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Much Polyphony but Little Harmony: Otto Sackur’s Groping for a Quantum Theory of Gases

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Abstract

The endeavor of Otto Sackur (1880–1914) was driven, on the one hand, by his interest in Nernst’s heat theorem, statistical mechanics, and the problem of chemical equilibrium and, on the other hand, by his goal to shed light on classical mechanics from the quantum vantage point. Inspired by the interplay between classical physics and quantum theory, Sackur chanced to expound his personal take on the role of the quantum in the changing landscape of physics in the turbulent 1910s. We tell the story of this enthusiastic practitioner of the old quantum theory and early contributor to quantum statistical mechanics, whose scientific ontogenesis provides a telling clue about the phylogeny of his contemporaries.

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Notes

  1. From entries in the Physikalische Zeitschrift and the Zeitschrift fu¨r Elektrochemie und angewandte Chemie it can be established that Sackur attended the meetings of the Bunsen Society for Physical Chemistry from 1906 until 1914, the year of his death. He also reported on the 1908 meeting for the Physikalische Zeitschrift.

  2. To a modern reader, Sackur’s talk about a tendency of molecules to clump in phase regions may suggest some, albeit naïve, anticipation of quantum statistics; see, for example, Darrigol, “Statistics and combinatorics II” (ref. 6). However, the main problem that quantum statistics had to deal with, namely, the extensivity of entropy, Sackur did not consider here. And when it came up, he did not seem to recognize it as a fundamental issue.

  3. From this argument one may discern another source of inspiration in Sackur’s work, for it closely resembles the 1905 light-quantum paper in which Einstein compares the probabilities for a gas to be in a given volume and in a part of that same volume.

  4. If S 1 = klnW 1 and S 2 = klnW 2 are the entropies of two subsystems and W 12 = W 1 W 2 is the probability of the composed system, then S 12 = S 1 + S 2, which is the definition of extensivity of entropy.

  5. The argument could be carried out for any point of oscillation, but Sackur chooses the equilibrium point since there the energy of the particle is purely kinetic.

References

  1. Max Planck, Vorlesungen über die Theorie der Wärmestrahlung (Leipzig: Johann Ambrosius Barth, 1906), pp. 140–148.

  2. A. Einstein, “Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme,” Annalen der Physik 22 (1907), 180–190; reprinted in John Stachel, ed., The Collected Papers of Albert Einstein. Vol. 2. The Swiss Years: Writings 1900–1909 (Princeton: Princeton University Press, 1989), Doc. 38, pp. 379–389; translated by Anna Beck (Princeton: Princeton University Press, 1989), Doc. 38, pp. 214–224.

  3. W. Nernst, “Zur Theorie der spezifischen Wärme und über die Anwendung der Lehre von den Energiequanten auf physikalisch-chemische Fragen überhaupt,” Zeitschrift für Elektrochemie und angewandte physikalische Chemie 17 (1911), 265–275.

  4. Clayton A. Gearhart, “‘Astonishing Successes’ and ‘Bitter Disappointment’: The Specific Heat of Hydrogen in Quantum Theory,” Archive for History of Exact Sciences 64 (2010), 113–202, on 118–122.

  5. A. Eucken, “Die Entwicklung der Quantentheorie vom Herbst 1911 bis Sommer 1913,” in A. Eucken, ed., Die Theorie der Strahlung und der Quanten: Verhandlungen auf einer von E. Solvay einberufenen Zusammenkunft (30. Oktober bis 3. November 1911) (Halle a. S.: Wilhelm Knapp, 1914), pp. 371–405.

  6. Olivier Darrigol, “Statistics and combinatorics in early quantum theory, II: Early symptoma of indistinguishability and holism,” Historical Studies in the Physical and Biological Sciences 21 (1991), 237–298, on 272–273, 276; Agostino Desalvo, “From the Chemical Constant to Quantum Statistics: A Thermodynamic Route to Quantum Mechanics,” Physis 29 (1992), 465–537, on 476–483.

  7. Of the few sources on Sackur’s life, we have relied especially on Alexander Kipnis, “Sackur, Otto,” Neue Deutsche Biographie, Zweiundzwantigster Band (Berlin: Dunker & Humblot, 2005), p. 344; Friedrich Auerbach, “Nekrologe Otto Sackur,” Jahres-Bericht der Schlesischen Gesellschaft für vaterländische Kultur. 1914. 1. Band (Breslau: G.P. Aderholz’ Buchhandlung, 1915), 35–37; W. Hertz, “Otto Sackur,” Physikalische Zeitschrift 16 (1915), 113–115; Hans Pick, “Otto Sackur,” Chemiker-Zeitung 39 (1915), 13.

