Abstract
After the birth of thermodynamics’ second principle—outlined in Carnot's Réflexions sur la puissance motrice du feu (1824)—several studies provided new arguments in the field. Mainly, they concerned the thermodynamics’ first principle—including energy conceptualisation—, the analytical aspects of the heat propagation, the statistical aspects of the mechanical theory of heat. In other words, the second half of nineteenth century was marked by an intense interdisciplinary research activity between physics and chemistry: new disciplines applied to the heat developed in the form of analytical, mechanical and statistical theories. Inside all these theories, entropy—the brand-new function that Clausius coined in his Mechanical theory of heat—started to play a central epistemic role. In the present paper, we analyse some steps of the historical process of conceptualisation of such function from 1850 to 1902. Particularly, we retrace the historical–foundational path that—starting from Clausius’ Second Law—lead Boltzmann and Gibbs to their distinguished formulations of statistical entropy. As usual, our research has been unrolled through the analyses of primary sources and by leaning on critical readings of the secondary literature. As for the methodological approach, text analysis of historical documents constituted our privileged modus operandi. This paper is the expression of a collaborative historical research program focused on the thermodynamic foundations of physics–chemistry relationship; early results have already been published by the same authors upon the concepts of reversibility––and––thermal equilibrium.
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Notes
On December 14th, 1900 Max Planck delivered his famous lecture before the Physicalische Gesellshaft, entitled “Zur Theorie des gesetzes der Energieverteilung im Normalspectrum” (Planck 1901).
L. Boltzmann, Vorlesungen uber Gastheorie 1, p. 21, 1896. Wiener Sitzungsberichte 78, Juni, 1878, at the end. Compare also S. H. Burbury, Nature, 51, p. 78, 1894.
Chaos is the translation of the german “unordnung”, found in Boltzmann’s papers.
Cfr. Carnot S. Carnot (1824, 1897, 1978, 1986), Gillipsie and Pisano (2014), Pellegrino 2019, Pisano 2010. Recently on Lazare Carnot's Essay on Machines in General see: Pisano, Coppersmith and Peake 2021. The other two pre-print volumes are: Principes fondamentaux de l’équilibre et du mouvement (1803) and Géométrie de position (1803), both by Pisano.
For the history of the conceptualization of heat and thermal equilibrium the reader can refer to Black (1807), Chang (2004, 2012), Hess (1842), Pisano, Anakkar, Pellegrino, Nagels (2019), Laplace (1822), Lavoisier and Laplace (1784), Lavoisier (1789) [1937], 1862–1893; See also: Lavoisier and Laplace (1784), Leicester (1951), Planck (1901, 1903, 1914, 1917), Kirchhoff (1860), Wien (1894).
There is a truly vast literature on the notion of entropy (and related statistical mechanics studies) in its several declinations. We report just a short selection: Buchdahl (1966), Čápek and Sheehan (2005), Kragh and Weininger (1996), Marcella (1992), Müller (2007), Nelson (1994), Pellegrino et al. (2014), Pisano (2004), Prigogine (1996, 1997), Prigogine and Kondepudi (1999), Prigogine and Stengers (1984, 1992), Strehlow (2005), Scerri (2001, 2013), Frigg (2008), Anakkar (2014).
Cfr. Sands and Dunning-Davies (2013).
Cfr. Clausius (1854).
In current notation this integral should be written as:\({\oint }\frac{\delta Q}{T}\).
Cfr. Clausius (1872).
In current notation this integral should be written as: \({\oint }\frac{\delta Q}{T}\)
The adjective Pfaffian comes from Johann Friedrich Pfaff (1765–1825) who, in 1814–1815, developed a general technique for integrating partial differential equations of the first order (Pogliani and Berberan-Santos 2000).
Karl Hermann Amandus Schwarz (1843–1921) was Carathéodory’s professor in Berlin.
In current notation this integral should be written as: \({\oint }\frac{\delta Q}{T}\).
The \(H\) is the Greek capital letter eta—traditionally used for entropy.
Original title Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen.
It corresponds to Eq. (20) of this paper.
