Environmental Engineering Reference
In-Depth Information
In cases where an electron has both orbital and spin angular momentum (e.g., the
electron in the
n ¼
1 state of the one-electron atomhas only
S
,no
L
), these two forms
of angular momentum combine as
J ¼L þ S
, which has a similar rule for its
p
(
j
(
j þ 1
))
magnitude:
J ¼
h
. The rules are required by the solutions of the
Schrodinger equation.
The wavefunctions
e
r
/2
exp(
i
Y
21,
1
¼C
2
r
sin
j
) are the
rst two states
having angular momentum. Apolar plot of
Y
21,
1
has a node along
z
, and resembles
a donut
flat in the
x
-,
y
-plane.
Linear combinations of states are very important in quantummechanics. Here, the
sum and difference of the
Y
21,
1
states are also solutions to Schrodingers equation,
for example,
e
r=
2
e
r=
2
2 cos
Y
211
þY
21
1
¼ C
2
r
sin
y
½
exp
ði
jÞþ
exp
ði
jÞ ¼ C
2
r
sin
y
j:
ð
3
:
31
Þ
This is twice the 2
p
x
wavefunction in Table 3.1. This linear combination, Equa-
tion 3.31, is exemplary of the real wavefunctions in Table 3.1, where linear combina-
tions have canceled the angular momenta to provide a preferred direction for the
wavefunction.
Apolar plot of the 2
p
x
wavefunction (3.31) shows a node in the
z
-direction from the
sin
, so it is a bit like a dumbbell
at the origin oriented along the
x
-axis. Similarly, the 2
p
y
resembles a dumbbell at the
origin oriented along the
y
-axis.
These real wavefunctions, in which the exp(
im
and amaximumalong the
x
-direction from the cos
j
j
) factors have been combined to
form sin
, are more suitable for constructing bonds between atoms in
molecules or in solids than are the equally valid (complex) angular momentum
wavefunctions. The complex wavefunctions that carry the exp(
im
j
and cos
j
) factors are
essential for describing orbital magnetic moments as occur in iron and similar
atoms. The electrons that carry orbital magnetic moments usually lie in inner shells
of their atoms.
The rules governing the one-electron atom wavefunction
j
Y
n
,
l
,
m
,
m
and the Pauli
exclusion principle, which states that only one electron can be accommodated in a
completely described quantum state, are the basis for Chemical Table of the
Elements. The number of electron states per atom is simply
Z
the nuclear charge.
As we saw at the end of Chapter 2, the maximum
Z
for any nucleus is set by Coulomb
repulsion among the
Z
protons.
As we have seen, the rules allow 2
n
2
distinct states for each value of the principal
quantum number,
n
. There are several notations to describe the
filled atomic shells
.
The K shell of an atom comprises the two electrons of
n¼
1 (1s
2
), followed by the L
shell with
n¼
2 (2s
2
2p
6
) (Ne) and the M shell with
n¼
3 (3s
2
3p
6
3d
10
) (Ar). These
closed shells contain, respectively, 2, 8, and 18 electrons. Completely filled electron
cores occur at
Z¼
2 (He),
Z¼
10 (Ne),
Z¼
18 (Ar),
Z¼
36 (Kr),
Z¼
54 (Xe), and
Z¼
86 (Rn).
