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The Wave Nature of
Matter
de Broglie, the Uncertainty Principle, and Quantum
Mechanics
Dual Nature of Matter?!

As a result of Planck’s and Einstein’s work, light
was found to have certain characteristics of
particulate matter



No longer purely wavelike
But is the opposite also true?
In 1923, Prince Louis de Broglie of France had
an idea

He proposed that everything propagates like a
wave, and that everything interacts like a
particle
de Broglie and Wave Nature
of Matter
 de
Broglie rewrote Einstein’s formula for the
momentum of a photon and applied it to any
object NOT traveling at the speed of light:
h
λ=
mν
λ
h
m
v
=
=
=
=
wavelength, meters
Planck’s constant
mass, kg
velocity, m/s
De Broglie Waves

h
λ=
mν
Using de Broglie’s equation and the fact that Bohr had
already reported the speed of an electron as 2.2 x 106
m s-1, we can calculate the wavelength of an electron:
λ=

This equation shows that the more massive the object,
the smaller its associated wavelength and vice versa!


6.6 x 10-34 kg m2 s-1
(9.1 x 10-31 kg)(2.2 x 106 m s-1)
For example, a thrown baseball only has a wavelength of
about 10-35 meters!
Hence the reason why on the macroscopic level,
objects do not seem to act as waves!
Wave-Particle
Duality of
Nature
• It is important to remember that Planck’s constant is very tiny
• Roughly speaking, this means that in our everyday world,
quantum effects like the wave-particle duality make a
difference only in the 34th decimal place when predicting the
behavior of a moving baseball
• Bottom line is large objects obey Newton’s laws and
subatomic particles defy classical physics and obey quantum
mechanics
Interpretation of de Broglie’s
Work

Electrons bound to the nucleus are similar to standing waves


Standing waves do not propagate through space
Standing waves are fixed at both ends
 Think of a guitar or violin
 A string is attached to both ends and vibrates to produce a
musical tone

Waves are “standing” because they are stationary – the wave does
not travel along the length of the string
The Dual Nature of Electrons and Its
Limitations


The velocity of an electron is
related to its wave nature
The position of an electron is
related to its particle nature


Particles have well-defined
positions; waves do not
We are unable to observe an
electron simultaneously as both
a particle and a wave


This is because in order to
observe an electron, one would
need to hit it with photons having
a very short wavelength and high
frequency
If one were to hit an electron, it
would cause the motion and the
speed of the electron to change

Lower energy photons would
have a smaller effect but would
not give precise information
The Electron and
Complementary Properties
 As
a result of this, we cannot
simultaneously measure its position AND
velocity

Position and velocity are complementary
properties
 Complementary
properties mean that the
more you know about one, the less you know
about the other
Heisenberg Uncertainty
Principle and Complementary
Properties

A physicist named Werner Heisenberg stated that
“[t]here is a fundamental limitation on how precisely
we can know both the position and momentum of a
particle at a given time”


In other words, it is impossible to know both the velocity
and location of an electron at the same time
He developed the following relationship:
h
∆x ∆(mv) >
4πm

As the mass of an object gets smaller, the product of
the uncertainty of its position (∆x) and speed (∆v)
increase
Again, How Can Something
be Both a Particle and a
Wave?
 Saying
that an object is both a particle
and a wave is saying that an object is
both a circle and a square – a
contradiction
 Complementary solves this problem

An electron is observed as either a particle
or a wave, but never both at once!
Energies and Electrons –
Introducing the Quantum
Mechanical Model
 Many
properties of an element depend on the
energies of its electrons

Remember, position and velocity of the electron are
complementary properties
 Since
velocity is directly related to energy via ½ mv2,
position and energy are also complementary properties

Therefore, we can specify the energy of the electron
precisely, but not its location at a given instant
 Instead,
the electron’s position is described as a
probable location where the electron is likely to be
found called an orbital
 Enter
Schrӧdinger…
Erwin Schrödinger and the
Quantum Model of the Atom
 Schrödinger
developed an equation to
describe the behavior and energies of electrons
in atoms
ℎ2
𝑑2Ψ
−
+ 𝑉Ψ = 𝐸Ψ
2
2
8𝜋 𝑚
𝑑𝑥
 General equation:
ĤΨ = 𝐸Ψ
Ĥ = set of mathematical instructions called an “operator” that represent the
total energy (kinetic and potential) of the electron within the atom
Ψ = Wave function that describes the wavelike nature of the electron

Orbitals
Orbitals…What?!