For the sake of completeness, the differential equation is:
\(\frac{{\partial f\left( {x,t} \right)}}{\partial t} = \mathop \smallint \limits_{0}^{\infty } \mathop \smallint \limits_{0}^{x + x^{\prime}} \left[ {\frac{{f\left( {\xi ,t} \right)}}{\sqrt \xi }\frac{{f\left( {x + x^{\prime} - \xi ,t} \right)}}{{\sqrt {x + x^{\prime} - \xi } }} - \frac{{f\left( {x,t} \right)}}{\sqrt x }\frac{{f\left( {x^{\prime},t} \right)}}{{\sqrt {x^{\prime}} }} } \right]\sqrt {xx^{\prime}} \psi \left( {x,x^{\prime},\xi } \right)dx^{\prime}d\xi .\)
Cfr. Boltzmann (1866).
For sake of brevity/calculus we choose to denote the quantity φ(t) which is not included in the primary sources text.
In current notation this integral should be written as: \({\oint }\frac{\delta Q}{T}\)
Concerning the influence of Boltzmann kinetic model on Planck’s works, Cerciniani comments: “Many historians of science have underlined the circumstance that these discrete models used by Boltzmann led Planck to the discovery of his energy quanta.” (Cercignani 1998, p. 89).
Cfr. Hill (1986).
Actually, Shannon refers to this same entity either as information or uncertainty.
In the history of science, we have significant examples of textbooks written by scholars, researchers during their teachings job (Mellone and Pisano, 2012; Pisano, 2007a, 2007b, 2009a, 2009b, 2011a, b, 2013a, b, 2015, 2016). One can consider the new born theories of heat and thermodynamics and related textbooks used at the end of the 19th century: Jean Baptiste JosephFourier’s (1768–1830) Théorie analytique de la chaleur (Fourier 1822) and the positivist Gabriel Lamé’s (1795–1870) Leçons sur la théorie analytique de la chaleur (Lamé1836,1861a,b) focusing the physical–mathematical relationship (Pisano 2013a,b); in the theory of heat and without considering experimental aspects of the scientific process of knowledge. For example, between chaleur and calorique and the second principle are avoided. Therefore, Reech’s théorie général des effets dynamiques de la chaleur (Reech 1853, 1854) in which he adopted and generalized Sadi Carnot’s (1796–1832)and Clapeyron’s (1799–1864)reasoning in order to obtain a general formula from which each of the two theories (on caloric and on heat) can be derived under the right conditions (Pisano, 2001, 2010). An Italian scholar Paolo Ballada (1815–1888) called Paul de Saint–Robert published (at the time into French language and translated in various languagesnot in Italian) a textbooks, Principes de thermodynamique. (Saint-Robert 1865, 1870a,b,c) whichfor the first timepresented a historical part concerning the first biographical notes on Sadi Carnot. In this textbook, the second principle is very much emphasized. Just to mention other: Zeuner, Verdet, Hirn, Combes, Clausius, Jacquier, Jamin (Gillispie and Pisano 2014, Table 10.2, pp. 326–327).
References
Anakkar, A.: Semantic difficulties in science and their implications for education. J. Balt. Sci. Educ. 13(4), 444–447 (2014)
Asimov, I.: Adding a Dimension: Seventeen Essays on the History of Science. Dobson, London (1966)
Balian, R.: Du microscopique au macroscopique, Tome 1. Ellipses, Paris (1982)
Berthollet, C.L.: Essai de statique chimique, vol. 2. Firmin Didot, Paris (1803)
Black, J.: Lectures on the Elements of Chemistry, vol. II. Carey, Philadelphia (1807)
Boltzmann, L.: Über die mechanische Bedeutung des zweiten Hauptsatzes der Wärmetheorie. Wiener Berichte 53, 195–220 (1866)
Boltzmann, L.: Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Sitzungsberichte Akademie der Wissenschaften 66, 275–370 (1872)
