Quantum Theory of The Atom

How are electrons distributed in space?

What are electrons doing in the atom?

The nature of the chemical bond must first be
approached by a closer examination of the
electrons

Electrons are involved in the formation of
chemical bonds between atoms

Quantum theory explains more about the
electronic structure of atoms
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1
Quantum Theory of The Atom

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Origin of Atomic Theory

When burned in a flame metals produce colors
characteristic of the metal

This process can be traced to the behavior of
electrons in the atom
2
Quantum Theory of The Atom

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Emission (line) Spectra of Some Elements
When elements are heated in a flame and their
emissions passed through a prism, only a few
color lines exist and are characteristic for each
element. Atoms emit light of characteristic
wavelengths when excited (heated)
3
Quantum Theory of The Atom

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Electrons and Light: Wave Nature of Light
 Light moves (propagates) along as a wave
(similar to ripples from a stone thrown in
water)
 Light consists of oscillations of electric and
magnetic fields that travel through space
 All electromagnetic radiation
Visible light, Microwaves, Radio Waves,
Ultraviolet Light, X-rays, Infrared Light
consists of energy propagated by means of
electric and magnetic fields that alternately
increase and decrease in intensity as they
move through space
4
Quantum Theory of The Atom
A Light Wave is Propagated as an Oscillating
Electric Field (Energy)
 The Wave properties of electromagnetic
radiation are described by two independent
variables: wavelength and frequency

Crest
Crest
 = wavelength, Distance from crest to crest
 = c/ = frequency – Speed light / wavelength
c = speed of electromagnetic radiation (3 x 108 m/s)
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Quantum Theory of The Atom

Wave Nature of Light

Wavelength – the distance between any two
adjacent identical points (crests) in a wave (given the
notation  (lamba)

Frequency – number of wavelengths that pass a
fixed point in one unit of time (usually per second,
given the notation (). The common unit of
frequency is hertz (Hz) – 1 cycle per second (1/sec)

Propagation (Velocity) of an Electromagnetic wave is
given as
c =  (1/sec * m) = m/sec
c = velocity of light = 3.0 x 108 m/s in a vacuum
c is independent of  or  in a vacuum
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Quantum Theory of The Atom

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Relationship Between Wavelength and Frequency
 Wavelength and frequency (c = ) are
inversely proportional
7
The electromagnetic spectrum
High
Frequency ()
Low
High
Energy (E)
Low
Short
Wavelength ()
Long
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8
Distinction between Energy & Matter

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At the macro scale level – everyday life – energy & matter behave
differently
Waves (Energy)
Particles (Matter)
Wave passing from air to water is
refracted (bends at an angle and slows
down). The angle of refraction is a
function of the density
White light entering a prism is dispersed
into its component colors – each
wavelength is refracted slightly differently
Particle entering a pond moves in a
curved path downward due to gravity
and slows down dramatically because of
the greater resistance (drag) of the water
A wave is diffracted through a small
opening giving rise to a circular wave on
the other side of the opening.
When a collection of particles encounter
a small opening, they continue through
the opening on a straight line along their
original path (until gravity pulls them
down
If light waves pass through two adjacent
slits, the emerging waves interact
(interference).
Constructive Interference – Crests
coincide in phase
Destructive Interference – Crests
coincide with troughs, cancelling out.
Particles passing through adjacent
openings continue on straight paths,
some colliding with each other moving at
different angles
9
Distinction between Energy & Matter
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10
Distinction between Energy & Matter
The diffraction pattern caused by waves passing through two adjacent slits
A. Constructive and destructive interference occurs as water waves viewed
from above pass through two adjacent slits in a ripple tank
B. As light waves pass through two closely spaced slits, they also emerge
as circular waves and interfere with each other
C. They create a diffraction (interference) pattern of bright regions where
crests coincide in phase and dark regions where crests meet troughs
(out of phase) cancelling each other out
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
11
Quantum Theory of The Atom

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Quantum Effects – Wave-Particle Duality
 Wave-Particle Duality is a central concept in
Chemistry & Physics
 All matter and energy exhibit both wave-like
and particle-like properties
 Duality applies to:
 macroscopic (large scale) objects
 microscopic objects (atoms and molecules)
 quantum objects (elementary particles –
protons, neutrons, quarks, mesons)
 As atomic theory evolved, matter was
generally thought to consist of particles
 At the same time, light was thought to be a
wave
12
Quantum Theory of The Atom


