Physical Chemistry Third Edition

23.3 Rotational and Vibrational Spectra of Diatomic Molecules 961

a.n3ton 2

b.n4ton 2

c.n5ton 2

d.n6ton 2

Compare these wavelengths with those of a normal

hydrogen atom.

23.11Calculate the frequency and the wavelength of the light
emitted in the following electronic transitions of a He+
ion. For each transition specify whether the radiation
is in the ultraviolet, visible, or infrared parts of the
spectrum.
a.n3ton 2
b.n2ton 1
c.n5ton 4

23.12Crudely approximate the electronic motion in a
hydrogen atom by assuming that the electron moves in
a cubical three-dimensional hard box of dimension
1.00 Å on a side. Calculate the wavelength of the light
emitted if the electron makes a transition from the first
excited level to the ground state, and compare this with
the wavelength of the light emitted in then2to
n1 transition using the correct energy levels. What
would the dimension of the hard box be to make the
transition have the same wavelength as the correct
n2ton1 transition?

23.13Tell whether each of the following transitions in a

hydrogen atom is allowed or forbidden:

a.n2,l1,m1ton1,l0,m 0

b.n4,l2,m1ton2,l1,m 1

c.n3,l2,m1ton2,l2,m 1

d.n6,l4,m1ton2,l2,m 1

23.14Tell whether each of the following transitions in a carbon

atom is allowed or forbidden:

a.^1 D 2 to^1 S 0

b.^3 P 2 to^3 P 1

23.15Tell whether each of the following transitions in a

selenium atom is allowed or forbidden:

a.^1 D 2 to^3 P 1

b.^3 P 0 to^1 S 0

23.16a.Consider the excited states of the helium atom that

arise from the (1s)(2s) and (1s)(2p) configurations.

These states were discussed in Chapters 18 and 19.

Draw a Grotrian diagram for these states and the

ground state of the He atom.

b. Using energies from Figure 19.3, estimate the

wavelengths at which spectral lines would be found

from the transitions of part a.

23.3 Rotational and Vibrational Spectra of

Diatomic Molecules

Rotational Spectra of Diatomic Molecules

If a diatomic molecule is represented as a rigid rotor, the transition dipole moment

integral for a rotational transition is

(μ)JM′′,J′M′

∫

Y∗J′′M′′μYJ′M′sin(θ)dθdφ (23.3-1)

TheYfunctions are the rotational wave functions (spherical harmonic functions) and

μμ(r,θ,φ) is the dipole moment operator of the molecule in the Born–Oppenheimer

approximation. The selection rules that result are^8

∆J±1 for a molecule with nonzero permanent dipole moment (23.3-2a)

(^8) I. N. Levine,Molecular Spectroscopy, Wiley, New York, 1975, p. 162ff.