Topic 7
Quantum Theory of the
Atom
Towards the end of the 19th century, experiments
involving light interacting with atoms and
molecules could not be explained fully by classical
physics; therefore, quantum mechanics and
relativity were developed to address failures of
classical physics in describing the current accepted
model of the atom.
1
The Wave Nature of Light
Light is known as electromagnetic radiation.
Much of the behavior of light can be explained
by thinking of it as a wave. Light can also be
thought of as a stream of particles called
photons.
A wave is a periodic (repeating) disturbance
that transfers energy from one place to another.
The term electromagnetic means that the disturbance
is due to oscillation of charged particles in electric and
magnetic fields and exerts a force on any charged
particle that is in its way.
– Visible light, X rays, and radio waves are all forms
of electromagnetic radiation.
2
The Wave Nature of Light
A wave can be characterized by its wavelength
and frequency.
The wavelength, l (lambda), is the distance (m or nm)
between any two adjacent identical points of a wave.
amplitude
The frequency, n (nu), of a wave is the number of
wavelengths that pass a fixed point in one second (1/s
or s-1 or Hz). The amplitude is the height of the wave.
3
The Wave Nature of Light
The product of the frequency, n (waves/sec) and the
wavelength, l (m/wave) would give the speed of the wave
in m/s.
Electromagnetic waves travel at the speed of light, c,
which is 3.00 x 108 m/s. Therefore,
n
c
l
So, given the frequency of light, its wavelength can be
calculated, or vice versa.
note: freq and wavelength are inversely proportional meaning
the shorter the wavelength, the higher the frequency.
4
The Wave Nature of Light
What is the wavelength of yellow light with a
frequency of 5.09 x 1014 s-1? (Note: s-1, commonly
referred to as Hertz (Hz) is defined as “cycles or
waves per second”.)
We simply rearrange the equation to solve for the wavelength by multiplying
both sides by lambda, l, and dividing both sides by nu, n, giving
n
c
l
l
c
n
Next, we plug in the values into the equation and cancel units.
l=
𝟑.𝟎𝟎 𝒙 𝟏𝟎𝟖 𝒎/𝒔
𝟓.𝟎𝟗 𝒙 𝟏𝟎𝟏𝟒 𝒔−𝟏
= 𝟓. 𝟖𝟗 𝒙 𝟏𝟎−𝟕 𝒎 = 𝟓𝟖𝟗 𝒏𝒎
5
Note: s-1 = 1/s
and
1 x 10-9 m = 1 nm
The Wave Nature of Light
What is the frequency of violet light with a
wavelength of 408 nm?
We simply plug the values into the equation and cancel units.
n
n=
Note:
𝟑.𝟎𝟎 𝒙 𝟏𝟎𝟖 𝒎/𝒔
𝟒𝟎𝟖 𝒙 𝟏𝟎−𝟗 𝒎
1/s = s-1 =
Hz
c
l
= 𝟕. 𝟑𝟓 𝒙 𝟏𝟎𝟏𝟒 𝒔−𝟏 = 𝟕. 𝟑𝟓 𝒙 𝟏𝟎𝟏𝟒 Hz
and
1 nm = 1 x 10-9 m
6
The range of frequencies or wavelengths of electromagnetic
radiation is called the electromagnetic spectrum and is
related to what we perceive as the color of light.
high E and v,
short l
low E and v,
long l
Visible light extends from the violet end of the spectrum at
about 400 nm (short l, high n) to the red end with
wavelengths about 800 nm (long l, low n) .
Beyond these extremes, electromagnetic radiation is not 7
visible to the human eye. HW 57 code: wave
Quantum Effects and Photons
By the end of 19th century, experiments dealing
with light showed behaviors that are inconsistent
with the notion of light as an electromagnetic wave.
These inconsistencies are resolved by assuming
that the energies of light and matter are quantized
(meaning values are restricted). Thus, the theory
eventually developed into what we call quantum
theory.
In the case of electromagnetic radiation, we have
to think of energy as being carried by a stream of
particles called photons.
8
Quantum Effects and Photons
The energy carried by each photon is given by:
𝑬𝒑𝒉𝒐𝒕𝒐𝒏 = 𝒉n 
𝒉𝒄
l
where h is called Planck’s constant = 6.626 x 10-34 J s.
Max Planck discovered this constant while trying to
explain blackbody radiation. Planck’s constant turned out
to be a universal constant used to explain other
phenomena as well.
Light energy is quantized because it is only available in
multiples of hn.
Note: energy is proportional to frequency and inversely
proportional to wavelength.