Orbitals are NOT circular orbits for electrons
Orbitals ARE areas of probability for locating
electrons


Square of absolute value of the wave function
gives a probability distribution
Ψ2
Electron density maps (probability distribution)
indicates the most probable distance from
the nucleus
Limitations of Orbitals
 Wave
functions and probability maps DO
NOT describe:



How an electron arrived at its location
Where the electron will go next
When the electron will be in a particular
location
More on the Schrödinger
Equation

While the equation is too complicated to actually
use in this class, we can still use the results!

His equation is used to plot the position of the
electron relative to the nucleus as a function of
time

The solution of the equation has demonstrated
that E (energy) must occur in integer multiples
(quanta) and that each electron can be
described in terms of its quantum numbers
What is a Quantum Number?
Think that each electron has a specific
“address” in the space around a nucleus

An electrons “address” is given as a set of
four quantum numbers

Each quantum number provides specific
information on the electrons location

Electron Configuration Analogy
state
town
house number
street
What are The 4 Quantum
Numbers?
 State
(energy level) - quantum number n
 Town
(shape of orbital) - quantum number l
 Street
(orbital room) - quantum number ml
 House
number (electron spin) - quantum
number ms
Principal Quantum Number (n)
 Same
as Bohr’s n
 Indicates probable distance from the
nucleus


Higher numbers = greater distance from
nucleus
Greater distance = less tightly bound =
higher energy
 Ranges
are integral values: 1, 2, 3, ….
Angular Momentum Quantum
Number (l)
 Indicates
the number of subshells that a
principal level contains
 It also tells the shape of the atomic
orbitals
 Ranges with integral values from 0 to n - 1
for each principal quantum number n
Value of l
0
1
2
3
4
Letter
used
s
p
d
f
g
More on Angular Quantum
Number, l

Size of orbitals is defined
as the surface that
contains 90% of the total
electron probability

Orbitals of the same
shape (s, for instance)
grow larger as principal
quantum number (n)
increases

# of nodes (areas in
which there is zero
electron probability)
increase as well
Why Do We Care About the
Shape of the Orbitals?
 Covalent
chemical bonds depend on the
sharing of the electrons that occupy
these orbitals

Shape of overlapping orbitals determine
the shape of the molecule!
The s Orbital
1s
The1s
2sorbital
orbitalhas
hasaamaximum
maximumprobability
probability
The
regionat
ataaradius
radiusof
of1~Å3 from
Å from
nucleus.
region
thethe
nucleus.
2s
1s
2s
0
1
2
3
4
5
Distance from nucleus, r (Å)
The s orbital is a sphere
Every level has one s orbital
S-Electron Probability Distributions
for First Three Energy Levels
 The
lower the energy level an electron is
located at, the higher chance it has of being
found near the nucleus
p sub-level (l = 1)
p (x)
y-axis
z-axis
p (y)
x-axis
There are three p orbitals: px,
py and pz
p (z)
Comparison of p- and sElectron Probability
Distributions
 Electrons
of the s-orbital are found closer to the
nucleus than p-orbital electrons
d sub-level (l = 2)
five clover-shaped orbitals
seen in all energy levels n=3 and above
Combined Orbitals - n = 1, 2 & 3
1s, 2s, 2p, 3s, 3p and 3d sublevels
Comparison of d-, p- and sElectron Probability Distributions
 As
electrons get to higher and higher energy
levels, the harder it is to locate it because the
radius of the orbital is greater
f sub-level (l = 3)
seven equal energy orbitals
shape is not well-defined
seen in all energy levels n=4 and above
Magnetic Quantum Number
(ml)
 Relates
to the orientation of the orbital in
space relative to an x, y, z plot

3-D orientation of each orbital
 Integral
values from l to -l, including zero
 For example, if l = 1, then ml would have
values of -1, 0, +1
 Knowing all three quantum numbers
provides us with a picture of all of the
orbitals
Magnetic Quantum Number
Electron Spin Quantum
Number (ms)

Wolfgang Pauli added one additional
quantum number that would allow only two
electrons to be in an orbital

This means that an orbital can hold only two
electrons, and they must have opposite spins


He also developed the Pauli Exclusion
Principle:


Spin can have two values, +1/2 and -1/2
"In a given atom no two electrons can have the
same set of four quantum numbers"
Therefore, each electron has its own specific
location around the nucleus of the atom
Orbitals and Their Energies
 For
the hydrogen atom, the principal
quantum number determines the energy
of the orbital


Electron closest to the nucleus (n=1) was
lowest in energy – the ground state
When an atom absorbs energy, electrons
may move to higher energy levels –the
excited state
 Things
get a bit more complex where
more than one electron is involved…
Orbital Energies in Polyelectronic
Atoms

Must make approximations with quantum mechanical
model to compensate for repulsions between electrons
 Variations in energy within the same quantum level
En-s < En-p < En-d < En-f
Orbital Energies
Work on Quantum
Numbers WS
Using the Periodic Table to
Predict Electron Locations
Electron Configurations
Using the Periodic Table to
Predict Electron Locations
 Use

the Aufbau principle
“aufbau”

 This
German for ‘build up or construct’
principle states that electrons are added one
at a time to the lowest energy orbitals available
until all the electrons of the atom have been
accounted for


If two or more orbitals exist at the same energy
orbital, do not pair the electrons until you first have
one electron per orbital
The lone electrons will have the same direction of
spin
 Called
Hund’s Rule
Hund’s Rule and Magnetism
 The
existence of unpaired electrons can
be tested for since each acts like a tiny
electromagnet

An atom that is paramagnetic is attracted
to magnetic field
 Indicates

the presence of unpaired electrons
An atom that is diamagnetic is unaffected
by a magnetic field
 Indicates
that all electrons are paired
The Aufbau
Principle
1s2
2s2
2p6
3s2
3p6
3d10
Blue sublevels
not occupied…
Not enough e-
4s2
4p6
4d10
4f14
5s2
5p6
5d10
5f14
5g18
6s2
6p6
6d10
6f14
6g18
6h22
7s2
7p6
7d10
7f14
7g18
7h22
7i26
s
The Periodic Table and Its
Classification by Sublevels
p
H
He
d
Li Be
Na Mg
K Ca Sc
Rb Sr
Y
Ti
V
B
C
N
O
F
Al
Si
P
S
Cl Ar
Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br
Zr Nb Mo Tc Ru Rh Pd Ag Cd In
Cs Ba Lu Hf
Fr Ra Lr
Ne
Ta W Re Os
Ir
Pt Au Hg
Tl
Sn Sb Te
I
Xe
Pb Bi Po At Rn
La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb
f
Kr
Ac Th Pa U Np Pu AmCm Bk Cf Es Fm Md No
Orbital Diagrams and Electron
Configurations




Electrons fill in order from lowest to highest energy
The Pauli exclusion principle holds. An orbital can hold
only two electrons
Two electrons in the same orbital must have opposite
signs
You must know how many electrons can be held by
each orbital





2 for s
6 for p
10 for d
14 for f
Hund’s rule applies. The lowest energy configuration for
an atom is the one having the maximum number of
unpaired electrons for a set of orbitals

By convention, all unpaired electrons are represented as
having parallel spins with the spin “up”
Writing Electron Configurations
 Electron
configurations can be written for
atoms or ions

Start with the ground-state configuration for
the atom
 For
positively-charged cations, remove a
number of the outermost electrons equal to
the charge
 For negatively-charged anions, add a
number of outermost electrons equal to the
charge
Practice!
Full configurations
O
1s 2 2s 2 2p 4
Ti
1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 2
Br
1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4p 5

Noble gas configurations
O
[He]
2s 2 2p 4
Ti
[Ar]
4s 2 3d 2
Br
[Ar]
4s 2 3d 10 4p 5

More Practice!

Example - Cl-

First, write the electron configuration for chlorine:
Cl
17 e1s 2 2s 2 2p 6 3s 2 3p 5 or [Ne] 3s 2 3p 5

Because the charge is 1-, add one electron
Cl-
1s 2 2s 2 2p 6 3s 2 3p 6
or [Ne] 3s 2 3p 6 or [Ar]
Orbital Notation
•
A simple method used to show only the
electrons in the highest main energy
level
•
•
The d and f sublevels are not shown
unless they are partially filled
Indicates which orbitals are occupied
and whether or not electrons are
paired with other electrons in the
orbitals
Practice!
Time for Electron
Configuration Battleship!