{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T00:26:20Z","timestamp":1790123180675,"version":"4.0.1"},"reference-count":104,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,1,7]],"date-time":"2020-01-07T00:00:00Z","timestamp":1578355200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We present a history of thermodynamics. Part 1 discusses definitions, a pre-history of heat and temperature, and steam engine efficiency, which motivated thermodynamics. Part 2 considers in detail three heat conservation-based foundational papers by Carnot, Clapeyron, and Thomson. For a reversible Carnot cycle operating between thermal reservoirs with Celsius temperatures t and     t + d t    , heat Q from the hot reservoir, and net work W, Clapeyron derived     W \/ Q = d t \/ C ( t )    , with     C ( t )     material-independent. Thomson used     \u03bc = 1 \/ C ( t )     to define an absolute temperature but, unaware that an additional criterion was needed, he first proposed a logarithmic function of the ideal gas temperature     T g    . Part 3, following a discussion of conservation of energy, considers in detail a number of energy conservation-based papers by Clausius and Thomson. As noted by Gibbs, in 1850, Clausius established the first modern form of thermodynamics, followed by Thomson\u2019s 1851 rephrasing of what he called the Second Law. In 1854, Clausius theoretically established for a simple Carnot cycle the condition      Q 1  \/  T 1  +  Q 2  \/  T 2  = 0    . He generalized it to      \u2211 i   Q i  \/  T  g , i   = 0    , and then     \u222e d Q \/  T g  = 0    . This both implied a new thermodynamic state function and, with appropriate integration factor     1 \/ T    , the thermodynamic temperature. In 1865, Clausius named this new state function the entropy S.<\/jats:p>","DOI":"10.3390\/e22010077","type":"journal-article","created":{"date-parts":[[2020,1,8]],"date-time":"2020-01-08T03:59:57Z","timestamp":1578455997000},"page":"77","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":39,"title":["A History of Thermodynamics: The Missing Manual"],"prefix":"10.3390","volume":"22","author":[{"given":"Wayne M.","family":"Saslow","sequence":"first","affiliation":[{"name":"Department of Physics, Texas A&amp;M University, College Station, TX 77843-4242, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,1,7]]},"reference":[{"key":"ref_1","unstructured":"ThomsonW.An Account of Carnot\u2019s Theory of the Motive Power of Heat, with numerical results deduced from Regnault\u2019s Experiments on SteamTrans. Edinb. R. Soc.18491654110.1017\/S0080456800022481\n\t\t  This paper has been published in numerous places, such as \n        Mathematical and Physical PapersCambridge University PressCambridge, UK1882Volume 1113114\n\t\tThis latter reference will be referred to as MPP1."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"ClausiusR.On the moving force of heat and the laws of heat which may be deduced therefromPoggendorf\u2019s Annalen185079368369\n\t\tAbout 50 pages. Translated Phil. Mag. 1851, 2, 8\u201321 and Phil. Mag. 1851, 2, 102\u2013119. Reprinted in Ref. [12]. These are the first two memoirs of nine in \n        The Mechanical Theory of Heat: With Its Applications to the Steam-Engine and to the Physical Properties of Bodies; Clausius, R., Ed.; Thomas Archer Hirst, Pub. J. Van Voorst, 1867.","DOI":"10.1080\/14786445108646840"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1017\/S0080456800033172","article-title":"On the dynamical theory of heat, with numerical results deduced from Mr. Joule\u2019s equivalent of a thermal unit, and M. Regnault\u2019s observations on steam","volume":"20","author":"Thomson","year":"1851","journal-title":"Trans. Edinb. R. Soc."},{"key":"ref_4","unstructured":"Carnot, Clapeyron, and Thomson (1849) all accepted the proportionality of the net work W to the heat QH from the hot reservoir. Clausius took the step of taking the heat to the cold reservoir QC as being QH diminished by W, or QC = QH \u2212 AW, where A is the heat equivalent of work."