I N T R O D U C T I O N T O V O L U M E 1 4 l x i x October 1924. In a letter to Elsa at the time, Einstein wrote: “It is also not true that I made a mistake in my paper, which is built on Bose’s papers. Ehrenfest only dis- putes a nuance Laue doesn’t dispute anything” (Doc. 334). By late November 1924, he reported to Ehrenfest that he had found another in- triguing consequence of the theory: “I examined with Grommer in more detail the degeneracy function. From a certain temperature on, the molecules ‘condense’ without attractive forces, i.e., they pile up at zero velocity. The theory is pretty, but is there any truth to it?” (Doc. 384). He added that he would try to investigate whether a connection to the behavior of thermo-forces at low temperatures could be found. Four days later, he again wrote to Ehrenfest: “The matter of the quantum gas is becoming very interesting. It seems to me more and more that there is much that is true and deep at the bottom of it. I look forward to our arguing about it” (Doc. 386). At the end of the year, he eventually wrote a second paper on the topic and pre- sented it to the Prussian Academy on 8 January (Einstein 1925f [Doc. 385]). The paper is written formally as a continuation of the first paper, but Einstein now goes back to an analysis of the statistical method that he had employed: “Herr Ehrenfest and other colleagues have found fault with Bose’s theory of radiation and my anal- ogous theory of the ideal gas in that these theories do not treat the quanta (or the molecules) as statistically independent entities, without our having especially ad- dressed this circumstance in our papers. This is entirely correct.” In the relevant section, Einstein now proceeds to compare explicitly the statistical assumptions un- derlying the different viewpoints. But this second paper on the quantum theory of the ideal gas contains yet another insight that would prove to be of great significance. In its first section, numbered §6, he observes that the equation of state that he had obtained does not allow for a straightforward limit in which, for a given number of molecules n and given tem- perature T, the volume may be made arbitrarily small. He asserts that “in this case, with increasing total density, a growing number of molecules go over into the 1st quantum state (a state without kinetic energy), while the rest of the molecules be- have according to the parameter value λ = 1.” Therefore, something occurs that is similar to the isothermal compression of vapor beyond the saturation volume. A separation takes place whereby one part “condenses” while the rest remains a “sat- urated ideal gas” (Doc. 385, p. 4). The predicted phenomenon of “condensation”—according to which, below a certain critical temperature and depending on the mass and density of the mole- cules, a finite fraction of the molecules condense into the cell corresponding to zero energy—is perhaps the most significant result of Einstein’s new theory. This frac- tion of condensed gas increases with decreasing temperature, and Einstein
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