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Chemistry
Second Edition
Julia Burdge
Lecture PowerPoints
Jason A. Kautz
University of Nebraska-Lincoln
6
Quantum Theory and the
Electronic Structure of
Atoms
Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
1
6
Quantum Theory and the Electronic Structure of
Atoms
6.1 The Nature of Light
Properties of Waves
The Electromagnetic Spectrum
The Double-Slit Experiment
6.2 Quantum Theory
Quantization of Energy
Photons and the Photoelectric Effect
6.3 Bohr’s Theory of the Hydrogen Atom
Atomic Line Spectra
The Line Spectrum of Hydrogen
6.4 Wave Properties of Matter
The de Broglie Hypothesis
Diffraction of Electrons
6.5 Quantum Mechanics
The Uncertainty Principle
The Schrödinger Equation
The Quantum Mechanical Description of the Hydrogen Atom
6
Quantum Theory and the Electronic Structure of
Atoms
6.6 Quantum Numbers
Principal Quantum Number (n)
Angular Momentum Quantum Number (l)
Magnetic Quantum Number (ml)
Electron Spin Quantum Number (ms)
6.7 Atomic Orbitals
s Orbitals
p Orbitals
d Orbitals and other High-Energy Orbitals
Energies of Orbitals
6.8 Electron Configuration
Energies of Atomic Orbitals in Many-Electron Systems
The Pauli Exclusion Principle
The Aufbau Principle
Hund’s Rule
General Rules for Writing Electron Configurations
6.9 Electron Configurations and the Periodic Table
6.1 The Nature of Light
Visible light is only a small component of the continuum of radiant energy
known as the electromagnetic spectrum.
The Nature of Light
All forms of electromagnetic
radiation travel in waves.
Waves are characterized by:
Wavelength (λ; lambda) – the
distance between identical points
on successive waves
Frequency (ν; nu) – the number of
waves that pass through a
particular point in 1 second.
Amplitude – the vertical distance
from the midline of a wave to the
top of the peak or the bottom of the
trough.
The Nature of Light
The speed of light (c) through a vacuum is a constant:
c = 3.00 x 108 m/s
Speed of light, frequency and wavelength are related:
c = λν
λ is expressed in meters
ν is expressed in reciprocal seconds (s−1)
The Nature of Light
An electromagnetic wave has both an electric field component and a
magnetic component.
The electric and magnetic components have the same frequency
and wavelength.
The Nature of Light
When light passes through two closely spaced slits, an interference
pattern is produced.
Constructive interference
is a result of adding
waves that are in phase.
Destructive interference
is a result of adding
waves that are out of
phase.
The Nature of Light
What is the wavelength of light (in meters) of an electromagnetic wave
whose frequency is 1.61 x 1012 s−1?
Solution:
Step 1: Rearrange the equation below to calculate wavelength:
c = λν
3.00  108 m/s
4
= =
=
1.86

10
m
12 1

1.61 10 s
c
6.2
Quantum Theory
When a solid is heated, it emits electromagnetic radiation, known as
blackbody radiation, over a wide range of wavelengths.
The amount of energy given off at a certain temperature depends on the
wavelength.
Classical physics failed to completely explain the phenomenon.
Max Planck suggested that radiant energy is only emitted or absorbed in
discrete quantities or bundles.
A quantum of energy is the smallest quantity of energy that can be
emitted (or absorbed).
E = hν
h is called Planck’s constant: 6.63 x 10−34 J•s
Quantum Theory
A quantum of energy is the smallest quantity of energy that can be
emitted (or absorbed).
E = hν
E is the energy (in Joules)
ν is the frequency
h is called Planck’s constant: 6.63 x 10−34 J•s
Energy is always emitted in whole-number multiples of hν.
Quantum Theory
Albert Einstein used Planck’s theory to
explain the photoelectric effect.
Electrons are ejected from the surface of
a metal exposed to light of a certain
threshold frequency.
The number of electrons ejected is
proportional to the intensity.
Einstein proposed that the beam of light
is really a stream of particles.
These particles of light are now called
photons.
Quantum Theory
Each photon (of the incident light) must
posses the energy given by the equation:
Ephoton = hν
where ν is at minimum the threshold
frequency.
hν = KE + W
KE is the kinetic energy of the ejected
electron
W is the binding energy of the electron
Quantum Theory
Calculate the wavelength (in nm) of light with an energy of 7.85 x 1019 J
per photon.
In what region of the electromagnetic radiation does this light fall?
Solution:
Step 1: Rearrange the equation below to calculate frequency.
Ephoton = hν
 =
Ephoton
h
7.85  1019 J
15
1
=
=
1.184