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1063\/1.3057388","article-title":"A Sketch for a History of Early Thermodynamics","volume":"2","author":"Mendoza","year":"1961","journal-title":"Phys. Today"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1992","DOI":"10.1103\/PhysRev.128.1992","article-title":"Osmotic Pressure of He3 in Liquid He4, with Proposals for a Refrigerator to Work below 1\u2218K","volume":"128","author":"London","year":"1962","journal-title":"Phys. Rev."},{"key":"ref_7","unstructured":"Cardwell, D.S.L. (1971). From Watt to Clausius: The Rise of Thermodynamics in the Early Industrial Age, Cornell University Press. For Dalton and Gay-Lussac, see pp. 129\u2013131. Cardwell questions Tait\u2019s crediting of Charles with this law, since Gay-Lussac notes that Charles gave results in agreement with Gay-Lussac for water-insoluble gases, but not for water-soluble gases."},{"key":"ref_8","unstructured":"Carnot, S. (1824). Reflexions sur la Puissance Motrice du feu, Bachelier."},{"key":"ref_9","unstructured":"Carroll, L. (1865). Alice\u2019s Adventures in Wonderland, MacMillan. It is well-known that Lewis Carroll is a pseudonym used by the mathematician Charles Dodgson."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Fox, R. (1971). The Caloric Theory of Gases from Lavoisier to Regnault, The Clarendon Press.","DOI":"10.1063\/1.3071049"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1111\/j.1600-0498.1976.tb00936.x","article-title":"Mayer\u2019s Hypothesis: A Study of the Early Years of Thermodynamics","volume":"20","author":"Hutchison","year":"1976","journal-title":"Centaurus"},{"key":"ref_12","unstructured":"Mendoza, E. (1960). Reflections on the Motive Power of Fire, and Other Papers on the Second Law of Thermodynamics by E. Clapeyron and R. Clausius, Dover. As the title indicates, it contains papers by Carnot, Clapeyron, and Clausius."},{"key":"ref_13","unstructured":"Fox, R. (1986). Reflections on the Motive Power of Fire: A Critical Edition with the Surviving Scientific Manuscripts, Lilian Barber Press."},{"key":"ref_14","first-page":"153","article-title":"Memoire sur la puissance motrice de la chaleur","volume":"14","author":"Clapeyron","year":"1834","journal-title":"Journal de l\u2019Ecole Polytechnique"},{"key":"ref_15","unstructured":"M\u00fcller, I. (2006). A History of Thermodynamics, Springer."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2576","DOI":"10.1080\/14786435.2013.784402","article-title":"The discovery of thermodynamics","volume":"93","author":"Weinberger","year":"2013","journal-title":"Philos. Mag."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Truesdell, C. (1980). The Tragicomical History of Thermodynamics 1822\u20131854, Springer. The Sources part of this book has been helpful, as well has his discussion of Mayer\u2019s reasoning process. The reader should be alerted, however, that Truesdell does not employ any modern histories of the subject (such as Cardwell, Mendoza, or Fox), and he writes in what I found to be an inscrutable and idiosyncratic style and notation.","DOI":"10.1007\/978-1-4613-9444-0"},{"key":"ref_18","unstructured":"Saslow, W.M. Entropy Uniqueness Determines Temperature, submitted for publication."},{"key":"ref_19","unstructured":"Planck, M. (1903). Treatise on Thermodynamics, Longmans. [1st ed.]. See Sect. 90."},{"key":"ref_20","unstructured":"ThomsonW.On an Absolute Thermometric scale founded on Carnot\u2019s Theory of the Motive Power of Heat, and Calculated from Regnault\u2019s observationsPhilos. Mag. Ser.18483313317\n\t\tSee Kelvin\u2019s \n\t\t  Mathematical and Physical PapersCambridge University PressCambridge, UK1882Volume I100106"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Young, T. (1807). A Course of Lectures on Natural Philosophy and the Mechanical Arts, J. Johnson. \u201cThe term energy may be applied, with great propriety, to the product of the mass or weight of a body, into the square of the number expressing its velocity.