Boltzmann, L.: Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung resp. den Sätzen über das Wärmegleichgewicht. Wiener Berichte 76, 373–435 (1877). Reprinted in F. Hasenöhrl (ed.): Wissenschaftliche Abhandlungen. Leipzig: J. A. Barth 1909, Vol. 2, 164–223
Boltzmann, L.: Further Studies on the Thermal Equilibrium of Gas Molecules. The Kinetic Theory of Gases. History of Modern Physical Sciences. 1. pp. 262–349 (2003)
Buchdahl, H.A.: The Concepts of Classical Thermodynamics. The Cambridge University Press, Cambridge (1966)
Bussotti, P., Pisano, R.: Newton’s Philosophiae Naturalis Principia Mathematica “Jesuit” edition: the tenor of a huge work. Accademia Nazionale Lincei-Rendiconti Matematica e Applicazioni 25, 413–444 (2014)
Bussotti, P., Pisano, R.: Historical and Philosophical Details on Leibniz’s Planetary Movements as Physical-Structural Model. In: Pisano, R., Fichant, M., Bussotti, P., Oliveira, A.R.E.: The Dialogue between Sciences, Philosophy and Engineering. New Historical and Epistemological Insights. Homage to Gottfried W. Leibniz 1646–1716. College London Publication, London, pp. 49–92 (2017)
Bussotti, P., Pisano, R.: Historical and foundational details on the method of infinite descent: every prime number of the form 4n+1 is the sum of two squares. Found. Sci. (Springer) 1, 1–32 (2020)
Callen, H.B.: Thermodynamics and an Introduction to Thermostatistics. Wiley, New York (1985)
Čápek, V., Sheehan, D.: Challenges to the Second Law of Thermodynamics. Theory and Experiment. Springer, Dordrecht (2005)
Carathéodory, C.: Untersuchungen über die Grundlagen der Thermodynamik. Math. Ann. 67, 355–386 (1909)
Carnot, S.: Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance. Bachelier, Paris (1824)
Carnot, S.: Reflections on the Motive Power of Heat. Wiley, Hoboken (1897)
Carnot, S.: Réflexions sur la puissance motrice du feu sur les machinés propre à développer cette puissance, édition critique par Fox Robert. Vrin J, Paris (1978)
Carnot, S.: Reflections on the Motive Power of Fire: A Critical Edition with the Surviving Scientific Manuscripts Translated and edited by Robert Fox. The Manchester University Press, Manchester (1986)
Cercignani, C.: Ludwig Boltzmann. The Man who Trusted Atoms. Oxford University Press, Oxford (1998)
Chang, H.: Inventing Temperature: Measurement and Scientific Progress. Oxford Studies in the Philosophy of Science. Oxford University Press, Oxford (2004)
Chang, H.: Is Water H2O? Evidence, Pluralism and Realism. Boston Studies in the Philosophy of Science. Springer, Dordrecht (2012)
Clapeyron, E.: Mémoire sur la puissance motrice de la chaleur. Journal de l‘École royale polytechnique 14, 153–190 (1834)
Clausius, R.J.E.: Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Warme Theorie. Annalen der Physik und Chemie 169(12), 481–506 (1854)
Clausius, R.J.E.: Über verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie. Annalen der Physik und Chemie 75(7), 353–400 (1865)
Clausius, R.J.E.: The Mechanical Theory of Heat: With Its Application to the Steam Engine and to the Physical Properties of Bodies. T.A. J. Van Voorst, London, Translated by Hirst (1867)
Clausius, R.J.E.: Zur Geschichte der Mechanischen Wärmetheorie. Annalen der Physik und Chemie 145, 132–146 (1872)
Clausius, R.J.E.: Über die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen. Annalen der Physik und Chemie, 79, 368–397, 500 [English trans: Id., On the Motive Power of Heat and on the Laws which can be deduced from it for the theory of heat itself. In: Mendoza 1960, pp. 73–74, 109–152] (1850)
Cochran, M.J., Heron, P.R.L.: Development and assessment of research-based tutorials on heat engines and the second law of thermodynamics. Am. J. Phys. 74(8), 734–741 (2006)