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Quantum Effects - Wave-Particle Duality

Christiaan Huygens proposed the wave theory of light

Huygen’s wave theory was displaced by Isaac Newton’s
view that light consisted of a beam of particles

In the early 1800s Young and Fresnel showed that light,
like waves, could be diffracted and produce interference
patterns, confirming Huygen’s view

In the late 1800s James Maxwell developed equations,
later verified by experiment, that explained light as a
propagation of electromagnetic waves
At the turn of the 20th century, physicists began to focus on 3
confounding phenomena to explain Wave-Particle Duality

Black Body radiation

The Photoelectric Effect

Atomic Spectra
13
Quantum Theory of The Atom
Quantum Effects – Wave-Particle Duality


Black Body Radiation - As the temperature of an
object changes, the intensity and wavelength of the
emitted light from the object changes in a manner
characteristic of the idealized “Blackbody” in which the
temperature of the body is directly related to the
wavelengths of the light that it emits

In 1901, Max Planck developed a mathematical model
that reproduced the spectrum of light emitted by
glowing objects

His model had to make a radical assumption (at that
time):
A given vibrating (oscillating) atom can have only certain
quantities of energy and in turn can only emit or
absorb only certain quantities of energy
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Quantum Theory of The Atom

Quantum Effects – Wave-Particle Duality

Planck’s Model:
E = nh
E (Energy of Radiation)
v (Frequency)
n (Quantum Number) = 1,2,3…
h (Planck’s Constant, a Proportionality Constant)
6.626 x 10-34 J  s)
6.626 x 10-34 kg  m2/s
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
Atoms, therefore, emit only certain quantities of
energy and the energy of an atom is described as
being “quantized”

Thus, an atom changes its energy state by emitting
(or absorbing) one or more quanta
15
Quantum Theory of The Atom

Wave-Particle Duality – The Photoelectric Effect

The Planck model views emitted energy as waves

Wave theory associates the energy of the light
with the amplitude (intensity) of the wave, not
the frequency (color)

Wave theory predicts that an electron would
break free of the metal when it absorbed enough
energy from light of any color (frequency)

Wave theory would also imply a time lag in the
flow of electric current after absorption of the
radiation

Both of these observations are at odds with the
Photoelectric Effect
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Quantum Theory of The Atom

Wave-Particle Duality – The Photoelectric Effect

Photoelectric Effect
Flow of electric current when monochromatic light of
sufficient frequency shines on a metal plate
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
Electrons are ejected from the metal surface, only
when the frequency exceeds a certain threshold
characteristic of the metal

Radiation of lower frequency would not produce any
current flow no matter how intense

Violet light will cause potassium to eject electrons, but
no amount of red light (lower frequency) has any
effect

Current flows immediately upon absorption of radiation
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Quantum Theory of The Atom

Wave-Particle Duality – The Photoelectric Effect

Einstein resolved these discrepancies

He reasoned that if a vibrating atom changed energy
from nhv to (n-1)hv, this energy would be emitted
as a quantum (hv) of light energy he called a photon

He defined the photon as a Particle of Electromagnetic
energy, with energy E, proportional to the observed
frequency of the light.
Ephoton = ΔEatom
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hc
= hν =
λ
 Δn
= 1

The energy (hv) of an impacting photon is taken up
(absorbed) by the electron and ceases to exist

The Wave-Particle Duality of light is regarded as
complimentary views of wave and particle pictures of
light
18
Quantum Theory of The Atom
In 1921 Albert Einstein received the Nobel Prize in
Physics for discovering the photoelectric effect
• Electrons in metals exist in different and
specific energy states
• Photons whose frequency matches or
exceeds the energy state of the electron
will be absorbed
• If the photon energy (frequency) is less
than the electron energy level, the photon
is not absorbed
• The electron moves to a higher energy
state and is ejected from the surface of the
metal
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• The electrons are attracted to the positive
anode of a battery, causing a flow of
current
19
Practice Problem
Light with a wavelength of 478 nm lies in the blue
region of the visible spectrum.
Calculate the frequency of this light
Speed of Light = 3 x 108 m/s
Ans:
c  λν
m
3 x 10
s
ν  c/λ 
-9
10 m
478 nm
nm
8
ν  6.28 x 10 / s  6.28 x 10 Hz
14
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14
20
Practice Problem
The green line in the atomic spectrum of Thallium
(Tl) has a wavelength of 535 nm.
Calculate the energy of a photon of this light?
Planck's Constant h = 6.626 x 10
-34
J •s
h•c
E =
λ
6.626 x 10-34 J  s x 3.00 x 108 m / s
E 
10-9 m
535 nm
nm
E  3.716 x 10-19 J
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21
Practice Problem
At its closest approach, Mars is 56 million km from
earth. How many minutes would it take to send a
radio message from a space probe of Mars to
Earth when the planets are at this closest
distance?
Velocity  Distance / Time
Time  Distance / Velocity
In a vacuum, all types of electromagnetic radiation travel at :
2.99792458  108 m / s
(3  108 m / s)