  8. Svante Arrhenius, “Richard Abegg,” Zeit. Elektrochemie u. an. phys. Chemie 16 (1910), 554–557; Th. Des Coudres, “Richard Abegg,” Phys. Zeit. 11 (1910), 425–429.

  9. For example, Otto Sackur, Lehrbuch der Thermochemie und Thermodynamik (Berlin: Julius Springer, 1912). During this period, Otto Stern earned his Ph.D. degree under Sackur’s supervision, on osmotic pressure of “generalized soda-water”; see Bretislav Friedrich and Dudley Herschbach, “Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics,” Physics Today 56 (December 2003), 53–59, on 53. Sackur, using Fritz Haber’s mediation, also helped Stern to his postdoctoral position with Albert Einstein in 1912; see Interview of Otto Stern by Thomas S. Kuhn, May 29 and 30, 1962, Niels Bohr Library and Archives, American Institute of Physics, College Park, MD USA, p. 3 of 8. On Sackur’s pedagogical activity, see Massimiliano Badino, “Dissolving the boundaries between research and pedagogy: Otto Sackur’s Lehrbuch der Thermochemie und Thermodynamik,” in Massimiliano Badino and Jaume Navarro, ed., Research and Pedagogy. A History of Quantum Physics through its Textbooks (Berlin: Max Planck Research Library for the History and Development of Knowledge, 2013).

  10. Jeremiah James, Thomas Steinhauser, Dieter Hoffmann, and Bretislav Friedrich, One Hundred Years at the Intersection of Chemistry and Physics: The Fritz Haber Institute of the Max Planck Society 1911–2011 (Berlin/Boston: Walter de Gruyter, 2011), p. 23.

  11. Bretislav Friedrich, Dieter Hoffmann, and Jeremiah James, “One Hundred Years of the Fritz Haber Institute,” Angewandte Chemie International Edition 50 (2011), 10022–10049, on 10031; idem, Angewandte Chemie 123 (2011), 10198–10225, on 10207.

  12. For a collection of their publications, see C.M. Guldberg und P. Waage, Untersuchungen über die chemischen Affinitäten. Abhandlungen aus den Jahren 1864, 1867, 1879. Uebersetzt und herausgegeben von R. Abegg [Ostwald’s Klassiker der Exakten Wissenschaften, Nr. 104] (Leipzig: Wilhelm Englemann, 1899).

  13. J.H. van’t Hoff, “L’équilibre chimique dans les systèms gazeux, ou dissous á l’état dilué,” Archives néerlandaises des Sciences exactes et naturelles 20 (1886), 239–302.

  14. Josiah Willard Gibbs, “On the equilibrium of heterogeneous substances,” Transactions of the Connecticut Academy 3 (1876), 108–248; ibid. (1878), 343–524; reprinted in Henry Andrews Bumstead, ed., The Scientific Papers of Josiah Willard Gibbs. Vol. I (New York: Longmans, 1906); reprinted as The Collected Works of J. Willard Gibbs). Vol. I. Thermodynamics (New York, London, Toronto: Longmans, Green and Co., 1928), pp. 55–353.

  15. Hermann Helmholtz, “Die Thermodynamik chemischer Vorgänge,” Sitzungsberichte der Akademie der Wissenschaften zu Berlin 1 (1882), 22–39; reprinted in Wissenschaftliche Abhandlungen. Zweiter Band (Leipzig: Johann Ambrosius Barth, 1883), pp. 958–978.

  16. A. Eucken, “Der Nernstsche Wärmesatz,” in Ergebnisse der exakten Naturwissenschaften. Erster Band (Berlin: Julius Springer, 1922), pp. 120–162, on p. 123 (italics in original).

  17. F. Haber, Thermodynamik technischer Gasreaktionen: Sieben Vorlesungen (München und Berlin: R. Oldenbourg, 1905), pp. 22–23.

  18. Edwin N. Hiebert, “The Energetics Controversy and the New Thermodynamics,” in Duane H. D. Roller, ed., Perspectives in the History of Science and Technology (Norman: University of Oklahoma Press, 1971), pp. 67–86; idem, “Walther Nernst and the Application of Physics to Chemistry,” in Rutherford Aris, H. Ted Davis and Roger H. Stuewer, ed., Springs of Scientific Creativity: Essays on Founders of Modern Science (Minneapolis: University of Minnesota Press, 1983), pp. 203–231; Diana Kormos Barkan, Walther Nernst and the Transition to Modern Physical Science (Cambridge, New York, Melbourne: Cambridge University Press, 1999), pp. 110–180; Patrick Coffey, “Chemical free energies and the third law of thermodynamics,” Hist. Stud. Phys. Bio. Sci. 36 (2006), 365–396.

