CHAPTER 7: QUANTUM
THEORY OF THE ATOM
Vanessa N. Prasad-Permaul
Valencia Community College
CHM 1045
1
THE WAVE NATURE OF LIGHT
Frequency, : The number of wave peaks that pass a
given point per unit time (1/s)
Wavelength, : The distance from one wave peak to
the next (nm or m)
Amplitude: Height of wave
Wavelength x Frequency = Speed
(m) x (s-1) = c (m/s)
The speed of light waves in a vacuum in a constant
c = 3.00 x 108 m/s
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THE WAVE NATURE OF LIGHT
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THE WAVE NATURE OF LIGHT
EXAMPLE 7.1 : WHAT IS THE WAVELENGTH OF
THE YELLOW SODIUM EMISSION, WHICH HAS A
FREQUENCY OF 5.09 X 1014 S-1?
c = 
= c

 = 3.00 x 108 m/s
5.09 x 1014 s-1
= 5.89 x 10 -7 m
= 589 x 10-9 m
= 589 nm
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THE WAVE NATURE OF LIGHT
EXERCISE 7.1 : The frequency of the strong red line in
the spectrum of potassium is 3.91 x 1014 s-1. What is the
wavelength of this light in nanometers?
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THE WAVE NATURE OF LIGHT
EXAMPLE 7.2 : WHAT IS THE FREQUENCY OF VIOLET
LIGHT WITH A WAVELENGTH OF 408nm?
c = 
= c

 = 3.00 x 108 m/s
408 X 10-9 m
= 7.35 x 1014 s-1
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THE WAVE NATURE OF LIGHT
EXERCISE 7.2 : The element cesium was discovered in
1860 by Robert Bunsen and Gustav Kirchoff, who found
to bright blue lines in the spectrum of a substance
isolated from a mineral water. One of the spectral lines
of cesium has a wavelength of 456nm. What is the
frequency?
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THE WAVE NATURE OF LIGHT
THE ELECTROMAGNETIC SPECTRUM
 Several types of electromagnetic radiation make up the
electromagnetic spectrum
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QUANTUM EFFECTS & PHOTONS
Atoms of a solid oscillate of vibrate with a
definite frequency
E=h
E = hc / 
h = Planck’s constant, 6.626 x 10-34 J s
E = energy
1 J = 1 kg m2/s2
When a photon hits the metal, it’s energy (h) is
taken up by the electron. The photon no longer
exists as a particle and it is said to be absorbed
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QUANTUM EFFECTS & PHOTONS
 Max Planck (1858–1947): proposed the energy
is only emitted in discrete packets called quanta
(now called photons).
The amount of energy depends on the frequency:
E = energy
 = frequency
 = wavelength c = speed of light
h = planck’s constant
E = h =
hc

h = 6.626  10 -34 J  s
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QUANTUM EFFECTS & PHOTONS
Albert Einstein (1879–1955):
Used the idea of quanta to explain the photoelectric effect.
 He proposed that light behaves as a stream of particles
called photons
 A photon’s energy must exceed a minimum threshold
for electrons to be ejected.
 Energy of a photon depends only on the frequency.
THE PHOTOELECTRIC EFFECT:
E=h
The ejection of electrons from the surface of a metal or
from a material when light shines on it
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QUANTUM EFFECTS & PHOTONS
EXAMPLE 7.3 : THE RED SPECTRAL LINE OF LITHIUUM
OCCURS AT 671nm (6.71 x 10-7m). CALCULATE THE
ENERGY OF ONE PHOTON OF THIS LIGHT.
 = c = 3.00 x 108 m/s = 4.47 x 1014 s-1

