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Chapter 7
Quantum Theory of Atom
Bushra Javed
1
Contents
1. The Wave Nature of Light
2. Quantum Effects and Photons
3. The Bohr Theory of the Hydrogen Atom
4. Quantum Mechanics
5. Quantum Numbers and Atomic Orbitals
6. Quantum Numbers and Atomic Orbitals
2
A Theory that Explains Electron Behavior
• The quantum-mechanical model explains the manner
in which electrons exist and behave in atoms
• It helps us understand and predict the properties of
atoms that are directly related to the behavior of the
electrons
3
The Nature of Light:
Its Wave Nature
• Light is a form of electromagnetic radiation
– composed of perpendicular oscillating waves, one
for the electric field and one for the magnetic field
• an electric field is a region where an electrically charged
particle experiences a force
• a magnetic field is a region where a magnetized particle
experiences a force
• All electromagnetic waves move through
space at the same, constant speed
– 3.00 x 108 m/s in a vacuum = the speed of light, c
4
Electromagnetic Radiation
5
Electromagnetic Spectrum
The range of frequencies and wavelengths of
electromagnetic radiation is called the
electromagnetic spectrum.
6
Characterizing Waves
• The amplitude is the height of the wave
– the distance from node to crest
• or node to trough
– the amplitude is a measure of how intense the light
is – the larger the amplitude, the brighter the light
• The wavelength (l) is a measure of the
distance covered by the wave
– the distance from one crest to the next
7
Wave Characteristics
8
Tro: Chemistry: A Molecular Approach, 2/e
Characterizing Waves
• The frequency (n) is the number of waves that pass a
point in a given period of time
– the number of waves = number of cycles
– units are hertz (Hz) or cycles/s = s−1
• 1 Hz = 1 s−1
• The total energy is proportional to the amplitude of
the waves and the frequency
– the larger the amplitude, the more force it has
9
Characterizing Waves
• The longer the wavelength of light, the lower
the frequency.
• The shorter the wavelength of light, the
higher the frequency.
10
Color
• The color of light is determined by
its wavelength or frequency
• White light is a mixture of all the
colors of visible light
– a spectrum
RedOrangeYellowGreenBlueViolet
• When an object absorbs some
of the wavelengths of white light
and reflects others, it appears
colored
11
Wavelength, Frequency & Speed of Light
Wavelength and Frequency are related as:
λ α 1/ v
c = λv
Where c is the speed of light
• Light waves always move through empty space at the
same speed.
• Therefore speed of light is a fundamental constant.
c = 3.00 x 108 m/s
12
The Wave Nature of Light
Example 1
Which of the following statements is incorrect?
a. The product of wavelength and frequency of
electromagnetic radiation is a constant.
b. As the energy of a photon increases, its frequency
decreases.
c. As the wavelength of a photon increases, its
frequency decreases.
d. As the frequency of a photon increases, its
wavelength decreases.
13
Units of Wavelength & Frequency
• Units of Wavelength:
nanometers, micrometers, meters and Angstrom
1 nm = 10-9m
1A° = 10-10m
• Units of Frequency:
1/second (per second) or 1s-1
Hertz = Hz, MHz and GHz
1-1s = 1 Hz
14
Wavelength & Frequency
Example 2
What is the wavelength of blue light with a
frequency of 6.4  1014/s?
n = 6.4  1014/s
c = 3.00  108 m/s
c = nl so
l = c/n
m
3.00 x 10
c
s
λ 
n
14 1
6.4 x 10
s
8
l = 4.7  10−7 m
7 | 15
Wavelength & Frequency
Example 3
What is the frequency of light having a
wavelength of 681 nm?
l = 681 nm = 6.81  10−7 m
c = 3.00  108 m/s
c = nl so
n = c/l
m
3.00  10
c
s
n 
7
l 6.81  10 m
8
l = 4.41  1014 /s
16
Electromagnetic Spectrum
Example 4
Which type of electromagnetic radiation has the
highest energy?
a.
b.
c.
d.
radio waves
gamma rays
blue light
red light
17
Wave Theory of light
One property of waves is that they can be
diffracted—that is, they spread out when they
encounter an obstacle about the size of their
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.
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Diffraction
19
The Wave/Particle Nature of Light
• Max Planck worked on the light of various
frequencies emitted by the hot solids at
different temperatures.
• In 1900, Max Planck proposed that radiant
energy is not continuous, but is emitted in
small bundles.
• Planck theorized that the atoms of an heated
object vibrate with a definite frequency.
• An individual unit of light energy is a photon.
20
Dual Nature of Light
• Einstein expanded on Planck’s work.
• Through Photoelectric Effect ,he showed that
when light of particular frequency shines on a
metal surface it knocks out electrons.
• This knocking out of electrons is dependent
only on the frequency of the shining light.
21
The Photoelectric Effect
22
Einstein’s Explanation
• Einstein proposed that the light energy was
delivered to the atoms in packets, called
quanta or photons
• The energy of a photon of light is directly
proportional to its frequency
– inversely proportional to its wavelength
– the proportionality constant is called Planck’s
Constant, (h) and has the value 6.626 x 10−34 J∙s
23
Einstein’s Explanation
Light, therefore, has properties of both waves
and matter.
Neither understanding is sufficient alone. This is
called the particle–wave duality of light.
24
Ejected Electrons
• One photon at the threshold frequency gives the
electron just enough energy for it to escape the
atom
– binding energy, f
• When irradiated with a shorter wavelength photon,
the electron absorbs more energy than is necessary
to escape
• This excess energy becomes kinetic energy of the
ejected electron
Kinetic Energy = Ephoton – Ebinding
KE = hn − f
25
The Photoelectric Effect
Example 5:
Suppose a metal will eject electrons from its surface
when struck by yellow light. What will happen if the
surface is struck with ultraviolet light?
a. No electrons would be ejected.
b. Electrons would be ejected, and they would have the
same kinetic energy as those ejected by yellow light.
c. Electrons would be ejected, and they would have
greater kinetic energy than those ejected by yellow
light.
d. Electrons would be ejected, and they would have lower
kinetic energy than those ejected by yellow light.
26
Photoelectric Effect
Example 6
The blue–green line of the hydrogen atom spectrum has a
wavelength of 486 nm. What is the energy of a photon of
this light?
E = hn and
l = 486 nm = 4.86  10−7 m
c = nl so
c = 3.00  108 m/s
E = hc/l
−34
h = 6.63  10 J  s

