Past exam of the chemistry A course of the University of Cambridge / 2022 / Paper 1 / A3 / a / iv

For a linear polyatomic molecule, the first-order polyatomic vibration-rotation interaction iswhere $d_i$ is the degeneracy of normal mode $i$. The bending mode has $d_2=2$, so the two measured states implyThe two stretching zero-point contributions are not supplied. The true all-mode equilibrium constant is therefore not uniquely determined by this band alone. Neglecting those unknown stretching corrections gives the bend-only estimateUsing $B=h/(8{\pi }^{2}cI)$ and converting inverse centimetres to inverse metres,If the intended course approximation instead treats the observed vibration as a single nondegenerate oscillator with $B_v=B_e-{\alpha }_e(v+1/2)$, its extrapolation givesThat is a useful effective one-coordinate estimate, but it should not be identified with an exact equilibrium constant of the doubly degenerate polyatomic bend. Both estimates give $I\simeq 1.90\times 10^{-46}\mathrm{kg}{\mathrm{m}}^2$ at the accuracy warranted here.