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We are told in my thermodynamics class that the ideal gas law is a classical law that follows from classical mechanics. However, when we derived it as follows, we computed the partition function $Z_1$ for one particle, by summing over quantized energy states $Z_1=\sum_{n_x, n_y, n_z} e^{-\frac{\hbar^2 \pi^2}{2 m L^2 \tau}\left(n_x^2+n_y^2+n_y^2\right)}$ where $n_i$ are translational degrees of freedom. This assumes quantum mechanics! Then the overall partition function was $Z = \frac{1}{N!}Z_1^N$ since they are non-interacting, and then we computed the free energy $F = -\tau \ln Z$, which gave us the ideal gas law for pressure using $p = -(\frac{dF}{dV})_\tau$.

The only assumptions I understand are made are that 1) gas particles are non-interacting point masses with 2) quantized energy that 3) depends only on translational motion. Due to the final assumption, and in particular, the presence of $\hbar$ in the derivation, this doesn't seem to be a "classical law" to me?

Tanishq Kumar
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It's been known for years that h shows up in purely classical (i.e., non-quantum mechanical and/or non-relativistic) derivations- which is a very cool thing. For example, there's a derivation of the strength of the Casimir effect (due to a guy named Boyer, I'll cite the full reference if I can find it in my archives) out of which h emerges when you do the algebra. Boyer explains that it appears there as a conversion factor between different ways of expressing energy.

Regarding why the computation of the partition function gets an h in it, it might be (that is, I'm waving my hands here) that the quantized energy states represent the statistical binning process i.e, the bins are discretized even though the energies of the particles grouped into any given bin are not.

niels nielsen
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  • https://doi.org/10.1103/PhysRev.182.1374 and 10.1088/0305-4470/36/26/313 – hyportnex Mar 04 '23 at 19:20
  • Boyer's derivations usually assume presence of zero point radiation with density that is function of frequency and $\hbar$, just like it comes out of quantum theory of EM radiation where canonical quantization leads to each oscillator having zero point energy $\hbar \omega/2$. So $h$ does not emerge in Boyer's papers, it is put in by hand as a physical assumption, motivated by the QT of radiation. – Ján Lalinský Mar 05 '23 at 01:19
  • @JánLalinský, is this true of his derivation based on the casimir effect? – niels nielsen Mar 05 '23 at 02:38
  • @nielsnielsen Not sure which derivation/paper you mean. – Ján Lalinský Mar 05 '23 at 04:12
  • @JánLalinský , it was in an article that ran in Scientific American many years ago. – niels nielsen Mar 05 '23 at 04:35