SECTION 4.2
The Quantum Model
of the Atom
The Bohr model of the atom seemed at first to contradict
common sense. Bohr’s model explained the experimental
evidence of the hydrogen emission-line spectrum, but Bohr
provided no reason for the atom’s orbital structure.
Electrons have wave-like properties.
To explain Bohr’s model, scientists had to change the way
they viewed the nature of the electron. Scientists had
thought that light behaved only as a wave. Now they were
beginning to understand that it had a dual wave-particle
nature. Electrons had always been thought of as particles.
In 1924, French scientist Louis de Broglie asked if electrons
could have a dual nature as well.
Key Terms
Heisenberg uncertainty principle
quantum theory
orbital
quantum number
principal quantum number
angular momentum quantum number
magnetic quantum number
spin quantum number
De Broglie pointed out the ways in which electrons in
Bohr’s model behaved like waves. For example, he knew
that any wave confined within a certain space can have only
certain frequencies. Electrons are confined to certain
frequencies within an atom.
De Broglie suggested that electrons be thought of as
waves confined to the space around an atomic nucleus.
These electron waves can only exist at certain frequencies,
and by the relationship E = hν, certain energies.
Scientists were soon able to confirm that electrons have
other wave-like properties. For example, electrons can be
diffracted. Diffraction is the bending of a wave through a
small opening. Interference occurs when waves that diffract
through a small opening overlap. In some areas of a target
screen, the overlapping waves increase the energy and
appear bright. In some areas the overlapping waves
decrease the energy and appear dark. The result is a pattern
of light and dark areas. A stream of particles would not be
affected in this way, and would appear as a solid circle
where the beam hits the screen.
READING CHECK
1. Name two properties of
electrons that indicate a
wave-like nature.
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99
The speed and position of an electron cannot be
measured simultaneously.
The idea of electrons having a dual wave-particle nature
troubled scientists. They wanted to be able to determine the
specific locations of electrons within an atom. In 1927, German
physicist Werner Heisenberg developed another principle that
changed how scientists viewed the subatomic world.
Heisenberg hypothesized that the act of observing a
­particle would itself change the behavior of the particle. For
example, to detect an electron, you use particles of light, or
photons. You locate the electron by its absorption, emission,
or other interaction with a photon. However, this interaction
will change the course of the electron’s movement. As a result,
there is always uncertainty in trying to locate an electron, or
any other particle.
LOOKING CLOSER
2. Define uncertainty and
principle separately.
uncertainty:
principle:
The Heisenberg uncertainty principle states that it is
impossible to determine the exact position and the exact
velocity of a particle at the same time. This principle is
now a foundation of current theories of the nature of light
and matter.
z
y
Orbitals indicate probable electron locations.
In 1926, Austrian physicist Erwin Schrödinger successfully
combined Bohr’s model of the atom and the dual waveparticle nature of electrons. Bohr’s model was a hypothesis
based on the assumption that energy levels in an atom were
fixed. Schrödinger used an equation to show that these fixed
energy levels resulted from the wave nature of electrons.
x
(a)
z
Together with the Heisenberg uncertainty principle,
Schrödinger’s equation formed the foundation of a new
atomic theory. Quantum theory describes mathematically
the wave properties of electrons and other small particles.
One result of quantum theory is that the position of an
electron in an atom cannot be determined precisely. Only the
probability of finding it in a certain region can be determined.
Electrons do not travel in specific orbits as in Bohr’s model of
the atom. Instead, they travel within orbitals, which are threedimensional regions around the nucleus of the atom that
indicate the probable location of an electron.
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y
x
(b)
Here are two ways of visualizing an
atomic orbital. (a) The electron is
likely to be found in the dense regions
of this cloud. (b) The electron is
located within this surface 90% of
the time.
The Four Quantum Numbers
Quantum Number
Symbol
Description
principal quantum
number
n
main energy level of the
electron
angular momentum
quantum number
l
shape of the orbital
magnetic quantum
number
m
orientation of the orbital
around the nucleus
1 ​or –​ __
1 ​
+ ​ __
2
2
spin state of the
electron
spin quantum
number
Quantum numbers describe atomic orbitals.
