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 R_N and R_O 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 R_N and R_O 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 R_NR_O term, it turns out that the R_N² is negligible. The resulting equation gives a value for R_O.

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

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.