De Donder, Th.: Leçons de thermodynamiques. Gauthier-Villars, Paris (1920)
De Donder, Th.: L’Affinité. Gauthier-Villars, Paris (1928)
De Donder, Th., Van Rysselberghe, P.: L’Affinité. Gauthier-Villars, Paris (1936)
De Donder, Th.: L’Affinité. (parte II). Gauthier-Villars, Paris (1931)
De Donder, Th.: L’Affinité. (parte III). Gauthier-Villars, Paris (1934)
Ebeling, W.: Max Planck on entropy and irreversibility. In: Hoffmann, D. (ed.) Max Planck: Annalen Papers EWILEY-VCH, pp. 29–166. Verlag GmbH & Co. KGaA, Weinheim (2008)
Fourier, J.: Théorie analytique de la chaleur. Firmin Didot, Paris (1822)
Fox, R.: James Prescott Joule, 1818–1889. In: North, J. (ed.) Mid-Nineteenth-Century Scientists, pp. 72–103. Pergamon Press, New York (1969)
Fox, R.: The Caloric Theory of Gases: From Lavoisier to Regnault. The Clarendon Press, Oxford (1971)
Frigg, R.: A field Guide to Recent Work on the Foundation of Statistical Mechanics. Ashgate, London (2008)
Gibbs, J.W.: Thermodynamischen Studien. Wilhelm Engelmann, Leipzig (1892)
Gibbs, J.W.: Équilibre des Système Chimiques. Carré et Naud, Paris (1899)
Gibbs, J.W.: Elementary Principles in Statistical Mechanics Developed with Especial Reference to the Rational Foundation of Thermodynamics. Charles Scribner’s Sons, New York (1902)
Gibbs, J.W.: On the Equilibrium of Heterogeneous Substances. The Scientific Papers of J. Willard Gibbs. Longmans, London (1906)
Gibbs, J.W.: Proceedings of the Gibbs Symposium. In: Caldi, D.G., Mostow, G.D. (eds.) Yale University, May 15–17, 1989. American mathematical Society and American Institute of Physics, New York (1990)
Gillispie, C.C., Pisano, R.: Lazare and Sadi Carnot. A Scientific and Filial Relationship, 2nd edn. Springer, Dordrecht (2014)
Hartely, H.: Studies in the History of Chemistry. The Clarendon Press, Oxford (1971)
Hertz, J.: Historique en grandes enjambées de la thermodynamique de l’équilibre. Journal de Physique 4, 122, 33–20; [see also Proceedings XXX JEEP. Journées d’Étude des Équilibres entre Phases, Saint-Avold, France 31 mars-1er avril 2004, EDP Science] (2004)
Hess, G.H.: Thermochemische Untersuchungen. Annalen de Physik un Chemie 56, 479 (1842)
Hill, T.L.: An Introduction to Statistical Thermodynamics. Dover Publications Inc., New York (1986)
Jaynes, F.T.: Gibbs vs Boltzmann entropies. Am. J. Phys. 5(33), 391–398 (1965)
Kapoor, S.: Entry: Berthollet, Claude Louis. Dictionary of Scientific Biography, vol. 2, pp. 73–82. Charles Scribner’s Sons, New York (1970–1980)
Kirchhoff, G.R.: Über die Verhältnis zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme und Licht. Ann. Phys. 109, 275–301 (1860)
Kragh, H., Weininger, S.J.: Sooner silence than confusion: the tortuous entry of entropy into chemistry. Hist. Stud. Phys. Biol. Sci. 27, 91–130 (1996)
Kuhn, T.S.: The Structure of Scientific Revolutions. The Chicago University Press, Chicago (1962)
Lamé, G.: Cours de physique de l’école polytechnique, vol. I. Bachelier, Paris (1836)
Lamé, G.: Leçons sur la théorie analytique de la chaleur. Mallet-Bachelier, Paris (1861b)
Lamé, G.: Discours préliminaire. In : Leçons sur la théorie analytique de la chaleur. Mallet–Bachelier, Paris (1861a)
Landau, L., Lifchitz, E.: Physique statistique. Editions Mir, Moscow (1984)
Laplace, P.S.: Notes sur la vitesse du son. Annales de chimie et de physique. 20, 266–268 (1822)
Lavoisier, A.L.: Œuvres de Lavoisier publiées par le soin de Son Excellence le Ministre de l’instruction publique et des cultes, 6 vols. Imprimerie Impériale, Paris (1862–1893)
Lavoisier, A.L.: Traité élémentaire de Chimie. Gauthier-Villars, Paris (1789 [1937])
