If you like us, please share us on social media.
The latest UCD Hyperlibrary newsletter is now complete, check it out.

GeoWiki.png
ChemWiki: The Dynamic Chemistry E-textbook > Physical Chemistry > Spectroscopy > Vibrational Spectroscopy > Vibrational Modes > Number of vibrational modes for a molecule

MindTouch
Copyright (c) 2006-2014 MindTouch Inc.
http://mindtouch.com

This file and accompanying files are licensed under the MindTouch Master Subscription Agreement (MSA).

At any time, you shall not, directly or indirectly: (i) sublicense, resell, rent, lease, distribute, market, commercialize or otherwise transfer rights or usage to: (a) the Software, (b) any modified version or derivative work of the Software created by you or for you, or (c) MindTouch Open Source (which includes all non-supported versions of MindTouch-developed software), for any purpose including timesharing or service bureau purposes; (ii) remove or alter any copyright, trademark or proprietary notice in the Software; (iii) transfer, use or export the Software in violation of any applicable laws or regulations of any government or governmental agency; (iv) use or run on any of your hardware, or have deployed for use, any production version of MindTouch Open Source; (v) use any of the Support Services, Error corrections, Updates or Upgrades, for the MindTouch Open Source software or for any Server for which Support Services are not then purchased as provided hereunder; or (vi) reverse engineer, decompile or modify any encrypted or encoded portion of the Software.

A complete copy of the MSA is available at http://www.mindtouch.com/msa

Number of vibrational modes for a molecule

All atoms in a molecule are constantly in motion while the entire molecule experiences constant translational and rotational motion. A diatomic molecule contains only a single motion. Polyatomic molecules have more than one type of vibration, known as normal modes.

Molecular Vibration

A molecule has translational and rotational motion as a whole while each atom has it's own motion. The vibrational modes can be IR or Raman active. For a mode to be observed in the IR spectrum, changes must occur in the permanent dipole (i.e. not diatomic molecules). Diatomic molecules are observed in the Raman spectra but not in the IR spectra. This is due to the fact that diatomic molecules have one band and no permanent dipole, and therefore one single vibration. An example of this would be O2 or N2. However, unsymmetric diatomic molecules (i.e. CN) do absorb in the IR spectra. Polyatomic molecules undergo more complex vibrations that can be summed or resolved into normal modes of vibration.

The normal modes of vibration are: asymmetric, symmetric, wagging, twisting, scissoring, and rocking for polyatomic molecules.

 

Symmetricical Stretching Asymmetrical Stretching Wagging
Symmetrical_stretching.gif Asymmetrical_stretching.gif Wagging.gif
Twisting Scissoring Rocking
Twisting.gif Scissoring.gif Modo_rotacao.gif

Figure 1: Six types of Vibrational Modes. Taken from publisher http://en.wikipedia.org/wiki/Infrared_spectroscopy with permission from copyright holder.

Calculate Number of Vibrational Modes

3n degrees of freedom describe the motion of a molecule in relation to the coordinates (x,y,z). The 3n degrees of freedom also describe the translational, rotational, and vibrational motions of the molecule. There are three degrees of freedom for translational, movement through space, and rotational motion, each for a nonlinear molecule. Therefore, translational and rotational can move and rotate around each of the three Cartesian axes. However, a nonlinear molecule can only rotate around 2 of the Cartesian axes because the rotation about the molecular axis does not represent a change of the nuclear coordinates. If you subtract the translational and rotational degrees of freedom, you obtain the following equations shown below for the degrees of vibrational freedom.

The degrees of vibrational modes for linear molecules can be calculated using the formula:

\[3n-5 \tag{1}\]

The degrees of freedom for nonlinear molecules can be calculated using the formula:

\[3n-6 \tag{2}\]

\(n\) is equal to the number of atoms within the molecule of interest. The following procedure should be followed when trying to calculate the number of vibrational modes:

  1. Determine if the molecule is linear or nonlinear (i.e. Draw out molecule using VSEPR). If linear, use Equation 1. If nonlinear, use Equation 2
  2. Calculate how many atoms are in your molecule. This is your n value.
  3. Plug in your \(n\) value and solve.

 

Example 1: \(CS_2\)
An example of a linear molecule would be \(CS_2\). There are a total of \(3\) atoms in this molecule. Therefore, to calculate the number of vibrational modes, it would be 3(3)-5 = 4 vibrational modes.
Example 2: \(CCl_4\)
CH4 is an example of a nonlinear molecule. In this molecule, there are a total of 5 atoms. Therefore, there are 3(5)-6 = 9 vibrational modes.
Example 3: \(POCl_3\)
A more complex example could be \(POCl_3\). The shape of this molecule dictates that this is a nonlinear molecule. It contains 5 atoms and therefore would have 9 degrees of vibrational freedom.

Why would CO2 and SO2 have a different number for degrees of vibrational freedom? Following the procedure above, it is clear that CO2 is a linear molecule while SO2 is nonlinear. SO2 contains a lone pair which causes the molecule to be bent in shape, whereas, CO2 has no lone pairs. It is key to have an understanding of how the molecule is shaped. Therefore, CO2 has 4 vibrational modes and SO2 has 3 modes of freedom.

References

  1. Harris, Daniel C., and Michael D. Bertolucci. Symmetry and Spectroscopy: an Introduction to Vibrational and Electronic Spectroscopy. New York: Dover Publications, 1989. Print.  
  2. Housecroft, Catherine E., and Alan G. Sharpe. Inorganic Chemistry. Harlow: Pearson Education, 2008. Print.
  3. http://en.wikipedia.org/wiki/Infrared_spectroscopy
You must to post a comment.
Last Modified
07:53, 23 Aug 2014

Tags

Classifications

(not set)
(not set)

Creative Commons License Unless otherwise noted, content in the UC Davis ChemWiki is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License. Permissions beyond the scope of this license may be available at copyright@ucdavis.edu. Questions and concerns can be directed toward Prof. Delmar Larsen (dlarsen@ucdavis.edu), Founder and Director. Terms of Use