1000 m 
6
 56  10 km

km
Time  

m
60
s

 3  108   


s
min
  


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Time  3.111 min
22
Quantum Theory of The Atom

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Atomic Line Spectra

When light from “excited” (heated) Hydrogen
atoms or other atoms passes through a prism,
it does not form a continuous spectrum, but
rather a series of colored lines (Line Spectra)
separated by black spaces

The wavelengths of these lines are
characteristic of the elements producing them

The spectra lines of Hydrogen occur in several
series, each series represented by a positive
integer, n
23
Quantum Theory of The Atom
n=1
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n=2
n=3
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Quantum Theory of The Atom

Atomic Line Spectra

In 1885, J. J. Balmer showed that the wavelengths,
, in the visible spectrum of Hydrogen could be
reproduced by a Rydberg Equation
1
λ
= R ( n12 - n12 ) = 1.096776×107 m -1 ( n12 - n12 )
1
2
1
2
where: R = The Rydberg Constant
 = wavelength of the spectral line
n1 & n2 are positive integers and n2 > n1
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Quantum Theory of The Atom

Atomic Line Spectra

For the visible series of lines the value of n1 = 2

The known wavelengths of the four visible lines for
hydrogen correspond to values of n2 = 3, n = 4,
n = 5, and n = 6

The Rydberg equation becomes
 1
1
1 
= R  2 - 2  with n2 = 3, 4, 5, 6….
λ
n2 
2
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
The above equation and the value of ”R” are based on
“data” rather than theory

The following work of Niels Bohr makes the
connection between the “data” model and Theory
26
Quantum Theory of The Atom

Bohr Theory of the Hydrogen Atom

Prior to the work of Niels Bohr, the stability of the
atom could not be explained using the then-current
theories, i.e.,
How can electrons (e-) lose energy and remain in orbit?

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Bohr in 1913 set down postulates to account for (1)
the stability of the hydrogen atom and (2) the line
spectrum of the atom

Energy level postulate:
An electron can have only specific energy levels in
an atom

Transitions between energy levels:
An electron in an atom can change energy levels
by undergoing a “transition” from one energy level
to another
27
Quantum Theory of The Atom
Energy x 10-20 (J/atom)
Transitions of the Electron
in the Hydrogen Atom
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28
Practice Problem
From the Bohr model of the Hydrogen atom we
can conclude that the energy required to excite an
Greater Than the
electron from n = 2 to n = 3 is ___________
energy to excite an electron from n = 3 to n = 4
a. less than
b. greater than
c. equal to
d. either equal to or less than
e. either equal to or greater than
Ans : b
E 2 3 > E 3 4
 1 1
E2 3 = hν = -R h  2 - 2  = - 2.179 x 10-18 J   -0.139  = 3.029 x 10-19 J
 3f 2i 
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 1 1
E34 = hν = -R h  2 - 2  = - 2.179 x 10-18 J  (-0.049) = 1.068 x 10-19 J
 4f 3i 
29
Practice Problem
An electron in a Hydrogen atom in the level n = 5
undergoes a transition to level n = 3.
What is the wavelength of the emitted radiation?
(R = 2.179 x 10-18 J)
E5 3
E53
 1 1
= hν = - R  2 - 2  = - 2.179 x 10-18 J *  0.071
 3f 5i 
= hν = - 1.547 x 10-19 J
Note: For computation of frequency and wavelength
the negative sign of the energy value can be ignored
-19
 E5 3   1.547 × 10 J 
14
ν = 
=
2.335
x
10
/ s  Hz 

 = 
-34
 h   6.626 × 10 J • s 
7/31/2017


8
 c   3.00 x 10 m / s 
-6
λ =   = 
=
1.285
×
10
m

14
ν
   2.335 × 10




s
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Quantum Theory of The Atom