\u201d The multi-talented Young, in addition to being a first-rate opthalmologist, and a talented mathematician and physicist (interference), was a scholar of many languages and made the first real progress in deciphering the Rosetta Stone.","DOI":"10.5962\/bhl.title.22458"},{"key":"ref_22","unstructured":"What we now know as nitrogen was called dephlogistated air, and carbon dioxide was called fixed air. That is, in the first case the \u2018oxygen principle\u2019 had been removed and in the second case it had been combined. See Ref. [10]."},{"key":"ref_23","unstructured":"Clagett, M. (1957). Energy conservation as an example of simultaneous discovery. Critical Problems in the History of Science, University of Wisconsin Press. (Proceedings of the Institute for the History of Science at the University of Wisconsin, 1\u201311 September 1957). Reprinted in Th. S. Kuhn, The essential tension, 1959, pp. 66\u2013104."},{"key":"ref_24","unstructured":"Smith, G. Personal communication."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1086\/350705","article-title":"Leibniz and the Vis Viva Controversy","volume":"62","author":"Iltis","year":"1971","journal-title":"Isis"},{"key":"ref_26","unstructured":"Jammer, M. (1957). Concepts of Force, Dover Publications. Footnotes 12 and 14."},{"key":"ref_27","first-page":"247","article-title":"On a General Method in Dynamics","volume":"124","author":"Hamilton","year":"1834","journal-title":"Philos. Trans. R. Soc. Part II"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1098\/rstl.1835.0009","article-title":"Second Essay on a General Method in Dynamics","volume":"125","author":"Hamilton","year":"1835","journal-title":"Philos. Trans. R. Soc. Part I"},{"key":"ref_29","unstructured":"Dugas, R. (1955). A History of Mechanics, Dover Publications. Originally published in 1950 in French. It is difficult to get definitions correct; in this work, Dugas incorrectly attributed to Helmholtz in 1847 the later-introduced terms kinetic energy and potential energy. Needless to day, Dugas did not have access to the modern tools of online access, interlibrary loan, copy machines, or cell phone photography."},{"key":"ref_30","unstructured":"Von HelmholtzH.Ueber die Erhaltung der Kraft. Eine physikalische AbhandlungG. ReimerBerlin, Germany1847\n          HermannHelmholtz, \u2019On the conservation of force; A Physical Memoir\u2019, in \n\t\t  Scientific Memoirs Selected from the Transactions of Foreign Academies of Science, and from Foreign Journals. Natural PhiliosophyJohn Tyndall and William FrancisTaylor & FrancisLondon, UK1853114162\n\t\t  Translation by John Tyndall. Among the references Helmholtz cites are Joule, J.P. Philos. Mag. 1845, 27, 205 for heating by friction of water in tubes, as well as Davy and Rumford. This is in Brush, S. The Nature of Gases and of Heat 1965; Volume 1, pp. 89\u2013110."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1080\/14786445308647205","article-title":"On the General Law of the Transformation of Energy","volume":"30","author":"Rankine","year":"1853","journal-title":"Phil. Mag. Ser. 4"},{"key":"ref_32","unstructured":"TaitP.G.On the Dissipation of EnergyPhil. Mag.1879VII34434610.1080\/14786447908639617ThomsonW.Note by Sir W. Thomson on the preceding Letter*Philos. Mag.1879VII346348\n\t\t  Both are included in \n        Mathematical and Physical PapersThomsonW.Cambridge, UK1911Volume V (MPP5)16."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Knowles Middleton, W.E. (1966). A History of the Thermometer and Its Use in Meteorology, The Johns Hopkins Press.","DOI":"10.56021\/9780801804625"},{"key":"ref_34","first-page":"442","article-title":"History of the Measurement of Heat I. Thermometry and Calorimetry","volume":"57","author":"Boyer","year":"1943","journal-title":"Sci. Mon."