10
s
6.63  1034 J  s
Quantum Theory
Calculate the wavelength (in nm) of light with an energy of 7.85 x 1019 J
per photon.
In what region of the electromagnetic radiation does this light fall?
Solution:
Step 2: Rearrange the equation below to calculate wavelength.
c = λν
3.00  108 m/s
7
= =
=
2.534

10
m
15
1

1.184  10 s
c
Quantum Theory
Calculate the wavelength (in nm) of light with an energy of 7.85 x 1019 J
per photon.
In what region of the electromagnetic radiation does this light fall?
Solution:
Step 3: Convert to nm.
2.534  10
7
1 nm
m
= 253 nm
9
1 10 m
Quantum Theory
Solution:
Step 4: Correlate the wavelength (or frequency) to the electromagnetic
spectrum.
235 nm is in the ultraviolet region of the electromagnetic spectrum
6.3
Bohr’s Theory of the Hydrogen Atom
Sunlight is composed of various color
components that can be recombined to produce
white light.
The emission spectrum of a substance can be
seen by energizing a sample of material with
some form of energy.
The “red hot” or “white hot” glow of an iron bar
removed from a fire is the visible portion of its
emission spectrum.
All wavelengths of visible light are present in
the emission spectra of sun light and a heated
solid.
Bohr’s Theory of the Hydrogen Atom
Line spectra are the emission of light only at specific wavelengths.
Bohr’s Theory of the Hydrogen Atom
Every element has its own unique emission spectrum.
Bohr’s Theory of the Hydrogen Atom
The Rydberg equation can be used to calculate the wavelengths of the
four visible lines in the emission spectrum of hydrogen.
 1
1
= R  2  2 

 n1 n2 
1
R∞ is the Rydberg constant (1.09737317 x 107 m−1)
λ the wavelength of a line in the spectrum
n1 and n2 are positive integers where n1 > n2.
Bohr’s Theory of the Hydrogen Atom
Neils Bohr attributed the emission of radiation by an energized hydrogen
atom to the electron dropping from a higher-energy orbit to a lower one.
As the electron dropped, it gave up a quantum of energy in the form of
light.
Bohr showed that the energies of the electron in a hydrogen atom are
given by the equation:
En =  2.18  10
En is the energy
n is a positive integer
18
 1
J 2 
n 
Bohr’s Theory of the Hydrogen Atom
As an electron gets closer to the nucleus, n decreases.
En =  2.18  10
18
 1
J 2 
n 
En becomes larger in absolute value (but more negative) as n gets
smaller.
En is most negative when n = 1
A free electron is considered to be infinitely far from the nucleus.
n = ∞ and E∞ = 0
Bohr’s Theory of the Hydrogen Atom
For hydrogen, the lowest energy (most stable) state occurs when n = 1
En =  2.18  10
18
 1
J 2 
n 
The lowest energy state is called the ground state.
The stability of the electron decreases as n increases.
Each energy state in which n > 1 is called an excited state.
Bohr’s Theory of the Hydrogen Atom
During an emission, an electron drops from an excited state to a lower
energy state.
nf is the final state
ni is the initial state
E = h =  2.18  10
18
 1
1
J 2  2 
 nf ni 
Bohr’s Theory of the Hydrogen Atom
E = h =  2.18  10
nf is the final state
ni is the initial state
18
 1
1
J 2  2 
 nf ni 
Bohr’s Theory of the Hydrogen Atom
To calculate wavelength, substitute c/λ for ν and rearrange:
E = h =  2.18  10
18
 1
1
J 2  2 
 nf ni 
2.18  1018 J  1
1
=
 2  2

hc
 nf ni 
1
Bohr’s Theory of the Hydrogen Atom
What is the wavelength (in nm) of a photon emitted during a transition
from the n = 3 state to the n = 1 state in the H atom?
Solution:
Step 1: Use the equation below to solve for λ.
2.18  1018 J  1
1
=
 2  2