6.71 x 10-7 m
E = h
= 6.63 x 10-34 J.s * 4.47 x 1014 s-1
= 2.96 x 10-19 J
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QUANTUM EFFECTS & PHOTONS
EXERCISE 7.3 : The following are representative
wavelengths in the infrared, ultraviolet and x-ray regions
of the electromagnetic spectrum, respectively:
1.0 x 10-6 m, 1.0 x 10-8 m and 1.0 x 10-10 m.
• What is the energy of a photon of each radiation?
• Which has the greatest amount of energy per photon?
• Which has the least?
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THE BOHR THEORY OF THE HYDROGEN ATOM
 Atomic spectra: Result from excited atoms
emitting light.
 Line spectra: Result from electron transitions
between specific energy levels.
 Blackbody radiation is the visible glow that solid
objects emit when heated.
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THE BOHR THEORY OF THE HYDROGEN ATOM
BOHR’S POSTULATE
• The stability of the atom (H2)
• The line spectrum of the atom
ENERGY-LEVEL POSTULATE: An electron can only
have specific energy level values in an atom called
ENERGY LEVELS
E = RH where n = 1, 2, 3
n2
RH = 2.179 x 10-18 J
n = principle quantum number
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THE BOHR THEORY OF THE HYDROGEN ATOM
BOHR’S POSTULATE
• The stability of the atom (H2)
• The line spectrum of the atom
TRANSITIONS BETWEEN ENERGY LEVELS: An
electron in an atom can change energy only by going
from one energy level to another energy level. By
doing so, the electron undergoes a transition.
An electron goes from a higher energy level (Ei) to a
lower energy level (Ef) emitting light:
-DE = -(Ef - Ei)
DE = Ei - Ef
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THE BOHR THEORY OF THE HYDROGEN ATOM
ENERGY LEVEL DIAGRAM OF THE HYDROGEN ATOM
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THE BOHR THEORY OF THE HYDROGEN ATOM
EXAMPLE 7.4 : WHAT IS THE WAVELENGTH OF THE
LIGHT EMITTED WHEN THE ELECTRON IN A
HYDROGEN ATOM UNDERGOES A TRANSITION
FROM ENERGY LEVEL n = 4 TO LEVEL n = 2.
Ei = -RH
42
Ef = - RH
22
DE = -RH - -RH
16
4
E = -4RH + 16RH = -RH + 4RH = 3RH = h
64
16
16
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THE BOHR THEORY OF THE HYDROGEN ATOM
EXAMPLE 7.4 : Cont…
 = E = 3RH = 3 * 2.179 x 10-18 J = 6.17 x 1014 s-1
h 16* h 16 * 6.626 x 10-34 J.s
 = c = 3.00 x 108 m/s = 4.86 x 10-7 m