},{"key":"ref_35","first-page":"355","article-title":"M\u00e9moires de la Acad\u00e9mie des Sciences","volume":"1780","author":"Lavoisier","year":"1783","journal-title":"M\u00e9moire sur la chaleur"},{"key":"ref_36","unstructured":"Calorie as defined at Wikipedia, 14 August 2019 from google search \u201ccalorie wikipedia\u201d."},{"key":"ref_37","unstructured":"Rankine, W.J.M. (1859). A Manual of the Steam Engine and other Prime Movers, Griffin and Co."},{"key":"ref_38","unstructured":"ClausiusR.Uber einige fur anwendung bequeme formen der hauptgleichungen der nischen warmetheorieAnnalen der Physik und der Chemie1865125353400\n\t\tTranslated in\n\t\tOn the Determination of the Energy and Entropy of a BodyPhil. Mag. S4186632117"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1086\/678102","article-title":"Measuring Fire: Herman Boerhaave and the Introduction of Thermometry into Chemistry","volume":"29","author":"Powers","year":"2014","journal-title":"Osiris"},{"key":"#cr-split#-ref_40.1","unstructured":"Dumas, J.-B. (1862). M\u00e9moire sur la combustion en g\u00e9n\u00e9ral. Oeuvres de Lavoisier, Imprimerie Imp\u00e9riale. First printed at 1777"},{"key":"#cr-split#-ref_40.2","unstructured":"Volume II, pp. 225-233. Originally published in: M\u00e9m. Acad. R. Sci. Paris ann\u00e9e 1777, 592-600 (1780)."},{"key":"ref_41","unstructured":"Lavoisier, A. (1864). Trait\u00e9e el\u00e9mentaire de chimie. (1789). Oeuvres de Lavoisier, Tome Premier, Imperial Press."},{"key":"ref_42","first-page":"505","article-title":"R\u00e9flexions sur le phlogistique, pour servir de suite \u00e1 la th\u00e9orie de la combustion et de la calcination, publi\u00e9e en 1777","volume":"1783","author":"Lavoisier","year":"1786","journal-title":"M\u00e9m. Acad. R. Sci. Paris Ann\u00e9e"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/s10698-015-9220-5","article-title":"Lavoisier\u2019s \u201cReflections on phlogiston\u201d I: Against phlogiston theory","volume":"17","author":"Best","year":"2015","journal-title":"Found. Chem."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s10698-015-9236-x","article-title":"Lavoisier\u2019s \u201cReflections on phlogiston\u201d II: On the nature of heat","volume":"18","author":"Best","year":"2016","journal-title":"Found. Chem."},{"key":"ref_45","first-page":"193","article-title":"Chemistry as a Branch of Physics: Laplace\u2019s Collaboration with Lavoisier","volume":"7","author":"Guerlac","year":"1976","journal-title":"Hist. Stud. Phys. Sci."},{"key":"ref_46","first-page":"238","article-title":"Sur la vitesse du son dans l\u2019air et dans l\u2019eau","volume":"3","author":"Laplace","year":"1816","journal-title":"Ann. Chim. Phys."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Benjamin Count of Rumford (1798). An inquiry concerning the source of the heat which is excited by friction. Philos. Trans. R. Soc. Lond., 88, 80\u2013102.","DOI":"10.1098\/rstl.1798.0006"},{"key":"ref_48","unstructured":"Davy, H. (1799). \u201cEssay on Heat, Light, and the Combinations of Light\u201d, with a discussion of heating and melting of ice by friction. Contributions to Physical and Medical Knowledge, BIGGS&COTTLE."},{"key":"ref_49","unstructured":"Rhodes, R. (2018). Energy: A Human History, Simon and Shuster."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Channell, D.F. (2019). The Rise of Engineering Science: How Technology Became Scientific, Springer.","DOI":"10.1007\/978-3-319-95606-0"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1086\/349358","article-title":"Latent Heat and the Invention of the Watt Engine","volume":"43","author":"Fleming","year":"1952","journal-title":"Isis"},{"key":"ref_52","unstructured":"Thompson, S.P. (1910). Life of Lord Kelvin, Macmillan."},{"key":"ref_53","unstructured":"Silvanus P. Thompson was a well-known applied physicist and technical author of the period around 1900. Nobel laureate in Physics Richard Feynman indicated that he learned calculus by reading Thompson\u2019s short book Calculus Made Easy, which is available online as a pdf and was reprinted in 1970 by St. Martin\u2019s Press, New York."