The Equation of State of a Non-ideal Einstein-Bose or Fermi-Dirac Gas

Abstract

With regard to the question if from isotherm measurements one can obtain an experimental test for the existence of Bose statistics in real gases, as is required by theory, we prove the following general theorem. The "Zustandsumme" of a non-ideal Bose or Fermi gas is given by the classical integral provided one replaces the Boltzmann $\exp (-\frac{{\phi }_{\mathrm{ij}}}{\mathrm{kT}})$ factor by: $e^{-\frac{{\phi }_{\mathrm{ij}}}{\mathrm{kT}}}(1\mathrm{}\pm \exp [-\frac{4{\pi }^2\mathrm{mkT}{r_{\mathrm{ij}}}^2}{h^2}]$ for each pair of molecules ($\mathrm{ij}$). For the second virial coefficient $B$, this has, i.e., in a Bose gas, as a consequence that: $B=B_{\mathrm{n}\mathrm{o}\mathrm{n}-\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s}}+B_{\mathrm{ideal}\mathrm{Bose}}+B^{\prime }$ where: $B^{\prime }=2\mathrm{}\pi N\int 0^{\infty }\mathrm{dr}{r}^2(1-e^{-\frac{\phi \mathrm{}\left(r\right)\mathrm{}}{\mathrm{kT}}})\exp [-\frac{4{\pi }^2\mathrm{mkT}{r}^2}{h^2}].$ Only at very low temperatures do the last two terms in (2) become appreciable. They are then of the same order of magnitude, but have opposite signs. Due to this fact, due to the lack of precise knowledge of the molecular forces, and due to the absence of accurate measurements of $B$ at very low temperatures, one can as yet not decide from isotherm measurements alone whether or not real gases obey the Bose statistics.