D O C . 2 8 3 Q U A N T U M T H E O R Y O F I D E A L G A S 4 4 1 Published in Preußische Akademie der Wissenschaften (Berlin). Physikalisch-mathematische Klasse. Sitzungsberichte 1924: 261–267. Submitted 10 July 1924, published 20 September 1924. A manu- script is also available ([1 040]). [1]Satyendra Nath Bose. Bose 1924a and Einstein 1924j (Doc. 278). Einstein may have errone- ously assigned the initial “D” to Bose because of a confusion with Debendra Mohan Bose, with whom he had corresponded earlier about an English translation of Lorentz et al. 1920 (see Doc. 261, note 4). In Bose 1924a (Doc. 278) as well as in its manuscript (in Einstein’s hand [1 045]), the author’s name is given as Bose without any initials. [2]For a critical discussion of Planck’s concept of probability and Boltzmann’s principle, see Einstein 1909b (Vol. 2, Doc. 56), pp. 187–188. [3]In the manuscript, “offenbar” is followed by “kontinuierliche,” then deleted. [4]Eq. (5) should read . [5]The constraint that the total number of atoms be conserved is a distinguishing feature of the material gas as compared to the case of a photon gas. [6]The exponent in the third expression should have an additional factor of r. [7]The summation term should read . [8]See Boltzmann 1898, §§48–52, and Einstein’s 1910 lecture notes on kinetic theory (Vol. 3, Doc. 4, pp. 180, 212–213). [9]For Js, , m–3, kg, J/K, K, the numerical value of is . [10]Einstein’s result is to be compared with the Tetrode-Sackur equation for the entropy of a mon- atomic ideal gas with quantum corrections (see Reiche 1921 for a contemporary review). Einstein’s expression can also be compared to Planck 1921, eq. (442), and Schrödinger 1924, p. 44, eq. (18). For Einstein’s earlier work on this topic, see “On the Theory of Tetrode and Sackur for the Entropy Constant” (Vol. 6, Doc. 26). [11]Nernst’s theorem was originally formulated for the chemical equilibrium at very low tempera- tures, and postulated that the curves of free energy and the total energy of chemical reactions between only solid or fluid bodies become tangent to one another at absolute zero (Nernst 1906). The hypoth- esis gained significance when interpreted in the framework of quantum theory, and was then reformu- lated in various ways. It was discussed at the 1911 and 1913 Solvay conferences (see, e.g., Einstein’s comments in Vol. 3, Doc. 25, pp. 513–514, and Vol. 4, Doc. 22, pp. 556–557). For a quantum- theoretical justification of Nernst’s theorem without invoking Boltzmann’s principle, see Einstein 1914n (Vol. 6, Doc. 5). In the present context, the theorem states that, for vanishing absolute tempera- ture, the entropy vanishes as well, as explicitly stated also in Einstein 1925i (Doc. 427), §2. For a his- torical discussion of Nernst’s theorem, see Hiebert 1978 and Barkan 1999, especially chaps. 8, 11. [12]The summation index in equations (18) and (19), and the σ in the paragraph following equation (19a), should be s, as per Einstein’s corrections in the manuscript [1 040]. [13]In the manuscript, “ausdrückt” is followed by the deleted sentence, “Wir wollen uns hier im Folgenden aber auf den Fall beschränken, dass wir gegen 1 vernachlässigen dürfen.” [14]The numerical factor should be , as in the manuscript, but it is crossed out and substituted by the wrong value of –0.0318. Apparently, the mistake arose from absorbing a factor of into the numerical factor (see Walther Meissner to Albert Einstein, 2 November 1925 [17 114]). It was corrected in Einstein 1925f (Doc. 385), p. 13. [15]For a brief reference to Gibbs’s paradox, mentioning the name of Bose, see an entry on p. 4 in Ehrenfest’s notebook, Ne-LeRM, ENB:1, 29. The entry is found next to entry 6072, which is dated 25 December 1924 (see Pérez and Sauer 2010, p. 604). S κ¦(pr s lgpr s) sr –= dαs eαs 1 – - s ¦--------------- h 6.626 10–34 ⋅ = n 6.02 1023 ⋅ = V 0.0224 = m 2 1.67 10–27 ⋅ ⋅ = k 1.38 1023 ⋅ = T 300 = eA 1.0 105 ⋅ λ2 2 5 2 /– 2 3 2 /– – 0.177 –= π 3 2 /–