},{"key":"ref_54","unstructured":"Had Thomson considered using that work to drive a paddle wheel and convert the work to heat he would have concluded that the net effect would be, with all other things unchanged, the production of heat. Thus heat would not be conserved."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1017\/S0080456800022493","article-title":"Theoretical Considerations on the Effect of Pressure in Lowering the Freezing Point of Water","volume":"16","author":"Thomson","year":"1849","journal-title":"Trans. R. Soc. Edinb."},{"key":"ref_56","first-page":"165","article-title":"The Effect of Pressure in Lowering the Freezing Point of Water Experimentally Demonstrated","volume":"37","author":"Thomson","year":"1850","journal-title":"Philos. Mag."},{"key":"ref_57","unstructured":"Wilson, D.B. (1990). The correspondence between Sir George Gabriel Stokes and Sir William Thomson, Baron Kelvin of Largs, Vol.1, Cambridge University Press."},{"key":"ref_58","unstructured":"MayerJ.R.Bemerkungen uber die Krafte der unbelebten NaturAnn. Chem. Pharm.18424223324010.1002\/jlac.18420420212\n\t\tTranslations into English include \n\t\tFosterG.C.Remarks on the Forces of Inorganic NaturePhilos. Mag.18624371377 \n\t\t  MagieW.F.A Source Book in PhysicsHarvard University PressNew York, NY, USA1935197203  \n\t\tBrushS.G.Kinetic TheoryPergamon PressOxford, UK1965Volume 17177"},{"key":"ref_59","unstructured":"Ref. [7], on pp. 133\u2013134, cites Gay-Lussac, J.-L. Mem. Soc. d\u2019Arcueil 18.7, 1, 221. Fox gives an extensive discussion on pp. 130\u2013134. See also  Gay-LussacJ.-L.\u201cRecherches sur la dilatation des gaz et des vapeurs\u201d (Researches on the expansion of gases and vapors)Annales de Chimie180243137175"},{"key":"ref_60","doi-asserted-by":"crossref","unstructured":"Lindsay, R.B. (1973). Men of Physics: Julius Robert Mayer, Pergamon. give an English translation of Mayer, J.R. Ann. Chem. Pharm. 1842, 42, 233\u2013240.","DOI":"10.1002\/jlac.18420420212"},{"key":"ref_61","unstructured":"Joule, J.P. (1884). The Scientific Papers of J. P. Joule, Physical Society."},{"key":"ref_62","first-page":"280","article-title":"On the production of heat by voltaic electricity","volume":"4","author":"Joule","year":"1840","journal-title":"Proc. R. Soc. Lond."},{"key":"ref_63","first-page":"369","article-title":"On the changes of temperature produced by the rarefaction and condensation of air","volume":"26","author":"Joule","year":"1845","journal-title":"Philos. Mag."},{"key":"ref_64","unstructured":"Gray, A. (1908). Lord Kelvin: An Account of his Scientific Life and Work, J. M. Dent. Gray Was Kelvin\u2019s successor at the University of Glasgow."},{"key":"ref_65","first-page":"56","article-title":"On the History of the Principle of the Conservation of Energy","volume":"47","author":"Colding","year":"1863","journal-title":"Lond. Edinb. Dublin J. Sci. S.4"},{"key":"ref_66","unstructured":"Holtzmann, K.A. (1845). Ueber die Warme und Elasticitat der Gase und Dampfe, Loeffler. \u2018On the Heat and Elasticity of Gases and Vapours.\u2019 See Ref. [7], Chapter 7, citation 46. I found this 40 page book online in German."},{"key":"ref_67","unstructured":"Joule, J.P.; Thomson, W. Phil. Trans. Roy. Soc. 1853, 143, 357\u2013365; this is part 1; for part 2, see ibid. 143 (1854); for part 3, see Trans. Roy. Soc, June, 1860 (MPP1, 400-414); for part 4 see Trans. Roy. Soc, June, 1862 (MPP1, 415\u2013431). There is an Appendix giving abstracts for all four Parts, in Proc. Roy. Soc. VI, June, 1853 (MPP1, 432\u2013455). Joule, J.P. and Thomson, W. On the Thermal Effects of Fluids in Motion, Part 4, in W. Thomson, Mathematical and Physical Papers, vol. 1 (MPP1) (Cambridge, Cambridge University Press, 1882), 415\u2013431; we reproduce here an equation from p. 430. This article was originally published in 1862 in Phil.Trans. Roy. Soc., vol. 152. J. P.Joule and W. Thomson, On the Thermal Effects of Fluids in Motion, in W. Thomson, Mathematical and Physical Papers, vol. 1 (Cambridge: Cambridge University Press, 1882), 346\u2013356. This article was originally published in 1853 in the Phil. Trans. Roy. Soc. Note that MPP1 contains a preliminary paper (1862), all four papers I (1863), II (1864), III (1860) IV (1862), and an Appendix in pp. 333\u2013455."