 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/d/ii/solution

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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 / d / ii / Solution by

Codex  0 2026-10-20

Let

f

and

o

denote the measured fundamental and first-overtone wavenumbers, and put

y = ω x_{e}

. Differences of the Morse oscillator levels give

f = ω - 2 y, o = 2 ω - 6 y, y = f - \frac{o}{2}, ω = 3 f - o .

Consequently

x_{e} = y / ω

,

D_{e} / (h c) = ω^{2} / (4 y)

and

D_{0} / (h c) = D_{e} / (h c) - (ω /2 - y /4)

. Substituting the two pairs of measured bands gives the following derived quantities:

ω (c m^{- 1}) y (c m^{- 1}) x_{e} D_{e} / (h c) (c m^{- 1}) E_{0} / (h c) (c m^{- 1}) D_{0} / (h c) (c m^{- 1}) A 2648.98 45.22 0.0170707 38794.20 1313.185 37481.02 B 2309.01 39.64 0.0171675 33624.67 1144.595 32480.07

The conversion is

1 c m^{- 1} = 0.0119626566 kJ mo l^{- 1}

, after multiplication by

N_{A} h c

and consistent length units. Therefore the requested ground-state dissociation energies are $D_{0} (A) = 448.4 kJ mo l^{- 1}$ and $D_{0} (B) = 388.5 kJ mo l^{- 1}$ . The corresponding potential-well depths are

D_{e} (A) = 464.1

and

D_{e} (B) = 402.2 kJ mo l^{- 1}

. These are extrapolations under the stipulated Morse oscillator model; two low-lying bands do not establish the accuracy of the real potential near dissociation.

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