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Conceptual polymorphism of entropy into the history: extensions of the second law of thermodynamics towards statistical physics and chemistry during nineteenth–twentieth centuries

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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 As for the methodological approach, text analysis of historical documents constituted our privileged modus operandi.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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Fig. 1

Source: Internet Archive (public domaine). https://archive.org/stream/annalenderphysi94unkngoog#page/n161/mode/2up

Fig. 2
Fig. 3

Source: BnF Gallica (public domain)

Fig. 4

Source: BnF Gallica (public domain)

Fig. 5

Source: BnF Gallica (public domain)

Fig. 6

Source: http://www.archive.org (public domain)

Fig. 7

adapted from Shannon (Shannon 1948, p. 381)

Fig. 8

Source: http://www.archive.org (public domain)

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Notes

  1. 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).

  2. 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.

  3. Chaos is the translation of the german “unordnung”, found in Boltzmann’s papers.

  4. 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.

  5. Cfr Mayer (1842a, b), Clausius (1854, 1865, 1867, 1872), Thomson W. (1848–1849, 1851a, b, 1852, 1882–1911).

  6. Cfr. Pellegrino (2019), Pellegrino et al. 2015, Pisano, Anakkar, Pellegrino, Nagels (2019).

  7. 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).

  8. Thermodynamic model is just one approach to chemical equilibrium (Hartely 1971; Hertz 2004). It is worth mentioning the « kinetic » approach that has its scientific root in Berthollet (1803; Kapoor 1970–1980) and Waage and Gulberg (1864, 1867).

  9. Cfr. Gibbs (1892,1902,1906, 1990), Pellegrino et al. (2016).

  10. 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).

  11. Cfr. Balian (1982), Cerciniani (1998), Jaynes (1965),Frigg (2008).

  12. Cfr. Sands and Dunning-Davies (2013).

  13. Cfr.: Carnot S. (1824), Clapeyron (1834), Fox (1969, 1971), De Donder (1920, 1928, 1931, 1934, 1936), Pisano, Anakkar, Pellegrino, Nagels (2019), Pellegrino et al. (2014), Pellegrino et al. (2016), Pellegrino et al. (2015), Cochran and Heron (2006), Pisano and Capecchi (2013, 2015, 2009).

  14. Cfr. Clausius (1854).

  15. In current notation this integral should be written as:\({\oint }\frac{\delta Q}{T}\).

  16. Cfr. Clausius (1872).

  17. In current notation this integral should be written as: \({\oint }\frac{\delta Q}{T}\)

  18. Cfr. Thomsen and Bers (1996), Tolmann and Fine (1948), Tishin and Spichkin (2016).

  19. Cfr. Nernst (1926, pp. 75–90), Nernst (1907, pp. 41–50), Strehlow (2005).

  20. 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).

  21. Karl Hermann Amandus Schwarz (1843–1921) was Carathéodory’s professor in Berlin.

  22. Pisano and Bussotti (2020a,b, 2017a, b, 2015, 2016, 2012, 2013, 2014a, b, c), Pisano (2020), Bussotti and Pisano (2020, 2014, 2017), Pisano and Sozzo (2020).

  23. Landau and Lifchitz (1984), Nash (1957).

  24. In current notation this integral should be written as: \({\oint }\frac{\delta Q}{T}\).

  25. In Boltzmann (Boltzmann 1866, p. 24) we read: “Für diesen Fall gilt in (20) das Gleichheitszeichen”. Equation 20 corresponds to Equation 23 of our paper.

  26. The \(H\) is the Greek capital letter eta—traditionally used for entropy.

  27. Original title Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen.

  28. It corresponds to Eq. (20) of this paper.

  29. For the sake of completeness, the differential equation is:

    \(\frac{{\partial f\left( {x,t} \right)}}{\partial t} = \mathop \smallint \limits_{0}^{\infty } \mathop \smallint \limits_{0}^{x + x^{\prime}} \left[ {\frac{{f\left( {\xi ,t} \right)}}{\sqrt \xi }\frac{{f\left( {x + x^{\prime} - \xi ,t} \right)}}{{\sqrt {x + x^{\prime} - \xi } }} - \frac{{f\left( {x,t} \right)}}{\sqrt x }\frac{{f\left( {x^{\prime},t} \right)}}{{\sqrt {x^{\prime}} }} } \right]\sqrt {xx^{\prime}} \psi \left( {x,x^{\prime},\xi } \right)dx^{\prime}d\xi .\)

  30. Cfr. Boltzmann (1866).

  31. For sake of brevity/calculus we choose to denote the quantity φ(t) which is not included in the primary sources text.

  32. In current notation this integral should be written as: \({\oint }\frac{\delta Q}{T}\)

  33. 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).

  34. Cfr. Hill (1986).

  35. Actually, Shannon refers to this same entity either as information or uncertainty.

  36. 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).

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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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