George Mason University
General Chemistry 211
Chapter 7
Quantum Theory and Atomic Structure
Acknowledgements
Course Text: Chemistry: the Molecular Nature of Matter and
Change, 7th edition, 2011, McGraw-Hill
Martin S. Silberberg & Patricia Amateis
The Chemistry 211/212 General Chemistry courses taught at
George Mason are intended for those students enrolled in a science
/engineering oriented curricula, with particular emphasis on
chemistry, biochemistry, and biology The material on these slides is
taken primarily from the course text but the instructor has modified,
condensed, or otherwise reorganized selected material.
Additional material from other sources may also be included.
Interpretation of course material to clarify concepts and solutions to
problems is the sole responsibility of this instructor.
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1
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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2
Quantum Theory of The Atom

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
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3
Quantum Theory of The Atom

Emission (line) Spectra of Some Elements
When elements are heated in a flame and their
emissions are 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)
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4
Quantum Theory of The Atom

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
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5
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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6
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 the
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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7
Quantum Theory of The Atom

Relationship Between Wavelength and Frequency
 Wavelength () and frequency () are inversely
proportional
 = c/ (frequency = Speed light / wavelength)
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8
The Electromagnetic Spectrum
High
Frequency ()
Low
High
Energy (E)
Low
Short
Wavelength ()
Long
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9
Distinction between Energy & Matter

At the macro scale level – everyday life – energy (waves) &
matter (particles) 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
A 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
Particles passing through adjacent openings
coincide in phase
continue on straight paths, some colliding
Destructive Interference – Crests coincide with each other moving at different angles
with troughs, cancelling out.
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10
Distinction Between Energy & Matter
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11
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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
12
Quantum Theory of The Atom

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
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13
Quantum Theory of The Atom

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
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14
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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15
Quantum Theory of The Atom

Quantum Effects – Wave-Particle Duality
 Planck’s Model:
E = nhν
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
 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
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16
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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17
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
 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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18
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
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
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19
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
20
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
21
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?
h•c
E =
λ
Planck's Constant h = 6.626 x 10
-34
J •s
6.626 x 10-34 J • s x 3.00 x 108 m / s
E =
10-9 m
535 nm
nm
-19
E = 3.716 x 10 J
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22
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
23
Quantum Theory of The Atom

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
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24
Quantum Theory of The Atom
n=1
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n=2
n=3
25
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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26
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
 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
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27
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?
 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
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● Transitions between energy levels:
An electron in an atom can change energy levels by
undergoing a “transition” from one energy level to
another
28
Quantum Theory of The Atom
Bohr Theory
Energy x 10-20 (J/atom)
Transitions of the Electron in
the Hydrogen Atom
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29
Quantum Theory of The Atom

Bohr Theory of the Hydrogen Atom
 Bohr’s Postulates
● Bohr derived the following formula for the
energy levels of the electron in the hydrogen
atom
2 

Z
E = - 2.18 × 10-18 J  2 
n 
n = 1, 2, 3 ...  (principal quantum no.s for Hydrogen)
Z = nuclear charge
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30
Quantum Theory of The At

Bohr Theory of the Hydrogen Atom
 Bohr’s Postulates
● For the Hydrogen atom, Z = 1
E = - 2.18 × 10
-18
 12 
J  2  = - 2.18 × 10-18
n 
 1 
J 2 
n 
● For the energy of the ground state (n =1)
E = - 2.18 × 10
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-18
 1
J  2  = - 2.18 × 10-18
1 
 1
J   = - 2.18 × 10-18 J
 1
31
Quantum Theory of The Atom

Bohr Theory of the Hydrogen Atom
 Bohr’s Postulates
● When an electron undergoes a transition from a
higher energy level (ni) to a lower one (nf), the
energy is emitted as a photon
Energy of emitted photon = hν = Ef - Ei = ΔE
-18
2.18×10 J
Ei = ni 2
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2.18×10-18 J
Ef = nf 2
 2.18 ×10-18 J   2.18 ×10-18 J 
ΔE = hν = Ef - Ei =   - 
2
2
n
n

 

f
i

1 
-18  1
ΔE = hν = - 2.18 × 10 J 2 - 2
n n 
i 
 f
32
Quantum Theory of The Atom

Bohr Theory of the Hydrogen Atom
 Bohr’s Theory vs. Rydberg Data model
● If we make a substitution into the previous equation
that states the energy of the emitted photon, h,
equals hc/
hc
ΔE = hν =
= - 2.18 × 10-18
λ
1
-2.18 × 10-18 J  1 1
=
 n2 n2
λ
hc
i
 f
-18

