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
Jesperson, Brady, Hyslop. Chemistry: The
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
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
7
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
1. Would you expect the waves above to increase or decrease in amplitude
when added together?
1. With the strong attractive force between the positively charged nucleus
and an electron, why doesn’t the nucleus capture electrons?
2. Determine quantum numbers for Phosphorous. It’s electron configuration
is 1(s)2 2(s)2 2(p)1
1. What is the electron configutration for Nitrogen? N-3?
1. Is Selenium (Se) paramagnetic or diamagnetic?
1. Draw an orbital diagram for Indium (In).