In Bohr’s model of the atom, electrons of increasing energy
occupy orbits farther and farther from the nucleus. According
to the Schrödinger equation, electrons in orbitals also have
quantized energies. Scientists can assign each orbital within an
atom a specific value of energy.
TIP
The words quantum, and
quantized, are related. A
quantum is a specific value. A
quantity is quantized if it is limited
to certain values.
Scientists need more numbers to completely describe the
properties of an electron. Quantum numbers specify the
properties of atomic orbitals and the properties of electrons in
these orbitals. The four quantum numbers are summarized in
the table above.
More than one electron can occupy the same energy level
in an atom. These electrons are sometimes said to be in the
same electron shell.
n=6
n=5
n=4
n=3
n=2
Energy
Principal Quantum Number
The principal quantum number, symbolized by n, indicates the
main energy level occupied by the electron. The first six
energy levels in an atom are shown at the right. If n = 1, an
electron occupies the first energy level and is located closest
to the nucleus. As n increases to 2, 3, 4, and so on, the electron
increases in energy and average distance from the nucleus.
n=1
READING CHECK
3.
An electron at the n = 3 level of an atom has
energy and is located
the nucleus than an electron at the n = 2 level.
The principal quantum number, n,
gives the main energy level occupied
by an electron.
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101
Angular Momentum Quantum Number
Not all orbitals are the same shape. At the n = 1 level, there is
just one orbital and it has a spherical shape. For higher main
energy levels, the orbitals can take on multiple shapes. The
angular momentum quantum number, symbolized by l,
indicates the shape of the orbital.
For a specific main energy level n, there are n possible
shapes for the orbitals. Each shape is called a sublevel. For
example, for the n = 2 level, there are two sublevels for
the orbitals: spherical (l = 0) and dumbbell-shaped (l = 1).
The n = 3 level includes three sublevels that include
spherical (l = 0), dumbbell-shaped (l = 1), and more
complex (l = 2) orbitals.
Each sublevel is also given a letter designation. The letter
designations are given in the table at the right. Every main
energy level includes an s orbital. Every energy level for n = 2
and higher also includes p orbitals. Every energy level for
n = 3 and higher includes d orbitals. So n = 3 includes
s orbitals, p orbitals, and d orbitals.
Every atomic orbital has a designation that includes a
number followed by a letter. The number is the main energy
level, or principal quantum number. The letter is the sublevel.
For example, the 1s orbital is the only orbital on the main
energy level n = 1. A 4d orbital is any one of the d orbitals on
energy level n = 4. The information given by these first two
quantum numbers for the first four main energy levels of an
atom is summarized in the table on the next page.
Orbital Letter Designations
According to Values of l
l
Letter
0
s
1
p
2
d
3
f
READING CHECK
4.
How would you designate an orbital in the p sublevel of the
third main energy level of an atom?
z
z
y
x
s orbital
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z
y
x
p orbital
y
x
d orbital
The orbitals s, p, and d have
different shapes.
Orbital Types for the First Four Main Energy Levels
Principal quantum
number, n (main
energy level)
Number of
orbital shapes
possible
Possible values of
angular momentum
quantum number, l
Possible
orbital
types
Orbital
designations
n=1
1
l=0
s
1s
n=2
2
l = 0, 1
s, p
2s, 2p
n=3
3
l = 0, 1, 2
s, p, d
3s, 3p, 3d
n=4
4
l = 0, 1, 2, 3
s, p, d, f
4s, 4p, 4d, 4f
Magnetic Quantum Number
The total number of orbitals that exist in a given main energy
level is equal to ​n2​ ​. Therefore, the first energy level (n = 1) of
an atom has one orbital. Every energy level higher than n = 1
has multiple orbitals. For example, the second energy level
(n = 2) has two sublevels that include four orbitals. The third
energy level (n = 3) has nine orbitals in three sublevels.
The orbitals of each main energy level are oriented so
that they do not overlap. At the n = 1 level, there is only
one orbital, so no overlap is possible. At the n = 2 level,
there are four orbitals. The 2s orbital is spherically shaped.
Three dumbbell-shaped 2p orbitals are oriented around the
2s orbital.
READING CHECK
5. Describe the main energy
level, shape, and orientation of
the 2​p​x​orbital.
