Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 2A / Section B / 22 / a / ii

Write the upper Morse oscillator term values in wavenumber units asThe spacing is $G(v+1)-G(v)={\tilde{\omega }}_e-2a(v+1)$. A Birge-Sponer plot therefore falls linearly in this model and can be extrapolated to zero spacing. If its horizontal coordinate is $x=v+1/2$, the plotted line is $y=({\tilde{\omega }}_e-a)-2ax$. Its area from $x=0$ to its zero givesEquivalently a plot against $v+1$ has intercept ${\tilde{\omega }}_e$ and triangle area ${\tilde{D}}_e={\tilde{\omega }}_e^2/(4a)$; subtracting $G(0)={\tilde{\omega }}_e/2-a/4$ recovers ${\tilde{D}}_0$. These conventions distinguish dissociation from the potential minimum and from the lowest vibrational level.

Measured consecutive members fix a local slope and intercept, so unobserved members can be supplied by extrapolation under an explicit spacing law. The upper-state limit on the absorption scale is ${\tilde{\nu }}_{\lim }={\tilde{\nu }}_{00}+{\tilde{D}}_0$. A local high-v fit can likewise estimate the remaining energy above an observed member without knowing its absolute v index. Near dissociation a real potential need not remain Morse-like, so an extrapolated triangle has model uncertainty; missing lines must not be silently assumed consecutive.