Ozone, , is bent (point group , molecular plane ) with three fundamentals: ( ,…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Atomic, Molecular & Optical Physics · Molecular structure and spectra

Deriving the Spectrum, Ch Polyatomic molecules: normal modes and symmetry

Problem

Ozone,

\mathrm{O_3}

, is bent (point group

C_{2v}

, molecular plane

yz

) with three fundamentals:

\nu_1=1103\,\mathrm{cm^{-1}}

(

A_1

, symmetric stretch),

\nu_2=701\,\mathrm{cm^{-1}}

(

A_1

, bend) and

\nu_3=1042\,\mathrm{cm^{-1}}

(

B_2

, antisymmetric stretch). A measured infrared spectrum of ozone shows strong lines at

701

1042

and

1103\,\mathrm{cm^{-1}}

, and a weaker line at

2110\,\mathrm{cm^{-1}}

. ((a)) State how many vibrational normal modes ozone has and check that the three strong lines account for all of them. ((b)) Show that

\nu_1+\nu_3

is close to the weak line at

2110\,\mathrm{cm^{-1}}

and explain why this line cannot be a fourth fundamental. ((c)) All three of ozone's fundamentals are infrared active, unlike carbon dioxide, where the symmetric stretch is silent in the infrared. Explain this difference from the point groups of the two molecules. ((d)) The

8

13\,\mu\mathrm m

range is called the atmospheric window because most gases do not absorb there. Convert

\nu_3=1042\,\mathrm{cm^{-1}}

to a wavelength using

\lambda(\mu\mathrm m)=10^4/\tilde\nu(\mathrm{cm^{-1}})

, and say what this implies for the window.

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