8 m
6.63  10 Js  3.00  10

hc
s


E

λ
4.86  107 m

34



E = 4.09  10−19 J
27
Problems with Rutherford’s Nuclear
Model of the Atom
• Electrons are moving charged particles
• According to classical physics, moving charged
particles give off energy
• Therefore electrons should constantly be
giving off energy
• The electrons should lose energy, crash into
the nucleus, and the atom should collapse!!
– but it doesn’t
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Continuous Vs Line Spectra
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.
7 | 29
Spectra
• When atoms or molecules absorb energy, that
energy is often released as light energy
– fireworks, neon lights, etc.
• When that emitted light is passed through a
prism, a pattern of particular wavelengths of
light is seen that is unique to that type of atom
or molecule – the pattern is called an emission
spectrum
– non-continuous
– can be used to identify the material
• flame tests
30
Exciting Gas Atoms to Emit Light
with Electrical Energy
31
Identifying Elements with
Flame Tests
Na
32
K
Li
Ba
Emission Spectra
33
Emission Line Spectra
• When an electrical voltage is passed across a
gas in a sealed tube, a series of narrow lines is
seen.
• These lines are the emission line spectrum.
The emission line spectrum for hydrogen gas
shows three lines: 434 nm, 486 nm, and 656
nm.
34
7 | 35
The Bohr Model of the Atom
• The nuclear model of the atom does not explain what
structural changes occur when the atom gains or loses
energy
• Bohr developed a model of the atom to explain how
the structure of the atom changes when it undergoes
energy transitions
• Bohr’s major idea was that the energy of the atom was
quantized, and that the amount of energy in the atom
was related to the electron’s position in the atom
– quantized means that the atom could only have
very specific amounts of energy
36
Bohr’s Model
• The electrons travel in orbits that are at a fixed
distance from the nucleus
– Energy Levels
– therefore the energy of the electron was
proportional to the distance the orbit was from the
nucleus
• Electrons emit radiation when they “jump” from an
orbit with higher energy down to an orbit with lower
energy
– the emitted radiation was a photon of light
– the distance between the orbits determined the
energy of the photon of light produced
37
Atomic Line Spectra
38
Bohr’s Model of Atom
Example 7
Which of the following energy level changes in a
hydrogen atom produces a visible spectral line?
a.
b.
c.
d.
3 →1
4→ 2
5→ 3
all of the above
39
Rydberg’s Spectrum Analysis
• Rydberg analyzed the spectrum of hydrogen and
found that it could be described with an
equation that involved an inverse square of
integers
40
Bohr’s Model of Atom
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
n2
RH = 2.179  10−18 J
n = principal quantum number
7 | 41
Bohr’s Model of Atom
The energy of the emitted or absorbed photon is
related to DE:
E photon  ΔE electron  hn
h  Planck' s constant
We can now combine these two equations:
 1

