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Apparently a molecule of N atoms has 3N degrees of freedom. How?
Shouldn't it be dependent on the structure of the molecule? How exactly do we derive this?

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1 Answer 1

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Bonds in a molecule are flexible. Thus, each atom can move in any possible direction in 3D space.

Note that they can't necessarily move any possible distance; indeed, if the molecule is stationary, the atoms may not be able to move much distance at all since, after all, they are bonded to other other atoms. In summmary, each atom can move a limited, non-zero distance in any possible direction.

Thus, to describe the motion of each atom, exactly 3 coordinates needed. Further, since each atom can move in any possible direction regardless of what the other atoms are doing (again, possible distances are constrained, not possible directions), we need exactly 3N coordinates to describe the motions of all atoms in a molecule with N atoms.

Since describing the motions of all atoms in a molecule also completely describes the motion of the molecule as a whole, no additional coordinates are needed. Hence the number of degrees of freedom (DOF's) in a molecule is equal to 3N, independent of the molecule's structure.

Structure does matter when it comes to vibrational and rotational DOF's, however. A linear molecule needs two angles to describe its orientation in space, while a non-linear molecule needs three. Hence these have 2 and 3 rotational DOF's, respectivey.

From this, and from the fact that a molecule as a whole has 3N DOF's, we can calculate the number of vibrational DOF's by simple subtraction:

Linear molecule:

Total DOF's = 3 N

Translational DOF's = 3

Rotational DOF's = 2

Vibrational DOF's = 3N – 5

Non-linear molecule:

Total DOF's = 3 N

Translational DOF's = 3

Rotational DOF's = 3

Vibrational DOF's = 3N – 6

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  • $\begingroup$ DOFs are defined as the number of ways a dynamic system can move.A linear diatomic molecule could translate along three axes, rotated about two, and vibrate about 3. So the number of DOFs I get is 8. Where am I going wrong? $\endgroup$ Commented Jan 18, 2022 at 5:33
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    $\begingroup$ Diatomic molecules have only two atoms. Thus they have only one vibrational mode. Hence DOF = 3 trans + 2 rot + 1 vib = 6 = 3*2 = 3N. Note also that you wrote "linear diatomic". Since diatomics have only two atoms, there was no need to add the "linear" qualifier, since they are all linear. $\endgroup$
    – theorist
    Commented Jan 18, 2022 at 5:44
  • $\begingroup$ So can't the molecule vibrate along all the three axes? $\endgroup$ Commented Jan 18, 2022 at 16:20
  • $\begingroup$ Its vibrational motion consists only of the atoms moving apart and coming together. So that's just one mode. You're arguing that the molecule can do this in any orientation in space, which is true. But that's already accounted for by its two rotational DOF's. I.e., those are not different vibrational modes, those are the same vibration taking place while the molecule is oriented in different directions in space. So when you get 8 DOF's you are double-counting the 2 rotational DOF's. $\endgroup$
    – theorist
    Commented Jan 18, 2022 at 16:56
  • $\begingroup$ ....Or you might be thinking of a movement in which the atoms move apart, but not along a straight line. Within the reference frame of the diatomic molecule, that can't happen—they can only move back and forth on a straight line, corresponding to the molecule's one vibrational mode. However, from the perspective of an external observer, the atoms can move apart along a curve. But that's not a 2nd vibrational mode. That's simply what happens, from the point of view of an external observer, when vibration and rotation occur at the same time. $\endgroup$
    – theorist
    Commented Jan 19, 2022 at 7:44

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