  19. Julius Thomsen (1826–1909) in 1852 and Marcellin Berthelot (1827–1907) in 1869 had speculated on the behavior of chemical reactions in the vicinity of absolute zero. See Kormos Barkan, Nernst (ref. 18), pp. 132–146, and Hans-George Bartel and Rudolf P. Huebener, Walther Nernst: Pioneer of Physics and of Chemistry (Singapore: World Scientific, 2007), pp. 150–198. For Nernst’s account, see W. Nernst, Die theoretischen und experimentellen Grundlagen des neuen Wärmesatzes (Halle (Saale): Wilhelm Knapp, 1918), “Historische Einleitung,” pp. 1–12.

  20. W. Nernst, “Ueber die Berechnung chemischer Gleichgewichte aus thermischen Messungen,” Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen 1 (1906), 1–39; idem, “Über die Beziehungen zwischen Wärmeentwicklung und maximaler Arbeit bei kondensierten Systemen,” Sitzungsberichte der königlich preussischen Akademie der Wissenschaften (1906), 933–940. For a modern treatment, see Philip M. Morse, Thermal Physics, Second Edition (New York: W.A. Benjamin, 1969), pp. 70–72.

  21. Walther Nernst, Experimental and Theoretical Applications of Thermodynamics to Chemistry (New Haven: Yale University Press, 1907), pp. 39–52, and Nernst, Die theoretischen und experimentellen Grundlagen (ref. 19), Kapitel V, X, pp. 58–65, 99–105.

  22. Nernst, “Ueber die Berechnung” (ref. 20), p. 22.

  23. Nernst, Experimental and Theoretical Applications (ref. 21), pp. 55–76.

  24. O. Sackur, “Die Anwendung der kinetischen Theorie der Gase auf chemische Probleme,” Ann. Phys. 36 (1911), 958–998, on 965.

  25. Walther Nernst, Theoretische Chemie vom Standpunte der Avogadro’schen Regel und der Thermodynamik (Stuttgart: Ferdinand Enke, 1893; Dritte Auflage, 1900).

  26. Nernst was reluctant to commit his conviction of the validity of the quantum hypothesis to paper, but there are at least two pieces of indirect evidence for it: First, around 1910 he attempted to prove his heat theorem by a purely thermodynamic argument independent of the nature of the substances involved; see A.J. Kox, “Confusion and clarification: Albert Einstein and Walther Nernst’s Heat Theorem, 1911–1916,” Studies in History and Philosophy of Modern Physics 37 (2006), 101–114. Second, after 1906 Nernst put his Berlin group to work on the measurement of the specific heats of gases. Prior to the Solvay Conference, he supervised three doctoral dissertations on the specific heats of gases, by F. Voller (1908), F. Keutel (1910), and R. Thibaut (1910). At the same time, Mathias Pier worked on the measurement of specific heats at high temperatures; see Mathias Pier, “Die Spezifischen Wärmen von Argon, Wasserdamf, Stickstoff, Wasserstoff bei sehr hohen Temperaturen,” Zeit. Elektrochemie u. an. phys. Chemie 15 (1909), 536–540; idem, “Spezifische Wärme und Gasgleichgewichte nach Explosionversuchen. II,” ibid. 16 (1910), 897–903), while Fritz Koref and Eucken concentrated on low temperatures; see F. Koref, “Messungen der spezifischen Wärme bei tiefen Temperaturen mit dem Kupferkalorimeter,” Ann. Phys. 36 (1911), 49–73, and A. Eucken, “Die Molekularwärme des Wasserstoffs bei tiefen Temperaturen,” Sitzungsber. k. preus. Akad. Wiss. (1912), 141–151.

  27. R. Abegg and O. Sackur, Physikalisch-Chemische Rechenaufgaben (Leipzig: G.J. Göschen’sche Verlagshandlung, 1909). Joseph Knox, Physico-Chemical Calculations (New York: D. Van Nostrand, 1916), was based upon Abegg and Sackur’s book.