Quantum Mechanics

Current ideas about atomic structure depend
on the principles of
quantum mechanics
7/31/2017

A theory that applies to subatomic particles
such as electrons

Electrons show properties of both waves
and particles
31
Quantum Theory of The Atom

Quantum Mechanics

The first clue in the development of quantum
theory came with the discovery of the
de Broglie relation

In 1923, Louis de Broglie reasoned that if light
exhibits particle aspects, perhaps particles of
matter show characteristics of waves
He postulated that a particle with mass m and
a velocity v has an associated wavelength
 The equation  = h/mv is called the

de Broglie relation
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Quantum Theory of The Atom

Quantum Mechanics
 If matter has wave properties, why are they
not commonly observed?
 The de Broglie relation shows that a
baseball (0.145 kg) moving at a velocity of
about 60 mph (27 m/s) has a wavelength of
about 1.7 x 10-34 m.
2
kg
•
m
-34
6.626
×
10
h
s
λ=
=
= 1.7 ×10-34 m
(0.145 kg)(27 m / s)
mv
This value is so incredibly small that such
waves cannot be detected.
 Electrons have wavelengths on the order of
a few picometers (1 pm = 10-12 m).

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33
Practice Problem
At what speed (v) must an neutron (1.67 x 10-27
kg) travel to have a wavelength of 10.0 pm?
λ = h / mv
(De Broglie Relation)




-34 kg • m
6.626 × 10
s


v = h/mλ =
-12


10
m
-27
 1.67 × 10 kg × 10 pm 

 pm  

2
v  3.97  10 5 m / s
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34
Quantum Theory of The Atom

7/31/2017
Quantum mechanics is the branch of physics that
mathematically describes the wave properties of
submicroscopic particles

We can no longer think of an electron as having a
precise orbit in an atom

To describe such an orbit would require knowing its
exact position and velocity, i.e., its motion (mv)

In 1927, Werner Heisenberg showed (from
quantum mechanics) that:
It is impossible to simultaneously measure the
present position while also determining the future
motion of a particle, or of any system small
enough to require quantum mechanical treatment
35
Quantum Theory of The Atom
7/31/2017

Mathematically, the uncertainty relation between
position and momentum, i.e., the variables, arises due
to the fact that the expressions of the wavefunction in
the two corresponding bases (variables) are Fourier
Transforms of one another

According to the Uncertainty Principle of Heisenberg ,
if the two operators representing a pair of variables
do not commute, then that pair of variables are
mutually complementary, which means they cannot
be simultaneously measured or known precisely

In the mathematical formulation of quantum
mechanics , changing the order of the operators
changes the end result, i.e., the operators are noncommuting, and are subject to similar uncertainty
limits
36
Quantum Theory of The Atom

Quantum Mechanics

Heisenberg’s uncertainty principle is a relation
that states that the product of the uncertainty in
position (Dx) and the uncertainty in momentum
(mDvx) of a particle can be no smaller than:
h
(Δ x)(m Δv x ) 
4π
h = Planck's constant - 6.626×10-34 J • s

7/31/2017
h
= 5.28  10-35 J • s
4π
When m is large (for example, a baseball) the
uncertainties are very small, but for electrons, high
uncertainties disallow defining an exact orbit
37
Practice Problem
Heisenberg's Uncertainty Principle can be expressed
mathematically as:
h
Δx × Δp =
4π
Where Dx is the uncertainty in Position
Dp (= mDv) is the uncertainty in Momentum
h is Planck's constant (6.626 x 10-34 kg  m2/s)
What would be the uncertainty in the position (∆x) of a fly
(mass = 1.245 g) that was traveling at a velocity of 3.024
m/s if the velocity has an uncertainty of 2.72%?
Uncertainty in velocity = 2.72 % = 0.0272
Planck’s Constant
 Δv  0.0272  3.024 m/s  8.225 x 10 m/s
-2
 6.626  10 J  s 1 kg  m / s 
 h 



 4π 
4

3.14159
J

Δx  
  
m
Δv
1
kg






1.245 g 
8.225  10-2 m / s 




 1000 g 


34
7/31/2017
Δx  5.157 x 10-33 m
2
2
h = 6.626 x 10-34 J  s
1 J = 1 kg  m2/s2
h = 6.626 x 10-34 kg  m2/s
38