13
Atomic structure and atomic spectra
Solutions to exercises
Discussion questions
E13.1(b)
(1) The principal quantum number,
n
, determines the energy of a hydrogenic atomic orbital through
eqn 13.13.
(2) The azimuthal quantum number,
l
, determines the magnitude of the angular momentum of a
hydrogenic atomic orbital through the relation
{
l(l
+
1
)
}
1
/
2
¯
h
.
(3) The magnetic quantum number,
m
l
, determines the z-component of the angular momentum of a
hydrogenic orbital through the relation
m
l
¯
h
.
(4) The spin quantum number,
s
, determines the magnitude of the spin angular momentum through
the relation
{
s(s
+
1
)
}
1
/
2
¯
h
. For a hydrogenic atomic orbitals,
s
can only be 1
/
2.
(5) The spin quantum number,
m
s
, determines the z-component of the spin angular momentum
through the relation
m
s
¯
h
. For hydrogenic atomic orbitals,
m
s
can only be
±
1
/
2.
E13.2(b)
(a)
A boundary surface for a hydrogenic orbital is drawn so as to contain most (say 90%) of the
probability density of an electron in that orbital. Its shape varies from orbital to orbital because
the electron density distribution is different for different orbitals.
(b)
The radial distribution function gives the probability that the electron will be found anywhere
within a shell of radius
r
around the nucleus. It gives a better picture of where the electron is
likely to be found with respect to the nucleus than the probability density which is the square of
the wavefunction.
E13.3(b)
The ±rst ionization energies increase markedly from Li to Be, decrease slightly from Be to B, again
increase markedly from B to N, again decrease slightly from N to O, and ±nally increase markedly
from N to Ne. The general trend is an overall increase of
I
1
with atomic number across the period.
That is to be expected since the principal quantum number (electron shell) of the outer electron
remains the same, while its attraction to the nucleus increases. The slight decrease from Be to B is
a re²ection of the outer electron being in a higher energy subshell (larger
l
value) in B than in Be.
The slight decrease from N to O is due to the half-±lled subshell effect; half-±lled sub-shells have
increased stability. O has one electron outside of the half-±lled
p
subshell and that electron must pair
with another resulting in strong electron–electron repulsions between them.
E13.4(b)
An electron has a magnetic moment and magnetic ±eld due to its orbital angular momentum. It also
has a magnetic moment and magnetic ±eld due to its spin angular momentum. There is an interaction
energy between magnetic moments and magnetic ±elds. That between the spin magnetic moment
and the magnetic ±eld generated by the orbital motion is called spin–orbit coupling. The energy of
interaction is proportional to the scalar product of the two vectors representing the spin and orbital
angular momenta and hence depends upon the orientation of the two vectors. See Fig. 13.29. The
total angular momentum of an electron in an atom is the vector sum of the orbital and spin angular
momenta as illustrated in Fig. 13.30 and expressed in eqn 13.46. The spin–orbit coupling results in

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