9
Quantum Effects and Photons
Photoelectric Effect is one phenomena that could not
be explained with the notion that light is a wave. A
photoelectron is an electron ejected from a metallic
surface when the surface is exposed to light.
It was found that:
• there is a minimum frequency of light (no) needed to cause
electrons to be ejected; the threshold frequency depends on the
nature of the metallic surface.
• the kinetic energy of the electrons ejected depends on the
frequency, but not on the intensity of light; this was puzzling
because the intensity of light is the amount of energy it delivers
per unit time per unit area.
• if electrons are ejected, more of them are ejected if the light is
made intense.
10
Quantum Effects and Photons
Albert Einstein explained the observations as follows:
• the kinetic energy of the ejected electron is the
difference between the energy of the photon and the
energy needed to dislodge the electron from the metal.
• if the individual photons do not have sufficient energy
to dislodge electrons, no photoelectrons will be
observed regardless of how intense the radiation is.
• if the photon energies are sufficient, then a higher
intensity would mean more photons and, consequently,
more photoelectrons produced.
• but the kinetic energy of the electrons depend only on
the photon energy (hence, on the frequency, not on the
intensity).
11
Quantum Effects and Photons
Based on the data, Einstein deduced:
• the photon energy is hn, where h is the same constant
that Planck earlier discovered when trying to explain
blackbody radiation.
• the minimum photon energy needed to dislodge the
electron, called the work function of the metal, hno.
12
Radio Wave Energy
What is the energy of a photon
corresponding to radio waves of frequency
1.255 x 10 6 Hz?
We simply plug the values into the energy equation and cancel units.
𝑬𝒑𝒉𝒐𝒕𝒐𝒏 = 𝒉n 
𝒉𝒄
l
𝑬𝒑𝒉𝒐𝒕𝒐𝒏 = (𝟔. 𝟔𝟐𝟔 𝒙 𝟏𝟎−𝟑𝟒 J s) (1.255 𝒙 𝟏𝟎𝟔 𝒔−𝟏 )  𝟖. 𝟑𝟏𝟔 𝒙 𝟏𝟎−𝟐𝟖 J
Hz
HW 58
code: energy
13
Atomic and Molecular Spectra
When light is passed through a prism, the prism causes the
waves corresponding to different wavelengths to bend at
different angles. If the light comes from a source where
there is a very high concentration of particles, we tend to get
a continuous band of colors (the familiar rainbow of colors).
Visible Light Spectrum
If we pass that light through a sample that has a very low
concentration of particles before we pass it through the
prism, we are likely to see a pattern of dark lines
interspersed within what would otherwise be a continuous
band of colors.
We say that the light waves (dark lines) were
Absorption Spectrum of Hydrogen
l’s absorbed by the particles in the sample.
14
Atomic and Molecular Spectra
If we were to examine light from a source where the
concentration of particles is very low, we are not likely going
to get a continuous band of colors. If we project the light that
passes through the prism onto a screen, we would see a
pattern of bright colored lines with dark regions in between.
The pattern depends on what atoms or molecules are
producing the light, and is called the atom’s or molecule’s
emission spectrum.
Emission spectrum of hydrogen
Each atom or molecule has a characteristic spectrum that
can serve as its “fingerprint” (unique) for identification
purposes. Spectra from atoms tend to have sharp lines and
15
are referred to as line spectra.
Figure:
Emission
(line)
spectra of
some
elements.
16
Atomic and Molecular Spectra
How come atoms and molecules do not absorb or emit a
continuous band of colors and give line spectra instead?
Niels Bohr provided the following explanation:
• Energies of atoms and molecules are quantized. They
cannot have just any amount of energy; there are
particular energies available to be absorbed or emitted for
each species.
• Transitions between allowed energy levels can occur
when a photon is absorbed or released.
• The energy of photon must be equal to the difference in
energy between two allowed levels (known as Bohr
frequency condition) and is associated with the
wavelengths absorbed or emitted in the spectrum thereby
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causing the line spectrum of a species.
The Bohr Theory of the
Hydrogen Atom
Atomic Line Spectra
– In 1885, J. J. Balmer showed that the allowed
transition wavelengths, l, in the visible spectrum of
hydrogen could be reproduced by a simple formula.
1
l
 1.097  10
7
1 1
m ( 22
 n1 )
2
– The known wavelengths of the four visible lines for hydrogen
correspond to values of n = 3, n = 4, n = 5, and n = 6
(electrons jump down from higher level to second level).
18
Figure :
Transitions of
the electron in
the hydrogen
atom.
There are names for
certain transitions:
Paschen series:
transition from higher
level to the n=3.
