Chemistry - A Molecular Science

2.6

ORBITAL SHAPES,

SIGNS, AND SIZES

1s

2s

3s

Figure 2.10 Relative sizes of 1s, 2s, and 3s orbitals

Solutions of the wave equations for the electr

ons in atoms result in mathematical functions

that can be plotted and whose sign depends up

on where in space the function is evaluated.

Plots of the functions describe the regions in

space in which the electrons are most likely

to be found. These regions of space are called ‘

orbitals

’, and orbitals lie at the heart of

chemistry. It is the interactions of orbitals and the electrons in them that lead to chemical bonding. Indeed, our discussion of bonding in Chapters 5 and 6 will rely on a good understanding of both the shape and sign of atomic orbitals.

l^ and m

dictate the shape and orientation of the orbital, respectively. The shapes l

represent regions of space around the nucleus

in which the electron can be found some

large fraction of the time (typically 90 - 95%). These

electron clouds

or regions of

electron density

are useful when considering interactions between atoms. The sign of the

function in a given region is designated by its shad

ing. In this text, we use blue shading in

regions where the sign is positive and red shad

ing in regions where it is negative.

z

x

y

pz

py

px

nodalplanes

pz

py

px

(a)

(b)

Figure 2.11 p orbitals (a) Truer representation of the shape (b) Commonly used representation

s orbitals (

l = 0)

l = 0 , so m

= 0 and there is only one s orbital in an s sublevel. An s orbital is spherical, l

which means that the electron has equal probabili

ty of being found in any direction and that

its charge is distributed equally in all

directions. As shown in Figure 2.10,

l dictates the

spherical shape of the orbital,

but n dictates its size. There are no nodal planes, so the

function has the same sign everywhere as indicated by the blue shading in Figure 2.10.

p orbitals (

l = 1)

l = 1, so there are three values of m

(-1, 0, +1) and three orbitals in a p sublevel - the pl

, px

(^) y
and p
orbitals. The shape of a p orbital is best rz
epresented by Figure 2.11a, but it is more
common to represent it as a “figure 8” as sh
own in Figure 2.11b. The electron density in
each orbital is distributed in two lobes, which
are of opposite sign. Where the sign changes,
the value of the function is zero, which means
that there is no electron density in the plane
that lies between the two lobes. This plane of zero electron density is called a
nodal plane
.
The number of nodal planes in an orbital is equal to the
l quantum number:
s orbitals have
no nodal planes (
l = 0), p orbitals have one (
l^ = 1), d orbitals have two (
l^ = 2),
etc
.
i
j
d=d,d,d
xy
yz
xz
ij
z
d^2 z
y
x
d^2 x-y
2
nodalplanes
Figure 2.12 d orbitals
d orbitals (
l = 2)
There are five possible values of
ml
in an
l = 2 sublevel (-2, -1, 0, +1, +2), so there are five
d orbitals. Figure 2.12 shows a representation of these orbitals. The d
2 (or simply zz
2 ) orbital
is directed along the z-axis, but there is a don
ut shaped region of electron density in the xy
plane. The d
(^2) x-y
2 ( or x
2 -y
2 ) orbital is directed along both the x- and the y-axes in both the
positive and negative directions. Much of the d
(xz) electron cloud lies in the xz plane 45xz
o^
from either axis. The shapes of the d
(yz) and dyz
(xy) are identical to that of the dxy
, only xz
the labels of the axes change.
Chapter 2 Quantum Theory
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State
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