 46.</ref>
:: ''The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.''<ref>Carnot, S. (1824/1986), p. 68.</ref>

In modern terms, Carnot's principle may be stated more precisely:
: The efficiency of a quasi-static or reversible [[Carnot cycle]] depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<ref>[[Clifford Truesdell|Truesdell, C.]] (1980), Chapter 5.</ref><ref>Adkins, C.J. (1968/1983), pp. 56–58.</ref><ref>Münster, A. (1970), p. 11.</ref><ref>Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998), pp.67–75.</ref><ref>Lebon, G., Jou, D., Casas-Vázquez, J. (2008), p. 10.</ref><ref>Eu, B.C. (2002), pp. 32–35.</ref>

=== Clausius statement ===
The German scientist [[Rudolf Clausius]] laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.{{sfnp|Clausius|1850}} His formulation of the second law, which was published in German in 1854, is known as the ''Clausius statement'':
<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.{{sfnp|Clausius|1854|p=86}}</blockquote>

The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.

Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of [[refrigeration]], for example. In a refrigerator, heat is transferred from cold to hot, but only when forced by an external agent, the refrigeration system.

=== Kelvin statements ===
[[William Thomson, 1st Baron Kelvin|Lord Kelvin]] expressed the second law in several wordings.
:: It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.
:: It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.{{sfnp|Thomson|1851}}

=== Equivalence of the Clausius and the Kelvin statements ===
[[Image:Deriving Kelvin Statement from Clausius Statement.svg|thumb|Derive Kelvin Statement from Clausius Statement]]
Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work (the drained heat is fully converted to work) in a cyclic fashion without any other result. Now pair it with a reversed [[Carnot engine]] as shown by the right figure. The [[Heat engine#Efficiency|efficiency]] of a normal heat engine is ''η'' and so the efficiency of the reversed heat engine is 1/''η''. The net and sole effect of the combined pair of engines is to transfer heat <math display="inline">\Delta Q = Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. This is a consequence of the [[first law of thermodynamics]], as for the total system's energy to remain the same; <math display="inline"> \text{Input}+\text{Output}=0 \implies (Q + Q_c) - \frac{Q}{\eta} = 0 </math>, so therefore <math display="inline"> Q_c=Q\left( \frac{1}{\eta}-1\right) </math>, where (1) the sign convention of heat is used in which heat entering into (leaving from) an engine is positive (negative) and (2) <math> \frac{Q}{\eta} </math> is obtained by [[Heat engine#Efficiency|the definition of efficiency]] of the engine when the engine operation is not reversed. Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.

=== Planck's proposition ===
Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.
:: It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the production of work and cooling of a heat reservoir.<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 86.</ref><ref>Roberts, J.K., Miller, A.R. (1928/1960), p. 319.</ref>

=== Relation between Kelvin's statement and Planck's proposition ===
It is almost customary in textbooks to speak of the "Kelvin–Planck statement" of the law, as for example in the text by [[Dirk ter Haar|ter Haar]] and [[Harald Wergeland|Wergeland]].<ref>[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966), p. 17.</ref> This version, also known as the '''heat engine statement''', of the second law states that 
:: It is impossible to devise a [[thermodynamic cycle|cyclically]] operating device, the sole effect of which is to absorb energy in the form of heat from a single [[heat reservoir|thermal reservoir]] and to deliver an equivalent amount of [[Work (physics)|work]].<ref name="Rao">{{cite book|last=Rao|first=Y. V. C.|title=Chemical Engineering Thermodynamics|publisher=Universities Press|isbn=978-81-7371-048-3|page=158|year=1997}}</ref>

=== Planck's statement ===
[[Max Planck]] stated the second law as follows.
:: Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<ref name="Planck 100">[[Max Planck|Planck, M.]] (1897/1903), p. 100.</ref><ref name="Planck 463">[[Max Planck|Planck, M.]] (1926), p. 463, translation by Uffink, J. (2003), p. 131.</ref><ref name="Roberts & Miller 382">Roberts, J.K., Miller, A.R. (1928/1960), p. 382. This source is partly verbatim from Planck's statement, but does not cite Planck. This source calls the statement the principle of the increase of entropy.</ref>

Rather like Planck's statement is that of [[George Uhlenbeck]] and G. W. Ford for ''irreversible phenomena''.

:: ... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<ref>[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963), p. 16.</ref>

=== Principle of Carathéodory ===
<!-- [[Caratheodory's principle]] redirects here -->
[[Constantin Carathéodory]] formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<ref>[[Constantin Carathéodory|Carathéodory, C.]] (1909).</ref>
<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.<ref>Buchdahl, H.A. (1966), p. 68.</ref></blockquote>
With this formulation, he described the concept of [[adiabatic accessibility]] for the first time and provided the foundation for a new subfield of classical thermodynamics, often called [[Ruppeiner geometry|geometrical thermodynamics]]. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic [[process function]], in other words, <math>\delta Q=TdS</math>.<ref name="Sychev1991">{{cite book |last=Sychev |first=V. V. |title=The Differential Equations of Thermodynamics |year=1991 |publisher=Taylor & Francis |isbn=978-1-56032-121-7}}</ref>

Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.<ref name="Munster 45">Münster, A. (1970), p. 45.</ref>{{sfnp|Lieb|Yngvason|1999|p=49}}<ref name="Planck 1926">[[Max Planck|Planck, M.]] (1926).</ref><ref>Buchdahl, H.A. (1966), p. 69.</ref>{{clarify|date=February 2014}}

=== Planck's principle ===
In 1926, Max Planck wrote an important paper on the basics of thermodynamics.<ref name="Planck 1926"/><ref>Uffink, J. (2003), pp. 129–132.</ref> He indicated the principle
:: The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<ref name="Munster 45"/>{{sfnp |Lieb|Yngvason|1999|p=49}}

This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work."<ref>[[Clifford Truesdell|Truesdell, C.]], Muncaster, R.G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 15.</ref> Planck wrote: "The production of heat by friction is irreversible."<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 81.</ref><ref>[[Max Planck|Planck, M.]] (1926), p. 457, Wikipedia editor's translation.</ref>

Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above.<ref>Lieb, E.H., Yngvason, J. (2003), p. 149.</ref> It is relevant that for a system at constant volume and [[Mole (unit)|mole numbers]], the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.

A statement that in a sense is complementary to Planck's principle is made by Claus Borgnakke and Richard E. Sonntag. They do not offer it as a full statement of the second law:
:: ... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<ref>Borgnakke, C., Sonntag., R.E. (2009), p. 304.</ref>

Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.

=== Relating the second law to the definition of temperature ===
The second law has been shown to be equivalent to the [[internal energy]] {{math|''U''}} defined as a [[convex function]] of the other extensive properties of the system.<ref>{{cite book |author-last=Grubbström | author-first=Robert W. |chapter=Towards a Generalized Exergy Concept |editor1-last=Van Gool |editor1-first=W. |editor2-last=Bruggink |editor2-first=J.J.C.  |title=Energy and time in the economic and physical sciences |publisher=North-Holland |year=1985 |pages=41–56 |isbn=978-0-444-87748-2}}</ref> That is, when a system is described by stating its [[internal energy]] {{math|''U''}}, an extensive variable, as a function of its [[entropy]] {{math|''S''}}, volume {{math|''V''}}, and mol number {{math|''N''}}, i.e. {{math|''U'' {{=}} ''U'' (''S'', ''V'', ''N''}}), then the temperature is equal to the [[partial derivative]] of the internal energy with respect to the entropy<ref name="Callen 146–148">[[Herbert Callen|Callen, H.B.]] (1960/1985), ''Thermodynamics and an Introduction to Thermostatistics'', (first edition 1960), second edition 1985, John Wiley & Sons, New York, {{ISBN|0-471-86256-8}}, pp. 146–148.</ref> (essentially equivalent to the first {{math|''TdS''}} equation for {{math|''V''}} and {{math|''N''}}  held constant):
: <math>T = \left ( \frac{\partial U}{\partial S} \right )_{V, N}</math>

=== Second law statements, such as the Clausius inequality, involving radiative fluxes ===
The Clausius inequality, as well as some other statements of the second law, must be re-stated to have general applicability for all forms of heat transfer, i.e. scenarios involving radiative fluxes. For example, the integrand (đQ/T) of the Clausius expression applies to heat conduction and convection, and the case of ideal infinitesimal blackbody radiation (BR) transfer, but does not apply to most radiative transfer scenarios and in some cases has no physical meaning whatsoever. Consequently, the Clausius inequality was re-stated<ref>{{Cite journal |last=Wright |first=S.E. |date=December 2007 |title=The Clausius inequality corrected for heat transfer involving radiation |url=http://dx.doi.org/10.1016/j.ijengsci.2007.08.005 |journal=International Journal of Engineering Science |volume=45 |issue=12 |pages=1007–1016 |doi=10.1016/j.ijengsci.2007.08.005 |issn=0020-7225}}</ref> so that it is applicable to cycles with processes involving any form of heat transfer. The entropy transfer with radiative fluxes (<math display> \delta S_\text{NetRad} </math>) is taken separately from that due to heat transfer by conduction and convection (<math display> \delta Q_{CC} </math>), where the temperature is evaluated at the system boundary where the heat transfer occurs. The modified Clausius inequality, for all heat transfer scenarios, can then be expressed as,<math display="block"> \int_\text{cycle} (\frac{\delta Q_{CC}}{T_b} + \delta S_\text{NetRad}) \le 0 </math>

In a nutshell, the Clausius inequality is saying that when a cycle is completed, the change in the state property S will be zero, so the entropy that was produced during the cycle must have transferred out of the system by heat transfer. The <math display> \delta </math> (or đ) indicates a path dependent integration.