Represents the four
lines of hydrogen:
transitions from 6 to 2,
5 to 2, 4 to 2, and 3 to 2.
Balmer series:
Transition from higher
level to n=2
Lyman series:
Transition from higher
level to n=1
19
The Bohr Theory of the
Hydrogen Atom
Prior to the work of Niels Bohr, the stability of
the atom could not be explained using the thencurrent theories. Classical mechanics predict
that a electron will crash into the nucleus
indicating that the nuclear structure is not
stable.
In 1913, using the work of Einstein and Planck, Bohr
applied a new theory to the simplest atom,
hydrogen, which involved the transition of electrons
between allowed energy levels.
20
The Bohr Theory of the
Hydrogen Atom
Bohr’s Postulates
Bohr set down postulates to account for (1) the stability
of the hydrogen atom and (2) the line spectrum of the
atom.
1. Energy level postulate An electron can have
only specific energy levels in an atom.
2. Transitions between energy levels An electron
in an atom can change energy levels by
undergoing a “transition” from one energy level to
another by either absorbing or emitting a photon
equal to the energy difference between the two
allowed energy levels.
3. Angular momentum keeps electrons in orbit.
21
The 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.
Rh
E 2
n
n  1, 2, 3 ..... (for H atom)
Rh is a constant (expressed in energy units) with a
value of 2.18 x 10-18 J.
22
The Bohr Theory of the
Hydrogen Atom
Bohr’s Postulates
When an electron undergoes a transition from a higher
energy level to a lower one, the energy is emitted as a
photon.
Energy of emitted photon  hn  Ei  Ef
Energy difference between two energy levels.
– From Postulate 1,
Rh
Ei   2
ni
Energy for initial level
Rh
Ef   2
nf
Energy for final level
23
The Bohr Theory of the
Hydrogen Atom
Bohr’s Postulates
If we make a substitution into the previous equation
that states the energy of the emitted photon, hn,
equals Ei - Ef,
hn  Ei  E f 
( )( )
( )
Rh
 2
ni

Rh
 2
nf
Rearranging, we obtain
Energy of photon emitted
or absorbed for electrons
changing energy levels
E  hn  Rh
1
1
 2
2
nf
ni
24
The 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 is emitted equal to the energy
difference between the levels (wavelength, 685 nm).
– When light of this same wavelength shines on a
hydrogen atom in the n = 2 level, the energy is
gained by the electron and undergoes a transition to
n = 3.
25
A Problem to Consider
Calculate the energy of a photon of light emitted from a
hydrogen atom when an electron falls from level n = 3 to
level n = 1.
This problem involves plugging the correct values into Bohr’s equation
where ni = 3 (initial energy level) and nf = 1 (final energy level):
(
E  hn  Rh
𝑬 = 𝟐. 𝟏𝟖
𝒙 𝟏𝟎−𝟏𝟖 𝑱
𝟏
𝟏𝟐
1
1
 2
2
nf
ni
𝟏
− 𝟐
𝟑
)
= 1.94 𝒙 𝟏𝟎−𝟏𝟖 𝑱
HW 59
code: bohr
26
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.
27
Matter Waves
Light exhibits wave-like behavior, as well as particle-like
behavior. The term wave-particle duality is used to
describe this dual nature of light.
Wave-Particle Duality of light - the “wave” and
“particle” pictures of light should be regarded as
complementary views of the same physical entity.
The equation E = hn displays this duality; E is the
energy of the “particle” photon, and n is the
frequency of the associated “wave.”
28
Matter Waves
The first clue in the development of waveparticle duality 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.
Momentum is the product of mass and speed.
– The equation l = h/mv is called the de Broglie
wavelength.
– It is now generally accepted that wave
characteristics apply to all matter; however, for
large masses, the De Broglie wavelength is too
small to detect.
29
Quantum Mechanics
A consequence of the wave-particle duality is the
Uncertainty Principle.
In 1927, Werner Heisenberg showed (from quantum
mechanics) that it is impossible to know both (position &
velocity) simultaneously – Heisenberg’s Uncertainty
Principle.
Electrons are moving; therefore, if we try to locate it by
bouncing a photon off it, the electron’s location would be
affected as well.
Wave-particle duality and the uncertainty principle resolve
the problem regarding the stability of the nuclear atom.
Classical mechanics predict that the electron will crash into
30
the nucleus indicating that the nuclear structure is not stable.
Quantum Mechanics
Therefore, classical mechanics could not explain
experimental results involving the nuclear structure.
We could no longer think of an electron as having a
precise orbit (spherical) in an atom.
To describe such an orbit would require knowing its
exact position and velocity.
