{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T23:45:39Z","timestamp":1791330339120,"version":"4.3.1"},"reference-count":36,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2015,6,25]],"date-time":"2015-06-25T00:00:00Z","timestamp":1435190400000},"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>The present work analyzes the cognitive process that led Clausius towards the translation of the Second Law of Thermodynamics into mathematical expressions. We show that Clausius\u2019 original formal expression of the Second Law was achieved by making extensive use of the concept of disgregation, a quantity which has subsequently disappeared from the thermodynamic language. Our analysis demonstrates that disgregation stands as a crucial logical step of such process and sheds light on the comprehension of such fundamental relation. The introduction of entropy\u2014which occurred three years after the first formalization of the Second Law\u2014was aimed at making the Second Law exploitable in practical contexts. The reasons for the disappearance of disgregation, as well as of other \u201cpre-modern\u201d quantities, from the thermodynamics language are discussed.<\/jats:p>","DOI":"10.3390\/e17074500","type":"journal-article","created":{"date-parts":[[2015,6,25]],"date-time":"2015-06-25T10:17:15Z","timestamp":1435227435000},"page":"4500-4518","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Clausius\u2019 Disgregation: A Conceptual Relic that Sheds Light on the Second Law"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9835-8128","authenticated-orcid":false,"given":"Emilio","family":"Pellegrino","sequence":"first","affiliation":[{"name":"Department of Chemistry, University of Torino, Via Giuria 7, I-10125, Torino, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1660-2482","authenticated-orcid":false,"given":"Elena","family":"Ghibaudi","sequence":"additional","affiliation":[{"name":"Department of Chemistry, University of Torino, Via Giuria 7, I-10125, Torino, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Luigi","family":"Cerruti","sequence":"additional","affiliation":[{"name":"Independent researcher, Viale delle acacie 12, I-10148, Vinovo, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2015,6,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"\u010c\u00e1pek, V., and Sheehan, D.P. (2005). Challenges to the Second Law of Thermodynamics, Springer.","DOI":"10.1007\/1-4020-3016-9"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3390\/e6010001","article-title":"The Second Law Mystique","volume":"6","author":"Nikulov","year":"2004","journal-title":"Entropy"},{"key":"ref_3","unstructured":"Larmor, J., and Bryan, G.H. (1891). Report of the Sixty-first Meeting held at Cardiff in August 1891, John Murray."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Bridgman, P.W. (1941). The Nature of Thermodynamics, Harvard University Press.","DOI":"10.4159\/harvard.9780674731356"},{"key":"ref_5","first-page":"305","article-title":"Bluff Your Way in the Second Law of Thermodynamics","volume":"32","author":"Uffink","year":"2001","journal-title":"Stud. Hist. Philos. Sci. B"},{"key":"ref_6","unstructured":"Clausius, R., and Hirst, T.A. (1867). The Mechanical Theory of Heat: With Its Application to the Steam Engine and to the Physical Properties of Bodies, J. Van Voorst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1002\/andp.18722210107","article-title":"Zur Geschichte der Mechanischen W\u00e4rmetheorie","volume":"145","author":"Clausius","year":"1872","journal-title":"Annalen der Physik und Chemie"},{"key":"ref_8","unstructured":"M\u00fcller, I. (2007). A History of Thermodynamics: The Doctrine of Energy and Entropy, Springer. Concerning Clausius\u2019 disputes with the scientific community, M\u00fcller reports that \u201cin his work on thermodynamics he had to fight off numerous objections and claims of priority by other people, who had thought, or said, or written something similar at about the same time\u201d."},{"key":"ref_9","unstructured":"M\u00fcller reports that the Newcomen\u2019s steam engine was actually developed by the engineer Thomas Savery (1650\u20131715).and the blacksmith Thomas Newcomen (1663\u20131729) Newcomen\u2019s machine worked cyclically by injection of cold water into the cylinder where the steam was condensed and a good vacuum was developed. The vacuum pulled down the piston in a powerful stroke and subsequently new steam pushed the piston back up. At this stage a new cycle began by injecting cold water into the cylinder. See [8] (p. 48)."},{"key":"ref_10","unstructured":"As regards the efficiency of the Newcomen\u2019s steam engine, M\u00fcller reports: \u201cActually, the efficiencies were all quite low: in Newcomen\u2019s case about 2% and 5\u20137% in Watt\u2019s case. A modern power station reaches between 45% and 50%. The engineers have done a good job indeed over the past 200 years\u201d [8] (Footnote in p. 50)."