Chapter 7: ELECTRONS IN
ATOMS AND PERIODIC
PROPERTIES
Problems: 7.1-7.16, 7.18-7.27, 7.31-7.56, 7.617.107, 7.109-7.119, 7.124-7.125, 7.128-7.134
Electromagnetic Radiation
Electromagnetic (EM) Spectrum: a continuum of the
different forms of electromagnetic radiation or radiant energy
Weather systems
Radar: Radio Waves
Radio telescopes.
This is the Very
Large Array (VLA)
in NM.
A galaxy imaged in the visible
spectrum.
The same galaxy imaged in the
radio spectrum at the VLA.
Thermal Imaging: Detecting IR radiation
Much of a person’s energy is radiated away from the body in the form
of infrared (IR) energy.
You can produce more IR energy to warm yourself by moving
around…this is why you shiver when you go outside in the cold with no
coat on. This process is called thermogenesis.
Why does your mother insist you wear a hat in the winter?
“IR” Photography
Image obtained with “IR” film,
which is really film that is
activated by light at 700-900 nm.
Electromagnetic Radiation
• Electromagnetic radiation or “light” is a form of
energy.
• Has both electric (E) and magnetic (H) components.
• Characterized by:
– Wavelength (λ)
– Amplitude (A)
Electromagnetic Radiation (cont.)
• Wavelength (l): The distance between two
consecutive peaks in the wave.
Increasing Wavelength
l1 > l2 > l3
Unit: length (m)
Electromagnetic Radiation (cont.)
• Frequency (n): The number of waves (or
cycles) that pass a given point in space per
second.
Decreasing Frequency
n1 < n2 < n3
Units: 1/time (1/sec)
or, Hertz (Hz)
Electromagnetic Radiation (cont.)
• The product of wavelength (l) and frequency
(n) is a constant.
(ν)(λ) = c
Speed of light
c = 3 x 108 m/s
c is a constant, independent of λ
Properties of Waves
• Wavelength (λ) is the distance
from peak-to-peak in a wave.
• Frequency (ν) is the number of
waves in a specific time frame
(usually per second = Hz)
• As wavelength goes up,
frequency goes down (and vice
versa)
• Electromagnetic waves travel at
the speed of light:
λν=c
c = speed of light = 3 x 108 m/s
What statement is true when comparing
red light to blue light?
A. Red light travels at a greater speed than blue light.
B. Blue light travels at greater speed than red light.
C. The wavelength of blue light is longer.
D. The wavelength of red light is longer.
Behavior of Waves
• Waves refract, diffract and interact…
• Refraction: The bending of light as it passes from
one medium to another of different density
Refraction throught a prism separates white light
into its separate components (light of different
wavelengths) without changing the light itself.
Light is also bent by
liquids, causing the straw
to appear disconnected.
Behavior of Waves
• Diffraction: The bending of electromagnetic
radiation as it passes around an edge of an object or
through a narrow opening.
What causes the bright and dark spots?
Behavior of Waves
• Waves interact by adding together or cancelling each
other out…
Light is a Wave, Right?
• Back in the old days…
– It was generally agreed that matter and light were
distinct.
– Matter was particulate in nature, light could be
described using waves.
– Physicists circa 1900 had it all figured out…
– One famous physicist asserted that within ten
years or so all the major problems in physics
would be solved.
– The only thing left, really, was this niggling little
problem with black-body radiation…
The State of Physics Before
1900
• Newton discovered light could be separated by a
prism in 170
• Heat, electricity and phlogiston were weightless,
“imponderable fluids” responsible for most observed
processes
• It wasn’t until the early 1800s that the following,
revolutionary ideas were put forth: light was a wave,
atoms existed, and air could trap heat (the
“greenhouse effect”).
Physics in the early 1900s
• The end of the 1800s saw an explosion of real
scientific progress.
• Thermodynamics, electromagnetism and the kinetic
molecular theory were well-developed.