  28. O. Sackur, “Zur kinetischen Begründung des Nernstschen Wärmetheorems,” Ann. Phys. 34 (1911), 455–468. Attempts along a similar direction also were made by Nernst, “Zur Theorie der spezifischen Wärme” (ref. 3); Ferenz Jüttner, “Über die Ableitung der Nernstschen Formeln für Reaktionen in kondensierten Systemen,” Zeit. Elektrochemie u. an. phys. Chemie 17 (1911), 139-143; Michael Polanyi, “Eine neue thermodynamische Folgerung aus der Quantenhypothese,” Verhandlungen der Deutschen Physikalischen Gesellschaft 15 (1913), 156–161.

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  29. Max Planck, Vorlesungen über Thermodynamik. Dritte, erweiterte Auflage (Leipzig: Veit & Comp., 1911), p. 268.

  30. Planck, Wärmestrahlung (ref. 1); Einstein, “Plancksche Theorie der Strahlung” (ref. 2).

  31. Sackur, “Zur kinetischen Begründung” (ref. 28), p. 467.

  32. Planck dwelled on the concept of absolute entropy both in the third edition of his Thermodynamik (ref. 29), pp. 268–269, and in the second edition; see Max Planck, Vorlesungen über die Theorie der Wärmestrahlung. Zweite, teilweise umbearbeitete Auflage (Leipzig: Johann Ambrosius Barth, 1913), pp. 115–118.

  33. Sackur, “Die Anwendung” (ref. 24), p. 960.

  34. In doing so, Sackur invoked the Stirling formula and replaced summation by integration. On the one hand, the applicability of the Stirling formula relies on the assumption that the elementary regions are large enough to contain many molecules. On the other hand, replacing summations by integration requires the regions to be infinitesimal. Classical kinetic theory hinges upon the balance between these two contravening requirements; see also Ulrich Hoyer, “Von Boltzmann zu Planck,” Arch. Hist. Exact Sci. 23 (1980), 47–86, and Olivier Darrigol, “Statistics and combinatorics in early quantum theory,” Hist. Stud. Phys, Bio. Sci. 19 (1988), 17–80.

  35. Sackur, “Die Anwendung” (ref. 24), pp. 965–975.

  36. Ibid., p. 968.

  37. Ibid., p 969.

  38. Ibid., p. 970.

  39. Planck, Wärmestrahlung (ref. 1), pp. 136–137.

  40. Max Planck, “Die gegenwärtige Bedeutung der Quantenhypothese für die kinetische Gastheorie,” in David Hilbert, ed., Vorträge über die kinetische Theorie der Materie und der Elektrizität gehalten in Göttingen auf Einladung der Kommission der Wolfskehlstiftung (Leipzig und Berlin: B.G. Teubner, 1914), pp. 1–16, on pp. 14–15.

  41. H. Tetrode, “Die chemische Konstante der Gase und das elementare Wirkungsquantum,” Ann. Phys. 38 (1912), 434–442, on 434; see also Darrigol, “Statistics and combinatorics II” (ref. 6), pp. 276–277, and Desalvo, “From the Chemical Constant” (ref. 6), pp. 481–482.

  42. O. Sackur, “Die Bedeutung des elementaren Wirkungsquantums für die Gastheorie und die Berechnung der chemischen Konstanten,” in Festschrift: W. Nernst zu seinem fünfundzwanzigjährigen Doktorjubiläum gewidmet von seinen Schülern (Halle a. d. S.: Wilhelm Knapp, 1912), pp. 405–423 (dated March 1912). Since Tetrode’s article was also published in March, it seems that Sackur worked out the consequences of his approach independently.

  43. Sackur, “Die Anwendung” (ref. 24), pp. 959–960.

  44. Sackur, “Die Bedeutung” (ref. 42), p. 406.

  45. Boltzmann considered molecular disorder as the condition for applying probability in gas theory, while Planck was convinced that disorder had the power to eliminate any violation of the second law of thermodynamics; see Massimiliano Badino, “The odd couple: Boltzmann, Planck and the application of statistics to physics (1900–1913),” Ann. Phys. 18 (2009), 81–101.

  46. Sommerfeld had proposed the interpretation of the elementary volume in terms of h as a general quantization condition for a periodic system at the first Solvay Conference and in a widely discussed paper, A. Sommerfeld, “Das Plancksche Wirkungsquantum und seine allgemeine Bedeutung für die Molekularphysik,” Phys. Zeit. 12 (1911), 1057–1068; reprinted in Gesammelte Schriften. Band III (Braunschweig: Friedr. Vieweg & Sohn, 1968), pp. 1–19.