},{"key":"ref_11","unstructured":"Watt\u2019s major invention was the rotative engine\u2014 where the up-and-down movement of the piston is converted into the rotation of a wheel. It opened to the use of steam machines as driving force for ships and locomotives. On this matter M\u00fcller reports: \u201cThus Watt\u2019s machine became the motor of the industrial revolution\u201d [8] (p. 49)."},{"key":"ref_12","unstructured":"Carnot, S. (1897). Reflections on the Motive Power of Heat, Wiley. [2nd ed.]."},{"key":"ref_13","unstructured":"Amoroso, R.L., Kauffman, L.H., and Rowlands, P. (2013). The Physics of Reality: Space, Time, Matter, Cosmos\u2014Proceedings of the 8th Symposium Honoring Mathematical Physicist Jean-Pierre Vigier, World Scientific."},{"key":"ref_14","unstructured":"M\u00fcller underlines the relevance of Carnot\u2019s work for thermodynamics and writes: \u201cHad he lived longer, it seems likely that he might have anticipated Clausius\u2019 work by nearly 30 years\u201d [8] (p. 55)."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1002\/andp.18571760302","article-title":"\u00dcber die Art der Bewegung, Welche Wir W\u00e4rme Nennen","volume":"176","author":"Clausius","year":"1857","journal-title":"Annalen der Physik und Chemie"},{"key":"ref_16","unstructured":"Brush, S.G. (1976). The Kind of Motion We Call Heat: A History of the Kinetic Theory of Gases in the 19th Century, North-Holland."},{"key":"ref_17","unstructured":"Brush, S.G. (1965). Kinetic Theory (The Commonwealth and International Library: Selected Readings in Physics), Pergamon Press."},{"key":"ref_18","unstructured":"The convention is to take the sign of heat (Q) positive when heat flows from the system towards the exterior. The German expression used for H is im K\u00f6rper vorhandenen W\u00e4rme,translated as the heat actually contained in the body,in the English version of the Abhandlungen edited in 1867. Appendix A of the Sixth Memoir, consecrated to terminology, reports: \u201cthe heat actually present in the body may be simply called the heat of the body or the thermal content of the body (W\u00e4rmeinhalt des K\u00f6rpers)\u201d [6] (p. 252) [Italics in the text]."},{"key":"ref_19","unstructured":"Equation (1) is reported in a footnote of Clausius\u2019 Sixth Memoir [6] (p. 225 footnote). Concerning the equivalence of heat and work, in Appendix A of the Sixth Memoir, Clausius writes: \u201cNow in the mechanical theory of heat, after admitting that heat can be transformed into work and work into heat, in other words, that either of these may replace the other, it becomes frequently necessary to form a magnitude of which heat and work are constituent parts. But heat and work being measured by different units, we cannot in such a case say, simply, the magnitude is the sum of heat and work; we are compelled to say either the sum of the heat and the heat-equivalent of the work,or the sum of the work and the work-equivalent of the heat\u201d [6] (pp. 252\u2013253) Within this frame of ideas, Clausius proposed the introduction of a new quantity, the ergon, that is the work measured according to the heat units. He writes: \u201cBut beside the work measured according to mechanical unit, let another magnitude be introduced denoting the work measured according to the thermal unit,that is to say, the numerical value of the work when the unit of work is that which is equivalent to the thermal unit.For the work thus expressed a particular name is requisite. I propose to adopt for it the Greek word (\u03ad\u03c1\u03b3\u03bf\u03bd) ergon [6] (pp. 252\u2013253)\u201d [Italics in the text]."},{"key":"ref_20","unstructured":"The denomination \u201cenergy\u201d for the quantity U is due to W. Thomson, as Clausius himself reports: \u201cLater, Thomson gave to this function\u2014designated by me with U\u2014the very appropriate name of energy of the body\u201d [7] (p. 136) (translated by the authors). It is worth reporting the justification given by Clausius for having defined the function U [6] (p. 225, footnote): \u201cSince, however, the increase in the quantity of heat actually contained in a body [ dH], and the heat consumed by an interior work during a change of condition [ AdI], are magnitudes of which we commonly do not know the individual values, but only the sum of those values, and which resemble each other in being fully determined as soon as we know the initial and final conditions of the body, without our requiring to know how it has passed from the one to the other, I have thought it advisable to introduce a function which shall represent the sum of these quantities and which I have denoted by U. Accordingly: dU=dH+AdI\u201d"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Knott, C.G. (1911). Life and Scientific Work of Peter Guthrie Tait Supplementing the Two Volumes of Scientific Papers Published in 1898 and 1900, Cambridge University Press.","DOI":"10.5962\/bhl.title.17932"},{"key":"ref_22","unstructured":"The sentence is quoted as it corresponds to the title of Clausius Sixth Memoir [6] (pp. 215\u2013266)."