# Quantum Theory and the Electronic

Quantum Theory and the Electronic
Structure of Atoms

Chapter 7

Our world until the end of the nineteenth century

The Macroscopic World

classical theory of
by Maxwell

Matter

Newtonian Mechanics

(Newton’s laws),

Thermodynamics

Studied and characterized by Classical Physics

The Microscopic World

For chemists it is the world of atoms and molecules

The end of the nineteenth century witnessed the birth of the
quantum theory and the discovery of the microscopic world

Studied and characterized by quantum physics

In order to understand the quantum theory and the microscopic

world, we need to Understand classical properties of matter

and waves

Waves

The distance between corresponding points

wavelength

(

The number of waves
passing a given point per
unit of time is the
frequency (

)
.

For waves traveling at
the same velocity, the
longer the wavelength,
the smaller the
frequency.

All electromagnetic radiation travels at the same velocity

the speed of light (
c
), 3.0

10
8

m/s.

Therefore,
c

=


The Nature of Energy of Radiation

The wave nature of light
does not explain how
an object can glow
when its temperature
increases.

Max Planck

explained it
by assuming that
energy comes in
packets called
quanta
.

Einstein used this
assumption to explain the
photoelectric effect.

He concluded that energy
is proportional to
frequency:

E

=
h

where
h

is Planck’s
constant, 6.63

10

34

J
-
s.

The Nature of Energy of Radiation

The Nature of Energy of Radiation

c

c
h
E

Therefore, if one knows the

Wavelength or the frequency of

light, one can calculate the energy

in one photon, or packet,

of that light:

photon
photon
h
E

or

where

Pay attention to units

Another mystery involved the emission spectra
observed from energy emitted by atoms and
molecules.

VIS

40 %

IR

51 %

UV

9 %

nm

500

1000

nm

500
max

The Nature of Energy of Radiation

Atomic spectra: http://astro.u
-
strasbg.fr/~koppen/discharge/index.html

Demo: H, He, and Ne atoms

Only a
line spectrum

of

discrete wavelengths is

observed.

Emission Spectrum of the Hydrogen Atom

Quantization of Energy of Electrons

assumption and explained

these phenomena in this way:

1. Electrons in an atom can
only occupy certain orbits

(corresponding to certain
energies).

1.
Electrons in an atom can only
occupy certain orbits
(corresponding to certain
energies).

2.
Electrons in permitted orbits
have specific, “allowed”
energies; these energies will
not be radiated from the atom.

Bohr Model for the energy Levels of the hydrogen atom

2
H
n
n
R
E

1
-
18
H
cm

737
,
109
J

10
18
.
2
R

,....
3
,
2
,
1
n

n

1

2

3

4

5

n
H
R

4
R
H

9
R
H

16
R
H

0

absorption

emission

2
i
2
f
H
2
i
H
2
f
H
n
1
n
1
R
n
R
n
R
E

2
i
2
f
H
n
1
n
1
R
h

h
E
E
E
i
f
2
H
n
n
R
E

-1
H
cm

737
,
109
R

,....
3
,
2
,
1
n

Frequencies of Absorption and Emission Transitions

In The hydrogen atom

Absorption

Emission

f
E
f
E
i
E
i
E

h
E
E
E
i
f

h
E
E
E
i
f

de Broglie’s Postulate (Wave
-
Particle Duality of Matter)

p
h

m

v

mv
P

Particle property

Wave property

De Broglie argued that both light and matter obey his equation

Why the wave property of matter is important for microscopic

objects while it is not important for macroscopic objects?

a. What is the value of

of an electron traveling at 1.00% of the speed
of light?

b. What is the value of

of a person (m = 62.6 kg) moving at 1m/s?

Let us carry out the following exercises

mv
P

;
p
h

This wavelength is about five times greater than the radius of the H atom

A

43
.
2
10
3
)
kg
(
10
1
.
9
s
.
J
10
626
.
6
6
31
34

a.

A

10
06
.
1
25

This wavelength is too small to be detected

b.

a. Electron diffraction (Al)

b. X
-
ray diffraction experiment (Al)

http://www.matter.org.uk/diffraction/introduction/what_is_diffraction.htm

de Broglie Waves Are Observed Experimentally

http://jchemed.chem.wisc.edu/JCEWWW/Articles/WavePacket/WavePacket.html

2
h

;
2
P
.
x

The Heisenberg Uncertainty Principle

2
)
mv
(
.
x

or

In many cases, our uncertainty of the whereabouts
of an electron is greater than the size of the atom
itself!

Quantum Mechanics

Erwin Schrödinger
developed a
mathematical treatment
into which both the
wave and particle nature
of matter could be
incorporated.

It is known as
quantum
mechanics
.

Quantum Mechanics

The wave equation is
designated with a lower
case Greek
psi

(

).