6.17 x 10 14 s-1
= 486 nm
(the color is blue-green)
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THE BOHR THEORY OF THE HYDROGEN ATOM
EXERCISE 7.4 : Calculate the wavelength of light
emitted from the hydrogen atom when the electron
undergoes a transition from level 3 (n = 3) to level 1
(n = 1).
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THE BOHR THEORY OF THE HYDROGEN ATOM
EXERCISE 7.5 : What is the difference in energy levels
of the sodium atom if emitted light has a wavelength
of 589nm?
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QUANTUM MECHANICS
 Louis de Broglie (1892–1987): Suggested
waves can behave as particles and particles can
behave as waves. This is called wave–particle
duality.
m = mass in kg
p = momentum (mc) or (mv)
h
h
For Light :  =
=
mc
p
h
h
For a Particle :  =
=
mv
p
The de Broglie relation
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QUANTUM MECHANICS
EXAMPLE 7.5 :
A) CALCULATE THE  (in m) OF THE WAVE
ASSOCIATED WITH A 1.00 kg MASS MOVING AT
1.00km/hr.
v = 1.00 km x 1000m x 1hr x 1min = 0.278m/s
hr
1km
60min 60 sec
=
h = 6.626 x 10-34 kg.m2/s2.s = 2.38 x 10-33m
mv
1.00kg * 0.278m/s
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QUANTUM MECHANICS
EXAMPLE 7.5 : cont…
B)
WHAT IS THE  (in pm) ASSOCIATED WITH AN
ELECTRON WHOSE MASS IS 9.11 x 10-31kg
TRAVELING AT A SPEED OF 4.19 X 106 m/s ?
 = h =
mv
6.626 x 10-34 kg.m2/s2.s
9.11 x 10-31kg * 4.19 x 106 m/s
= 1.74 x 10-10 m
= 174pm
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QUANTUM MECHANICS
EXERCISE 7.6 : Calculate the  (in pm) associated with
an electron traveling at a speed of 2.19 x 106 m/s.
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QUANTUM MECHANICS
QUANTUM MECHANICS ( WAVE MECHANICS):
The branch of physics that mathematically describes
the wave properties of submicroscopic particles
UNCERTAINTY PRINCIPLE:
A relation that states that the product of the
uncertainty in position and the uncertainty in
momentum (mass times speed) of a particle can be no
smaller than Planck’s constant divided by 4p.
SCHRODINGER’S EQUATION:
Y2 gives the probability of finding the particle within a
region of space
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Quantum Mechanics
 Niels Bohr (1885–1962): Described atom as
electrons circling around a nucleus and
concluded that electrons have specific energy
levels.
 Erwin Schrödinger (1887–1961): Proposed
quantum mechanical model of atom, which
focuses on wavelike properties of electrons.
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Quantum Mechanics
 Werner Heisenberg (1901–1976): Showed that
it is impossible to know (or measure) precisely
both the position and velocity (or the
momentum) at the same time.
 The simple act of “seeing” an electron would
change its energy and therefore its position.
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Quantum Mechanics
 Erwin Schrödinger (1887–1961): Developed a
compromise which calculates both the energy of
an electron and the probability of finding an
electron at any point in the molecule.
 This is accomplished by solving the Schrödinger
equation, resulting in the wave function
29
QUANTUM NUMBERS
According to QUANTUM MECHANICS:
Each electron in an atom is described by 4 different
quantum numbers: (n, l, m1 and ms).
The first 3 specify the wave function that gives the
probability of finding the electron at various points in
space.
The 4th (ms) refers to a magnetic property of electrons
called spin
ATOMIC ORBITAL: A wave function for an electron in
an atom
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Quantum Numbers
 Wave functions describe the behavior of electrons.
 Each wave function contains four variables called
quantum numbers:
• Principal Quantum Number (n)
• Angular-Momentum Quantum Number (l)
• Magnetic Quantum Number (ml)
• Spin Quantum Number (ms)
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QUANTUM NUMBERS
PRINCIPLE QUANTUM NUMBERS (n):
This quantum number is the one on which the energy
of the electron in an atom principally depends; it can
have any positive value (1, 2, 3 etc..)
•The smaller n, the lower the energy.
•The size of an orbital depends on n; the larger the
value of n, the larger the orbital.
•Orbitals of the same quantum number (n) belong
to the same shell which have the following letters:
Letter: K L M N
n: 1 2 3 4
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Quantum Numbers
 ANGULAR MOMENTUM QUANTUM NUMBER (l):
Defines the three-dimensional shape of the orbital.
 For an orbital of principal quantum number n, the
value of l can have an integer value from
0 to n – 1.
 This gives the subshell notation:
Letter:
s
p
d
f
g
l:
0
1
2
3
4
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Quantum Numbers
 Magnetic Quantum Number (ml): Defines the
spatial orientation of the orbital.
 For orbital of angular-momentum quantum
number, l, the value of ml has integer values from
–l to +l.
 This gives a spatial orientation of:
l = 0 giving ml = 0
l = 1 giving ml = –1, 0, +1
l = 2 giving ml = –2, –1, 0, 1, 2, and so on…...
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Quantum Numbers
 Magnetic Quantum Number (ml): –l to +l
S orbital
0
P orbital
-1
0
1
-2
-1
0
D orbital
1
2
F orbital
-3
-2
-1
0
1
2
3
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Quantum Numbers
Table of Permissible Values of Quantum Numbers
for Atomic Orbitals
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Quantum Numbers
 Spin Quantum Number: ms
 The Pauli Exclusion Principle states that no
two electrons can have the same four quantum
numbers.
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QUANTUM MECHANICS
EXAMPLE 7.6 : State whether each of the following
sets of quantum numbers is permissible for an electron
in an atom. If a set is not permissible, explain.
a) n = 1, l = 1, ml = 0, ms = +1/2
NOT permissible: The l quantum number is equal to n.
IT must be less than n.
b) n = 3, l = 1, ml = -2, ms = -1/2
NOT permissible: The magnitude of the ml quantum
number (that is the ml value, ignoring it’s sign) must be
greater than l.
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QUANTUM MECHANICS
EXAMPLE 7.6 : cont…
c) n = 2, l = 1, ml = 0, ms = +1/2
Permissible
d) n = 2, l = 0, ml = 0, ms = +1
NOT permissible: The ms quantum number can only be
+1/2 or -1/2.
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QUANTUM MECHANICS
EXERCISE 7.7 : Explain why each of the following sets
of quantum numbers is not permissible for an orbital:
a) n = 0, l = 1, ml = 0, ms = +1/2
b) n = 2, l = 3, ml = 0, ms = -1/2
c) n = 3, l = 2, ml = +3, ms = +1/2
d) n = 3, l = 2, ml = +2, ms = 0
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Electron Radial Distribution
 s Orbital Shapes: Holds 2 electrons
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Electron Radial Distribution
 p Orbital Shapes: Holds 6 electrons,
degenerate
42
Electron Radial Distribution
 d and f Orbital Shapes: d holds 10 electrons
and f holds 14 electrons, degenerate
43
Effective Nuclear Charge
 Electron shielding leads to energy differences
among orbitals within a shell.
 Net nuclear charge felt by an electron is called
the effective nuclear charge (Zeff).
 Zeff is lower than actual nuclear charge.
 Zeff increases toward nucleus
ns > np > nd > nf
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Effective Nuclear Charge
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Example1: Light and Electromagnetic Spectrum
 The red light in a laser pointer comes from a
diode laser that has a wavelength of about 630
nm. What is the frequency of the light? c = 3 x
108 m/s
46
Example 2: Atomic Spectra
 For red light with a wavelength of about 630 nm, what is
the energy of a single photon and one mole of photons?
47
Example 3: Wave–Particle Duality
 How fast must an electron be moving if it has a
de Broglie wavelength of 550 nm?
me = 9.109 x 10–31 kg
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Example 4: Quantum Numbers
 Why can’t an electron have the following quantum
numbers?
(a) n = 2, l = 2, ml = 1
(b) n = 3, l = 0, ml = 3
(c) n = 5, l = –2, ml = 1
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Example 5: Quantum Numbers

Give orbital notations for electrons with the
following quantum numbers:
(a) n = 2, l = 1
(b) n = 4, l = 3
(c) n = 3, l = 2
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