Lavoisier, A.L., Laplace, P.S.: Mémoires sur la Chaleur. Histoire de l’Académie Royale des Sciences, Paris (1784)
Leicester, H.M.: German Henri Hess and the foundations of thermochemistry. J. Chem. Educ. 28(11), 581–583 (1951)
Marcella, T.V.: Entropy production and the second law of thermodynamics: an introduction to second law analysis. Am. J. Phys. 60(10), 888–895 (1992)
Mayer, R.J.: Bemerkungen über die Kräfte der unbelebten Natur. Weyrauch, Die Mechanik Warme, Stuttgart (1842)
Mayer, R.J.: Bemerkungen über die Kräfte der unbelebten Natur. Annalen der Chemie und Pharmazie 42, 233–240 (1842)
Mellone, M., Pisano, R.: Learning Mathematics in Physics Phenomenology and Historical Conceptual Streams. Problems of education in the 21st century, 46, 93–100 (2012)
Müller, I.: A History of Thermodynamics. The Doctrine of Energy and Entropy. Springer, Berlin (2007)
Nash, L.K.: The atomic-molecular theory. In: Conant, B.J. (ed.) Harvard Case Histories in Experimental Science, vol. 1. The Harvard University Press, Cambridge-MA (1957)
Nelson, P.G.: Statistical mechanical interpretation of entropy. J. Chem. Educ. 71, 103–104 (1994)
Nernst, W.: Experimental and Theoretical Applications of Thermodynamics to Chemistry. Charles Scribner’s Sons, New York (1907)
Nernst, W.: Studies in Chemical Thermodynamics, Nobel Lecture, 12 Dec 1921 (1921)
Nernst, W.: The New Heat Theorem. Methuen and Company Ltd, York (1926). (Reprinted in 1969 by Dover)
Pellegrino, et al., E.: Entropy from Clausius to Kolmogorov: historical evolution of an open concept. In: Tucci, P (ed.) SISFA Proceedings 2014, pp. 75–84. Pavia University Press, Pavia (2014)
Pellegrino, E.M., Cerruti, L., et al.: Clausius’ disgregation: a conceptual relic that sheds light on the second law. Entropy 17, 4500–4518 (2015)
Pellegrino, E.M., Cerruti, L., et al.: From steam engines to chemical reactions: Gibbs’ contribution to the extension of the second law. Entropy 18(5), 162 (2016)
Pellegrino, E.M.: Lazare and Sadi Carnot: A Scientific and Filial Relationship by Charles Coulston Gillispie and Raffaele Pisano. The Math Intelligencer (2019)
Pisano, R.: Interpretazione della nota matematica nelle Réflexions sur la Puissance Motrice du Feu di Sadi Carnot. In: Schettino, E. (ed.) SISFA Proccedings 2000, pp. 205–230. Bibliopolis, Napoli (2001)
Pisano, R.: A history of chemistry à la Koyré? Introduction and setting of an epistemological problem. Khimiya 17(2), 143–161 (2007a)
Pisano, R.: Brief historical notes on the theory of centers of gravity. In: Kokowski, M, (ed.). The Global and the Local: The History of Science and the Cultural Integration of Europe—Proceedings of the 2nd International Conference of the European Society for the History of Science, Polish Academy of Arts and Sciences, Krakow, pp. 934–941 (2007b)
Pisano, R.: Towards high qualification for science education. The loss of certainty. Journal Baltic of Science Education 8(2), 64–68 (2009a)
Pisano, R.: Continuity and discontinuity. On method in Leonardo da Vinci’ mechanics. Organon 41, 165–182 (2009b)
Pisano, R.: On principles in Sadi Carnot’s thermodynamics (1824). Epistemological Reflections. Almagest 2(1), 128–179 (2010)
Pisano, R.: Physics–mathematics relationship. Historical and epistemological notes. In: Barbin E., Kronfellner M., Tzanakis C., (eds.) Proceedings of the ESU 6 European Summer University History and Epistemology in Mathematics, pp. 457–472 . Verlag Holzhausen GmbH–Publishing Ltd., Vienna (2011a)
Pisano, R.: Textbooks, Foundations, History of Science and Science education. Problems of education in the 21st century, vol. 35, pp. 5–10 (2011b)
