D O C . 3 8 5 Q U A N T U M T H E O R Y O F I D E A L G A S I I 5 9 3 Published in Preußische Akademie der Wissenschaften (Berlin). Physikalisch-mathematische Klasse. Sitzungsberichte (1925): 3–14. A manuscript is also available (NL-LeUL, BPL 3545 [95 381]). [1]Einstein 1924o (Doc. 283). [2]Satyendra Nath Bose for his derivation of Planck’s formula, see Bose 1924a (in Doc. 278). For Einstein’s use of the initial “D” in Bose’s name, see Einstein 1924o (Doc. 283), note 1. [3]Einstein 1924o (Doc. 283) ends with §5 and equation (23). [4]For a discussion of Planck’s function, see Planck 1911, §287, and Planck 1913, §133. [5]In the manuscript, the footnote starts with the deleted text “Die gesamte Substanz.” [6]After submitting Einstein 1924o (Doc. 283) on 10 July 1924, Einstein had met with Ehrenfest during his stay in Leyden from 4 to 24 October. From Leyden, Einstein wrote to Elsa Einstein that Ehrenfest “disputes only a nuance” (“bestreitet nur eine Nuance” see Doc. 334). [7]The reference to eq. (13) should be to eq. (18). [8]In the manuscript, the phrase in brackets is interlineated. [9]In the manuscript, “Die” replaces the deleted text “Zu Ungunsten der” and “führt nach dem … muss” replaces the deleted phrase “spricht ferner, dass im Sinne der Quantentheorie kein klarer Grund dafür besteht, (29b) durch (29c) zu ersetzen.” [10]For this hypothesis, see also Einstein 1925i (Doc. 427), p. 21. [11]The method to calculate fluctuations employed here was first developed by Einstein in Einstein 1909b (Vol. 2, Doc. 56), pp. 188–189. [12]In the first equality, the exponent should be . [13]Broglie 1924b. [14] should be . [15]For Einstein’s comment on Gibbs’s paradox, see Einstein 1924o (Doc. 283), p. 267. In the man- uscript, the paragraph begins as follows: “Auf Grund dieser Überlegungen können wir auch das Para- doxon lösen, auf welches….” After corresponding with Einstein about a draft of his review of Einstein 1924o (Doc. 283), Adolf Smekal deleted a critical comment to the effect that Gibbs’s paradox cannot be solved using Bose-Einstein counting (see Doc. 434, note 3). [16]In the manuscript, “Wellenfelder” replaces the deleted “Strahlungen.” [17]In 1919, Walther Nernst had derived, on the basis of a quantum version of the equation of state of gases, a repulsive force between molecules proportional to the inverse cube of the intermolecular distance (Nernst 1919). Ernst Günther, his Assistent, had made measurements of the viscosity of hydrogen to test his hypothesis (Günther 1920). His conclusion was that the viscosity is linearly dependent in the temperature range between 80K and 273K, and follows a dependence, as pre- dicted by Nernst, for temperatures below 20K. In Günther 1924, improved measurements, including observations for helium, were reported. [18]For the summations in eq. (22) reduce to and , respectively, with ] the Riemann zeta function. More accurate values for these quantities than the ones used by Einstein are 1.342 and 2.612. [19]Nernst 1919, Günther 1920. Copies of these papers had been sent to Einstein by Nernst (see Walther Nernst to Einstein, 28 November 1920 [Vol. 10, Doc. 213]). [20]For a reference to the possible role of quantum corrections for the explanation of deviations from the law of corresponding states, see Doc. 361. [21]In the manuscript, “auf Grund dieser Theorie… behandelt” replaces the deleted phrase “darauf beruhen, dass das für die Agitation der Elektronen zur Verfügung stehende Volumen von der Natur des Metalles abhänge.” [22]In the manuscript, “Vergleich mit der Erfahrung gefunden habe” replaces the deleted text “Rechnung gefunden habe.” Σ κ --- Zν zν T1.5 λ 1 = ζ( 5 2) ⁄ ζ( 3 2) ⁄