The Quantum
Mechanical Atom
CHAPTER 7
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
Electromagnetic
Radiation
Light Energy is a Wave
Electromagnetic Spectrum
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
3
Electromagnetic
Radiation
Light Energy is a Wave
Waves travel through space at speed of light in vacuum
c = speed of light = 2.9979 × 108 m/s
Can define waves as systematic fluctuations in
intensities of electrical and magnetic forces that vary
regularly with time and exhibit a wide range of energy.
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Electromagnetic
Radiation
Light Energy is a Wave
Wavelength ()
– Distance between two successive peaks or troughs
– Units are in meters, centimeters, nanometers
Frequency ()
– Number of waves per second that pass a given
point in space
– Units are in Hertz (Hz = cycles/sec = 1/sec = s–1)
Related by    = c
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Electromagnetic
Radiation
Light Energy is a Wave
Amplitude
– Maximum and minimum height
– Intensity of wave, or brightness
– Varies with time as travels through space
Nodes
– Points of zero amplitude
– Place where wave goes though axis
– Distance between nodes is constant
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
6
Electromagnetic
Radiation
Ex: Converting between
Wavelengths and Frequency
Example: The bright red color in fireworks is due to emission of light
when Sr(NO3)2 is heated. If the wavelength is ~650 nm, what is the
frequency of this light?
c 3.00 ´ 108 m/s
n= =
l
650 ´ 10-9 m
 = 4.61 × 1014 s–1 = 4.6 × 1014 Hz
Example: WCBS broadcasts at a frequency of 880 kHz. What is the
wavelength of their signal?
c 3.00 ´ 10 m/s
l= =
n
880 ´ 103 / s
8
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
= 341 m
7
Electromagnetic
Radiation
Electromagnetic Spectrum
low energy, long waves
high energy, short waves
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
8
Electromagnetic
Radiation
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
Electromagnetic Spectrum
9
Electromagnetic
Radiation
Electromagnetic Spectrum
Visible light
• Band of wavelengths that human eyes can see
• 400 to 700 nm make up spectrum of colors
• White light is a combination of all these colors and can be
separated into individual colors with a prism.
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Electromagnetic
Radiation
Particle Theory of Light
Max Planck and Albert Einstein (1905)
• Electromagnetic radiation is stream of small
packets of energy
• Quanta of energy or photons
• Each photon travels with velocity = c
• Waves with frequency = 
Energy of photon of electromagnetic radiation is
proportional to its frequency
• Energy of photon
E=h
• h = Planck’s constant
= 6.626 × 10–34 J s
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Electromagnetic
Radiation
Ex: Determining Energy from
Frequency
Example: A microwave oven uses radiation with a frequency
of 2450 MHz (megahertz, 106 s–1) to warm up food. What is
the energy of such photons in joules?
E = hn
æ 1 ´ 106 s-1 ö
÷÷
E = 6.626 ´ 10-34 J s ´ 2450 MHz ´ çç
è MHz ø
(
) (
)
= 1.62 × 10–24 J
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Electromagnetic
Radiation
Photoelectric Effect
If shine light on a
metal surface:
• Below a certain frequency
nothing happens
• Above a certain frequency
electrons are ejected
• Increasing intensity
increases # of
electrons ejected
• Increasing frequency
increases KE of
electrons
KE = h – BE
h = energy of light shining on surface
BE = binding energy of electron
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E
http://hyperphysics.phy-astr.gsu.edu/hbase/mod1.html
13
Electromagnetic
Radiation
Photoelectric Effect
Therefore Energy is Quantized
• Can occur only in discrete units of size h
• 1 photon = 1 quantum of energy
• Energy gained or lost in whole number multiples of h
E = nh
• If n = NA, then one mole of photons gained or lost
E = 6.02 × 1023 h
If light is required to start reaction
• Must have light above certain frequency to start reaction
• Below minimum threshold energy, intensity is NOT important
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E
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Electromagnetic
Radiation
Ex: Energy, Frequency & Moles
Example: If a mole of photons has an energy of 1.60 × 10–3
J/mol, what is the frequency of each photon? Assume all
photons have the same frequency.
E
n=
N Ah
n=
1.60 ´ 10-3 J/mol
(6.02 ´ 1023 mol-1 )(6.626 ´ 10-34 J s)
= 4.01 × 106 Hz
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Atomic
Spectra
Electronic Structure of the Atom
excited state
Because energy is quantized we can
study the electronic structure of an
atom the frequency of light it absorbs or
emits:
1. Study of light absorption
+h
ground state
• Electron absorbs energy
• Moves to higher energy “excited state”
excited state
h
2. Study of light emission
• Electron loses photon of light
• Drops back down to lower energy “ground
state”
Jesperson, Brady, Hyslop. Chemistry: The
ground state
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Atomic
Spectra
Spectrum of Light
A continuous spectrum of light is an unbroken spectrum of
all colors
• i.e., visible light through a prism; sunlight; incandescent light
bulb; or a very hot metal rod
An atomic spectrum or the light emitted by an atom is a
discontinuous (or line) spectrum of light
• A discontinuous spectrum has only a few discrete lines
• Each element has a unique emission spectrum
Jesperson, Brady, Hyslop. Chemistry: The
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Atomic
Spectra
Spectrum of Light
Jesperson, Brady, Hyslop. Chemistry: The
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Atomic
Spectra
Electronic Structure of the Atom
Hydrogen is the simplest atomic spectra with only one electron
Emission: (Hydrogen, Mercury, Neon)
Absorption: (Hydrogen)
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Mattexr, 6E
http://facstaff.cbu.edu/~jvarrian/252/emspex.html
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Atomic
Spectra
Rydberg Equation
 1

