Degrees of freedom of a molecule of N atoms [closed]








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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?




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


 5