= Solution
Successive differences remove the unknown absolute origin and the unknown initial vibrational quantum number. The five spacings are $329.3$, $296.0$, $261.7$, $226.7$ and $190.7\ \mathrm{cm^{-1}}$. Plot them at relative mid-indices $x=n+1/2$. A straight-line fit gives
$$
\Delta\widetilde\nu(x)\simeq347.505-34.650x\quad\mathrm{cm^{-1}},\qquad x_*\simeq10.029.
$$
The curve reaches zero spacing at $x_*$. Since a quadratic Morse progression has a spacing equal to its derivative at the interval midpoint, integrating the fitted line from the first observed relative level $n=0$ to $x_*$ estimates the energy still needed to reach the continuous limit:
$$
\widetilde\nu_{\rm lim}\simeq56298.9+\int_0^{x_*}(347.505-34.650x)\,dx
=56298.9+\frac{347.505^2}{2(34.650)}
\simeq58041.5\ \mathrm{cm^{-1}}.
$$
Thus the appropriate graphical conclusion is \b[$\widetilde\nu_{\rm lim}\approx5.80\times10^4\ \mathrm{cm^{-1}}$], not a many-digit threshold. Directly fitting a quadratic to the six measured transition positions puts its vertex at about $58042.8\ \mathrm{cm^{-1}}$, which checks the arithmetic but not the physical extrapolation law. The slight variation of second differences shows that extrapolating a constant anharmonicity beyond the observed range is approximate. No assignment of the arbitrary index n to an absolute upper-state v is needed.
\Image[/past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2a-dissociation.png]
{title=Original relative-index Birge–Sponer extrapolation and the inferred progression limit}
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