2.18
×
10
J
 = (-1) ×

(6.626 × 10-34 J • s) (3.00 × 108 m / s)


1
1
7
-1  1
= - 1.10 × 10 m
 n2 n2
λ
Bohr (theory)  f
i
1/13/2015
 1 1 
J 2 - 2 
n n 
i 
 f




 1 1 

- 2
2
n n 
i 
 f
7
-1 1
1
1
versus λ = - 1.096776 ×10 m ( 2 - 2 )
n
n
Rydberg (data)
1
2
● Thus, from the classical relationships of charge and
motion combined with the concept of discreet
energy levels – theory matches data
33
Practice Problem
From the Bohr model of the Hydrogen atom we can conclude
that the energy required to excite an electron from n = 2 to
Greater Than the energy to excite an electron
n = 3 is ___________
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
 1 1
E2 3 = hν = -R h  2 - 2  = - 2.179 x 10-18 J ×  -0.139  = 3.029 x 10-19 J
 3f 2 i 
 1 1
E34 = hν = -R h  2 - 2  = - 2.179 x 10-18 J × (-0.049) = 1.068 x 10-19 J
 4f 3 i 
Ans : b
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E 2 3 > E 3®4
34
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)
E 5 3
 1 1
= hν = - R  2 - 2  = - 2.179 x 10-18 J *  0.071
 3f 5 i 
E5 3 = 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 
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

8
 c   3.00 x 10 m / s 
-6
λ =   = 
=
1.285
×
10
m

14
ν
   2.335 × 10




s
35
Quantum Theory of The Atom

Bohr Theory of the Hydrogen Atom
Bohr’s Postulates
● Bohr’s theory explains not only the emission
of light, but also the absorption of light
● When an electron falls from n = 3 to n = 2
energy level, a photon of red light
(wavelength, 685 nm) is emitted
● When red light of this same wavelength
shines on a hydrogen atom in the n = 2 level,
the energy is gained by the electron that
undergoes a transition to n = 3
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36
Quantum Theory of The Atom

Quantum Mechanics
 Bohr’s theory established the concept of atomic
energy levels but did not thoroughly explain the
“wave-like” behavior of the electron
 Current ideas about atomic structure depend on
the principles of
quantum mechanics
● A theory that applies to subatomic particles
such as electrons
● Electrons show properties of both waves and
particles
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37
Quantum Theory of The Atom

Quantum Mechanics
 The first clue in the development of quantum
theory came with the discovery in 1923 by
Louis de Broglie who 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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38
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
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● Electrons have wavelengths on the order of a
few picometers (1 pm = 10-12 m)
39
Practice Problem
At what speed (v) must an neutron (1.67 x 10-27 kg)
travel to have a wavelength of 10.0 pm?
1 pm (picometer) = 10-12 m = 10-10 cm = 0.01 Å)
λ = h / mv
(De Broglie Relation)


2


-34 kg • m
6.626 × 10
s


v = h/mλ =
-12


10
m
-27
 1.67 × 10 kg × 10 pm 

 pm  

v = 3.97 × 10 5 m / s
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40
Quantum Theory of The Atom

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:
1/13/2015
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
41
Quantum Theory of The Atom
 Max Born stated in his Nobel Laureate speech:
To measure space coordinates and instants of
time, rigid measuring rods and clocks are
required
On the other hand, to measure momenta and
energies, devices are necessary with movable
parts to absorb the impact of the test object and
to indicate the size of its momentum (mass x
velocity)
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Paying regard to the fact that quantum
mechanics is competent for dealing with the
interaction of object and apparatus, it is seen
that no arrangement is possible that will fulfill
both requirements simultaneously
42
Quantum Theory of The Atom
 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 non-commuting, and are
subject to similar uncertainty limits
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43
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 ) ³
4π
2


kg
•
m
-34
h = Planck's constant - 6.626×10 J • s 

s


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
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44
Practice Problem
Heisenberg's Uncertainty Principle can be expressed
mathematically as:
h
Planck’s Constant
Δx × Δp =
h = 6.626 x 10 J  s
4π
-34
Where Dx is the uncertainty in Position
1 J = 1 kg  m2/s2
h = 6.626 x 10-34 kg  m2/s
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
 Δv = 0.0272 * 3.024 m / s = 8.225 x 10-2 m / s
 6.626×10-34 J • s 1 kg • m 2 / s 2 
 h 
×


