 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 $190.7\mathrm{c}{\mathrm{m}}^{-1}$. Plot them at relative mid-indices $x=n+1/2$. A straight-line fit givesThe curve reaches zero spacing at $x_{\ast }$. 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 $x_{\ast }$ estimates the energy still needed to reach the continuous limit:Thus the appropriate graphical conclusion is ${\tilde{\nu }}_{\lim }\approx 5.80\times 10^4\mathrm{c}{\mathrm{m}}^{-1}$, not a many-digit threshold. Directly fitting a quadratic to the six measured transition positions puts its vertex at about $58042.8\mathrm{c}{\mathrm{m}}^{-1}$, 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.