Rotational Spectra and Structure of a Linear Molecule

The frequencies of microwave absorption transitions can be used to calculate molecular structure parameters. The structure of nitrous oxide is one which can be readily calculated from the measurement of molecular rotational spectra. The structure of this linear molecule involves only two parameters RN and RO as illustrated.

The rotational energy of the molecule is given by

The J=0 to J=1 transition of NNO can be measured to determine the moment of inertia I of the molecule.

But if only one moment of inertia is measured, there is not sufficient data to calculate the structure since it involves two unknown parameters. If the isotope 15N is substituted at position m1, then another moment of inertia is determined and the two structural parameters can be determined. This strategy of isotopic substitution to gain structural information is a common one, depending upon the fact that the structure of the molecule is determined by the electric properties of the atoms and unaffected by substitution of a different isotope.

The moment of inertia in terms of the molecular bond lengths RN and RO is

where M is the total mass of the molecule. The moments of inertia of the two isotopic species can be obtained from the following spectral measurements.

Atomic masses

16O: 15.99491 amu
14N: 14.00307 amu
15N: 15.00011 amu
M: 44.00105 amu
M':44.99809 amu

Substituting mass values into the moment of inertia equation

When moment of inertia values are inserted and the pair of equations factored to eliminate the RNRO term, it turns out that the RN2 is negligible. The resulting equation gives a value for RO.

Substituting this back into the moment of inertia equation for the 14N14NO species gives a quadratic equation for the bond length RN.

This algebraic solution for the structural parameters of a molecule shows the utility of rotational spectra for determining molecular structure. In general, you will need as many equations as there are unknown parameters in the structure. Additional equations can be obtained by measuring the spectra of the same molecule with different isotopes substituted.

Rotational Spectra
Determining the structure of a diatomic molecule
Example of the structure of a polyatomic molecule

Molecular spectra concepts
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