Lecturer : Dr K. Makgopa
Email
: MakgopaK@tut.ac.za
Office No. : Building 3, Room 614D
Department : Chemistry (Arcadia Campus)
Ext.
: 6406
Lecture Hall : 3-308
Consultation Time Mon, Wed - Thurs: 13:00 – 15.00
“I have never met a man so ignorant that I couldn't learn something from him” - (Galileo Galilei)
Learning Objectives;
After completing this study unit theme, the student must be able to:
• Describe how different radiations are produced.
• Describe how energy contents of different radiations compare.
• Interpret conversions among 𝜆𝜆, 𝜈𝜈, 𝜈𝜈̅ and energy for a given radiation.
• Write and be able to convert the units of all properties mentioned above.
• Describe how atomic spectra are produced.
• Describe the six spectral series of the hydrogen atom:
• Lyman, Balmer, Paschen, Bracket, Pfund and Humphreys with
wavelength
• Interpret 𝜈𝜈 of each spectral line of atomic hydrogen using Rydberg formula.
• Describe and demonstrate the concept, Series limit.
• Calculate 𝜆𝜆, 𝜈𝜈, 𝜈𝜈̅ of lines belonging to a given series limit.
• Calculate energy required to promote an electron from given energy levels.
• Calculate ionisation energy of the hydrogen-like a species
QUANTUM CHEMISTRY
Quantum Chemistry
~concerns the restriction of quantum mechanics to
the electromagnetic force
~leads to a treatment of the structure and
properties of atoms and molecules where
Quantum mechanics
~involves the duality of light
~i.e. electromagnetic radiation to have a particle
nature and classical particles to have a wave nature
 A wave is a continuously repeating change or
oscillation in matter or in a physical field.
 Classically, electromagnetic waves (which
make up the electromagnetic radiation) are
synchronized oscillations of electric and
magnetic fields that propagate at the speed of
light through a vacuum
 Light is an electromagnetic wave, consisting of
oscillations in electric and magnetic fields
traveling through space.
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A wave can be characterized by its wavelength
and frequency.
Wavelength, symbolized by the Greek letter
lambda, λ, is the distance between any two
identical points on adjacent waves.
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Wave Nature of light
c= λ𝑣𝑣
Interference of light:
Interaction of two waves in space
Frequency, symbolized by the Greek letter nu, ν,
is the number of wavelengths that pass a fixed
point in one unit of time (usually a second). The
unit is 1/S or s-1, which is also called the Hertz (Hz).
Wavelength and frequency are related by the wave
speed, which for light is c, the speed of light, 3.00
x 108 m/s.
c = νλ
The relationship between wavelength and
frequency due to the constant velocity of light is
illustrated on the next slide.
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When the
wavelength is
reduced by a
factor of two,
the frequency
increases by
a factor of
two.
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?
What is the wavelength of blue light
with a frequency of 6.4 × 1014/s?
ν = 6.4 × 1014/s
c = 3.00 × 108 m/s
c = νλ so
λ = c/ν
m
3.00 x 10
c
s
λ= =
ν
14 1
6.4 x 10
s
8
λ = 4.7 × 10-7 m
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The range of frequencies and wavelengths of
electromagnetic radiation is called the
electromagnetic spectrum.
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Electromagnetic Spectrum
One property of waves is that they can be diffracted—that
is, they spread out when they encounter an obstacle about
the size of the wavelength.
In 1801, Thomas Young, a British physicist, showed that
light could be diffracted. By the early 1900s, the wave
theory of light was well established.
Planck’s Quantization of Energy (1900)
– According to Max Planck, the atoms of a solid,
oscillate with a definite frequency, ν.
– He proposed that an atom could have only certain
energies of vibration, E, those allowed by the
formula
E = nhν
where h (Planck’s constant) is assigned a value of
6.63 x 10-34 J. s and n must be an integer.
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Failures of Classial Physics
Line spectra of atoms
Black body radiation
Heat capacity of solids
Photoelectric effect
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Emission
(line)
spectra of
some
elements.
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Blackbody Radiation:
All bodies at a temperature above absolute zero emit radiation over a range of wavelengths
A blackbody is defined as a
perfect absorber because it
aborbs all the radiation that
strikes it.
It is also a perfect emitter of