},{"key":"ref_68","unstructured":"There was not yet a notation among mathematicians for dQ\/dt at constant p. My mathematics colleague Phil Yasskin, trained as a physicist, has informed me that mathematicians still do not employ such a notation!"},{"key":"ref_69","unstructured":"Thomson, in Ref. [3], indicates that Rankine communicated his result on February 4 of 1850. This corresponds to J. M. C. Rankine, Trans. Roy. Soc. Edinburgh, vol. XX, which is reprinted in Ref. [88], 234\u2013284."},{"key":"ref_70","unstructured":"Hirst, T.A. (1867). The Mechanical Theory of Heat: With Its Applications to the Steam-engine and to the Physical Properties of Bodies, J. Van Voorst. See the footnote on p. 65, in 1864 added to the English translation. Clausius notes that Rankine \u201cwas not yet in possession of the necessary second fundamental theorem of the mechanical theory of heat\u201d."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"1839","DOI":"10.1103\/PhysRevE.48.1839","article-title":"Specific Heats of saturated water vapor and liquid","volume":"48","author":"Eisner","year":"1993","journal-title":"Phys. Rev. E"},{"key":"ref_72","unstructured":"ThomsonW.Thermo-electric currentsTrans. R. Soc. Edinb.1854126\n\t\tThis is simply a page number within an article that goes from \n        ThomsonW.Thermo-electric currentsTrans. R. Soc. Edinb.18542112317210.1017\/S0080456800032014\n\t\treprinted in Math. and Phys. Papers, I (XLVIII), pp. 232\u2013291, 1882. I attach Ref71ThomsonThermoelectric from the reprint."},{"key":"ref_73","doi-asserted-by":"crossref","unstructured":"Thomson, W. (1911). Mathematical and Physical Papers, Cambridge University Press. Original: On the Discovery of the True Form of Carnot\u2019s Function. Philos. Mag. 1856, XI, 447\u2013449.","DOI":"10.1080\/14786445608642099"},{"key":"#cr-split#-ref_74.1","doi-asserted-by":"crossref","unstructured":"Clausius, R. Poggendoff's Annalen 1854, 93, 481-506","DOI":"10.1002\/andp.18541691202"},{"key":"#cr-split#-ref_74.2","unstructured":"translated in the Journal de Mathematiques 1855, XX Paris, and in Philos. Mag. 1856, XII, 81. I found this available online at wikisource."},{"key":"ref_75","unstructured":"I was unable to understand the argument given in the (confusing) translation of Clausius that I employed. (Perhaps an indication of the confusing nature of the derivation and\/or translation is that Ref. [7] avoids discussion of the derivation.) However, Clausius\u2019s definitions form the basis of the argument presented here."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1080\/00033790410001712246","article-title":"The Absolute and Its Measurement: William Thomson on Temperature","volume":"62","author":"Chang","year":"2005","journal-title":"Ann. Sci."},{"key":"ref_77","unstructured":"Thomson, W. (1911). Mathematical and Physical Papers, Cambridge University Press. Original: Philos. Mag. 1856, XI, 449\u2013450."},{"key":"ref_78","first-page":"458","article-title":"Rudolf Julius Emanuel Clausius","volume":"16","author":"Gibbs","year":"1889","journal-title":"Proc. Am. Acad."},{"key":"ref_79","unstructured":"Clausius, R. Communicated to the Naturforschende Gesellschaft of Zurich, 27 January 1862; published in the Viertaljahrschrift of this Society, VII, 48; in Poggendorff\u2019s Annalen, 1862, CXVI, 73; in the Philosophical Magazine, 1862, xxiv, 81; and in the Journal des Mathematiques of Paris, S.2. VII, 209. On the Application of the Theorem of the Equivalence of Transformations to Interior Work."