Quantum Theory and Atomic Structure

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Chapter 7
Quantum Theory and Atomic Structure
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Quantum Theory and Atomic Structure
7.1 The Nature of Light
7.2 Atomic Spectra
7.3 The Wave-Particle Duality of Matter and
Energy
7.4 The Quantum-Mechanical Model of the Atom
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Electromagnetic Radiation

• electromagnetic radiation aka radiant energy aka
  light
• Energy propagated by means of electric and
  magnetic fields that alternately increase and
  decrease in intensity as they travel through
  space
• includes visible, IR, UV, gamma rays, X-rays,
  etc forms of light.
• Our eyes only equipped to see visible light,
  all other types are just different colors
• white light mixture of all colors in rainbow

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Electromagnetic Radiation

• electromagnetic radiation travels as a wave and
  as such is characterized by wavelength and
  frequency
• wavelength (?)- length of a complete cycle
  (distance between peaks)(nm)
• frequency (?)- of cycles past a point in 1
  second (s-1)
• for electromagnetic radiation
• ?? c, or ?? 3.00 x 108 m/s
• Speed of light (c) - in vacuum is constant for
  all types.
• 3.00 x 108 m/s

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Figure 7.1
Frequency and Wavelength
c l n
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Figure 7.2
Amplitude (Intensity) of a Wave
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Electromagnetic Radiation

• colors of light depend on wavelength or
  frequency,
• the different kinds of light can be arranged by
  wavelength into an electromagnetic spectrum
• different colors of light have different
  energies, violet more than red
• visible light 400nm 750 nm

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Regions of the Electromagnetic Spectrum
Figure 7.3
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Sample Problem 7.1
Interconverting Wavelength and Frequency
SOLUTION
PLAN
Use c ln
1.00x10-10m
wavelength in units given
3x108m/s
3x1018s-1
n
1.00x10-10m
325x10-2m
3x108m/s
wavelength in m
n
9.23x107s-1
325x10-2m
473x10-9m
frequency (s-1 or Hz)
6.34x1014s-1
n
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Distinction Between Energy and Matter

• Energy
• Speed of light only constant in a vacuum, light
  slows down when it travels through other media.
• When light changes speed, it also changes
  direction.
• Refraction immediate change in direction of
  light when it passes from one media to another.
• Angle of refraction depends on material and
  wavelength of the light
• Dispersion separation of light mixtures into
  component colors caused by wavelength dependence
  of refraction.
• Matter
• When a particle impacts another fluid, its
  change in speed is more gradual and its change
  in direction only influenced by gravity.

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Distinction Between Energy and Matter

• Energy
• Diffraction - When waves pass through a hole in a
  barrier, they bend and form semicircular waves on
  the other side of the opening.
• When waves pass through 2 adjacent holes, they
  form an interference pattern on the other side.
  Some waves add up, and some cancel out.
• Matter
• When particles pass through a hole in a barrier,
  ones that pass are unaffected by its presence

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Figure 7.4
Different behaviors of waves and particles.
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The interference pattern caused by light passing
through two adjacent slits.
Figure 7.5
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Particle Nature of Light

• 3 observations noted in the early 20th century
  cannot be explained based on the wave nature of
  light
• 1) Blackbody radiation light emitted as a
  normally dark object gets hot
• 2) photoelectric effect electrons ejected by
  metals as they absorb light
• 3) atomic spectra arent continuous spectra

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Figure 7.6
DE h n
Blackbody Radiation
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Particle Nature of Light

• To explain blackbody radiation mathematically
  Planck had to quantize energy.
• In effect Planck said atoms could only have
  certain energies and to change from one to
  another it had to absorb or emit packets of
  energy. These packets can only have certain
  energies, not a continuous amount, and are
  therefore quantized.

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Particle Nature of Light

• Photoelectric effect - Electrons ejected from
  metals when they absorb light
• Minimum frequency required,intensity of light had
  no effect - shouldnt an electron eject when it
  absorbs enough energy of any color?
• has no lag time shouldnt it take time for the
  electrons to absorb enough energy?
• Einstein proposed light is particulate photons
• Photons must have enough energy to eject e-,
  depends on n not intensity
• An e- breaks free when a single photon of enough
  energy is absorbed.
• Dim light has fewer photons, not lower energy.
• Energy of a photon of light
• Ephoton hn
• where
• Ephoton is the energy of each photon (packet) of
  light.
• h is Plancks constant 6.626x10-34 Js
• n is the frequency of light in Hz (s-1).