  47. Sackur, “Die Bedeutung” (ref. 42), p. 409.

  48. O. Sackur, “Die universelle Bedeutung des sog. elementaren Wirkungsquantums,” Ann. Phys. 40 (1912), 67–86.

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  49. The energy-time uncertainty relation has a more genuine predecessor in the work of Niels Bohr, “Anwendung der Quantentheorie auf den Atombau. I. Die Grundpostulate der Quantentheorie,” Zeit. für Phys. 13 (1923), 117–165; translated as “On the Application of the Quantum Theory to Atomic Structure. Part I. The Fundamental Postulates,” Proceedings of the Cambridge Philosophical Society (Supplement) (Cambridge: At the University Press, 1924), 42 pages; reprinted in Niels Bohr Collected Works. Vol. 3. The Correspondence Principle (1918–1923), edited by J. Rud Nielsen (Amsterdam, New York, Oxford: North-Holland Publishing Company, 1976), pp. 457–499.

  50. Sackur, “Die universelle Bedeutung” (ref. 48), p. 85.

  51. Ibid., pp. 75–76.

  52. Sackur’s idea of using the mean free path as a parameter to quantize the gas was adopted the following year by Sommerfeld and his collaborator Wilhelm Lenz; see A. Sommerfeld, “Probleme der freien Weglänge,” in Hilbert, Vorträge (ref. 40), pp. 125–166; reprinted in Gesammelte Schriften. Band II (Braunschweig: Friedr. Vieweg & Sohn, 1968), pp. 287–308.

  53. O. Sackur, “Die ‘Chemischen Konstante’ der zwei- und dreiatomigen Gase,” Ann. Phys. 40 (1912), 87–106. Although Tetrode’s letter to Sackur has been lost, we know about its existence from Sackur’s paper, p. 87, footnote 2.

  54. O. Sackur, “Die Zustandgleichung der Gase und die Quantentheorie,” Zeit. Elektrochemie u. an. phys. Chemie 20 (1914), 563–570. Because of his intention to move on to the experimental determination of the gas law at very low temperature, Sackur applied for a position in Kamerlingh Onnes’s institute in Leiden. However, Onnes dismissed his application with scorn; see, for example, Dirk van Delft, Freezing Physics: Heike Kamerlingh Onnes and the quest for cold (Amsterdam: Koninklijke Nederlandse Akademie van Wetenschappen, 2007), p. 477.

  55. Otto Stern, “Zur kinetischen Theorie des Dampfdruckes einatomiger fester Stoffe und über die Entropiekonstante einatomiger Gase,” Phys. Zeit. 14 (1913), 629–632; idem, “Zusammenfassender Bericht über die Molekulartheorie des Dampfdruckes fester Stoffe und ihre Bedeutung für die Berechnung chemischer Konstanten,” Zeit. Elektrochemie u. an. phys. Chemie 25 (1919), 66–80.

  56. Enrico Fermi, “Sopra la teoria di Stern della costante assoluta dell’entropia di un gas perfetto monoatomico,” Rendiconti dell’Accademia dei Lincei 32 (1923), 395–398; reprinted in Collected Papers (Note e Memorie). Vol. I. Italy 1921–1938 (Chicago: University of Chicago Press and Roma: Accademia Nazionale dei Lincei, 1962), pp. 114–117.

  57. Suman Seth, Crafting the Quantum: Arnold Sommerfeld and the Practice of Theory, 1890-1926 (Cambridge, Mass. and London: The MIT Press, 2010), pp. 2–4.

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Acknowledgment

We are grateful to Roger H. Stuewer for greatly improving our manuscript by correcting many errors and inaccuracies it was fraught with at the time of submission.

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  1. Centre d’Història de la Ciència, Facultat de Ciències, Universitat Autonoma de Barcelona, Cerdanyola del Valles, 08193, Bellaterra (Barcelona), Spain

    Massimiliano Badino

  2. Abteilung Molekülphysik, Fritz Haber Institute, Faradayweg 4-6, 14195, Berlin, Germany

    Bretislav Friedrich

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Correspondence to Massimiliano Badino.

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Massimiliano Badino (corresponding author) has worked on the History of Quantum Physics Project at the Max Planck Institute for the History of Science in Berlin, Germany, and is currently postdoctoral research fellow in the Centre d’Història de la Ciència at the Universitat Autonoma de Barcelona, Spain. Bretislav Friedrich is a Research Group Leader at the Fritz Haber Institute of the Max Planck Society and Honorarprofessor at the Technische Universität in Berlin, Germany.

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Badino, M., Friedrich, B. Much Polyphony but Little Harmony: Otto Sackur’s Groping for a Quantum Theory of Gases. Phys. Perspect. 15, 295–319 (2013). https://doi.org/10.1007/s00016-013-0110-8

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