hc
 nf ni 
1
1 1
2.18  1018 J
1
=

=
9742572
m


 (6.63  1034 J  s)(3.00  108 m/s)  12 32 
1
λ = (9742572 m−1)−1 = 1.03 x 10−7 m
Bohr’s Theory of the Hydrogen Atom
What is the wavelength (in nm) of a photon emitted during a transition
from the n = 3 state to the n = 1 state in the H atom?
Solution:
Step 2: Convert to nm.
1.03  10
7
1 nm
m 
= 103 nm
9
1 10 m
6.4
Wave Properties of Matter
Louis de Broglie reasoned that if light can behave like a stream of
particles (photons), then electrons could exhibit wavelike properties.
According to deBroglie,
electrons behave like
standing waves.
Only certain wavelengths
are allowed.
At a node, the amplitude
of the wave is zero.
Wave Properties of Matter
De Broglie deduced that the particle and wave properties are related by
the following expression:
=
h
mu
λ is the wavelength associated with the particle
m is the mass (in kg)
u is the velocity (in m/s)
The wavelength calculated from this equation is known as the de Broglie
wavelength.
Wave Properties of Matter
Calculate the de Broglie wavelength (in nm) of a hydrogen atom (m =
1.674 x 10‒27 kg) moving at 15.0 m/s.
Solution:
Step 1: Use the equation below to calculate wavelength:
h
=
mu
6.63  1034 kg  m2 / s
4

=
2.64

10
m
31
(1.674  10 kg)(15.0 m/s)
Step 2: Convert m to nm:
2.64  104 m 
1 nm
5
=
2.64

10
nm
9
1 10 m
Wave Properties of Matter
Experiments have shown that electrons do indeed posses wavelike
properties:
X-ray diffraction pattern of
aluminum foil
Electron diffraction pattern of
aluminum foil.
6.5
Quantum Mechanics
The Heisenberg uncertainty principle states that it is impossible to
know simultaneously both the momentum and the position of a particle
with certainty.
x  p 
h
4
Δx is the uncertainty in position in meters
Δp is the uncertainty in momentum
x  mu 
Δu is the uncertainty in velocity in m/s
m is the mass in kg
h
4
Quantum Mechanics
If an electron in the hydrogen atom has a velocity of 5 x 106 m/s +− 1
percent. What is the uncertainty in position?
Solution:
Step 1: Calculate Δu:
Δu = 0.01 x (5 x 106 m/s) = 5 x 104 m/s
Quantum Mechanics
If an electron in the hydrogen atom has a velocity of 5 x 106 m/s +‒ 1
percent. What is the uncertainty in position?
Solution:
Step 2: Rearrange the equation below to solve for Δx:
h
x  mu 
4
h
x 
4  mu
The minimum uncertainty in
position is 1 x 10‒9 m = 10 Å.
This is about 10 times larger than
the atom!
6.63  1034 kg  m2 /s
9
x 