radiation because it is in
thermal equilibrium with its
surroundings.
Quantum Effects and Photons
Planck’s Quantization of Energy.
– The only energies a vibrating atom can have are
hν, 2hν, 3hν, and so forth.
– The numbers symbolized by n are quantum
numbers.
– The vibrational energies of the atoms are said to
be quantized.
– Solved the ultraviolet catastrophe in blackbody
radiation
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The photoelectric
effect.
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Photoelectric Effect
– The photoelectric effect is the ejection of electrons from the
surface of a metal when light shines on it.
– Electrons are ejected only if the light exceeds a certain
“threshold” frequency.
– Violet light, for example, will cause potassium to eject
electrons, but no amount of red light (which has a lower
frequency) has any effect.
By the early part of twentieth century, the wave theory of light
seemed to be well entrenched.
– In 1905, Albert Einstein proposed that light had both wave
and particle properties to explain the observations in the
photoelectric effect.
– Einstein based this idea on the work of a German physicist,
Max Planck.
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Photoelectric Effect
– The energy of the photons proposed by Einstein would be
proportional to the observed frequency, and the
proportionality constant would be Planck’s constant.
E = hν
– In 1905, Einstein used this concept to explain the
“photoelectric effect.”
– Einstein’s assumption that an electron is ejected when
struck by a single photon implies that it behaves like a
particle.
– When the photon hits the metal, its energy, hν is taken up
by the electron.
– The photon ceases to exist as a particle; it is said to be
“absorbed.”
Photoelectric Effect
– The “wave” and “particle” pictures of light should be
regarded as complementary views of the same physical
entity.
– This is called the wave-particle duality of light.
– The equation E = hν displays this duality; E is the energy
of the “particle” photon, and ν is the frequency of the
associated “wave.”
– For a given metal, a certain amount of energy is needed to
eject the electron
– This is called the work function
– Since E=hν, the photons must have a frequency higher
than the work function in order to eject electrons
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Photoelectric effect
 In the early 1900s, the atom was understood to consist of a
positive nucleus around which electrons move (Rutherford’s
model).
 This explanation left a theoretical dilemma: According to the
physics of the time, an electrically charged particle circling a
center would continually lose energy as electromagnetic
radiation. But this is not the case—atoms are stable.
In addition, this understanding could not explain the observation
of line spectra of atoms.
 A continuous spectrum contains all wavelengths of light.
 A line spectrum shows only certain colors or specific
wavelengths of light. When atoms are heated, they emit light.
This process produces a line spectrum that is specific to that
atom. The emission spectra of six elements are shown on the
next slide.
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In 1913, Neils Bohr, a Danish scientist, set down postulates
to account for
1. The stability of the hydrogen atom
2. The line spectrum of the atom
Bohr Model
 The Bohr model of the hydrogen atom (Z = 1) or a hydrogenlike ion (Z > 1), where the negatively- charged electron
confined to an atomic shell encircles a small, positively
charged atomic nucleus and where an electron jumps between
orbits it is accompanied by an emitted or absorbed amount of
electromagnetic energy (hν).
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Shortcomings of Bohr Model
 Much of the spectra of larger atoms
 the relative intensities of spectral lines
 The Zeeman effect – changes in spectral lines due
to external magnetic fields;
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Energy-Level Postulate
An electron can have only certain energy values,
called energy levels. Energy levels are quantized.
For an electron in a hydrogen atom, the energy is
given by the following equation:
E=−
RH
n
2
RH = 2.179 x 10-18 J
n = principal quantum number
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Transitions Between Energy Levels
 An electron can change energy levels by absorbing
energy to move to a higher energy level or by
emitting energy to move to a lower energy level.
For a hydrogen electron the energy change is
given by
ΔE = E f − E i
 1

