Why 3n 6 vibrational modes

Molecular Vibrations A molecule has translational and rotational motion as a whole while each atom has it's own motion. Calculate Number of Vibrational Modes Degree of freedom is the number of variables required to describe the motion of a particle completely.

The following procedure should be followed when trying to calculate the number of vibrational modes: Determine if the molecule is linear or nonlinear i. Answer In this molecule, there are a total of 5 atoms. Answer In this molecule, there are a total of 60 carbon atoms. References Harris, Daniel C. New York: Dover Publications, Housecroft, Catherine E. Viewed 7k times. Improve this question. And how many degrees of freedom does each atom have? Add a comment. Active Oldest Votes.

Improve this answer. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Similarly a rotational motion can also be resolved into its components. Likewise the same is true for vibrational motion. Degree of freedom is the number of variables required to describe the motion of a particle completely. For an atom moving in 3-dimensional space, three coordinates are adequate so its degree of freedom is three.

Its motion is purely translational. If we have a molecule made of N atoms or ions , the degree of freedom becomes 3N, because each atom has 3 degrees of freedom. Furthermore, since these atoms are bonded together, all motions are not translational; some become rotational, some others vibration.

For non-linear molecules, all rotational motions can be described in terms of rotations around 3 axes, the rotational degree of freedom is 3 and the remaining 3N-6 degrees of freedom constitute vibrational motion.

For a linear molecule however, rotation around its own axis is no rotation because it leave the molecule unchanged. So there are only 2 rotational degrees of freedom for any linear molecule leaving 3N-5 degrees of freedom for vibration.

The number of vibrational normal modes can be determined for any molecule from the formula given above. A linear molecule will have another bend in a different plane that is degenerate or has the same energy.

This accounts for the extra vibrational mode. It is important to note that there are many different kinds of bends, but due to the limits of a 2-dimensional surface it is not possible to show the other ones.

The frequency of these vibrations depend on the inter atomic binding energy which determines the force needed to stretch or compress a bond. The point group is also C 2v but the molecule has 11 atoms. In this case we would have to draw up a C 2v character table showing the symmetries of all 27 vibrations. For this reason vibrations are not normally included in character tables. Instead it is up to the user you to determine the number and type of vibrations as defined by their symmetry species for any molecules that are encountered.

How we do this is described below. Reducible Representations. As previously noted one of the important properties of character tables is that the symmetry operations are members of a mathematical group and important relationships exist between them.

In general the same can be said about the symmetry species or irreducible representations of the molecular properties. The new representations are no longer the most simple possible and are referred to as "reducible representations" because they can be "reduced" down to their component parts. For example, for water the symmetries of the translations T x , T y , T z can be gleaned directly from the C 2v character table.

We know that A 1 , B 1 and B 2 are merely the shorthand notations for the symmetry species irreducible representations. This is a valid reducible representation and a member of the C 2v group. In a similar manner we can write down the reducible representation for all three rotations and all the vibrations. The representation for ALL the degrees of freedom for the molecule 3N can be written as.

Make a note of this - it will come up later! Although the set of characters 9 -1 3 1 is a valid member of the C 2v group it is not particularly useful when discussing molecular properties. What is really required is the language of the shorthand symmetry species Mulliken symbols, A 1 , B 1 etc. From the information covered above it can be easily shown by inspection that:. The good news is that for other more complex molecules we do not have to perform this task by inspection because:.