The square of the wave
equation,

2
, gives a
probability density map of
where an electron has a
certain statistical likelihood
of being at any given instant
in time.

Quantum Numbers

Solving the wave equation gives a set of wave
functions, or
orbitals
, and their corresponding
energies.

Each orbital describes a spatial distribution of
electron density.

An orbital is described by a set of three
quantum
numbers
.

Principal Quantum Number,
n

The principal quantum number,
n
, describes the
energy level on which the orbital resides.

The values of
n

are integers ≥ 0.

Azimuthal Quantum Number,
l

This quantum number defines the shape of
the orbital.

Allowed values of
l

are integers ranging from
0 to
n

1.

We use letter designations to communicate
the different values of
l

and, therefore, the
shapes and types of orbitals.

Azimuthal Quantum Number,
l

Value of
l

0

1

2

3

Type of orbital

s

p

d

f

Magnetic Quantum Number,
m
l

Describes the three
-
dimensional orientation
of the orbital.

Values are integers ranging from
-
l

to
l
:

l

m
l

l.

Therefore, on any given energy level, there
can be up to 1
s

orbital, 3
p

orbitals, 5
d

orbitals, 7
f

orbitals, etc.

Magnetic Quantum Number,
m
l

Orbitals with the same value of
n

form a
shell
.

Different orbital types within a shell are
subshells
.

s

Orbitals

Spherical in shape.

with increasing value of
n.

l

= 0

m
l
= 0

s
Orbitals

Observing a graph of
probabilities of finding
an electron versus
distance from the
nucleus, we see that
s

orbitals possess
n

1
nodes
, or regions
where there is 0
probability of finding an
electron.

p

Orbitals

Have two lobes with a node between them.

l

= 1

m
l
= +1, 0,
-
1

d

Orbitals

Four of the five orbitals

have 4 lobes;

the other resembles a
p

orbital with a doughnut

around the center.

l

= 2

m
l
= +2, +1, 0,
-
1,
-
2

Energies of Orbitals

For a one
-
electron
hydrogen atom,
orbitals on the same
energy level have the
same energy.

That is, they are
degenerate
.

Energies of Orbitals

As the number of
electrons increases,
though, so does the
repulsion between
them.

Therefore, in many
-
electron atoms,
orbitals on the same
energy level are no
longer degenerate.

Spin Quantum Number,
m
s

In the 1920s, it was
discovered that two
electrons in the same
orbital do not have
exactly the same energy.

The “spin” of an electron
describes its magnetic
field, which affects its
energy.

Spin Quantum Number,
m
s

This led to a fourth
quantum number, the
spin quantum number,
m
s
.

The spin quantum
number has only 2
allowed values: +1/2
and

1/2.

Pauli Exclusion Principle

No two electrons in the
same atom can have
exactly the same energy.

For example, no two
electrons in the same
atom can have identical
sets of quantum
numbers.

Electron Configurations

Distribution of all electrons in an atom consist of

5
p
4
4
-
Number denoting the energy level

p
-
Letter denoting the type of orbital

5
-
Superscript denoting the number of electrons in those

orbitals

Orbital Diagrams

Each box represents one
orbital.

Half
-
arrows represent
the electrons.

The direction of the
arrow represents the
spin of the electron.

Hund’s Rule

“For degenerate
orbitals, the lowest
energy is attained
when the number of
electrons with the
same spin is
maximized.”

Periodic Table

We fill orbitals in
increasing order of
energy.

Different blocks on
the periodic table,
then correspond to
different types of
orbitals.

Some Anomalies

Some irregularities occur when there are enough
electrons to half
-
fill
s

and
d

orbitals on a given row.

For instance, the electron
configuration for copper is

[Ar] 4
s
1

3
d
5

rather than the expected

[Ar] 4
s
2

3
d
4
.

These anomalies occur

in
f
-
block atoms, as well.

This occurs because the 4
s

and 3
d

orbitals are very
close in energy.

8
O

11
Na

19
K

Detailed and Abbreviated Electron Configurations

Of Metals and Nonmetals

16
S

Detailed and Abbreviated Electron Configurations

Of Transition Metals

21
Sc

24
Cr

25
Mn

29
Cu

Diamagnetism and Paramagnetism

Paramagnetic substances

Any substance that possesses
net nonzero

electron spin

quantum number and therefore attracted by a magnet

6
2
18
26
d
3
[Ar]4s

:
Fe
1
2
10
13
p
3
[Ne]3s

:
Al
Diamagnetic substances

Any substance that possesses
net zero

electron spin

and therefore not attracted by a magnet

10
2
36
48
d
4
5s

:
[Kr]

:
Cd
10
2
18
30
d
3
[Ar]4s

:
Zn
A Very Useful Tool For Understanding Electron

Configuration of Atoms

Electron Configuration, From Prentice Hall

Just click on the following link and you love what you find