The Quantum
Mechanical Atom
CHAPTER 8
Chemistry: The Molecular Nature of Matter, 6th edition
By Jesperson, Brady, & Hyslop
CHAPTER 8: Quantum Mechanical Atom
Learning Objectives
 Light as Waves, Wavelength and Frequency
 The Photoelectric Effect, Light as Particles and the Relationship between
Energy and Frequency
 Atomic Emission and Energy Levels
 The Bohr Model and its Failures
 Electron Diffraction and Electrons as Waves
 Quantum Numbers, Shells, Subshells, and Orbitals
 Electron Configuration, Noble Gas Configuration and Orbital Diagrams
 Aufbau Principle, Hund’s Rule, and Pauli Exclusion Principle, Heisenberg
Uncertainty Principle
 Valence vs Inner Core Electrons
 Nuclear Charge vs Electron Repulsion
 Periodic Trends: Atomic Radius, Ionization Energy, and Electron Affinity
2
Particle-Wave
Duality
Light Exhibits Interference
Constructive interference
– Waves “in-phase” lead to greater amplitude
– They add together
Destructive interference
– Waves “out-of-phase” lead to lower amplitude
– They cancel out
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Particle-Wave
Duality
Are Electrons Waves or Particles?
Light behaves like both a particle and a wave:
– Exhibits interference
– Has particle-like nature
When studying behavior of electrons:
– Known to be particles
– Also demonstrate interference
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Particle-Wave
Duality
Standing vs Traveling Waves
Traveling wave
– Produced by wind on surfaces
of lakes and oceans
Standing wave
– Produced when guitar string
is plucked
– Center of string vibrates
– Ends remain fixed
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Molecular Nature of Matter, 6E
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Particle-Wave
Duality
Standing Wave on a Wire
• Integer number (n) of peaks and troughs is required
• Wavelength is quantized:
• L is the length of the string
l=
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
2L
n
6
Particle-Wave
Duality
Standing Wave on a Wire
• Has both wave-like and particle-like properties
• Energy of moving electron on a wire is E =½ mv 2
• Wavelength is related to the quantum number, n, and the wire
length:
l=
2L
n
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Molecular Nature of Matter, 6E
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Particle-Wave
Duality
Electron on a Wire
Standing wave
• Half-wavelength must occur integer number of
times along wire’s length
l=
2L
n
de Broglie’s equation relates the mass and speed of
the particle to its wavelength
l= h
mv
v= h
lm
• m = mass of particle
• v = velocity of particle
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
8
Particle-Wave
Duality
Electron on a Wire
Starting with the equation of the standing wave and the de Broglie
equation
l=
2L
n
v= h
lm
Combining with E = ½mv 2, substituting for v and then λ, we get
1
h2
1 h2
E = m 2 2=
2 lm
2 ml 2
Combining gives:
n 2h 2
E =
2
8mL
æ
1ç
h2
E = ç
2 ç m 2L / n
è
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
(
)
ö
÷ n 2h 2
=
÷
2
2
8
mL
÷
ø
9
Particle-Wave
Duality
De Broglie & Quantized Energy
• Electron energy quantized
– Depends on integer n
• Energy level spacing
changes when positive
charge in nucleus changes
– Line spectra different for
each element
• Lowest energy allowed is
for n =1
• Energy cannot be zero, hence atom cannot collapse
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
10
Particle-Wave
Duality
Ex: Wavelength of an Electron
What is the de Broglie wavelength associated with an
electron of mass 9.11 × 10 –31 kg traveling at a velocity
of 1.0 × 107 m/s?
6.626  10 J s
1 kg m /s