 4π 
4×
3.14159
J
 = 5.149×10-33 meters
Δx = 
= 

 1 kg 


 m Δv 
1.245
g
8.225 × 10-2 m / s 





 1000 g 


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Quantum Theory of The Atom

Quantum Mechanics
 Acceptance of the dual nature of matter and energy (E
= mc2) and the “Uncertainty” Principle culminated in the
field of Quantum Mechanics:
Wave Nature of objects on the Atomic Scale
 Erwin Schrodinger developed quantum mechanical
model of the Hydrogen atom, where
● An Atom has certain allowed quantities of energy
● An Electron’s behavior is wavelike, but its exact
location is impossible to know
● The Electron’s Matter-Wave occupies 3-dimentional
space near nucleus
● The Matter-Wave experiences continuous, but varying
influence from the nuclear charge
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Quantum Theory of The Atom

Quantum Mechanics
 Schrodinger Equation:


H
H = 
= Energy of the atom
= Wave Function
= Hamiltonian Operator – Mathematical
operations that when carried out on a
particular wave yields the allowed energy
value

Each solution of the wave equation is associated with a
given “atomic orbital”, which bears no resemblance to
an orbit in the Bohr model

An “Orbital” is a mathematical function, which like a Bohr
Orbit, represents a particular energy level of the orbiting
electron, but it has no direct physical meaning
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Quantum Theory of The Atom

Quantum Mechanics
 Heisenberg's uncertainty principle says we
cannot precisely define an electron’s orbit
 The wave function (atomic orbital) has no direct
physical meaning
 The square of the wave function,  2, however,
is defined as the probability density, a measure
of the probability that the electron can be found
within a particular tiny volume of the atom
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Quantum Theory of The Atom
Probability of Finding an Electron in a Spherical Shell About
the Nucleus
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Quantum Theory of The Atom

Quantum Numbers and Atomic Orbitals

According to quantum mechanics each electron is
described by 4 quantum numbers
 Principal Quantum Number
(n)
 Angular Momentum Quantum Number
(l)
 Magnetic Quantum Number
(ml)
 Spin Quantum Number
(ms)
The first three quantum numbers define the wave
function of the electron’s atomic orbital
The fourth quantum number refers to the spin orientation
of the 2 electrons that occupy an atomic orbital
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Quantum Theory of The Atom

Quantum Numbers and Atomic Orbitals
 The Principal Quantum Number (n) represents
the “Shell Number” in which an electron “resides”
● It represents the relative size of the orbital
● Equivalent to periodic chart Period Number
● Defines the principal energy of the electron
● The smaller “n” is, the smaller the orbital
● The smaller “n” is, the lower the energy of the
electron
● n can have any positive value from
1, 2, 3, 4 … 
(Currently, n = 7 is the maximum known)
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Quantum Theory of The Atom

Quantum Numbers and Atomic Orbitals (Con’t)
 The Angular Momentum Quantum Number (l)
distinguishes “sub shells” within a given shell
● Each main “shell,” designated by quantum number
“n,” is subdivided into:
l = n - 1 “sub shells”
● (l) can have any integer value from 0 to n - 1
● The different “l” values correspond to the
s, p, d, f designations used in the electronic
configuration of the elements
l
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Letter
s
p
d
f
value
0
1
2
3
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Quantum Theory of The Atom

Quantum Numbers and Atomic Orbitals (Con’t)
 The Magnetic Quantum Number (ml) defines atomic
orbitals within a given sub-shell
● Each value of the angular momentum number (l)
determines the number of atomic orbitals
● For a given value of “l,” ml can have any integer
value from -l to +l
ml = -l to +l
(-2 -1 0 +1 +2)
● Each orbital has a different shape and orientation (x,
y, z) in space
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● Each orbital within a given angular momentum
number sub shell (l) has the same energy
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Quantum Theory of The Atom