Pisano, R.: Historical reflections on physics mathematics relationship in electromagnetic theory. In: Barbin, E., Pisano, R. (eds.) The Dialectic Relation between Physics and Mathematics in the XIXth Century, pp. 31–57. Springer, Dordrecht (2013a)
Pisano, R.: On Lagrangian in Maxwell’s electromagnetic theory. Scientiatum VI. História das Ciências e das Técnicas e Epistemologia, pp. 44–59. The University of Federate University of Press, Rio de Janeiro (2013b)
Pisano, R. (ed.): A Bridge Between Conceptual Frameworks, Science. Society and Technology Studies. Springer, Dordrecht (2015)
Pisano, R.: What kind of Mathematics in Leonardo da Vinci and Luca Pacioli? Bull. Br. Soc. Hist. Math. 31, 104–111 (2016)
Pisano, R.: A Tale of Tartaglia’s Libro Sesto & Gionta in Quesiti et Inventioni diverse (1546–1554). Exploring the Historical and Cultural Foundations. Foundations of Science (Springer) 25/2:477–505 (2020)
Pisano, R., Bussotti, P.: Galileo and Kepler. On Theoremata Circa Centrum Gravitates Solidorum and Mysterium Cosmographicum. Hist. Res. 2(2), 110–145 (2012)
Pisano, R., Bussotti, P.: Open problems in mathematical modelling and physical experiments: exploring exponential function. Problems of education in the 21st century, vol. 50, pp. 56–69 (2013)
Pisano, R., Bussotti, P.: Notes on mechanics and mathematics in Torricelli as physics mathematics relationships in the history of science. Problems of education in the 21st century, vol. 61, pp. 88–97 (2014a)
Pisano, R., Bussotti, P.: Fibonacci and the reception of the abacus schools in Italy. Mathematical conceptual streams and their changing relationship with v. Almagest 6(2), 127–165 (2014b)
Pisano, R., Bussotti, P.: Historical and Philosophical Reflections on the Culture of Machines around the Renaissance. How Science and Technique Work? Acta Baltica Historiae et Philosophiae Scientiarum 2(2), 20–42 (2014c)
Pisano, R., Bussotti, P.: The emergencies of mechanics and thermodynamics in the western technoscience society during 18th–19th century. Pisano 2015, 399–436 (2015)
Pisano, R., Bussotti, P.: A Newtonian Tale details on notes and proofs in Geneva edition of Newton’s Principia. Bull. J. Br. Soc. Hist. Math. 31(3), 160–178 (2016)
Pisano, R., Bussotti, P.: The fiction of the infinitesimals in Newton’s Works: a note on the metaphoric use of infinitesimals. Spec. Issue Isonomia 9, 141–160 (2017a)
Pisano, R., Bussotti, P. (eds.): Homage to Galileo Galilei 1564–2014. Reading Iuvenilia Galilean Works within History and Historical Epistemology of Science. Philos. Sci. 21(1) (2017b)
Pisano, R., Bussotti, P.: Newton’s Geneva Edition: the Notes on Integral Calculus. In: La Rana, A., Rossi, P. (eds.) Proccedings International SISFA 2019, pp. 127–134. University Press, Pavia (2020a)
Pisano, R., Bussotti, P.: Conceptual Frameworks on Mathematics Applied to Physics in the Newton Principia Geneva edition (1822). Submitted to European Journal for Philosophy of Science Special Issue, Dimensions of Applied Mathematics Springer (2020b)
Pisano, R., Capecchi, D.: La Théorie Analytique de La Chaleur. Notes On Fourier And Lamé. Bulletin de la Sabix 44, 83–90 (2009)
Pisano, R., Capecchi, D.: Conceptual and mathematical structures of mechanical science in the western civilization around the 18th century. Almagest 4(2), 86–121 (2013)
Pisano, R., Capecchi, D.: Tartaglia’s science weights and Mechanics in XVI century. Selections from Quesiti et invention diverse; Books VII–VIII. Springer, Dordrecht (2015)
Pisano R., Sozzo, S.: A Unified Theory of Human Judgements and Decision-making under Uncertainty. Entropy. Special Issue Quantum Models of Cognition and Decision-Making 22/7:1–34 (2020)