},{"key":"ref_80","first-page":"127","article-title":"Gibbs on Clausius","volume":"1","author":"Klein","year":"1969","journal-title":"Hist. Stud. Phys. Sci."},{"key":"ref_81","doi-asserted-by":"crossref","unstructured":"Darrigol, O. (2018). The Gibbs paradox: Early history and solutions. Entropy, 20, To compare with Clausius [79] note that Clausius uses dQ > 0 when heat goes to a reservoir. Using the usual sign convention for heat (change the sign of Clausius\u2019s dQ), and taking dQ = TdS, causes Clausius\u2019s (II.) to become dS = dH\/T + dZ, where he lets \u201cthe quantity of heat contained in [a body] be expressed by H\u201d, and Z is the \u201cdisgregation\u201d. Darrigol\u2019s (4) is dS = dY + dK\u00af\/T, where Y is Darrigol\u2019s notation for disgregation and he employs the \u201cfree heat\u201d K\u00af, \u201cwhich is the average kinetic energy of the molecular system\u201d.","DOI":"10.3390\/e20060443"},{"key":"ref_82","unstructured":"Maxwell, J.C. (1890). The Scientific Letters and Papers of James Clerk Maxwell, Cambridge University Press. It was republished by Cambridge in 2009, and edited by P. M. Harman. p. 16 of Harman\u2019s Introduction notes that Numbers 356, 402, and 403 contain negative comments about Clausius\u2019s terminiology, but that by the time of Number 483 he has corrected his misconception about entropy."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S0007087400018264","article-title":"Rankine and the Rise of Thermodynamics","volume":"14","author":"Hutchison","year":"1981","journal-title":"Brit. J. Hist. Sci."},{"key":"ref_84","first-page":"72","article-title":"Rankine, Atomic Vortices, and the Entropy Function","volume":"31","author":"Hutchison","year":"1981","journal-title":"Archives internationales d\u2019histoire des Sciences"},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1177\/007327531305100403","article-title":"Ranking Rankine: W. J. M. Rankine (1820-72) and the Making of \u2018Engineering Science\u2019 Revisited","volume":"51","author":"Marsden","year":"2013","journal-title":"Hist. Sci."},{"key":"ref_86","unstructured":"Hutchinson, K. Personal communication."},{"key":"ref_87","doi-asserted-by":"crossref","unstructured":"Maxwell, J.C. Tait\u2019s \u2018Thermodynamics\u2019 Nature. 1878, 17, 257-259, 278-280. In Volume 2 of Ref. [82] pp. 660\u2013671. This is a review of Tait\u2019s Thermodynamics. It should not be assumed that Maxwell had only criticism for Rankine; many instances in Maxwell\u2019s works on Statics have positive comments about Rankine\u2019s work on force diagrams. See p. 163 of Ref. [82].","DOI":"10.1038\/017257a0"},{"key":"ref_88","unstructured":"Tait, P.G. (1881). Miscellaneous Scientific Papers, C. Griffin and Co.. Available online: https:\/\/archive.org\/details\/miscellaneoussci00rank."},{"key":"ref_89","unstructured":"Rankine, J.M.C. on p. 302 of Ref. [37] uses the phrase \u201ca curve of no transmission, or adiabatic curve.\u201d A few lines later he discusses \u201ca curve whose coordinates represent the pressures corresponding to various volumes when the substance is absolutely destitute of heat, then this curve, which may be called the line of absolute cold, is at once an isothermal curve and an adiabatic curve.\u201d"},{"key":"ref_90","doi-asserted-by":"crossref","unstructured":"Sears, F.W. (1953). An Introduction to Thermodynamics, the Kinetic Theory of Gases, and Statistical Mechanics, Addison-Wesley. [2nd ed.].","DOI":"10.1119\/1.1933565"},{"key":"ref_91","unstructured":"Smith, C. (1998). The Science of Energy, University of Chicago Press. This work gives a history of energy, largely for Joule\u2019s Manchester and Thomson\u2019s Glasgow, with a sociological view of the physics and engineering, including religious motivations. Despite citing Gibbs\u2019s very favorable view of Clausius\u2019s contributions to thermodynamics [78], in my opinion this work undervalues Clausius\u2019s contributions, specifically, it does not cite Clausius as one of the very few true founders of thermodynamics, yet it places Rankine in that category. On p. 263 Smith notes that Maxwell declined to celebrate Rankine when \u201cTait and others struggled to edit a memorial volume of Rankine\u2019s Miscellaneous scientific papers [88]. Nevertheless, that volume does include an extract from a mixed review by Maxwell of Rankine\u2019s work [87]."