1
ΔE = − R H  2 − 2 
n

n
i 
 f
RH = 2.179 × 10-18 J, Rydberg constant
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The energy of the emitted or absorbed photon is
related to ∆E:
E photon = ΔE electron = hν
h = Planck' s constant
We can now combine these two equations:
 1

1
hν = − R H  2 − 2 
n

n
i 
 f
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Light is absorbed by an atom when the electron transition is
from lower n to higher n (nf > ni). In this case, ∆E will be positive.
Light is emitted from an atom when the electron transition is
from higher n to lower n (nf < ni). In this case, ∆E will be negative.
An electron is ejected when nf = ∞.
Energy-level diagram for
the hydrogen atom.
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7 | 30
Electron transitions for an
electron in the hydrogen atom.
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?
What is the wavelength of the light emitted
when the electron in a hydrogen atom
undergoes a transition from n = 6 to n = 3?
 1
ΔE = − R H  2 −
n
 f
hc
ΔE =
so λ =
λ
ni = 6
nf = 3
RH = 2.179 × 10-18 J
(
1 
2 
ni 
hc
ΔE
)
1 
 1
ΔE = − 2.179 x 10 −18 J  2 − 2  = -1.816 x 10-19 J
6 
3
(
)

8 m
6.626 x 10 J • s  2.998 x 10

s

λ=
= 1.094 × 10-6 m
−19
- 1.816 x 10 J
− 34
(
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)
Planck
Vibrating atoms have only certain energies:
E = hν or 2hν or 3hν
Einstein
Energy is quantized in particles called photons:
E = hν
Bohr
Electrons in atoms can have only certain values of
energy. For hydrogen:
E=−
RH
n2
RH = 2.179 x 10 −18 J, n = principal quantum number
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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.
 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 λ = h/mv is called the de Broglie
relation.
de Broglie Relation
For a photon that has both wave and particle
characteristics:
E = hν = hc/λ (recall c= νλ)
E = mc2
mc2 = hc/λ or λ = h/mc
Since mc is the momentum of a photon, can we
replace this with the momentum of a particle?
λ = h/mu (u= velocity)
This suggests that particles have wave-like
characteristics!
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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 about 60 mph (27 m/s) has a
wavelength of about 1.7 x 10-34 m.
=
λ
−34 kg⋅m 2
10
s
6.63 ×
(0.145 kg )(27 m / s )
= 1.7 ×10
−34
m
– This value is so incredibly small that such waves
cannot be detected.
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Quantum Mechanics
If matter has wave properties, why are they not
commonly observed?
– Electrons have wavelengths on the order of a few
picometers (1 pm = 10-12 m).
– Under the proper circumstances, the wave character
of electrons should be observable.
– Molecules are of the dimension of a few pm, so the
wave character of electrons is very important in
molecules
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DE BROGLIE: WAVE PROPERTIES OF ELECTRONS
Quantum Mechanics
 If matter has wave properties, why are they not
commonly observed?
– In 1927, Davisson and
Germer was
demonstrated that a
beam of electrons,
just like X rays, could
be diffracted by a
crystal.
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 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.
– In 1927, Werner Heisenberg showed (from quantum
mechanics) that it is impossible to know both
simultaneously.
 Heisenberg’s uncertainty principle is a relation that states
that the product of the uncertainty in position (∆x) and the
uncertainty in momentum (m∆vx) of a particle can be no larger
than h/4π.
h
(∆x)(m∆vx ) ≥
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4π
– When m is large (for example, a baseball) the uncertainties
are small, but for electrons, high uncertainties disallow
defining an exact orbit.
 According to quantum mechanics,
described by four quantum numbers:
1.
2.
3.
4.
each
electron
is
Principal quantum number (n)
Angular momentum quantum number (l)
Magnetic quantum number (ml)
Spin quantum number (ms)
 The first three define the wave function for a particular
electron. The fourth quantum number refers to the magnetic
property of electrons.
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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.
The wave equation is designated
with a lower case Greek psi (ψ).
The square of the wave equation, ψ2, 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.
Letter designations are used to communicate the different
values of l and, therefore, the shapes and types of orbitals.
Value of l
0
1
2
3
Type of orbital
S
P
d
f
Magnetic Quantum Number, ml
Describes the three-dimensional orientation of the orbital.
Values are integers ranging from -l to l:
−l ≤ ml ≤ 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.
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
Value of l = 1.
Have two lobes with a node between them.
d Orbitals
Value of l is 2.
Four of the five orbitals have
4 lobes; the other resembles
a p orbital with a doughnut
around the center.
Magnetic Quantum Number, ml
Orbitals with the same value of n form a shell.
Different orbital types within a shell are subshells.
Energies of Orbitals
For a one-electron
hydrogen atom, orbitals on
the same energy level
have the same energy.
That is, they are
degenerate.
•As the number of electrons
increases, so does the
repulsion between them.
•Therefore,
in
manyelectron atoms, orbitals on
the same energy level are
no longer degenerate.
Spin Quantum Number, ms
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.
This led to a fourth quantum number, the spin quantum
number, ms.
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
Hund’s Rule
“For degenerate orbitals, the lowest energy is attained
when the number of electrons with the same spin is
maximized.”
Simply put!!
Hund's rule: every orbital in a subshell is singly
occupied with one electron before any one orbital is
doubly occupied, and all electrons in singly occupied
orbitals have the same spin