Pisano, R., Agassi, J., Drozdova, D.: Hypothesis and Perspective Within History and Philosophy of Science. Homage to Alexandre Koyré 1964–2014. Springer, Dordrecht (2017a)
Pisano, R., Fichant, M., Bussotti, P., Oliveira, A.R.E.: The Dialogue between Sciences, Philosophy and Engineering. New Historical and Epistemological Insights. Homage to Gottfried W. Leibniz 1646–1716. College London Publication, London (2017b)
Pisano, R., Anakkar, A., E.M. Pellegrino, Nagels, M.: Thermodynamic foundations of physical chemistry: reversible processes and thermal equilibrium into the history. Foundation of Chemistry 21, 297–323 (2019)
Pisano R., Vincent, P., Dolenc, K., Ploj-Virtič, M.: Historical Foundations of Physics & Applied Technology as Dynamic Frameworksin Pre-Service STEM. Foundations of Science. https://doi.org/10.1007/s10699-020-09662-4 (2020a)
Pisano R., Mauricio, P., Vincent P.: Introduction. Joule’s Bicentenary: History of Science, Foundations and Nature of Science. Foundations of Science. https://doi.org/10.1007/s10699-020-09680-2 (2020b)
Pisano, R., Coopersmith, J., Peake M.: Essay on Machines in General (1786) Text, Translations and Commentaries. Lazare Carnot's Mechanics - Volume 1. Springer, Dordrecht (2021)
Planck, M.: Über das Gesetz der Energieverteilung im Normalspectrum. Ann. Phys. 309(3), 353–363 (1901)
Planck, M.: Vorlesungen ̈über Thermodynamik. Berlin: Walter de Gruyter (Reprinted in English translation as Treatise on Thermodynamics. Dover, New York) ([1897] 1903)
Planck, M.: The Theory of Heat Radiation. Translation by Masius M. of the second German edition. P. Blakiston’s Son & Co., Philadelphia (1914)
Planck, M.: Treatise on Thermodynamics. Translated with the Author’s sanction by A. Ogg. Dover, New York (1917)
Planck, M.: Autobiographie scientifique. Edition Flammarion, Paris (1991)
Pogliani, L., Berberan-Santos, M.N.: Constantin Carathéodory and the axiomatic thermodynamics. J. Math. Chem. 28, 313–324 (2000)
Prigogine, I.: Introduction to Thermodynamics of Irreversible Processes. Thomas Publisher, Springfield (1955)
Prigogine, I.: Introduction to Thermodynamics of Irreversible Processes. Interscience Publishers, New York (1967)
Prigogine, I.: La fin de Certitude. Odile Jacob, Paris (1996)
Prigogine, I.: The End of Certainty. The free Press, New York (1997)
Prigogine, I., Kondepudi, D.: Modem Thermodynamics. From Heat Engines to Dissipative Structures Wiley, New York (1999)
Prigogine, I., Stengers, I.: Order Out of Chaos: Man’s New Dialogue with Nature. Bantam, New York (1984)
Prigogine, I., Stengers, I.: Entre le temps et l’éternité. Edition Flammarion, Paris (1992)
Reech, F.: Théorie général des effets dynamiques de la chaleur. Journal de Mathématiques pures et appliquées 18, 357–378 (1853)
Reech, F.: Théorie générale des effets dynamiques de la chaleur. Extrait du Journal de Mathématiques pures et appliquées. Bachelier, Paris (1854)
Rosenfeld, L.: La genèse des principes de la thermodynamique. Bulletin de la Société Royale de Liège 10, 197–212 (1941)
Rosenfeld, L.: Four Centuries of Clinical Chemistry, pp. 72–75. CRC Press, New York (1999)
Saint-Robert, P.: Principes de Thermodynamique. Cassone, Torino (1865)
Saint-Robert, P.: Note II. Jules–Robert Mayer. In: Saint-Robert 1870c, pp. 451–470 (1870a)
Saint-Robert, P.: Note I. Sadi Carnot. In: Saint-Robert 1870c, pp. 431–450 (1870b)
Saint-Robert, P.: Principes de Thermodynamique, 2nd ed. Loecher, Turin-Florence (1870c). [see also idem edition printed by Gauthier-Villars, Paris]
