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For {molecule}, calculate the number of rotational modes:

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For {molecule}, calculate the number of vibrational modes:

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Is {molecule} microwave active?

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For {molecule}, calculate the number of infrared active modes:

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select

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a) It is always straightforward to calculate the number of rotational modes for a molecule. If the molecule is linear (and note that all diatomic molecules are linear) then it has 2 rotational modes. If the molecule is non-linear then it has three rotational modes.

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\\[\\rm {For}~{\\var {molecule}}~{\\rm the~number~of~rotational~modes~is~equal~to~} {\\var {get_rot}}\\] 

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b) The overall motion of a molecule can be described using 3N coordinates (where N is the number of atoms in the molecule). This number arises because each individual atom has three degrees of freedom (ie, three coordinates) of translational motion. Simply stated, each individual atom moves in three dimensional space (x,y,z). A detailed analysis (which we won't do here) would confirm that; (i) three of the coordinates that describe the movement of the overall molecule turn out to be translations (the molecule moves in 3 dimensional space, along  x,y and z coordinates) and two or three (depending on whether or not a molecule is linear, see part (a) above for explanation) of the coordinates that describe the movement of the overall molecule turn out to be rotations. When we subtract the number of translational coordinates and the number of rotational coordinates from 3N for a particular molecule, we calculate the number of vibrational coordinates that a molecule has. Note that normal modes can also be referred to as \"coordinates\" or \"degrees of freedom\". For all practical purposes, these three terms are interchangeable. Expressing the information above in a succinct, mathematical form, we can say that; 

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For a linear molecule\\[{\\rm Number~of~vibrational~modes}=3N-5\\]

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For a non-linear molecule\\[{\\rm Number~of~vibrational~modes}=3N-6\\]

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These descriptions yield very simple results, though they are only easy to visualise for small, highly symmetrical molecules. Diatomic molecules have only one vibrational mode (the symmetric stretch). Linear triatomic molecules have four vibrational modes (a degenerate bending motion (\"degenerate\" means that the molecule bends in two  planes, each of which counts as a different mode), a symmetric stretch and an asymmetric stretch) while non-linear triatomics have three vibrational modes (the two stretches and a single, non-degenerate bend). These can be visualised using the animation at this University of Liverpool web site (select from the panel on the left hand side and \"click to see vibration list\" to animate the modes); 

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<http://www.chemtube3d.com/vibrationsH2O.htm>

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So;

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\\[\\rm {For}~{\\var {molecule}}~{\\rm the~number~of~vibrational~modes~is~equal~to~} {\\var {get_vib}}\\] 

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(c) If a molecule has a permanent electric dipole moment, then it absorbs microwave radiation and a microwave spectrum can be measured for it. If it doesn't have an electric dipole moment, the opposite is true. 

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So;

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\\[\\rm {It~is}~{\\var {get_mic}}~that~{\\var {molecule}}~is~microwave~active.\\] 

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(d)

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In order to answer this part, you must decide which of the vibrations of the molecule cause the electric dipole moment of the molecule to oscillate in phase with the vibration.

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For a diatomic molecule, this is easy. If it is homonuclear, then the single vibrational mode is infrared active. If it is heteronuclear, then the single vibrational mode is not infrared active. 

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Larger molecules require more detailed examination. While referring to this link; 

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<http://www.chemtube3d.com/vibrationsH2O.htm>

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select the CO2 molecule in the left hand panel. Then click \"CLICK HERE to show frequency list\". The degenerate ${\\Pi}_{\\rm u}$ vibrations (there are two different ${\\Pi}_{\\rm u}$ modes) and the ${\\Sigma}_{\\rm u}$ vibration each cause the electric dipole moment of the molecule to oscillate in phase with the vibration. The ${\\Sigma}_{\\rm g}$ vibration does not cause the electric dipole moment to oscillate in phase with the vibration. CO2 therefore has three infrared-active vibrational modes and one vibrational mode (the ${\\Sigma}_{\\rm g}$ mode) that is not infrared active.

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If performing the analysis for a linear molecule that does not have a centre of inversion (such as OCS), the molecule will have two ${\\Pi}_{\\rm u}$ modes (ie bending) and two ${\\Sigma}_{\\rm u}$ modes (stretching) such that all 4 modes will be IR active. Performing the analysis for a non-linear molecule such as H2O will reveal 1 bending motion, 1 symmetric stretch and 1 antisymmetric stretch which will all be infrared-active. 

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\\[\\rm {The~number~of~infrared~active~modes~of}~{\\var {molecule}}~is~therefore~{\\var {get_ir}}.\\] 

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Point groups, character tables and symmetry analysis greatly simplify the analysis for molecules with more than 3 atoms. The ${\\Pi}$ and ${\\Sigma}$ symbols in the feedback above are symmetry labels. You will not learn about these in CHY1201 but it will be a topic of future studies. If you wish a more sophisticated explanation of symmetry point groups and character tables now, this is a good source; 

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<https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Spectroscopy/Vibrational_Spectroscopy/Vibrational_Modes/Normal_Modes>

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