• There were still some problems, though, that needed
explaining…
– Blackbody radiation, new particle discoveries, what
was the “ether”?, radioactivity, the instability of the
atom, the photoelectric effect.
• There had to be one theory to explain them all…
The big problem
• Black body radiation: When a metal object is
heated, it begins to glow. If it is heated hot enough, it
glows “white hot”, emitting all the wavelengths of
visible light – but little to none in the UV range or
higher.
• Current (at the time) theories by Maxwell could not
explain this common phenomenon.
• Max Planck proposed a new theory of light – that it
behaved as a wave, but with particle-like properties.
– Light traveled in particle-like packets he called
“quanta” (a single one is called a “quantum”).
– Each quantum was the smallest amount of energy
found in nature.
Light as Energy
• Planck found that in order to model this behavior, one
has to envision that energy (in the form of light) is lost
in integer values according to:
DE = nhn
Energy Change
frequency
n = 1, 2, 3 (integers)
h = Planck’s constant = 6.626 x 10-34 J•s
Light as Energy (cont.)
• In general the relationship
between frequency and
“photon” energy is
Ephoton = hν
h = 6.636 x 10-34 J•s
c = 2.9979 x 108 m/s
1 Hz = 1 s-1
• Example: What is the energy of a 500 nm photon?
ν = c/λ = (3x108 m/s)/(5.0 x 10-7 m)
ν = 6 x 1014 1/s
E = hν =(6.626 x 10-34 J•s)(6 x 1014 1/s) = 4 x 10-19 J
Energy Quantization
The student can stop at any point on
the ramp. Her distance from the
ground changes continuously.
The student can stop only at certain points
on a flight of stairs. Her distance from the
ground is quantized.
Similarly, atomic energy levels are like steps…the energies
available to an atom do not form a continuum, they are
quantized.
Evidence of Quantization
• Black-body Radiation (Planck)
– A system can transfer energy in “packets” of size hn.
– These packets are called quanta
– Prior to this discovery it was thought that systems could absorb or emit
any amount of energy.
Other observations supported this “quantum” view:
• Atomic Emission spectra (Balmer, Rydberg, Bohr)
– Light emitted from excited atoms occurs in discrete lines rather than a
continuum.
• Photoelectric Effect (Einstein)
– Energy itself is actually quantized into packets called photons.
– This means energy has a particle-like nature as well as a wave-like
nature.
• Electron Diffraction Patterns (Davisson & Germer)
– Matter also has a wave-like nature!!
Atomic Emission
When we heat a sample of an element, the atoms become excited.
When the atom relaxes it emits visible light. The color of the light
depends on the element.
Li
Na
When the light emitted
from excited atoms is
passed through a
prism, we see discrete
bands of color at
specific wavelengths.
K
Ca
Sr
H
Li
Ba
Photon Emission
We can determine the energy
difference (ΔE) between levels by
measuring the wavelength of the
emitted photon.
ΔE = hc/λ  λ = hc/ΔE
If λ = 410 nm, ΔE = 4.52 x 10-19 J
Emission
of photon
An excited atom relaxes from high E
to low E by emitting a photon.
The Photoelectric Effect
Observed by Albert Einstein in 1905: Light shining on a clean metal surface
causes the emission of electrons but only when the light has a minimum
threshold frequency (ν0)
• When ν<ν0  no electrons are emitted
• When ν>ν0  electrons are emitted, more e– emitted with greater intensity
of light
ν < ν0
ν > ν0
Einstein applied Planck's
quantum theory to light:
light exists as a stream
of "particles" called
photons.
The Photoelectric Effect (cont)
Frequency: determines
whether e- are ejected, and
their KE (velocity).
Intensity: determines the
number of e- that are
ejected…but they will all
have the same velocity!!
The Photoelectric Effect (cont.)
As frequency of incident light is
increased, kinetic energy of
emitted e- increases linearly.
1
2
men  hn photon  
2
0
n0
Frequency (n)