Sands, D., Dunning-Davies, J.: Clausius' Concepts of 'Aequivalenzwerth' and Entropy: A Critical Appraisal, The Physics of Reality: Space, Time Matter, Cosmos. Amoroso, R., et al. (eds.). Proceedings of the 8th Symposium Honuring Mathematical Physicist Jean-Pierre Vigier, World Scientific, Singapore, pp. 207–2017 (2013)
Scerri, E.: The periodic table: the ultimate paper tool in chemistry. In: Klein, U. (ed.) Tools and Modes of Representation in the Laboratory Sciences. Boston Studies in the Philosophy of Science, vol. 222, pp. 163–177. Kluwer Academic Press, Dordrecht (2001)
Scerri, E.: A Tale of seven elements. Oxford-New York: The Oxford University Press (2017. Italian translation by Raffaele Pisano and Paolo Bussotti, with collaboration of Emilio Marco Pellegrino. Aracne, Roma) (2013)
Shannon, C.E.: The mathematical theory of communication. Bell Syst. Tech. J. 3(27), 379–423 (1948)
Smith, C., Wise, M.N.: Energy and Empire: A Biographical Study of Lord Kelvin. Cambridge University Press, Cambridge (1989)
Strehlow, P.: Die Kapitulation der Entropie. Physik Journal. 4, 45–51 (2005)
Tachray, A.: Atoms and Powers. The Harvard University Press, Cambridge (1970)
Thomsen, J.S., Bers, H.C.: The reversible process: a zero-entropy-production limit. Am. J. Phys. 64(5), 580–583 (1996)
Thomson, W.: An account of Carnot’s theory of the motive power of heat with numerical results deduced from Regnault’s experiments on steam. Mathematical and Physical Papers by Sir William Thomson, vol. I, pp. 113–155. Cambridge University Press, Cambridge (1882–1911)
Thomson, W.: On an absolute thermometric scale founded on Carnot’s theory of the motive power of heat, and calculated from Regnault’s observations. Proceedings of Cambridge Philosophical Society. 1(5), 66–71 (1848–1849)
Thomson, W.: On an absolute thermometric scale founded on Carnot’s theory of the motive power of heat, and calculated from Regnault’s observations. Mathematical and Physical Papers, vol. I, pp. 100–106. Cambridge University Press, Cambridge (1851a)
Thomson, W.: On the dynamical theory of heat, with numerical results deduced from Mr. Joule’s equivalent of a thermal unit, and M. Regnault’s observations on steam. Mathematical and Physical Papers, vol. I, pp. 175–183. Cambridge University Press, Cambridge (1851b)
Thomson, W.: An account of Carnot’s theory of the motive power of heat with numerical results deduced from Regnault’s experiments on steam. Annales de Chimie et de Physique 35, 248–255 (1852)
Tishin, A.M., Spichkin, Y.I.: The Magneto caloric Effect and its Applications. Condensed Matter Physics. CRC Press, Boca Raton (2016)
Tolman, R.C., Fine, P.C.: On the irreversible production of entropy. Rev. Mod. Phys. 20(1), 51–77 (1948)
Truesdell, C.A.: The Tragicomical History of Thermodynamics 1822–1854. Springer, Berlin (1980)
Uffink, J.: Compendium of the Foundations of Classical Statistical Physics. In: Butterfield, J., Eraman, J. (eds.) Philosophy of Physics, pp. 923–1074. Elsevier, Oxford (2007)
Waage, P., Guldberg, C.M.: Studier over Affiniteten, Forhandlinger: Videnskabs-Selskabet i Christiana, 35–45, 92–94, 111–120 (1864)
Waage, P., Guldberg, C.M.: Études sur les affinités chimiques. Brøgger and Christie, Christiania (1867)
Wien, W.: Temperatur und Entropie der Strahlung. Ann. Phys. 52, 132–165 (1894)
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Pisano, R., Pellegrino, E.M., Anakkar, A. et al. Conceptual polymorphism of entropy into the history: extensions of the second law of thermodynamics towards statistical physics and chemistry during nineteenth–twentieth centuries. Found Chem 23, 337–378 (2021). https://doi.org/10.1007/s10698-021-09401-y
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DOI: https://doi.org/10.1007/s10698-021-09401-y