1
hn   R H  2  2 
n

n
i 
 f
42
Bohr’s Model of Atom
Light is absorbed by an atom when the electron
transition is from lower n to higher n (nf > ni). In
this case, DE 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, DE will be negative.
An electron is ejected when nf = ∞.
7 | 43
Electron transitions
for an electron in the
hydrogen atom.
7 | 44
Example 8.
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

1
ΔE  RH  2  2 
ni = 6
n

n
f
i 

nf = 3
hc
hc
−18
ΔE 
so λ 
RH = 2.179  10 J
λ

ΔE  2.179  10
18
ΔE
1 
 1
J  2  2 = −1.816  10−19 J
6 
3


8 m
6.626  10 J  s  3.00  10

s

λ

−6 m
1.094

10
1.816  1019 J

34



7 | 45
Bohr’s Model of Atom
Example 9
What is the frequency (in kHz ) of light emitted when
the electron in a hydrogen atom undergoes a transition
from level n = 6 to level n = 2?
h = 6.63 × 10-34 J . s), RH = 2.179 × 10-18 J)
a.
b.
c.
d.
5.49 × 105 kHz
7. 31× 1011 kHz
7. 31 × 1014 kHz
3.64 × 10–28 kHz
46
Wave properties of Matter
In 1923, Louis de Broglie, a French physicist,
reasoned that particles (matter) might also have
wave properties.
The wavelength of a particle of mass, m (kg),
and velocity, v (m/s), is given by the de Broglie
relation:
h
λ
mv
h  6.626  1034 J  s
7 | 47
Wave properties of Matter
Example 10
Compare the wavelengths of
(a) an electron traveling at a speed that is onehundredth the speed of light and (b) a baseball of mass
0.145 kg having a speed of 26.8 m/s (60 mph).
Baseball
Electron
me = 9.11  10−31 kg
v = 3.00  106 m/s
m = 0.145 kg
v = 26.8 m/s
h
λ
mv
48
Electron
me = 9.11  10−31 kg
v = 3.00  106 m/s
6.63  1034 J  s
λ
 2.43  10−10 m