Due to the inherent emission of radiation from all matter, most entropy flux calculations involve incident, reflected and emitted radiative fluxes. The energy and entropy of unpolarized blackbody thermal radiation, is calculated using the spectral energy and entropy radiance expressions derived by Max Planck<ref>{{Cite journal |last=Planck |first=Max |date=1914 |title=Translation by Morton Mausius, The Theory of Heat Radiation |journal=Dover Publications, NY}}</ref> using equilibrium statistical mechanics,<math display="block">
K_\nu = \frac{ 2 h }{c^2} \frac{\nu^3}{\exp\left(\frac{h\nu}{kT}\right) - 1}, 
</math>
<math display="block">
L_\nu = \frac{ 2 k \nu^2 }{c^2} ((1+\frac{c^2 K_\nu}{2 h \nu^3})\ln(1+\frac{c^2 K_\nu}{2 h \nu^3})-(\frac{c^2 K_\nu}{2 h \nu^3})\ln(\frac{c^2 K_\nu}{2 h \nu^3}))
</math>
where ''c'' is the speed of light, ''k'' is the Boltzmann constant, ''h'' is the Planck constant, ''ν'' is frequency, and the quantities ''K''<sub>v</sub> and ''L''<sub>v</sub> are the energy and entropy fluxes per unit frequency, area, and solid angle. In deriving this blackbody spectral entropy radiance, with the goal of deriving the blackbody energy formula, Planck postulated that the energy of a photon was quantized (partly to simplify the mathematics), thereby starting quantum theory.
	
A non-equilibrium statistical mechanics approach has also been used to obtain the same result as Planck, indicating it has wider significance and represents a non-equilibrium entropy.<ref>{{cite journal |last1=Landsberg |first1=P T |last2=Tonge |first2=G |date=April 1979 |title=Thermodynamics of the conversion of diluted radiation |url=http://dx.doi.org/10.1088/0305-4470/12/4/015 |journal=Journal of Physics A: Mathematical and General |volume=12 |issue=4 |pages=551–562 |doi=10.1088/0305-4470/12/4/015 |bibcode=1979JPhA...12..551L |issn=0305-4470}}</ref> A plot of ''K''<sub>v</sub> versus frequency (v) for various values of temperature (''T)'' gives a family of blackbody radiation energy spectra, and likewise for the entropy spectra. For non-blackbody radiation (NBR) emission fluxes, the spectral entropy radiance ''L''<sub>v</sub> is found by substituting ''K''<sub>v</sub> spectral energy radiance data into the ''L''<sub>v</sub> expression (noting that emitted and reflected entropy fluxes are, in general, not independent). For the emission of NBR, including graybody radiation (GR), the resultant emitted entropy flux, or radiance ''L'', has a higher ratio of entropy-to-energy (''L/K''), than that of BR. That is, the entropy flux of NBR emission is farther removed from the conduction and convection ''q''/''T'' result, than that for BR emission.<ref>{{Cite journal |last=Wright |date=2001 |title=On the entropy of radiative heat transfer in engineering thermodynamics |journal=Int. J. Eng. Sci. |volume=39 |issue=15 |pages=1691–1706|doi=10.1016/S0020-7225(01)00024-6 }}</ref> This observation is consistent with Max Planck's blackbody radiation energy and entropy formulas and is consistent with the fact that blackbody radiation emission represents the maximum emission of entropy for all materials with the same temperature, as well as the maximum entropy emission for all radiation with the same energy radiance.

