 Solution

ID: past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2b/section-b/34/c/solution

The ground-state dissociation energy starts at the lowest vibrational level, rather than the minimum of the potential. The zero-point energy of the Morse oscillator is

E_{0} = h c ω (\frac{1}{2} - \frac{x _{e}}{4}) = D_{e} (2 x_{e} - x_{e}^{2}) .

Subtracting it from the bond potential-well depth gives

D_{0} = D_{e} - E_{0} = D_{e} (1 - 2 x_{e} + x_{e}^{2}) = D_{e} (1 - x_{e})^{2} .

This distinction matters whenever a dissociation energy is inferred from a vibrational spectrum.

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