 = hn0
Workfunction: energy
needed to release e-
Light apparently behaves as a particle.
The Photoelectric Effect (cont.)
For Na with Φ = 4.4 x 10-19 J,
what wavelength corresponds to νo?
0
1
2
men  hn photon  
2
hν = Φ = 4.4 x 10-19 J
0
hc/λ = 4.4 x 10-19 J
n0

Frequency (n)


hc
6.626 x10 34 J .s 3 x108 m / s
l

19
4.4 x10 J
4.4 x10 19 J


λ = 4.52 x 10-7 m = 452 nm

Electron Diffraction Patterns
Light is shined through a crystal, and its
waveforms are “scattered.” When they come out
the other side, they create interference patterns
on a detector plate.
Diffraction can only be explained by treating light as a
wave instead of a particle.
Diffraction of Particles?
• Turns out we can get
similar interference
patterns by bombarding
crystals with beams of
high energy electrons
also…
• This can only be
explained by treating
matter as a wave.
de Broglie Wavelength
• If matter exhibits wave-like properties, we should be able
to determine the wavelength of a particle.
• Recall the energy of a photon, and the definition of the
speed of light:
Ephoton = hν
• Substituting,
c = λν  ν = c/λ
E = hc/λ
• Using Einstein’s relationship, E = mc2,
hc
= mc2
λ
hc
h
λ = 2=
mc
mc
h
λ=
mc
de Broglie Wavelength
• We can generalize this relationship to any velocity (v):
h
λ=
mc
h
λ=
mv
• What is the de Broglie wavelength of an electron
traveling at the speed of light? (melectron = 9.31 x 10-31 kg)
λ = (6.626 x 10-34 J•s)/[(9.31x10-31 kg)(3x108 m/s)] = 2.37 x 10-12 m
• What is the de Broglie wavelength of a 80 kg student
walking across campus at 3 m/s?
λ = (6.626 x 10-34 J•s)/[(80 kg)(3 m/s)] = 2.76 x 10-36 m
You are a wave, too, but you have a VERY small
wavelength!!
What does all this mean for
matter?
• Scientists needed to find a way to explain how light,
usually thought of as a wave, could behave like a
particle AND how matter, which was usually thought
of as a particle, could behave as a wave.
• They started with the hydrogen emission spectrum…
• Balmer and Rydberg developed equations that
predicted where these lines should appear (even
before anyone had observed them).
 1
1 
n  Ry  2  2 
n1 n 2 
The Bohr Model
• Balmer and Rydberg didn’t know why their
equations worked.
• Niels Bohr used their equations and observations
to develop a quantum model for H.
• Central idea: electron circles the “nucleus” in only
certain allowed circular orbitals.
• Bohr postulated that there is Coulombic attraction
between e- and nucleus. However, classical
physics is unable to explain why an H atom
doesn’t simply collapse. Why doesn’t the electron
just spiral into the nucleus?
The Bohr Model
of the atom
Principle Quantum number: n
An “index” of the energy levels
available to the electron.
The Bohr Model (cont.)
• Bohr model for the H atom is capable of
reproducing the energy levels given by the
empirical formulas of Balmer and Rydberg.
2 

Z
18
E  2.178x10 J 2 
n 
Ry x h = -2.178 x 10-18 J
Z = atomic number (1 for H)
n = integer (1, 2, ….)
The Bohr Model (cont.)
2 

Z
18
E  2.178x10 J 2 
n 
Energy levels get closer
together as n increases

At n = infinity, E = 0  the
electron is free (not a part of
the atom)
The Bohr Model (cont.)
We can use the Bohr model to predict what DE
is for any two energy levels
DE  E final  E initial
18
DE  2.178x10

 1 
 1 
18
J
n 2 
 (2.178x10 J)n 2 
 initial 
 final 
18
DE  2.178x10
 1
1 
J
n 2  n 2 

 final
initial 
The Bohr Model (cont.)
Example: At what wavelength will emission from
n = 4 to n = 1 for the H atom be observed?
 1

1
DE  2.178x1018 J
n 2  n 2 

 final
initial 
1
4
 1 
DE  2.178x10 J1  2.04x1018 J
 16 
18

18

DE  2.04 x10 J 
hc
l
l  9.74 x108 m  97.4nm
The Bohr Model (cont.)
Example: What is the longest wavelength of
light that will result in removal of the e- from H?
 1

1
DE  2.178x1018 J
n 2  n 2 

 final
initial 
1



DE  2.178x1018 J0 1  2.178x1018 J
18
DE  2.178x10 J 
hc
l
l  9.13x108 m  91.3nm
Extension to Higher Z
The Bohr model can be extended to any single
electron system….must keep track of Z (atomic
number).
2 

Z
18
E  2.178x10 J 2 
n 
Z = atomic number
n = integer (1, 2, ….)
Examples: He+ (Z = 2), Li+2 (Z = 3), etc.
Extension to Higher Z (cont.)
Example: At what wavelength will emission from n = 4
to n = 1 for the He+ atom be observed?
 1