=== Generalized conceptual statement of the second law principle ===
Second law analysis is valuable in scientific and engineering analysis in that it provides a number of benefits over energy analysis alone, including the basis for determining energy quality (exergy content<ref>{{Cite journal |last1=Wright |first1=S.E. |last2=Rosen |first2=M.A. |last3=Scott |first3=D.S. |last4=Haddow |first4=J.B. |date=January 2002 |title=The exergy flux of radiative heat transfer for the special case of blackbody radiation |url=http://dx.doi.org/10.1016/s1164-0235(01)00040-1 |journal=Exergy |volume=2 |issue=1 |pages=24–33 |doi=10.1016/s1164-0235(01)00040-1 |issn=1164-0235}}</ref><ref>{{Cite journal |last1=Wright |first1=S.E. |last2=Rosen |first2=M.A. |last3=Scott |first3=D.S. |last4=Haddow |first4=J.B. |date=January 2002 |title=The exergy flux of radiative heat transfer with an arbitrary spectrum |url=http://dx.doi.org/10.1016/s1164-0235(01)00041-3 |journal=Exergy |volume=2 |issue=2 |pages=69–77 |doi=10.1016/s1164-0235(01)00041-3 |issn=1164-0235}}</ref><ref>{{Cite journal |last1=Wright |first1=Sean E. |last2=Rosen |first2=Marc A. |date=2004-02-01 |title=Exergetic Efficiencies and the Exergy Content of Terrestrial Solar Radiation |url=http://dx.doi.org/10.1115/1.1636796 |journal=Journal of Solar Energy Engineering |volume=126 |issue=1 |pages=673–676 |doi=10.1115/1.1636796 |issn=0199-6231}}</ref>), understanding fundamental physical phenomena, and improving performance evaluation and optimization. As a result, a conceptual statement of the principle is very useful in engineering analysis. Thermodynamic systems can be categorized by the four combinations of either entropy (S) up or down, and uniformity (Y) – between system and its environment – up or down. This 'special' category of processes, category IV, is characterized by movement in the direction of low disorder and low uniformity, counteracting the second law tendency towards uniformity and disorder.<ref name="Wright 12–18">{{Cite journal |last=Wright |first=S.E. |date=February 2017 |title=A generalized and explicit conceptual statement of the principle of the second law of thermodynamics |url=http://dx.doi.org/10.1016/j.ijengsci.2016.11.002 |journal=International Journal of Engineering Science |volume=111 |pages=12–18 |doi=10.1016/j.ijengsci.2016.11.002 |issn=0020-7225}}</ref>
[[File:Exergysun1.png|thumb|Four categories of processes given entropy up or down and uniformity up or down]]

The second law can be conceptually stated<ref name="Wright 12–18"/> as follows: Matter and energy have the tendency to reach a state of uniformity or internal and external equilibrium, a state of maximum disorder (entropy). Real non-equilibrium processes always produce entropy, causing increased disorder in the universe, while idealized reversible processes produce no entropy and no process is known to exist that destroys entropy. The tendency of a system to approach uniformity may be counteracted, and the system may become more ordered or complex, by the combination of two things, a work or exergy source and some form of instruction or intelligence. Where 'exergy' is the thermal, mechanical, electric or chemical work potential of an energy source or flow, and 'instruction or intelligence', although subjective, is in the context of the set of category IV processes.

Consider a category IV example of robotic manufacturing and assembly of vehicles in a factory. The robotic machinery requires electrical work input and instructions, but when completed, the manufactured products have less uniformity with their surroundings, or more complexity (higher order) relative to the raw materials they were made from. Thus, system entropy or disorder decreases while the tendency towards uniformity between the system and its environment is counteracted. In this example, the instructions, as well as the source of work may be internal or external to the system, and they may or may not cross the system boundary. To illustrate, the instructions may be pre-coded and the electrical work may be stored in an energy storage system on-site. Alternatively, the control of the machinery may be by remote operation over a communications network, while the electric work is supplied to the factory from the local electric grid. In addition, humans may directly play, in whole or in part, the role that the robotic machinery plays in manufacturing. In this case, instructions may be involved, but intelligence is either directly responsible, or indirectly responsible, for the direction or application of work in such a way as to counteract the tendency towards disorder and uniformity.

There are also situations where the entropy spontaneously decreases by means of energy and entropy transfer. When thermodynamic constraints are not present, spontaneously energy or mass, as well as accompanying entropy, may be transferred out of a system in a progress to reach external equilibrium or uniformity in intensive properties of the system with its surroundings. This occurs spontaneously because the energy or mass transferred from the system to its surroundings results in a higher entropy in the surroundings, that is, it results in higher overall entropy of the system plus its surroundings. Note that this transfer of entropy requires dis-equilibrium in properties, such as a temperature difference. One example of this is the cooling crystallization of water that can occur when the system's surroundings are below freezing temperatures. Unconstrained heat transfer can spontaneously occur, leading to water molecules freezing into a crystallized structure of reduced disorder (sticking together in a certain order due to molecular attraction). The entropy of the system decreases, but the system approaches uniformity with its surroundings (category III).

On the other hand, consider the refrigeration of water in a warm environment. Due to refrigeration, as heat is extracted from the water, the temperature and entropy of the water decreases, as the system moves further away from uniformity with its warm surroundings or environment (category IV). The main point, take-away, is that refrigeration not only requires a source of work, it requires designed equipment, as well as pre-coded or direct operational intelligence or instructions to achieve the desired refrigeration effect.