},{"key":"ref_92","unstructured":"Gibbs, J.W. Graphical Methods in the Thermodynamics of Fluids, Transactions of (he Connecticut Academy, II, pp. 309\u2013342, April\u2013May, 1873. This is reprinted in J. W. Gibbs, Collected Works. Vol. I, Thermodynamics (Longmans, New York, 1928). The quote is from the very last line of the long footnote on p. 52. The introduction of this volume notes that one of Gibbs\u2019s articles is on electromagnetic radiation in matter, and shows that a work of Kelvin\u2019s on the same subject is in error. In general, Gibbs seems to be aware of who did what and when, and who did it correctly and who didn\u2019t, but he does not present any history, merely final results, so that much of his work reads like pure mathematics. As far as I can tell, he does not bother to indicate how temperature and entropy arose. His thermodynamics text emphasizes the properties of matter rather than the efficiency of steam engines."},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1086\/350264","article-title":"Atomism and Thermodynamics","volume":"58","author":"Daub","year":"1967","journal-title":"Isis"},{"key":"ref_94","first-page":"321","article-title":"Entropy and Dissipation","volume":"2","author":"Daub","year":"1970","journal-title":"Hist. Stud. Phys. Sci."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1103\/PhysRev.37.405","article-title":"Reciprocal Relations in Irreversible Processes. I","volume":"37","author":"Onsager","year":"1931","journal-title":"Phys. Rev."},{"key":"ref_96","unstructured":"Callen, H.B. (1961). Thermodynamics, Wiley. [1st ed.]."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"4959","DOI":"10.1103\/PhysRevB.35.4959","article-title":"Thermodynamic analysis of interfacial transport and of the thermomagnetoelectric system","volume":"35","author":"Johnson","year":"1987","journal-title":"Phys. Rev. B"},{"key":"ref_98","doi-asserted-by":"crossref","unstructured":"Kjelstrup, S., and Bedeaux, D. (2008). Non-Equilibrium Thermodynamics of Heterogeneous Systems, World Scientific.","DOI":"10.1142\/9789812779144"},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"1041","DOI":"10.1139\/p11-093","article-title":"Irreversible thermodynamics of transport across interfaces","volume":"89","author":"Sears","year":"2011","journal-title":"Can. J. Phys."},{"key":"ref_100","doi-asserted-by":"crossref","unstructured":"Flood, R., McCartney, M., and Whitaker, A. (2008). Kelvin: Life, Labours, and Legacy, Oxford University Press. Kelvin and Statistical Mechanics, discusses Kelvin\u2019s work on the thermoelectric effect in terms of Onsager\u2019s more recent work. It considers only the off-diagonal terms in the thermodynamic flux vs. thermodynamic force relations.","DOI":"10.1093\/acprof:oso\/9780199231256.001.0001"},{"key":"ref_101","unstructured":"Philosophical MagazineAvailable online: https:\/\/onlinebooks.library.upenn.edu\/webbin\/serial?id=philosmag(accessed on 4 January 2020) \n\t\tSee also \n        https:\/\/archive.org\/details\/texts(accessed on 4 January 2020)"},{"key":"ref_102","unstructured":"Tuckerman, A. (1890). Index on the Literature of Thermodynamics, Smithsonian Institution."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/1\/77\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:18:58Z","timestamp":1760361538000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/1\/77"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,7]]},"references-count":104,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["e22010077"],"URL":"https:\/\/doi.org\/10.3390\/e22010077","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1,7]]}}}