1

10
m
31
4
4 (9.11 10 kg)(5  10 m/s)
Quantum Mechanics
Erwin Schrödinger derived a complex mathematical formula to incorporate
the wave and particle characteristics of electrons.
Wave behavior is described with the wave function ψ.
The probability of finding an electron in a certain area of space is
proportional to ψ2 and is called
electron density.
Quantum Mechanics
The Schrödinger equation specifies possible
energy states an electron can occupy in a
hydrogen atom.
The energy states and wave functions are
characterized by a set of quantum numbers.
Instead of referring to orbits as in the Bohr
model, quantum numbers and wave
functions describe atomic orbitals.
6.6
Quantum Numbers
Quantum numbers are required to describe the distribution of electron
density in an atom.
There are three quantum numbers necessary to describe an atomic
orbital.
The principal quantum number (n) – specifies size
The angular moment quantum number (l) – specifies shape
The magnetic quantum number (ml) – specifies orientation
Quantum Numbers
The principal quantum number (n) designates the size of the orbital.
Larger values of n correspond to larger orbitals.
The allowed values of n are integral numbers: 1, 2, 3 and so forth.
The value of n corresponds to the value of n in Bohr’s model of the
hydrogen atom.
A collection of orbitals with the same value of n is frequently called a
shell.
Quantum Numbers
The angular moment quantum number (l) describes the shape of the
orbital.
The values of l are integers that depend on the value of the principal
quantum number
The allowed values of l range from 0 to n – 1.
l
0
1
2
3
Orbital designation
s
p
d
f
A collection of orbitals with the same value of n and l is referred to as a
subshell.
Quantum Numbers
The magnetic quantum number (ml) describes the orientation of the
orbital in space.
The values of ml are integers that depend on the value of the angular
moment quantum number:
– l,…0,…+l
Quantum Numbers
Quantum numbers designate shells, subshells, and orbitals.
Quantum Numbers
The electron spin quantum number (ms) is used to specify an electron’s
spin.
There are two possible directions of
spin.
Allowed values of ms are +½ and −½.
Quantum Numbers
A beam of atoms is split by a magnetic field.
Statistically, half of the electrons spin clockwise, the other half spin
counterclockwise.
Quantum Numbers
To summarize quantum numbers:
principal (n) size
Required to describe an atomic orbital
angular (l) shape
magnetic (ml) orientation
principal (n = 2)
2px
related to the
magnetic quantum
number (ml )
angular momentum (l = 1)
electron spin (ms) direction of spin
Required to describe an electron
in an atomic orbital
Quantum Numbers
Which of the following are possible sets of quantum numbers?
Quantum number
(a)
(b)
(c)
Principal (n)
1
2
3
Angular moment (l)
1
0
2
Magnetic (ml)
0
0
–2
Electron spin (ms)
+½
+½
–½
Solution:
Step 1: The principle quantum number must be a positive integral
number.
(a) n = 1 
(b) n = 2 
(c) n = 3 
Quantum Numbers
Which of the following are possible sets of quantum numbers?
Quantum number
(a)
(b)
(c)
Principal (n)
1
2
3
Angular moment (l)
1
0
2
Magnetic (ml)
0
0
–2
Electron spin (ms)
+½
+½
–½
Solution:
Step 2: The angular momentum quantum number has allowed values of
0 to n–1.
(a) n = 1, l = 1 not allowed  set (a) is not possible
(b) n = 2, l = 0 
(c) n = 3, l = 2 
Quantum Numbers
Which of the following are possible sets of quantum numbers?
Quantum number
(a)
(b)
(c)
Principal (n)
1
2
3
Angular moment (l)
1
0
2
Magnetic (ml)
0
0
–2
Electron spin (ms)
+½
+½
–½
Solution:
Step 3: The magnetic quantum number has allowed values of –l,...0,...+l.
(b) n = 2, l = 0, ml = 0 
(c) n = 3, l = 2, ml = –2 
Quantum Numbers
Which of the following are possible sets of quantum numbers?
Quantum number
(a)
(b)
(c)
Principal (n)
1
2
3
Angular moment (l)
1
0
2
Magnetic (ml)
0
0
–2
Electron spin (ms)
+½
+½
–½
Solution:
Step 4: Electron spin has two allowed values +½ and –½.
(b) n = 2, l = 0, ml = 0, ms = +½  (b) is a possible set
(c) n = 3, l = 2, ml = –2, ms = –½  (c) is a possible set
6.7
Atomic Orbitals
All s orbitals are spherical in shape but differ in size:
1s < 2s < 3s
principal quantum
number (n = 2)
2s
angular momentum
quantum number (l = 0)
ml = 0; only 1 orientation
possible
Atomic Orbitals
The p orbitals:
Three orientations:
l = 1 (as required for a p orbital)
ml = –1, 0, +1
Atomic Orbitals
The d orbitals:
Five orientations:
l = 2 (as required for a d orbital)
ml = –2, –1, 0, +1, +2
Atomic Orbitals
The energies of orbitals in the hydrogen atom depend only on the
principal quantum number.
the 3d subshell (n = 3; l = 2)
the n = 2 shell
6.8
Electron Configuration
The electron configuration describes how the electrons are distributed
in the various atomic orbitals.
In a ground state hydrogen atom, the electron is found in the 1s orbital.
Ground state electron
configuration of hydrogen
Energy
principal (n = 1)
2s
1s
2p
2p
2p
1
1s
number of electrons in
the orbital or subshell
angular momentum (l = 0)
The use of an up arrow indicates and electron
with ms = + ½
Electron Configuration
If hydrogen’s electron is found in a higher energy orbital, the atom is in an
excited state.
A possible excited state electron
configuration of hydrogen
Energy
1
2s
2s
1s
2p
2p
2p
Electron Configuration
