Editing Entropy (section)

=== Carnot cycle ===
The concept of entropy arose from [[Rudolf Clausius]]'s study of the [[Carnot cycle]] which is a [[thermodynamic cycle]] performed by a Carnot heat engine as a reversible heat engine.<ref>{{cite book| last1=Lavenda| first1=Bernard H.| title=A new perspective on thermodynamics| date=2010|publisher=Springer|location=New York|isbn=978-1-4419-1430-9|edition=Online-Ausg.|chapter=2.3.4}}</ref> In a Carnot cycle, the heat <math display="inline">Q_\mathsf{H}</math> is transferred from a hot reservoir to a working gas at the constant temperature <math display="inline">T_\mathsf{H}</math> during [[Isothermal process|isothermal]] expansion stage and the heat <math display="inline">Q_\mathsf{C}</math> is transferred from a working gas to a cold reservoir at the constant temperature <math display="inline">T_\mathsf{C}</math> during [[Isothermal process|isothermal]] compression stage. According to [[Carnot's theorem (thermodynamics)|Carnot's theorem]], a heat engine with two thermal reservoirs can produce a [[work (physics)|work]] <math display="inline">W</math> if and only if there is a temperature difference between reservoirs. Originally, Carnot did not distinguish between heats <math display="inline">Q_\mathsf{H}</math> and <math display="inline">Q_\mathsf{C}</math>, as he assumed [[caloric theory]] to be valid and hence that the total heat in the system was conserved. But in fact, the magnitude of heat <math display="inline">Q_\mathsf{H}</math> is greater than the magnitude of heat <math display="inline">Q_\mathsf{C}</math>.<ref>{{cite book|last1=Carnot|first1=Sadi Carnot|editor1-last=Fox|editor1-first=Robert |title=Reflexions on the motive power of fire|url=https://archive.org/details/reflexionsonmoti0000carn|url-access=registration| date=1986|publisher=Lilian Barber Press| location=New York|isbn=978-0-936508-16-0|pages=[https://archive.org/details/reflexionsonmoti0000carn/page/26 26]}}</ref><ref>{{cite book|last1=Truesdell|first1=C.|title=The tragicomical history of thermodynamics 1822–1854|url=https://archive.org/details/tragicomicalhist18221854iiic|url-access=limited|date=1980|publisher=Springer|location=New York|isbn=978-0-387-90403-0|pages=[https://archive.org/details/tragicomicalhist18221854iiic/page/n85 78]–85}}</ref> Through the efforts of [[Rudolf Clausius|Clausius]] and [[Lord Kelvin|Kelvin]], the work <math display="inline">W</math> done by a reversible heat engine was found to be the product of the Carnot efficiency (i.e., the efficiency of all reversible heat engines with the same pair of thermal reservoirs) and the heat <math display="inline">Q_\mathsf{H}</math> absorbed by a working body of the engine during isothermal expansion:<math display="block">W = \frac{ T_\mathsf{H} - T_\mathsf{C} }{ T_\mathsf{H} } \cdot Q_\mathsf{H} = \left( 1 - \frac{ T_\mathsf{C} }{ T_\mathsf{H} } \right) Q_\mathsf{H}</math>To derive the Carnot efficiency Kelvin had to evaluate the ratio of the work output to the heat absorbed during the isothermal expansion with the help of the Carnot–Clapeyron equation, which contained an unknown function called the Carnot function. The possibility that the Carnot function could be the temperature as measured from a zero point of temperature was suggested by [[James Prescott Joule|Joule]] in a letter to Kelvin. This allowed Kelvin to establish his absolute temperature scale.<ref>{{cite book |last1=Clerk Maxwell|first1=James|editor1-last=Pesic|editor1-first=Peter|title=Theory of heat|date=2001|publisher=Dover Publications |location=Mineola|isbn=978-0-486-41735-6|pages=115–158}}</ref>

