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I am trying to understand the degres of freedom for gases. Air has 5 degrees of freedom at room temperature but why does Argon, $Ar$ have 3 degrees of freedom while $O_2$ has 5 at the same temperature (at 1 atm) and 3 at 100K?

I looked up the boiling temperature for oxygen and it is $-183^oC=90K$ and it is well below room temperature for both. If the boiling point is well below the room temperature for both $Ar$ and $O_2$, I thought that all atoms now have so much energy that they move (3 degrees), rotate (2 degrees) and vibrate (2 degrees) = 7.

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A single atom can't vibrate, because vibration implies a change in size, which an atom can't really do. You could consider moving electron to larger orbits to be kind of a vibration, but that takes much higher temperatures. For vibration you need multiple atoms: a two-atom molecule has a single vibration degree of freedom, which is just the distance between the atoms.

An atom can rotate about itself; however, as explained in the answers here and here, the moment of inertia of such a rotation, as well as that of a linear molecule about its axis, is very small, which for a fixed angular momentum means a very large energy and hence very large temperature, so we don't count it either.

At everyday temperatures, you can essentially think of atoms as points and think geometrically. A point cannot rotate or vibrate, you need two or more points for that.

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  • $\begingroup$ Thank you! So I get that Ar is one atomic gas and hence can only have 3 degrees of freedom at maximum. But what about oxygen? Why does it have 5 degrees of freedom at room temperature but only 3 at 100K? Is there some point that this goes from 5 to 3? $\endgroup$
    – Clone
    Commented Oct 26, 2020 at 1:45
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    $\begingroup$ @Clone Oh, I missed that part of your question. The number of degrees of freedom depends on the temperature, because of quantum mechanics: there is a minimum amount of energy to excite a degree of freedom. Below that energy, the degree of freedom "doesn't count". Of course, the transition is smooth, not sharp: as the temperature rises, there are more and more atoms with enough energy to excite that degree of freedom. $\endgroup$
    – Javier
    Commented Oct 26, 2020 at 1:51
  • $\begingroup$ So is there some point like "boiling point" or "critical point" that dictates when oxygen goes from 5 to 3 degrees that I can look up in a table somehwere? $\endgroup$
    – Clone
    Commented Oct 26, 2020 at 2:02
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    $\begingroup$ @Clone You could look at a plot of the specific heat (at constant volume) as a function of temperature, since $c_v = Nk/2$, where $N$ is the number of degrees of freedom. But I don't know of any definitive reference. $\endgroup$
    – Javier
    Commented Oct 26, 2020 at 2:13
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For oxygen the transition to rotational freedom is about 2.07 K, for nitrogen about 2.86 K but hydrogen is a warm 85.4 K. There is one more degree of freedom for diatomic molecules and that is vibration, the centers of the atoms moving away and towards each other. That transition is at much higher temperatures and as Javier said comes in gradually. The Gas Dynamics text by Owczarek I used for those numbers used Statistical Thermodynamics by Fowler and Guggenhein as his reference.

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