The helium emission spectrum is more complex than the hydrogen
spectrum.
There are more possible energy transitions in a helium atom.
Electron Configuration
In a multi-electron atoms, the energies of the atomic orbitals are split.
Splitting of energy levels refers to
the splitting of a shell into
subshells of different energies
Electron Configuration
According to the Pauli exclusion principle, no two electrons can have
the same four quantum numbers.
The ground state electron
configuration of helium
Energy
2p
2p
2p
2
1s
2s
Quantum number
Principal (n)
1s
describe the 1s orbital
Angular moment (l)
Magnetic (ml)
describes the electrons in the 1s orbital
Electron spin (ms)
1
0
0
+½
1
0
0
‒½
Electron Configuration
The Aufbau principle states that electrons are added to the lowest
energy orbitals first before moving to higher energy orbitals.
The third electron must go in the
next available orbital with the
lowest possible energy.
The ground state electron
configuration of Li
2
1
1s 2s
Energy
2p
2p
2p
Li has a total of 3 electrons
2s
1s
The 1s orbital can only accommodate 2
electrons (Pauli exclusion principle)
Electron Configuration
The Aufbau principle states that electrons are added to the lowest
energy orbitals first before moving to higher energy orbitals.
The ground state electron
configuration of Be
2
2
1s 2s
Energy
2p
2s
1s
2p
2p
Be has a total of 4 electrons
Electron Configuration
The Aufbau principle states that electrons are added to the lowest
energy orbitals first before moving to higher energy orbitals.
The ground state electron
configuration of B
2
2
1
1s 2s 2p
Energy
2p
2s
1s
2p
2p
Electron Configuration
According to Hund’s rule, the most stable arrangement of electrons is
the one in which the number of electrons with the same spin is
maximized.
The ground state electron
configuration of C
2
2
2
1s 2s 2p
Energy
2p
2p
2p
2s
The 2p orbitals are degenerate.
1s
Put 1 electron in each before pairing (Hund’s rule).
Electron Configuration
According to Hund’s rule, the most stable arrangement of electrons is
the one in which the number of electrons with the same spin is
maximized.
The ground state electron
configuration of N
2
2
3
1s 2s 2p
Energy
2p
2s
2p
2p
The 2p orbitals are degenerate.
Put 1 electron in each before pairing (Hund’s rule).
1s
Electron Configuration
According to Hund’s rule, the most stable arrangement of electrons is
the one in which the number of electrons with the same spin is
maximized.
The ground state electron
configuration of O
2
2
4
1s 2s 2p
Energy
2p
2s
1s
2p
2p
Once all the 2p orbitals are singly occupied, additional
electrons will have to pair with those already in the
orbitals.
Electron Configuration
General rules for writing electron
configurations:
1) Electrons will reside in the available
orbitals of the lowest possible energy.
2) Each orbital can accommodate a
maximum of two electrons.
3) Electrons will not pair in degenerate
orbitals if an empty orbital is available.
4) Orbitals will fill in the order indicated in
the figure.
Electron Configuration
Write the electron configuration for a Si atom (Z = 14).
Solution: Fill in the energy diagram following the rules for electron
configurations.
3p
3p
3p
2p
2p
2p
Energy
3s
2s
Electron configuration of Si
1s22s22p63s23p2
1s
6.9
Electron Configurations and the Periodic Table
The electron configurations of all elements except hydrogen and helium
can be represented using a noble gas core.
The ground state electron configuration of K:
1s22s22p63s23p64s1
[Ar]
1
[Ar]4s
Electron Configurations and the Periodic Table
Elements in Group 3B through Group 1B are the transition metals.
Electron Configurations and the Periodic Table
There are several notable exceptions to the order of electron filling for
some of the transition metals.
Cr
[Ar]
4s
3d
3d
3d
3d
3d
Greater stability with half-filled
3d subshell
Cu
[Ar]
4s
3d
3d
3d
3d
3d
Greater stability with filled 3d
subshell
Electron Configurations and the Periodic Table
Write the electron configuration for an iron atom (Z = 26)
Solution:
Step 1: Locate the noble gas in the period above Fe. This is the noble
gas core.
Argon core
Electron Configurations and the Periodic Table
Write the electron configuration for an iron atom (Z = 26)
Solution:
Step 2: Write the remaining electron configuration
[Ar] 4s23d5
Electron Configurations and the Periodic Table
Electron Configurations and the Periodic Table
6
Chapter Summary: Key Points
The Nature of Light
Properties of Waves
The Electromagnetic Spectrum
The Double-Slit Experiment
Quantum Theory
Quantization of Energy
Photons and the Photoelectric
Effect
Bohr’s Theory of the Hydrogen
Atom
Atomic Line Spectra
The Line Spectrum of Hydrogen
Wave Properties of Matter
The de Broglie Hypothesis
Diffraction of Electrons
Quantum Mechanics
The Uncertainty Principle
The Schrödinger Equation
The Quantum Mechanical
Description of the Hydrogen Atom
Quantum Numbers (n,l,ml,ms)
Atomic Orbitals
s orbitals, p orbitals, d orbitals and
other High-Energy Orbitals
6
Chapter Summary: Key Points
Energies of Orbitals
Electron Configuration
Energies of Atomic Orbitals in Many-Electron Systems
The Pauli Exclusion Principle
The Aufbau Principle
Hund’s Rule
General Rules for Writing Electron Configurations
Electron Configurations and the Periodic Table