},{"key":"ref_23","unstructured":"Clausius clarified his idea of compensation in a footnote of his Fourth Memoir that we quote here below: \u201cNow it is to these compensations that our principle [i.e.,Clausius\u2019 Second Law] refers; and with the aid of this conception the principle may also be expressed thus: an uncompensated transmission of heat from a colder to a warmer body can never occur.The term \u2018uncompensated\u2019 here expresses the same idea as that which was intended to be conveyed by the words \u2018by itself\u2019 in the previous enunciation of the principle [see Clausius\u2019 statement of the Second Law at p. 4]\u201d [6] (p. 118, footnote)."},{"key":"ref_24","unstructured":"Sands and Dunning-Davies remark: \u201cMathematically, Clausius treated compensation through the concept of Aequivalenzwerth or equivalence value, which was a quantity with the value Q\/T, Q being a quantity of heat transmitted at a temperature T\u201d [13] (p. 5)."},{"key":"ref_25","unstructured":"Clausius, R. (1854). Annalen der Physik und Chemie, 499."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Clausius, R. (1862). Annalen der Physik und Chemie, 73.","DOI":"10.1002\/andp.18621920506"},{"key":"ref_27","unstructured":"The extension of the Second Law to non-cyclic processes is treated in Clausius\u2019 Sixth Memoir [6] (pp. 215\u2013266)."},{"key":"ref_28","unstructured":"The definition of entropy is reported in the Ninth Memoir, On Several Convenient Forms of the Fundamental Equations of the Mechanical Theory of Heat.According to the footnote reported in its incipit,this memoir was the object of a lecture \u201cread at the Philosophical Society of Zurich on the 24th of April. 1865 and published in the Vierteljahrsschrift of this Society, Bd. X. S. 1.; Pogg. Ann. July, 1865, Bd. Cxxv. S. 353; Jour. De Liouville, 2\u00b0 s\u00e9r. T. x. P. 361\u201d [6] (p. 327)."},{"key":"ref_29","unstructured":"The fundamental equations correspond to the analytical expressions of the two fundamental theorems whereon the whole mechanical theory of heat rests: \u201cthat of the equivalence of heat and work and that of the equivalence of transformations\u201d [6] (p. 327)."},{"key":"ref_30","unstructured":"The ultimate implication of the definition of Entropy was the object of the Appendix to the Ninth Memoir published in 1866\u2014whose aim was to \u201cshow how the equations which serve for the determination of energy and entropy may be derived directly from the fundamental equations of the mechanical theory of heat\u201d. In this context Clausius could \u201cbring the two equations [i.e.,dQ = dU + dw and dQ = TdS] into conjunction\u201d\u2014under the supposition of reversible change of a body [6] (p. 366). It is remarkable that the result attained has the form of what today is known as the \u201cfirst Gibbs equation\u201d and that would have constituted the foundational element of Gibbs thermodynamics [34] (pp. 33\u201354). It is worth to quote Clausius on this particular matter: \u201cIn this case [i.e.,reversible change] the thermal element dQ is the same in both equations, hence we can eliminate it from the equations, whereby we obtain TdS = dU + dw\u201d [6] (p. 366)."},{"key":"ref_31","unstructured":"Clausius assumed the opposite convention; in fact, as regard to the heat element, he specified that \u201cdQ is the heat element that flows from the body under transformation towards a heat reservoir\u201d [7] (p. 144) (translated by the authors)."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/BF00248155","article-title":"Processes in Simple Thermodynamic Materials","volume":"33","author":"Meixner","year":"1969","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_33","first-page":"127","article-title":"Gibbs on Clausius","volume":"1","author":"Klein","year":"1969","journal-title":"Hist. Stud. Phys. Sci."},{"key":"ref_34","unstructured":"Bumstead, H.A., and van Name, R.G. (1906). The Scientific Papers of J. Willard Gibbs, Longmans."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"91","DOI":"10.2307\/27757770","article-title":"Sooner Silence than Confusion: The Tortuous Entry of Entropy into Chemistry","volume":"27","author":"Kragh","year":"1996","journal-title":"Hist. Stud. Phys. Biol. Sci."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1261","DOI":"10.1119\/1.18389","article-title":"Thermodynamic Entropy: The Spreading and Sharing of Energy","volume":"64","author":"Leff","year":"1996","journal-title":"Am. J. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/7\/4500\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:48:23Z","timestamp":1760215703000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/7\/4500"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,6,25]]},"references-count":36,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2015,7]]}},"alternative-id":["e17074500"],"URL":"https:\/\/doi.org\/10.3390\/e17074500","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,6,25]]}}}