31
6 m
9.11  10 kg  3.00  10

s




Baseball
m = 0.145 kg
v = 26.8 m/s
34
6.63  10 J  s
λ
 1.71  10−34 m
m

 0.145 kg  26.8 
s

49
Quantum (or Wave) mechanics
Building on de Broglie’s work, Erwin
Schrodinger devised a theory that could be used
to explain the wave properties of electrons in
atoms and molecules.
The branch of physics that mathematically
describes the wave properties of submicroscopic
particles is called or wave mechanics.
50
Uncertainty Principle
• Heisenberg stated that the product of the
uncertainties in both the position and speed of a
particle was inversely proportional to its mass
– x = position, Dx = uncertainty in position
– v = velocity, Dv = uncertainty in velocity
– m = mass
51
Uncertainty Principle
• This means that the more accurately you know the
position of a small particle, such as an electron, the
less you know about its speed
– and vice-versa
7 | 52
Schrodinger’s Equation
• Schodinger’s Equation allows us to calculate the
probability of finding an electron with a
particular amount of energy at a particular
location in the atom
• Solutions to Schödinger’s Equation produce
many wave functions, A plot of distance vs. Y2
represents an orbital, a probability distribution
map of a region where the electron is likely to be
found
53
Solutions to the Wave Function, Y
• Calculations show that the size, shape, and
orientation in space of an orbital are
determined to be three integer terms in the
wave function
– added to quantize the energy of the electron
• These integers are called quantum numbers
– principal quantum number, n
– angular momentum quantum number, l
– magnetic quantum number, ml
54
Principal Quantum Number, n
• Characterizes the energy of the electron in a particular
orbital
– corresponds to Bohr’s energy level
• n can be any integer  1
• The larger the value of n, the more energy the orbital
has
• Energies are defined as being negative
– an electron would have E = 0 when it just escapes
the atom
• The larger the value of n, the larger the orbital
• As n gets larger, the amount of energy between orbitals
gets smaller
55
Angular Momentum Quantum
Number, l
• The angular momentum quantum number
determines the shape of the orbital
• l can have integer values from 0 to (n – 1)
• Each value of l is called by a particular letter that
designates the shape of the orbital
56
Angular Momentum Quantum
Number,
s orbitals are spherical
– p orbitals are like two balloons tied at the knots
– d orbitals are mainly like four balloons tied at the
– Knot
– f orbitals are mainly like eight balloons tied at the
– knot
57
Magnetic Quantum Number, ml
• The magnetic quantum number is an integer that
specifies the orientation of the orbital
– the direction in space the orbital is aligned relative
to the other orbitals
• Values are integers from −l to +l
– including zero
– gives the number of orbitals of a particular shape
• when l = 2, the values of ml are
• −2, −1, 0, +1,+2
• which means there are five orbitals with l = 2
58
Describing an Orbital
• Each set of n, l, and ml describes one orbital
• Orbitals with the same value of n are in the
same principal energy level
– aka principal shell
• Orbitals with the same values of n and l are
said to be in the same sublevel
59
Energy Shells and Subshells
60
Quantum Numbers
Spin Quantum Number, ms
This quantum number refers to the two possible
orientations of the spin axis of an electron.
It may have a value of either +1/2 or -1/2.
7 | 61
The Quantum Numbers
Example 11
Give the notation (using letter designations for 1) of
subshells for the following quantum numbers.
n. = 3, l = 1
n. = 4, l = 0
n. = 5,1 = 3
n. = 6,1 = 2
n. = 2,1 = 1
62
The Quantum Numbers
Example 12
Determine how many valid sets of quantum numbers
exist for a 3p orbital ; Give two examples.
Strategy:
Write the the number of possible values for each
quantum number.
Quantum number n
1
m
ms
Possible values
3
1
1, 0, -1
+1/2, -1/2
# of possible values 1
1
3
2
Possible sets of values for a 3p electron: (1)(1)(3)(2)
=6
63
The Quantum Numbers
There are six sets of quantum numbers that describe
a 3p electron.
Examples:
n
= 3
n = 3
1=1
m1= 0
ms = +1/2
1=1
m1 = -1
ms = +1/2
64
The Quantum Numbers
Example 13
Which of the following sets of quantum numbers (n, l,
ml, ms) is not permissible?
a.
b.
c.
d.
1
4
2
3
0
0
2
1
0
0
1
0
+½
-½
+½
-½
65
The Quantum Numbers
Example 14
Determine how many valid sets of quantum
numbers exist for 4d orbitals ;Give three examples.
66
The Quantum Numbers
Example 15
Determine how many valid sets of quantum
numbers exist for 5f orbitals ;Give three
examples.
67
The Shapes of Orbitals
s Orbitls :
1. All s orbitals are spherical
2. The probability of finding an s electron depends only
on distance from the nucleus, not on direction
3. There is only one orientation of a sphere in space
Therefore S orbital has ml = 0
4. The value of ψ2 for an s-orbital is greatest near the
nucleus
5. The size of the s orbital increases as the number of
shell increases i.e. 3s > 2s>1s
68
S Orbitals
69
Probability Density Function
The probability density function represents the total
probability of finding an electron at a particular point in space
70
Tro: Chemistry: A Molecular Approach, 2/e
p Orbitals
1. p orbitlas are dumbbell –shaped.
2. The electron distribution is concentrated in identical
lobes on either side of the nucleus and separated
by a planar node cutting through the nucleus
3. Probability of finding a p electron near the nucleus
is zero.
4. The two lobes of a p orbital have different phases of
a wave (different algebraic signs)
5. The size of p orbitals increases in the order
4p>3p>2p
71
7 | 72
d Orbitals
1. The third and higher shells contain each contain five
d orbitals
2. d orbitals have two shapes
3. Four of the five d orbitals are cloverleaf –shaped and
have four lobes of maximum electron probability
4. The fifth d orbital is similar in shape to a pz orbital
but has an additional donut –shaped region of
electron probability centered in the xy plane
5. All five d orbitals have the same energy
6. Alternating lobes of the d orbitals have different
phases
73
d Orbitals
74
f orbitals
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Tro: Chemistry: A Molecular Approach, 2/e