It is known that a work <math display="inline">W > 0</math> produced by an engine over a cycle equals to a net heat <math display="inline"> Q_\Sigma = \left\vert Q_\mathsf{H} \right\vert - \left\vert Q_\mathsf{C} \right\vert </math> absorbed over a cycle.<ref name="PlanckBook">{{cite book |last=Planck |first=M. |title=Treatise on Thermodynamics |page=§90 & §137|quote=eqs.(39), (40), & (65) |publisher=Dover Publications |year=1945}}.</ref> Thus, with the sign convention for a heat <math display="inline"> Q</math> transferred in a thermodynamic process (<math display="inline"> Q > 0</math> for an absorption and <math display="inline"> Q < 0</math> for a dissipation) we get:<math display="block">W - Q_\Sigma = W - \left\vert Q_\mathsf{H} \right\vert + \left\vert Q_\mathsf{C} \right\vert = W - Q_\mathsf{H} - Q_\mathsf{C} = 0</math>Since this equality holds over an entire Carnot cycle, it gave Clausius the hint that at each stage of the cycle the difference between a work and a net heat would be conserved, rather than a net heat itself. Which means there exists a [[state function]] <math display="inline">U</math> with a change of <math display="inline">\mathrm{d} U = \delta Q - \mathrm{d} W</math>. It is called an [[internal energy]] and forms a central concept for the [[first law of thermodynamics]].<ref name="Clausius1867">{{cite book |author=Rudolf Clausius |title=The Mechanical Theory of Heat: With Its Applications to the Steam-engine and to the Physical Properties of Bodies|url=https://books.google.com/books?id=8LIEAAAAYAAJ |year=1867 |publisher=J. Van Voorst |isbn=978-1-4981-6733-8|page=28}}</ref>

Finally, comparison for both the representations of a work output in a Carnot cycle gives us:<ref name="PlanckBook" /><ref name="FermiBook">{{cite book |last=Fermi |first=E. |title=Thermodynamics |page=48 |quote=eq.(64) |publisher=Dover Publications (still in print) |year=1956}}.</ref><math display="block">\frac{\left\vert Q_\mathsf{H} \right\vert}{T_\mathsf{H}} - \frac{\left\vert Q_\mathsf{C} \right\vert}{T_\mathsf{C}} = \frac{Q_\mathsf{H}}{T_\mathsf{H}} + \frac{Q_\mathsf{C}}{T_\mathsf{C}} = 0</math>Similarly to the derivation of internal energy, this equality implies existence of a [[state function]] <math display="inline">S</math> with a change of <math display="inline">\mathrm{d} S = \delta Q / T</math> and which is conserved over an entire cycle. Clausius called this state function ''entropy''.

In addition, the total change of entropy in both thermal reservoirs over Carnot cycle is zero too, since the inversion of a heat transfer direction means a sign inversion for the heat transferred during isothermal stages:<math display="block">- \frac{ Q_\mathsf{H} }{ T_\mathsf{H} } - \frac{ Q_\mathsf{C} }{ T_\mathsf{C} } = \Delta S_\mathsf{r, H} +  \Delta S_\mathsf{r, C} = 0</math>Here we denote the entropy change for a thermal reservoir by <math display="inline">\Delta S_{\mathsf{r}, i} = - Q_i / T_i</math>, where <math display="inline">i</math> is either <math display="inline">\mathsf{H}</math> for a hot reservoir or <math display="inline">\mathsf{C}</math> for a cold one.

If we consider a heat engine which is less effective than Carnot cycle (i.e., the work <math display="inline"> W</math> produced by this engine is less than the maximum predicted by Carnot's theorem), its work output is capped by Carnot efficiency as:<math display="block"> W < \left( 1 - \frac{T_\mathsf{C}}{T_\mathsf{H}} \right) Q_\mathsf{H} </math>Substitution of the work <math display="inline">W</math> as the net heat into the inequality above gives us:<math display="block">\frac{Q_\mathsf{H}}{T_\mathsf{H}} + \frac{Q_\mathsf{C}}{T_\mathsf{C}} < 0</math>or in terms of the entropy change <math display="inline">\Delta S_{\mathsf{r}, i}</math>:<math display="block">\Delta S_\mathsf{r, H} + \Delta S_\mathsf{r, C} > 0</math>A [[Carnot cycle]] and an entropy as shown above prove to be useful in the study of any classical thermodynamic heat engine: other cycles, such as an [[Otto cycle|Otto]], [[Diesel cycle|Diesel]] or [[Brayton cycle]], could be analysed from the same standpoint. Notably, any machine or cyclic process converting heat into work (i.e., heat engine) that is claimed to produce an efficiency greater than the one of Carnot is not viable — due to violation of [[Second law of thermodynamics|the second law of thermodynamics]].

For further analysis of sufficiently discrete systems, such as an assembly of particles, [[Statistical mechanics|statistical thermodynamics]] must be used. Additionally, descriptions of devices operating near the limit of [[Matter wave|de Broglie waves]], e.g. [[Solar cell|photovoltaic cells]], have to be consistent with [[Quantum statistical mechanics|quantum statistics]].