1
1

 RH 

n2 n2 

2 
 1
RH = 109,678 cm–1 = Rydberg constant
 = wavelength of light emitted
n1 and n2 = whole numbers (integers) from 1 to 
where n2 > n1
If n1 = 1, then n2 = 2, 3, 4, …
• Can be used to calculate all spectral lines of hydrogen
• The values for n correspond to allowed energy levels for atom
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Mattexr, 6E
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Atomic
Spectra
Ex: Rydberg Equation
Example: Consider the Balmer series where n1 = 2 Calculate  (in
nm) for the transition from n2 = 6 down to n1 = 2.
æ1 1ö
æ
ö
1
1
= RH çç 2 - 2 ÷÷ = 109,678 cm-1 çç - ÷÷ = 24,373 cm–1
l
è2 6 ø
è 4 36 ø
1
l=
1
-5
-1
24,372.9 cm
= 4.1029 ´ 10
 = 410.3 nm
1m
1 nm
cm ´
´
100 cm 1 ´ 10-9 m
Violet line in spectrum
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Mattexr, 6E
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Atomic
Spectra
Ex: Rydberg Equation
Example: A photon undergoes a transition from nhigher
down to n = 2 and the emitted light has a wavelength of
650.5 nm?
-7
1 ´ 10 cm
l = 650.5 nm ´
= 650.5 ´10-7 cm
1 nm
-1 1
1
1 )
=
109,678
cm
(
650.5 ´ 10-7 cm
22 (n )2
2
1
=(1 - 12 )
7.13455
4 (n2 )
(n )
2
2
1
=
= 9.10
0.110
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Mattexr, 6E
1 =11
= 0.110
2
(n2 ) 4 7.13455
n2 = 3
22
Atomic
Spectra
Understanding Atomic Structure
Atomic line spectra tells us when excited atom loses energy
• Only fixed amounts of energy can be lost
• Only certain energy photons are emitted
• Electron restricted to certain fixed energy levels in atoms
Atomic line spectra tells us Energy of electron is quantized and is the
simple extension of Planck's Theory
Therefore any theory of atomic structure must account for
• Atomic spectra
• Quantization of energy levels in atom
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Mattexr, 6E
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“Quantum”
What do we mean by “Quantized”
• Energy is quantized if only
certain discrete values are
allowed
• Presence of discontinuities
makes atomic emission
quantized
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Bohr Model
Bohr Model of an Atom
First theoretical model of atom to successfully
account for Rydberg equation
• Quantization of energy in hydrogen atom
• Correctly explained atomic line spectra
Proposed that electrons moved around nucleus
like planets move around sun
• Move in fixed paths or orbits
• Each orbit has fixed energy
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Bohr Model
Energy Level Diagram for a Hydrogen Atom
• Absorption of photon
– Electron raised to
higher energy level
• Emission of photon
– Electron falls to lower
energy level
Energy levels are quantized
•
•
•
Every time an electron drops from one
energy level to a lower energy level
Same frequency photon is emitted
Yields line spectra
RH hc
b
E =- 2 =- 2
n
n
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Bohr Model
Bohr model of the Hydrogen Atom
• n = 1 First Bohr orbit
– Most stable energy state equals the ground state which is
the lowest energy state
– Electron remains in lowest energy state unless disturbed
How to change the energy of the atom?
– Add energy in the form of light: E = h
– Electron raised to higher n orbit n = 2, 3, 4, … 
– Higher n orbits = excited states = less stable
– So electron quickly drops to lower energy orbit and emits
photon of energy equal to E between levels
E = Eh – El
h = higher l = lower
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Bohr Model
Bohr’s Model Fails
• Theory could not explain spectra of multi-electron atoms
• Theory doesn’t explain collapsing atom paradox
• If electron doesn’t move,
atom collapses
• Positive nucleus should
easily capture electron
• Vibrating charge should
radiate and lose energy
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Bohr Model
Ex: Bohr’s Model of Energy Levels
Example: In Bohr's atomic theory, when an electron moves
from one energy level to another energy level more distant
from the nucleus,
A.
B.
C.
D.
E.
energy is emitted
energy is absorbed
no change in energy occurs
light is emitted
none of these
Jesperson, Brady, Hyslop. Chemistry: The
Molecular Nature of Matter, 6E
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Problem
Set A
1. Which electromagnetic radiation has a higher energy? Radio waves or
microwaves? UV light or X rays?
2. How does thermal imaging work? (Use what you have learned about the
electromagnetic spectrum to briefly explain).
3. Blue, red, and green lasers have wavelengths of 445 nm, 635 nm, and 532
nm respectively what are their frequencies, and what is the energy in Joules
of a photon from each laser?
4. In Neon there is a line with the frequency of 4.546 x1014 Hz. What is its
wavelength and color of the line? And what is the energy of each of its
photons?
5. What is the wavelength of light (in nm) that is emitted when an excited
electron in the hydrogen atom falls from n = 5 to n = 3? Would you expect to
be able to see the light emitted?
6. How many grams of water could have its temperature raised by 7°C by a
mole of photons that have a wavelength of 450 nm?