It became evident that classical mechanics is just an
approximation on a macroscopic level; a more general
theory was needed in order to account for
observations on a microscopic level which lead to the
birth of Quantum Mechanics.
31
Quantum Mechanics
Quantum mechanics is the branch of physics that
mathematically describes the wave properties of
submicroscopic particles.
This is a probabilistic theory as opposed to a
deterministic one.
In quantum mechanics, we do not try to locate particles;
we only try to calculate probabilities or likelihood of
finding particles in whatever region we are interested in.
Erwin Schrodinger defined this probability in a
mathematical expression called a wavefunction.
32
Quantum Numbers and
Atomic Orbitals
An orbital is a wavefunction that gives us
information about an electron; a description of
the region around the nucleus where we are
most likely to find the electron.
For each atom, ion, or molecule, quantum
theory gives us an infinite set of orbitals that
we can use to describe the electrons.
It is convenient to refer to these orbitals by a
set of numbers called quantum numbers that
is unique for each orbital.
33
Quantum Numbers and Atomic Orbitals
According to quantum mechanics, each orbital
describing electrons has four quantum numbers:
– Principal quantum number (n)
Can only be a positive integer: n = 1, 2, 3,…..
– Orbital Angular momentum quantum number (l)
Can only be a positive integer that is less than n
l = 0, 1, …, n-1
– Magnetic quantum number (ml or m)
Can only be an integer from –l to + l
– Spin quantum number (ms)
Can only be -½ or + ½
34
Quantum Numbers and Atomic
Orbitals
Orbitals can be classified into shells or levels.
The principal quantum number (n) represents the
“shell number or energy level” in which an electron
“resides.”
– The smaller n is, the smaller the orbital.
– The smaller n is, the lower the energy of the
electron.
– May be an integer from 1 to  , although greater
than 7 as far as today, are unimportant.
– n determines the size and energy of the orbital.
The effective volume of space electron is moving
about.
35
Quantum Numbers and
Atomic Orbitals
Orbitals can also be classified into subshells or sublevels.
The orbital angular momentum quantum number (l)
distinguishes “sub shells” within a given shell that have
different shapes (shape of region (volume) that electron
occupies).
– Each main “shell - n” is subdivided into “sub shells.”
Within each shell of quantum number n, there are
“l” sub shells, each with a distinctive shape.
– l can have any integer value from 0 to (n - 1)
– The different subshells are denoted by letters.
Letter
s p d f g …
l
=
0 1 2 3 4 ….
36
Quantum Numbers and Atomic
Orbitals
In general, the number of orbitals that belong to the
same subshell is equal to 2l + 1:
If l = 0, 2(0) + 1 = 1; there is only one orbital in an s subshell.
If l = 1, 2(1) + 1 = 3; there are three orbitals in an p subshell.
If l = 2, 2(2) + 1 = 5; there are five orbitals in an d subshell.
If l = 3, 2(3) + 1 = 7; there are seven orbitals in an f subshell.
We can say that s orbitals comes in a set of 1, p orbitals come in
sets of 3, d orbitals come in sets of 5, f orbitals come in sets of 7,
etc. Each orbital in a subshell has a unique ml quantum number.
37
Quantum Numbers and Atomic Orbitals
The magnetic quantum number (ml or m) distinguishes
orbitals within a given sub-shell that have different shapes
and magnetic orientations in space. It is the magnetism
caused by the orbital motion of an electron around the
nucleus that affects the orientation.
– Each subshell (s, p, d, f) is subdivided into “orbitals,”
each capable of holding a pair of electrons (max 2
electrons/orbital orientation).
– ml can have any integer value from -l to 0 to +l.
– Each orbital within a given subshell has the same
energy (all “nl” have same energy for all orientations
but not each “l” – 4s<4p<4d<4f).
– So, when we say “3d orbital”, we mean any one of the
five orbitals in the 3d subshell (-2, -1, 0, +1, or +2)
d: l = 2 and ml = -2 to 0 to +2 meaning -2, -1, 0, +1, +2 38
Figure : Cutaway diagrams showing the
spherical shape of S orbitals.
n=1
n=2
l = 0 are referred to as “s” subshells and are spherical in shape.
There is only 1 magnetic orientation (ml: 0) for the “s” subshell.
39
Figure : The 2p orbitals.
l = 1 are referred to as “p” subshells and are dumbbell in shape.
40
There are 3 magnetic orientations (ml: -1, 0, +1) for the “p” subshell.
Figure : The five 3d orbitals.
l = 2 are referred to as “d” subshells and are cloverleaf in shape. There
are 5 magnetic orientations (ml: -2, -1, 0, +1, +2) for the “d” subshell.
