# Why do methane molecules have 15 degrees of freedom?

I've just been learning a little thermal chemistry including the equipartition theorem. As part of this, my textbook discusses how to figure out the degrees of freedom for different chemicals.

The answer to one of the exercises says methane has 15 degrees of freedom: 3 for translation, 3 for rotations and 9 for vibrations. I am confused by two of these.

Firstly the text says that diatomic molecules like oxygen gas only have 2 rotational degrees of freedom, because they can't rotate about their axis of symmetry. However methane has an axis of symmetry, so why does it have 3 rotational degrees?

Secondly I don't understand where all the vibration degrees of freedom come from. The molecule has 4 bonds, so wouldn't each bond have one degree for kinetic energy and one for potential energy, for a total of 8 degrees, not 9?

• Sorry for my bad english, but did you learn about wagging and twisting? Because those can account for the "missing" degrees of freedom. Commented Sep 25, 2017 at 14:37

## 1 Answer

15 is there because you have 5 atoms which can move independently, and we live in a 3D space, and $3\ \times\ 5\ =\ 15$.

3 of these are translational degrees of freedom, which you seem to agree with, so I'll just leave it at that.

3 are for rotations, because the methane molecule is not linear. Axes of symmetry are irrelevant: indeed, you may rotate methane around the axis of symmetry as well as around any other axis. Being or not being diatomic is hardly relevant either, except for the fact that all diatomic molecules are necessarily linear; however, the converse is not true.

Subtracting 3 and 3 from 15, we get 9, and that's how many vibrational degrees of freedom we have, because they can't be anything else. To visualize them is another story, and a great deal more difficult. Counting bonds doesn't help much. The separation of degrees into kinetic and potential is not a thing at all. You don't have to do that anyway. Write $15\ -\ 3\ -\ 3\ =\ 9$ and call it a day.

• So, there's 9 vibrational degrees not because we counted them, but because there has to be 15 total and we only have one category left? Sure, it works, but that doesn't sound right...
– Mast
Commented Sep 25, 2017 at 10:55
• My understanding is that the remaining 9 don't refer to any one atom, but interactions involving pairs (or even larger groups?) of atoms. This page had some helpful visualizations. Within the constraints of the molecule, the motion of any individual molecule has some affect on the others. Commented Sep 25, 2017 at 12:31
• @leob That's plain nonsense. There is no such thing as a degree of freedom for kinetic or potential energy; instead, every single vibrational degree of freedom represents a motion in which potential energy changes to kinetic and back. Moreover, a diatomic molecule has but one vibration. Commented Sep 25, 2017 at 12:38
• The axis of rotation for diatomics only matters because the symmetry is $C_{\infty}$, i.e., rotation by any amount gives the exact same thing. Methane's axes are $C_2$ and $C_3$.
– Zhe
Commented Sep 25, 2017 at 12:42
• @IvanNeretin I believe the idea of two degrees of freedom for a vibration is a poor attempt by Schroeder to explain that the average energy of an oscillator is $k_bT$ using equipartition because the mean value of any quadratically varying term in the energy is $\frac{1}{2}k_bT$.
– Tyberius
Commented Sep 25, 2017 at 14:45