Symmetry of Cyclohexane: How Many Vibrations in Its Spectrum?

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SUMMARY

The chair conformer of cyclohexane belongs to the D3h point group. In its vibrational spectrum, there are 18 total vibrational modes, with 6 being Raman active and 12 being infrared active. The confusion arises from miscalculating the total number of vibrations, which should not exceed the initial 18 vectors derived from the three vibrational modes per carbon atom. Proper application of symmetry operations is crucial for accurate vibrational analysis.

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Homework Statement


Determine the point group of the chair conformer of cyclohexane. How many Raman active vibrations and how many infrared active vibrations might you expect to detect in its spectrum?


Homework Equations



(see link below)

The Attempt at a Solution



http://img442.imageshack.us/img442/240/symmetrycw.jpg

Im rubbish with LaTex so I've hosted a picture of what I've done in word. I've determined the point group to be D3h but when I use the equation to get the values for the different vibrations, I get a total of 24 when it should be 18 because I started with 18 vectors (3 on each carbon). I am not sure where I am going wrong. Thanks.
 
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