Chapter 7: ELECTRONS IN ATOMS AND PERIODIC PROPERTIES

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Chapter 7 ELECTRONS IN ATOMS AND PERIODIC
PROPERTIES

• Problems 7.1-7.16, 7.18-7.27, 7.31-7.56,
  7.61-7.107, 7.109-7.119, 7.124-7.125, 7.128-7.134

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Electromagnetic Radiation

• Electromagnetic (EM) Spectrum a continuum of the
  different forms of electromagnetic radiation or
  radiant energy

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Radar Radio Waves
Weather systems
A galaxy imaged in the visible spectrum.
Radio telescopes. This is the Very Large Array
(VLA) in NM.
The same galaxy imaged in the radio spectrum at
the VLA.
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Thermal Imaging Detecting IR radiation
Much of a persons 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 aroundthis 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?
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IR Photography
Image obtained with IR film, which is really
film that is activated by light at 700-900 nm.
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Electromagnetic Radiation

• Electromagnetic radiation or light is a form of
  energy.

• Has both electric (E) and magnetic (H) components.

• Characterized by
• Wavelength (?)
• Amplitude (A)

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Electromagnetic Radiation (cont.)

• Wavelength (l) The distance between two
  consecutive peaks in the wave.

Increasing Wavelength
l1 gt l2 gt l3
Unit length (m)
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Electromagnetic Radiation (cont.)

• Frequency (n) The number of waves (or cycles)
  that pass a given point in space per second.

Decreasing Frequency
n1 lt n2 lt n3
Units 1/time (1/sec) or, Hertz (Hz)
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Electromagnetic Radiation (cont.)

• The product of wavelength (l) and frequency (n)
  is a constant.

(?)(?) c
Speed of light
c 3 x 108 m/s
c is a constant, independent of ?
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Properties of Waves

• Wavelength (?) is the distance from peak-to-peak
  in a wave.
• Frequency (?) 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
• ? ? c
• c speed of light 3 x 108 m/s

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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.
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Behavior of Waves

• Waves refract, diffract and interact
• Refraction The bending of light as it passes
  from one medium to another of different density

Light is also bent by liquids, causing the straw
to appear disconnected.
Refraction throught a prism separates white light
into its separate components (light of different
wavelengths) without changing the light itself.
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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?
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Behavior of Waves

• Waves interact by adding together or cancelling
  each other out

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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

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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 wasnt 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).

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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

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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.

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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
frequency
Energy Change
n 1, 2, 3 (integers)
h Plancks constant 6.626 x 10-34 Js
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Light as Energy (cont.)

• In general the relationship between frequency and
  photon energy is

h 6.636 x 10-34 Js c 2.9979 x 108 m/s 1 Hz
1 s-1
Ephoton h?
Example What is the energy of a 500 nm
photon?
? c/? (3x108 m/s)/(5.0 x 10-7 m)
? 6 x 1014 1/s
E h? (6.626 x 10-34 Js)(6 x 1014 1/s) 4 x
10-19 J
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Energy Quantization
The student can stop only at certain points on a
flight of stairs. Her distance from the ground is
quantized.
The student can stop at any point on the ramp.
Her distance from the ground changes continuously.
Similarly, atomic energy levels are like
stepsthe energies available to an atom do not
form a continuum, they are quantized.
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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!!

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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.

When the light emitted from excited atoms is
passed through a prism, we see discrete bands of
color at specific wavelengths.
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Photon Emission
An excited atom relaxes from high E to low E by
emitting a photon. We can determine the energy
difference (?E) between levels by measuring the
wavelength of the emitted photon.
Emission of photon
?E hc/? ? ? hc/?E If ? 410 nm, ?E
4.52 x 10-19 J
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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 (?0)
• When ?lt?0 ? no electrons are emitted
• When ?gt?0 ? electrons are emitted, more e
  emitted with greater intensity of light

? lt ?0
? gt ?0
Einstein applied Planck's quantum theory to
light light exists as a stream of "particles"
called photons.
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The Photoelectric Effect (cont)
Frequency determines whether e- are ejected, and
their KE (velocity).
Intensity determines the number of e- that are
ejectedbut they will all have the same velocity!!
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The Photoelectric Effect (cont.)
As frequency of incident light is increased,
kinetic energy of emitted e- increases linearly.

• hn0
• Workfunction energy needed to release e-

Light apparently behaves as a particle.
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The Photoelectric Effect (cont.)
For Na with F 4.4 x 10-19 J, what wavelength
corresponds to ?o?
0
h? F 4.4 x 10-19 J
hc/? 4.4 x 10-19 J
? 4.52 x 10-7 m 452 nm
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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.
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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.

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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
• Substituting, E hc/?
• Using Einsteins relationship, E mc2,

Ephoton h? c ?? ? ? c/?
 
 
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de Broglie Wavelength

• We can generalize this relationship to any
  velocity (v)
• What is the de Broglie wavelength of an electron
  traveling at the speed of light? (melectron
  9.31 x 10-31 kg)
• What is the de Broglie wavelength of a 80 kg
  student walking across campus at 3 m/s?

 
 
? (6.626 x 10-34 Js)/(9.31x10-31 kg)(3x108
m/s) 2.37 x 10-12 m
? (6.626 x 10-34 Js)/(80 kg)(3 m/s) 2.76 x
10-36 m
You are a wave, too, but you have a VERY small
wavelength!!
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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).

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The Bohr Model

• Balmer and Rydberg didnt 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 doesnt simply collapse. Why doesnt the
  electron just spiral into the nucleus?

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The Bohr Model of the atom
Principle Quantum number n An index of the
energy levels available to the electron.
656
486
434
410
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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.

Z atomic number (1 for H)
n integer (1, 2, .)
Ry x h -2.178 x 10-18 J
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The Bohr Model (cont.)
Energy levels get closer together as n
increases At n infinity, E 0 ? the electron
is free (not a part of the atom)
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The Bohr Model (cont.)
We can use the Bohr model to predict what DE is
for any two energy levels
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The Bohr Model (cont.)
Example At what wavelength will emission from n
4 to n 1 for the H atom be observed?
1
4
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The Bohr Model (cont.)
Example What is the longest wavelength of light
that will result in removal of the e- from H?
?
1
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Extension to Higher Z
The Bohr model can be extended to any single
electron system.must keep track of Z (atomic
number).
Z atomic number
n integer (1, 2, .)
Examples He (Z 2), Li2 (Z 3), etc.
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Extension to Higher Z (cont.)
Example At what wavelength will emission from n
4 to n 1 for the He atom be observed?
2
1
4
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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!
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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.

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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.

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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 half- integer
values of the wavelength correspond to
standing waves.
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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.

not allowed
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Quantum Concepts (cont.)

• Erwin Schrodinger developed a mathematical
  formalism that incorporates the wave nature of
  matter
• H, the Hamiltonian, is a special kind of
  function that gives the energy of a quantum
  state, which is described by the wavefunction, Y.

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Quantum Concepts (cont.)

• What is a wavefunction?

a probability amplitude

• Probability of finding a particle in space

The probability distribution for the hydrogen 1s
orbital in three-dimensional 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.
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Hydrogens 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)
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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.

where
Uncertainty in position
Uncertainty in momentum
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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)

?x (1 Å)(0.01) 1 x 10-12 m
HUGE!
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Quantum Concepts (cont.)

• Example (youre 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.

?p m?v (80kg)(0.013 m/s) 1.04 kg.m/s
Very smallwe know where you are.