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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.

Clone
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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.

Javier
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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.

W H G
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