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252 7 QUANTUM THEORY
(c) Comparing it to the result from Problem P7E.1, there is a 1.5% di�erence,
which is relatively small. �e assumption of replacing the e�ective mass
by the mass of the lighter nucleus is quite a good approximation.
(d) A C–H vibration in an organic molecule can be thought of as the H atom
moving against the rest of the molecule, which has a much greater mass.
�erefore, to some rough approximation such vibrations can bemodelled
by a diatomic X–1H in which mX ≫ mH. In such a situation the analysis
in part (a) applies and the vibrational frequency of X–2H is well approxi-
mated as (1 mu/2 mu)1/2 = 0.7 times that of X–1H.
P7E.4 �e vibrational frequency in Hz, ν is given by ν = (1/2π) (kf/µ)1/2; using this
the following table is drawn up
12C16O 13C16O 12C18O 13C18O
µ/mu 6.85... 7.17... 7.2 7.54...
ν/THz 64.3 62.8 62.7 61.3
P7E.6 As in Problem P7E.5, the force constant can be found from the wavenumber of
the υ = 0 → 1 transition, kf = µ(2πν̃c)2. �is allows the following table to be
drawn up
1H35Cl 1H81Br 1H127I 12C16O 14N16O
ν̃/ cm−1 2990 2650 2310 2170 1904
µ/ mu 0.972... 0.987... 0.992... 6.85... 7.46...
kf/ Nm−1 512 409 312 1902 1595
Hence, the order of increasing sti�ness is HI