(1.0  10 m/s)(9.11  10 kg)
1J
34
2
2
31
7
 = 7.27 × 10–11 m
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Particle-Wave
Duality
Ex: Wavelength of an Electron
Calculate the de Broglie wavelength of a baseball with a
mass of 0.10 kg and traveling at a velocity of 35 m/s.
A. 1.9 × 10–35 m
B. 6.6 × 10–33 m
æ 6.626 ´ 10-34 J s ö æ 1 kg m2 /s2 ö
C. 1.9 × 10–34 m
÷÷ ´ çç
÷÷
l = çç
D. 2.3 × 10–33 m
35 m/s ´ 0.10 kg ø è
1J
è
ø
–31
E. 2.3 × 10 m
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
12
Particle-Wave
Duality
Wave Functions
Schrödinger’s equation
– Solutions give wave functions and energy levels of
electrons
Wave function
– Wave that corresponds to electron
– Called orbitals for electrons in atoms
Amplitude of wave function squared
– Can be related to probability of finding electron at
that given point
Nodes
– Regions where electrons will not be found
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Quantum
Numbers
Orbitals Characterized by 3 Quantum #’s
Quantum Numbers:
– Shorthand
– Describes characteristics of electron’s position
– Predicts its behavior
n = principal quantum number
– All orbitals with same n are in same shell
ℓ = secondary quantum number
– Divides shells into smaller groups called subshells
mℓ = magnetic quantum number
– Divides subshells into individual orbitals
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Quantum
Numbers
Principal Quantum Number (n)
• Allowed values: positive integers from 1 to 
– n = 1, 2, 3, 4, 5, … 
• Determines:
– Size of orbital
E =-
– Total energy of orbital
Z 2RH hc
n2
• RHhc = 2.18 × 10–18 J/atom
• For given atom,
– Lower n = Lower (more negative) E
= More stable
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Quantum
Numbers
Orbital Angular Momentum (ℓ)
– Allowed values: 0, 1, 2, 3, 4, 5…(n – 1)
– Letters:
s, p, d, f, g, h
Orbital designation
number
nℓ
letter
• Possible values of ℓ depend on n
– n different values of ℓ for given n
• Determines
• Shape of orbital
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Quantum
Numbers
Magnetic Quantum Number (mℓ)
• Allowed values: from –ℓ to 0 to +ℓ
– Ex. when ℓ=2 then mℓ can be
• –2, –1, 0, +1, +2
• Possible values of mℓ depend on ℓ
– There are 2ℓ+1 different values of mℓ for given ℓ
• Determines orientation of orbital in space
• To designate specific orbital, you need three quantum
numbers
– n, ℓ, mℓ
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Quantum
Numbers
n, ℓ, and mℓ
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Quantum
Numbers
Multiple Electrons in Orbitals
Orbital Designation
 Based on first two
quantum numbers
 Number for n and
letter for ℓ
 How many electrons
can go in each
orbital?
 Two electrons
 Need another
quantum number
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Quantum
Numbers
Spin Quantum Number (ms)
• Arises out of behavior of electron
in magnetic field
• electron acts like a top
• Spinning charge is like a magnet
– Electron behave like tiny
magnets
• Leads to two possible directions of
electron spin
– Up and down
– North and south
Possible Values:
+½

Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
½

20
Quantum
Numbers
Pauli Exclusion Principle
• No two electrons in same atom can have same set of all
four quantum numbers (n, ℓ, mℓ , ms)
Can only have two electrons per orbital
• Two electrons in same orbital must have opposite spin
– Electrons are said to be paired
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Quantum
Numbers
Number of Orbitals
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Quantum
Numbers
Magnetic Properties
• Two electrons in same orbital have different spins
– Spins paired—diamagnetic
– Sample not attracted to magnetic field
– Magnetic effects tend to cancel each other
• Two electrons in different orbital with same spin
– Spins unpaired—paramagnetic
– Sample attracted to a magnetic field
– Magnetic effects add
• Measure extent of attraction
– Gives number of unpaired spins
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Quantum
Numbers
Ex: Number of Electrons
What is the maximum number of electrons
allowed in a set of 4p orbitals?
A. 14
B.
6
C.
0
D.
2
E. 10
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Electron
Configurations
Ground State
Electron Configurations
•
Distribution of electrons among orbitals of atom
1. List subshells that contain electrons
2. Indicate their electron population with superscript
e.g. N is 1s 2 2s 2 2p 3
Orbital Diagrams
•
Way to represent electrons in orbitals
1. Represent each orbital with circle (or line)
2. Use arrows to indicate spin of each electron
e.g. N is
1s
2s
2p
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Electron
Configurations
Energy Level Diagram
4f
6s
5p
4d
5s
4p
3d
4s
Energy
3p
3s
2p
2s


How to put electrons into a diagram?
Need some rules
1s
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Electron
Configurations
Aufbau Principle
• Building-up principle
Pauli Exclusion Principle
• Two electrons per orbital
• Fill following the order suggested by
the periodic table
• Spins must be paired
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Electron
Configurations
Hund’s Rule
• If you have more than one orbital all at the same
energy
– Put one electron into each orbital with spins
parallel (all up) until all are half filled
– After orbitals are half full, pair up electrons
Why?
• Repulsion of electrons in same region of space
• Empirical observation based on magnetic properties
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Electron
Configurations
Orbital Diagram & Electron
Configurations: e.g. N, Z = 7
4p
3d
4s
Energy
3p
3s
2p
2s
Each arrow represents electron
1s 2 2s 2 2p 3
1s
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Electron
Configurations
Orbital Diagram and Electron
Configurations: e.g. V, Z = 23
4p
3d
4s
Energy
3p
3s
2p
2s
Each arrow represents an electron
1s 2 2s 2 2p 2 3s 2 3p 2 4s 2 3d 3
1s
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Electron
Configurations
Ex: Orbital Diagrams &
Electron Configurations
Give electron configurations and orbital diagrams for Na and As
6s
5p
4d
5s
4p
3d
4s
Energy
3p
3s
2p
2s
Na Z = 11
1s 2 2s 2 2p 2 3s 1
As Z = 33
1s
1s 22s 22p 63s 23p 64s 23d 104p 3
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Electron
Configurations
Ex: Ground State Electron
Configurations
What is the correct ground state electron
configuration for Si?
A. 1s 22s 22p 63s 23p 6
B. 1s 22s 22p 63s 23p 4
C. 1s 22s 22p 62d 4
D. 1s 22s 22p 63s 23p 2
E. 1s 22s 22p 63s 13p 3
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Molecular Nature of Matter, 6E
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Problem
Set B