Geology Reference
In-Depth Information
The shapes of electron orbitals
probability is less than one at every individual point
(that is, there is no one point at which we can locate the
electron with certainty), but considering the atom as a
whole it must add up to one for each of the electrons
present.
Solutions to the Schrödinger equation for the elec-
tron in an atom are independent of time (since, like
Equation 5.1, they contain no
t
terms), and so they
cannot indicate the precise trajectory or orbit fol-
lowed by the electron, showing how its
x
,
y
and
z
co-
ordinates vary as a function of time. The electron
must be visualized as a constant but diffuse cloud
extending throughout the volume of the orbital, the
'
electron density
' of the cloud varying from point to
point according to the magnitude of
ψ
2
.
For the vibrating string, one quantum number
n
is
sufficient to enumerate the various types of stationary
wave observed. Unsurprisingly the electron wave in
the three-dimensional atom presents a more compli-
cated picture, and four quantum numbers (Table 5.3)
are required to encompass all possible stationary states
that the electron wave can adopt. For most purposes,
we shall only need to consider two of these four quan-
tum numbers:
n
, known as the
principal quantum num-
ber
, and
l
, the
angular-momentum quantum number
. (The
physical origins of these names need not concern us.)
The significance of each quantum number, summa-
rized in Table 5.3, will become clear presently.
Atomic orbitals come in all shapes and sizes, and
some acquaintance with their geometry will be help-
ful in understanding the shapes of molecules and the
internal structures of crystals (Chapter 8). The inter-
nal structure of the mineral diamond, for instance, is
a direct expression of how electron density is arranged
within each constituent carbon atom. In a similar
way, the disposition of electron orbitals in the oxygen
atom is responsible for the bent shape of the water
molecule, upon which the unique solvent power of
water depends (Box 4.1).
The
symmetry
of an orbital is determined by the
value of the quantum number
l
, in the manner out-
lined in Table 5.3. Orbitals for which
l
is zero have
simple spherical symmetry, but as
l
increases we
encounter progressively more complex symmetry.
We shall begin with the simplest type of orbital sym-
metry, in which the parallel with the vibrating string
is most apparent.
s-orbitals
The simplest solutions to the Schrödinger wave equa-
tion, for which
l
is zero, all possess spherical symmetry
and therefore
ψ
and
ψ
2
can be depicted simply in terms
of a radial coordinate
r,
representing the distance from
Table 5.3
Physical significance of quantum numbers
Quantum number
Permitted values
Influence on geometry of orbital
Name
Symbol
Principal quantum
number
n
integer 1, 2, 3 ..
(a)
Determines the
size
of the orbital:
low
n
: compact orbital
high
n
: spread-out orbital (Figures 5.3 and 5.4)
(
n
-1) is the number of nodal surfaces, where
ψ
2
= 0.
(b)
Angular momentum
quantum number
l
integer 0 to
n
-1
Determines the
shape
of the orbital:
l
= 0
l
= 1
s-orbital:
s
pherical symmetry
p-orbital:
p
olar symmetry - electron density forms 2 balloons on opposite sides
of the nucleus (Figure 5.4).
d-orbital: electron density forms 4 balloons (Figure 5.4)
f-orbital : still more complex
l
= 2
l
= 3
Magnetic quantum
number
m
integer −
l
to +
l
Determines the
orientation
of the orbital: e.g. indicates whether a p-orbital is
aligned along the
x, y
or
z
axis.
Note:
The spin quantum number,
s
, only becomes relevant when multi-electron atoms are concerned. It has only 2 permitted values, −½
and + ½.
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