Quantum Numbers and Atomic Orbitals (Con’t)
 The Spin Quantum Number (ms) refers to the
two possible spin orientations of the electrons
residing within a given atomic orbital
● Each atomic orbital can hold only two (2)
electrons
● Each electron has a “spin” orientation value
● The spin values must oppose one another
● The possible values of ms spin values are:
+1/2 and –1/2
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Summary of Quantum Numbers
Name
principal
Symbol
n
l
angular
momentum
Permitted
Values
positive integers
(1, 2, 3, …)
Property
orbital energy
(size)
integers from
0 to n -1
orbital shape
The l values
0, 1, 2, and 3
correspond to
s, p, d, and f
orbitals, respectively
magnetic
ml
integers from
-l to 0 to +l
orbital (x,y,z)
orientation
spin
ms
+1/2 or -1/2
e- spin orientation
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Quantum Numbers and Atomic Orbitals
The
Hierarchy
of
Quantum
Numbers
for
Atomic
Orbitals
l=0
l=1
l=0
l=1
l=2
(1s)
(2s)
(2p)
(3s)
(3p)
(3d)
ml = 0
ml =0
ml = -1 0 +1
0
-1 0 +1
-2 -1 0 +1 +2
n=4
l=0
ml = 0
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n=3
l=0
(4s)
Note:
n>7&l>3
not defined
for the current
list of
elements in
the Periodic
Table
n=2
n=1
l=1
(4p)
-1 0 +1
l=2
l=3
(4d)
(4f)
-2 -1 0 +1 +2
-3 -2 -1 0 +1 +2 +3
n=5
l=0
(5s)
ml = 0
l=1
(5p)
-1 0 +1
l=2
l=3
(5d)
(5f)
-2 -1 0 +1 +2
-3 -2 -1 0 +1 +2 +3
n=6,7
l=0
(6s,7s)
l=1
(6p,7p)
ml = 0
-1 0 +1
l=2
l=3
(6d)
(6f)
-2 -1 0 +1 +2
-3 -2 -1 0 +1 +2 +3
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Quantum Numbers and Atomic Orbitals

Using calculated probabilities of electron
“position,” the shapes of the orbitals can be
described
 The s (n = 1) sub shell orbital (there is only
one) is spherical
 The p (n = 2) sub shell orbitals (there are
three) are dumbbell shape
 The d (n = 3) sub shell orbitals (there are five)
are a mix of cloverleaf (pear-shaped
lobes) and dumbbell shapes
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Quantum Numbers and Atomic Orbitals
Cross-sectional Representations
of the Probability Distributions
of “s” Orbitals (spherical)
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Quantum Numbers and Atomic Orbitals
Cutaway Diagrams Showing the
Spherical Shape of “s” Orbitals
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Quantum Numbers and Atomic Orbitals
Radial Probability Distribution
of the Three 2p Orbitals (dumbell shapes)
n=2
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l = 2 – 1= 1 (p)
ml = -1 0 +1
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Quantum Numbers and Atomic Orbitals
Radial Probability Distribution
of the Five 3d Orbitals (Cloverleaf & Dumbells)
n=3
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l = n -1 = 3 – 1= 2 (d)
ml = -2 -1 0 +1 +2
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Quantum Numbers and Atomic Orbitals
Radial Probability Distribution of the
Seven 4f Orbitals
n=4
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l = 4 – 1= 3 (f)
ml = -3 -2 -1 0 +1 +2 +3
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Quantum Numbers and Atomic Orbitals
Orbital Energies of the Hydrogen Atom
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Practice Problems
If the n quantum number of an atomic orbital is 4,
what are the possible values of the l quantum
number?
Ans: (l) can have any integer value from 0 to n –1
l = n-1 = 4 –1=3
 Values of l = 0 1
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2
3
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Practice Problem
If the l quantum number is 3, what are the possible
values of ml?
Ans: ml can have any integer value from -l to +l
Since l = 3
ml = -3 -2 -1
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0
+1
+2
+3
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Practice Problem
State which of the following sets of quantum numbers
would be possible and which impossible for an electron
in an atom?
a. n = 0,
l=0,
ml = 0,
ms = +1/2
b. n = 1,
l=0,
ml = 0,
ms = +1/2
c. n = 1,
l=0,
ml = 0,
ms = -1/2
d. n = 2,
l=1,
ml = -2,
ms = +1/2
e. n = 2,
l=1,
ml = -1,
ms = +1/2
Ans: Possible
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b
c e
Impossible
a “n ” must be positive 1, 2, 3...
Impossible
d ml can only be -1 0 +1
66
Summary Equations
Light
c = 
Planck’s Model
nhc
E = nhν =
λ
Photoelectric
Balmer Rydberg
Bohr Model
c = 3 x 108 m/s
Ephoton = ΔEatom
h = 6.626 ×10-34 J • S
2
kg
•
m
h = 6.626 ×10-34
s
hc
= hν =
λ
1
= 1.096776×107 m -1 ( 21 - n1 )
λ
2 

Z
E = - 2.18 ×10-18 J  2 
n 
2
Bohr Postulate
De Broglie
h
λ =
mν
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= 1
2
hc
ΔE = hν =
= - 2.18 × 10-18
λ
Heisenberg
 Δn
 1 1 
J 2 - 2 
n n 
i 
 f
h
Δx • mΔu =
4π
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