# At what temperature do quantum effects become significant for a gas

Hi, I was wondering how to calculate a rough estimate of the temperature at which quantum effects begin to become important for a gas of interacting particles. Let's say the particles interact via a spherically symmetric potential with an equilibrium distance R and a well-depth E (such as the Lennard-Jones potential). Can I use the mass of the particles and the interaction energy parameters (R & E) to estimate a temperature around which quantum effects become important?

By quantum effects becoming important, I mean they begin to significantly affect thermodynamic properties of the gas, such as the magnitude of the second virial coefficient B(T).

I can calculate things like the thermal de Broglie wavelength, the de Boer parameter, etc., but I was wondering if there was some straightforward way to estimate this. Presumably for rare gas atoms with weaker interactions (Helium) the "quantum temperature" should be larger than that for rare gas atoms with strong interactions (eg, argon).

Thank you - I hope I am clear!

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The derivation is not exactly trivial, and it depends on whether your gas consists of bosons or fermions (you get the same general formula but with different constants in front). Interparticle interactions do not really matter too much. You can find it in most statistical physics textbooks. The critical temperature is on the order of

$$\frac{\hbar^2}{2m} n^{2/3}$$,

where n is particle number density. For a typical gas at atmospheric pressure, it's on the order of a few kelvin.

Hm... a very interesting result. I have been calculating interaction parameters for the rare gas atoms. Following Pahl, Calvo, Kocˇi, and Schwerdtfeger (Angew. Chem. Int. Ed. 2008, 47, 8207 –8210), I've calculated the de Boer parameter for the different RG atoms, getting for instance 0.027 for Ar and 0.079 for Neon.

I am far from an expert but surely there must be some relationship between well depth, width and position and the temperature at which quantum effects begin to become important? Your result tells me there is no difference between the temp for Ar and the temp for He? But there are definite differences between the interaction energies of these atoms...

These quantum effects are present even if the interaction is zero. The dominant effect for fermions, for example, is Pauli exclusion principle. Interactions affect some thermodynamic properties, but I don't think that they significantly change the critical temperature.

Maybe this thread should be moved into the atomic physics subforum. It's a condensed-matter problem. Personally, I'm not an expert on condensed matter.

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