41
In general, the nth shell has n2 orbitals:
For n = 1, n2 = 1; there is only one orbital;
1 in the 1s (n = 1; l = 0; ml = 0)
For n = 2, n2 = 4; there are 4 orbitals;
1 in the 2s (n = 2; l = 0; ml = 0)
3 in the 2p (n = 2; l = 1; ml = -1, 0, +1)
For n = 3, n2 = 9; there are 9 orbitals;
1 in the 3s (n = 3; l = 0; ml = 0)
3 in the 3p (n = 3; l = 1; ml = -1, 0, +1)
5 in the 3d (n = 3; l = 2; ml = -2, -1, 0, +1, +2)
For n = 4, n2 = 16; there are 16 orbitals;
1 in the 4s (n = 4; l = 0; ml = 0)
3 in the 4p (n = 4; l = 1; ml = -1, 0, +1)
5 in the 4d (n = 4; l = 2; ml = -2, -1, 0, +1, +2)
7 in the 4f (n = 4; l = 3; ml = -3, -2, -1, 0, +1, +2, +3)
42
Energy level
shape
orientations
0 to (n-1)
-l…0…+l
Max #es
s
p
s
p
d
s
p
d
f
px, py, pz
2
2
6
2
6
10
2
6
10
14
43
Quantum Numbers and Atomic Orbitals
The magnetic properties of atoms suggest that electrons
have an intrinsic magnetism due to their spinning motion
in addition to the magnetism that is generated by their
motion around the nucleus. The motion of an electron
around the nucleus, ml, is analogous to that of a planet
revolving around the sun, while electron spin is analogous
to the rotation of the planet around its own axis, ms.
The spin quantum number, ms, refers to the two
possible spin orientations (spin up, +1/2, and spin down,
-1/2 ) of the electrons residing within a given orbital.
Each orbital can hold only two electrons whose spins
must oppose one another.
44
Quantum Numbers and Atomic Orbitals
Thus, an electron in an atom is completely described by
specifying four quantum numbers (n, l, ml, ms). The
quantum numbers n and l tell us the shell and subshell,
the quantum number ml tells us which orbital, and ms tells
us the spin. No two electrons can have the same four
quantum numbers; If two electrons have the same n, l, ml
(same orbital), they must have opposite spins.
Pauli Exclusion Principle – no two electrons may have the
same set of 4 quantum numbers. Where two electrons
occupy the same orbital, they must have opposite spins:
ms = +1/2, ms = –1/2.
The maximum number of electrons that we can assign to
an orbital is 2; meaning one with spin up, +1/2, and the
45
other with spin down, -1/2 .
Questions
How many electrons can be assigned to a 4p orbital?
The key to this question is that we are referring to “a” single 4p
orbital. For any single orbital, the limit is 2 electrons; one electron
with spin up (ms = +1/2) and the other with spin down (ms = -1/2).
How many electrons can be assigned to 4p orbitals
(meaning 4p subshells)?
The key to this question is that we are referring to “all” the 4p
orbitals or subshells. The 4p orbitals come in a set of 3 (ml: -1, 0, 1)
with a maximum of 2 electrons in each orbital. All 6 electrons will
have n = 4, l = 1. They will differ in ml and ms values; the six
possible (ml, ms) values:
(-1, +1/2), (0, +1/2), (1, +1/2) spin up
(-1, -1/2), (0, -1/2), (1, -1/2) spin down
46
Questions
How many electrons can be assigned to the n = 3 shell?
In the n = 3 shell, there are 3 subshells (3s, 3p, 3d).
The 3s subshell consists of 1 orbital with 2 e-.
The 3p subshell consists of 3 orbitals with 2 electrons in each giving 6 e-.
The 3d subshell consists of 5 orbitals with 2 electrons in each giving 10 e-.
The total number of e- that can be assigned in the n = 3 shell is 2 + 6 + 10 = 18.
In general, since there are n2 orbitals in the nth shell, the number of electrons
that can be assigned to a shell is 2n2.
Which of the following can accommodate the most number of e-?
A. a 4f orbital
B. the 3d subshell
C. the n =2 shell
D. a 3d orbital
A. a 4f orbital is a single orbital accommodating a maximum of 2e-  2eB. the 3d subshell has 5 orbitals (l = 2, ml = -2, -1, 0, +1, +2) with 2e- each  10eC. the n =2 shell has 4 orbitals (l = 0, ml = 0; l = 1, ml = -1, 0, +1) with 2e- each  8eD. a 3d orbital is a single orbital accommodating a maximum of 2e-  2e47
The correct answer is B, the 3d subshell.
HW 60
code: quantum