The magnetic quantum number, symbolized by m, indicates
the orientation of an orbital around the nucleus. Values of m
are whole numbers that range from –l to +l. All s orbitals are
spherical, so they can only have one orientation. Since the s
orbital has an angular magnetic quantum number of l = 0,
all s orbitals have the magnetic quantum number m = 0.
The p orbitals have three possible orientations, depending
on the axis with which the dumbbell is aligned. These
​ ​z​. Because l = 1 for
p orbitals are designated as p
​ x​ ​, ​p​y​, and p
p orbitals, they can be assigned the magnetic quantum
numbers m = –1, m = 0, or m = +1.
z
z
y
x
x
px orbital
z
y
y
py orbital
x
pz orbital
The three p orbitals in any main
energy level are oriented as shown.
The letters x, y, and z on each orbital
name describe the axis on which the
dumbbell is oriented. The nucleus is at
the intersection of the three axes.
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103
z
z
z
y
y
y
x
x
x
dx2 – y 2 orbital
dxy orbital
z
z
dyz orbital
y
y
x
x
The five d orbitals in any main energy
level are oriented as shown. Each
occupies a different region of space,
but is centered around the nucleus.
dz 2 orbital
dxz orbital
The d sublevel in every main energy level has five different
orientations. The value of l for d orbitals is l = 2. This means
that the five different orientations correspond to values of
|m = –2, m = –1, m = 0, m = +1, and m = +2.
With each move from simple to more complex orbitals,
two more orientations in space become available. For
example, the n = 4 level, has one s orbital, three p orbitals,
five d orbitals, and seven f orbitals. The number of orbitals,
1 + 3 + 5 + 7 = 16, is equal to the number of sublevels as
defined by n​2​= (4​)2​ ​= 16.
PRACTICE
A.
Complete this chart defining the 16 sublevels in the n = 4
energy level of an atom.
Designation
angular momentum
quantum number, l
magnetic quantum
number, m
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4s
4p
4p
4p
4d
4d
4d
4d
4d
4f
4f
4f
4f
4f
4f
4f
Spin Quantum Number
An electron in an orbital behaves in some ways like Earth
spinning on its axis. Earth’s spinning generates a magnetic
field. An electron exists in one of two possible spin states.
Each spin state creates a different magnetic field. To account
for the magnetic properties of the electron, scientists assign
electrons a spin quantum number.
The spin quantum number has only two possible values,
+1/2 and –1/2, which indicate the two possible spin states of
an electron in an orbital. Each orbital of an atom can contain
up to two electrons. However, the electrons in the orbital must
have opposite spin states.
LOOKING CLOSER
6. Which quantum numbers
define the properties of electrons
in an orbital?
7. Which quantum numbers define
the properties of the orbitals?
For example, examine the n = 2 main energy level. It has
two sublevels. The s sublevel includes one s orbital. The
p sublevel includes three p orbitals. Each one of these four
orbitals can contain two electrons if the electrons have
opposite spin states. Therefore, the n = 2 level of an atom can
hold up to 8 electrons. The structure of the atom given by
these quantum numbers is summarized in the table below.
READING CHECK
8.
What is the difference between an orbital and a sublevel?
Quantum Number Relationships in Atomic Structure
Principal quantum Sublevels in main Number of
Number of
Number of
Number of
number: main
energy level
orbitals per orbitals per main electrons
electrons per main
energy level (n)
(n sublevels)
sublevel
energy level (​n2​ ​) per sublevel energy level (2​n2​ ​)
1
s
1
1
2
2
2
s
p
1
3
4
2
6
8
3
s
p
d
1
3
5
9
2
6
10
18
4
s
p
d
f
1
3
5
7
16
2
6
10
14
32
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105
SECTION 4.2 REVIEW
VOCABULARY
1. Define each of the following.
a. main energy level
b. quantum number
REVIEW
2. Identify the four quantum numbers by name and symbol.
3. What general information about atomic orbitals is provided by the
quantum numbers?
4. Describe briefly what specific information is given by each of the four
quantum numbers.
Critical Thinking
5. INFERRING RELATIONSHIPS
a.What are the possible values of the magnetic quantum number m
for f orbitals?
b.What is the maximum number of electrons that can exist in the
orbitals in the 4f sublevel?
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