1
DE  2.178x1018 J Z 2 
n 2  n 2 

 final
initial 
2
1
4
 1 
DE  2.178x10 J41  8.16x1018 J
 16 
hc
18
l  2.43x108 m  24.3nm
DE  8.16x10 J 
l
l H  l He
18



So what happened to the Bohr Model?
• Although it successfully described the line spectrum of
hydrogen and other one-electron systems, it failed to
accurately describe the spectra of multi-electron atoms.
• The Bohr model was soon scrapped in favor of the
Quantum Mechanical model, although the vocabulary of
the Bohr model persists.
• However, Bohr pioneered the idea of quantized
electronic energy levels in atoms, so we owe him big.
Thanks Niels Bohr!
Quantum Concepts
• The Bohr model was capable of describing the
discrete or “quantized” emission spectrum of H.
• But the failure of the model for multielectron
systems combined with other issues (the
ultraviolet catastrophe, workfunctions of metals,
etc.) suggested that a new description of atomic
matter was needed.
Quantum Concepts (cont.)
• This new description was known as wave
mechanics or quantum mechanics.
• Recall, photons and electrons readily
demonstrate wave-particle duality.
• The idea behind wave mechanics was that the
existence of the electron in fixed energy levels
could be though of as a “standing wave”.
Quantum Concepts (cont.)
• What is a standing wave?
• A standing wave is a motion in
which translation of the wave does
not occur.
• In the guitar string analogy
(illustrated), note that standing
waves involve nodes in which no
motion of the string occurs.
• Note also that integer and halfinteger values of the wavelength
correspond to standing waves.
Quantum Concepts (cont.)
• Louis de Broglie suggested that for the e- orbits
envisioned by Bohr, only certain orbits are allowed
since they satisfy the standing wave condition.
h
h
l 
p mv
l  wavelength
m  mass
v  velocity
h  Planck's constant
not allowed
Quantum Concepts (cont.)
• Erwin Schrodinger developed a mathematical
formalism that incorporates the wave nature
of matter:
Hˆ   E
• H, the “Hamiltonian,” is a special kind of
function that gives the energy of a quantum
state,
which is described by the wavefunction,
Y.
Quantum Concepts (cont.)
• What is a wavefunction?

= a probability amplitude
• Probability of finding a
particle in space:
Probability =

*
With the wavefunction, we can
describe spatial distributions.

The probability of finding the
electron at points along a line
drawn from the nucleus
outward in any direction for
the hydrogen 1s orbital.
The probability distribution for
the hydrogen 1s orbital in
three-dimensional space
Hydrogen’s Electron
Cross section of the hydrogen 1s
orbital probability distribution
divided into successive thin
spherical shells (b) The radial
probability distribution
The surface contains 90%
of the total electron
probability (the size of the
orbital, by definition)
Quantum Concepts (cont.)
• Another limitation of the Bohr model was that it
assumed we could know both the position and
momentum of an electron exactly.
• Werner Heisenberg observed that there is a
fundamental limit to how well one can know both
the position and momentum of a particle.
h
DxDp 
4
Uncertainty in
position
Uncertainty in
momentum
where…
Dp  D(mv)  mDv

Quantum Concepts (cont.)
Example:
What is the uncertainty in velocity for an electron in a
1 Å radius orbital in which the positional uncertainty
is 1% of the radius. (1 Å = 10-10 m)
Δx = (1 Å)(0.01) = 1 x 10-12 m
34
6.626x10
J.s

h
23
Dp 


5.27x10
kg.m /s
12
4 Dx
4 1x10 m
Dp 5.27x1023 kg.m /s
7m
Dv 


5.7x10
s
m
9.11x1031 kg
HUGE!
Quantum Concepts (cont.)
Example (you’re quantum as well):
What is the uncertainty in position for a 80 kg student
walking across campus at 1.3 m/s with an uncertainty
in velocity of 1%.
Δp = mΔv = (80kg)(0.013 m/s) = 1.04 kg.m/s
34
6.626x10
J.s

h
Dx 

 5.07x1035 m
4Dp 4 1.04kg.m /s
Very small……we know where you are.
