So, while reading over equations of states, I learned that quantum gases do not conform to the same corresponding state behavior as normal fluids do. Why are these known as quantum gases and why do they not conform to the same corresponding state behavior as normal fluid?

One example of this language, appearing in Introduction To Chemical Engineering Thermodynamics by JM Smith, is as follows:

The Lee/Kessler correlation provides reliable results for gases which are nonpolar or only slightly polar; for these, errors of no more than 2 or 3 percent are indicated. When applied to highly polar gases or to gases that associate, larger errors can be expected.

The quantum gases (e.g., hydrogen, helium, and neon) do not conform to the same corresponding-states behaviour as do normal fluids. Their treatment by the usual correlations is sometimes accommodated by use of temperature-dependent effective critical parameters.18 For hydrogen, the quantum gas most commonly found in chemical processing, the recommended equations are: \begin{align} T_c/\mathrm{K} = \frac{43.6}{1+\frac{21.8}{2.016 T}} \quad (\text{for H}_2) \tag{3.58} \\ P_c/\mathrm{bar} = \frac{20.5}{1+\frac{44.2}{2.016 T}} \quad (\text{for H}_2) \tag{3.59} \\ V_c/\mathrm{cm}^3\:\mathrm{mol}^{-1} = \frac{51.5}{1-\frac{9.91}{2.016 T}} \quad (\text{for H}_2) \tag{3.60} \end{align}


The usage you have found is at odds with the modern understanding of the term, which (as explained in the existing answer) tends to revolve around low-temperature behaviour, and can include all sorts of gases (say, all the way up to rubidium).

The passage you've quoted seems to be looking at different behaviour, and its meaning becomes clearer in the related paper

Vapor-Liquid Equilibria at High Pressures. Vapor-Phase Fugacity Coefficients in Nonpolar and Quantum-Gas Mixtures P. L. Chueh, and J. M. Prausnitz. Ind. Eng. Chem. Fundamen. 6, 492 (1967),

available as a pdf here, which makes the claim much more clear:

Quantum Gases

The configurational properties of low-molecular-weight gases (hydrogen, helium, neon) are described by quantum, rather than classical, statistical mechanics.

(The rest of that passage looks eerily similar to the one in your textbook. Is the Chueh & Prausnitz paper the reference 18 cited in your book? If it isn't, there's some pretty flagrant behaviour there.)

Basically, what they're claiming is that if you're studying the dynamics of a gas molecule leaving the liquid phase and into more open space, then classical mechanics is a good approximation so long as the molecule is massive enough, and that this approximation works well for all but the very lightest of molecules.

That's where your listing comes in: H$_2$, He and Ne are the lightest possible constituents of reasonable gases, as most everything in between will coalesce into diatomics that are heavier than neon. Presumably the claim goes that by the time you get to N$_2$ at mass 14 then the quantum mechanical effects become effectively negligible.

(And there are, of course, unreasonable gases ─ HF in particular, but also potentially Li$_2$ and Be$_2$ ─ which lie below that mass-$10$ cutoff, so presumably the fugacity calculations would need to be repeated for them, but I don't think that studying the equilibrium gas and liquid fractions of hydrofluoric acid as a function of temperature is a particularly appealing experiment.)

  • $\begingroup$ Lithium forms a diatomic gas? Never heard of that before, though presumably the formation requires extreme-ish conditions. I don't see any reason why that would be stable, though chem's notoriously strange. Do you know of any good literature about it? $\endgroup$
    – user191954
    Jul 17 '18 at 16:15
  • $\begingroup$ @Chair Wikipedia pegs it as stable, but it does also put it at about a 1% mass fraction of vapor-phase lithium (without a reference), though presumably that should depend on the temperature. I don't see what's particularly surprising about it - if the temperature is high enough and the pressure is just right, why wouldn't gas-phase lithium form dimers? $\endgroup$ Jul 17 '18 at 16:19
  • $\begingroup$ Ah, never mind. I somehow had a very distinct impression that $\text{Na}$ exists as a monomer in the gaseous state, so I was expecting $\text{Li}$ to show a similar trend. Turns out that wikipedia says $\text{Na}_2$ and $\text{Li}_2$ are the stable ones. $\endgroup$
    – user191954
    Jul 17 '18 at 16:25
  • $\begingroup$ Well, as a rule of thumb, it's a reasonable approximation that the alkali metals are roughly hydrogenic on their own in vacuum, so dimers are generally my initial guess. The Wikipedia claim that Li forms clusters is not unreasonable, but then again so does H at low enough temperatures, so presumably you just have to crank the temperature up and the clusters might start to break apart. $\endgroup$ Jul 17 '18 at 16:29
  • $\begingroup$ @EmilioPisanty the book does indeed refers to a paper from Prausnitz et al, but it is not the one you mentioned. It is dated 1999 and the co authors are different $\endgroup$ Jul 17 '18 at 16:50

At high temperature, all of the elements you said will closely follow the behavior of ideal gas.

Those gases reach quantum degeneracy when temperature becomes cold enough such that the thermal de Broglie wavelength (inversely proportional to standard deviation in momentum - as temperature goes down, momentum spread decreases) starts to become comparable to interparticle spacing (see the Wikipedia article on Thermal de Broglie wavelength).

Another way to say the same thing is that the phase space density of the gas starts to approximate unity.

At this point, the quantum statistics of particles become important, and the quantum degenerate gas can be classified as Bose-Einstein condensate or Fermi degenerate gas, depending on whether the element is boson or fermion. I think all of the elements you listed only have bosonic isotopes.

Interacting Bose-Einstein condensates display superfluid behavior. Look up any popular article on BEC.

Now some miscellaneous points:

  1. The first hydrogen BEC was created at MIT by Dan Kleppner and Tom Greytak. Most elements form solid at low temperature, but hydrogen stays gaseous. Actually you can make BEC with other elements (e.g. alkali), but they are metastable.

  2. helium is special since it stays as fluid at low temperature. You need high pressure to make it solidify at low temperature. Superfluid helium is an example of strongly interacting superfluid, whereas other quantum degenerate gases are typically weakly interacting, unless you modify the interparticle scattering behavior using external fields.

  • 2
    $\begingroup$ This is the modern understanding of the term, but it seems to have little to do with the usage as in the example in the question. $\endgroup$ Jul 17 '18 at 15:56
  • 1
    $\begingroup$ Indeed. After the OP added more information to the question, it seems like the relevant topic is the virial expansion of the gas equation of state (power series expansion of the deviation from the ideal gas law), and what principles (quantum-mechanical vs classical) are used to calculate the virial coefficients. $\endgroup$
    – wcc
    Jul 17 '18 at 16:30
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    $\begingroup$ It's still worthwhile to have the modern usage of the term make a presence here, though. $\endgroup$ Jul 17 '18 at 16:31
  • $\begingroup$ @IamAStudent Not specifically virial gas equation of state, but it may well apply to the general cubic equation of state as well! $\endgroup$ Jul 17 '18 at 16:53
  • 1
    $\begingroup$ @ShahMHasan A virial expansion is a generic approach, and the cubic equation of state is just one example. $\endgroup$
    – wcc
    Jul 17 '18 at 17:04

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