Equipartition Theorem and Number of Modes

[Opus_723]

I was looking at this page:

http://en.wikipedia.org/wiki/Molecular_vibration

And saw that they have six different vibrational modes listed for the water molecule. Elsewhere on they page they discuss the "3N-6" rule for determining the number of vibrational modes of a polyatomic molecule. That seems contradictory to me. There seems to be clearly six possible modes.

I tried to check around the web for an answer to the contradiction, but everything else I find only lists the first three modes (the ones pictured on the wikipedia page as "symmetrical stretching," "asymmetrical stretching," and "scissoring.") And make no mention of the other three. If you only count these three, the 3N-6 rule works, of course. But I don't see how you can simply ignore the others, particularly since "scissoring" appears to be simply "asymmetrical rocking."

Is this just a disconnect between classical and quantum mechanics? Are some of those six "modes" not actually possible, or do they just not contribute to the heat capacity for some other reason?

 

[Simon Bridge]

Not sure I see the contradiction: can you provide an example?

3N-6 is for normal modes of vibration
It is only a rule of thumb for larger molecules - it just kinda averages out that way to a good approximation.
Clearly the rule does not work is N<3 for example.

The wikipedia page does not mention "water".

There are 6 modes illustrated in an animation for the -CH₂ group.
... suggests that you need N>3 doesn't it?

 

[Opus_723]

Sorry, it says -CH2 group, but if you view the Simple English version of the page it says something to the effect of "or water molecule," concerning the same diagram. I must have been looking at the other version first and not realized.

So is that the only issue? That 3N-6 is simply of rough rule of thumb for large N? Because I have seen it used all over the place to justify the water molecule only having 3 vibrational modes, as I noted above. For example, here:

http://books.google.com/books?id=6KDwy4SKYpIC&pg=PA218&lpg=PA218&dq=water+molecule+3n-6+vibrational+modes&source=bl&ots=eDBY_okWsU&sig=6zMLfsoq2ihCBYAGxVggfShExWE&hl=en&sa=X&ei=viBFUrjmIqzWiAKogoHoCg&ved=0CGcQ6AEwCA#v=onepage&q=water%20molecule%203n-6%20vibrational%20modes&f=false

Or here:

http://www.stanford.edu/~hkulik/www/Tutorials/Entries/2011/12/27_Vibrational_properties_of_molecules.html

That wikipedia page is the only thing I can find that even acknowledges the other three modes. I've only seen 3N-6 used as an exact answer, even for small N.

 

[Simon Bridge]

The -CH₃ group in the example is attached to something - which is why that group gets six modes.
If it were free, then three of the modes would become rotations: as is with water.

 

[Opus_723]

This is a bit late, but thanks!

 

[AlephZero]

That wikipedia page is the only thing I can find that even acknowledges the other three modes. I've only seen 3N-6 used as an exact answer, even for small N.

The "6" comes form the number of motions of the molecule, considering it as a rigid object. In special cases it can be fewer than 6 - for example on your Stamford link, the CO2 molecule has 3N-5 modes, because the three atoms lie in a straight line, and if you consider them to be "points" that can move in any direction but not rotate, you can't represent the rotation of the line along its own length.

In a water molecule the 3 atoms are not in a straight line, so the 3N translations of the atoms can represent all 6 rigid body motions of the molecule.

The more atoms there are in a the molecule, the less likely it is to have any "special" geometry that changes the 6 to a smaller number.

Of course molecules with 2 atoms are a special case, because the two atoms always lie on a straight line!