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Figure 7.7
Demonstration of the photoelectric effect
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Sample Problem 7.2
Calculating the Energy of Radiation from Its
Wavelength
PLAN
After converting cm to m, we can use the energy
equation, E hn combined with n c/l to find
the energy.
SOLUTION
E hc/l
6.626X10-34Js
3x108m/s
x
E
1.66x10-23J
1.20cm
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Line Spectra

• line spectra - light emitted or absorbed by
  gaseous atoms of an element consist of only very
  specific colors, not a rainbow.
• Each element has a unique line spectra
• An equation that predicts these colors was
  developed, called the Rydberg equation.

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Figure 7.8
The line spectra of several elements
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R
Rydberg equation
-
R is the Rydberg constant 1.096776 m-1
Three series of spectral lines of atomic hydrogen
Figure 7.9
for the visible series, n1 2 and n2 3, 4, 5,
...
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Line Spectra

• Bohr used line spectra to explain electron
  structure in atoms.
• Each element has only certain allowable energy
  levels in the form of circular electron orbits of
  different distances from the nucleus
• different distances are different potential
  energies
• An electron can move from one orbit to another
  only by absorbing/emitting a photon of exactly
  the energy difference between the two states.
• light is emitted/absorbed as electrons in atoms
  move from one potential energy to another.
• Ground State The lowest energy, the one closest
  to the nucleus
• Excited State any other energy level

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Figure 7.10
Quantum staircase
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Figure 7.11
The Bohr explanation of the three series of
spectral lines.
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Figure B7.1
Flame tests
strontium 38Sr
copper 29Cu
Figure B7.2
Emission and absorption spectra of sodium atoms.
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Wave-Particle Duality

• Energy, originally thought to be a wave, also
  shows some particle like behavior.
• Maybe matter behaves somewhat like a wave?
• De Broglie proposed that electrons could only
  have certain energy levels because they had
  wavelike properties.
• ? h/mu
• These wavelike properties of matter only
  observable in very small particles.
• Later shown to be true by electron diffraction
  patterns

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Figure 7.13
Wave motion in restricted systems
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Figure 7.14
Comparing the diffraction patterns of x-rays and
electrons
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CLASSICAL THEORY
Figure 7.15
Matter particulate, massive
Energy continuous, wavelike
Summary of the major observations and theories
leading from classical theory to quantum theory.
Observation
Theory
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Figure 7.15 continued
Observation
Theory
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The Heisenberg Uncertainty Principle

• Because an electron behaves like a particle and a
  wave, it is impossible to know both its position
  and momentum.
• D x m D u h/4?
• Result circular electron orbits impossible

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Quantum-Mechanical Model

• In 1926 Erwin Schrödinger derived an equation
  using wave mechanics that predicts the
  probabilities of an electrons location.
• Result not orbits, but 3-D areas of probability
  called orbitals.
• Orbitals are drawn representing 90 probability
  of finding an electron.

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The Schrödinger Equation
HY EY
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Figure 7.16
Electron probability in the ground-state H atom
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Quantum-Mechanical Model

• the quantum mechanical model uses three numbers
  to describe an orbital
• The principal quantum number, n
• can have any integer value 1,2,3,...
• the higher the n, the farther on average the e is
  from the nucleus and the higher the energy level
• The angular momentum quantum number, l
• can have values from 0 to n-1
• this defines the shape of the orbital (spherical,
  dumbell, etc)
• l's of 0,1,2, 3 are called s,p,d,f
• The magnetic quantum number, ml
• has values of l to -l including zero
• this defines the orientation in space of the
  orbital (x, y, z)

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Quantum-Mechanical Model

• energy level(shell) - all orbitals with the same
  n
• sublevel(subshell) - orbitals with the same n and
  l values
• Basically each energy level is broken into
  sublevels, which are made up of orbitals, each
  orbital can hold 2 electrons
• 1. any level, n, has exactly n subshells in it
• 2. each sublevel has a specific number of
  orbitals 2l1, s has 1, p has 3, etc
• 3. the total number of orbitals in a shell is n2
• The maximum electrons per level is 2n2

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Table 7.2 The Hierarchy of Quantum Numbers for
Atomic Orbitals
Name, Symbol (Property)
Allowed Values
Quantum Numbers
Principal, n (size, energy)
Positive integer (1, 2, 3, ...)
1
2
3
Angular momentum, l (shape)
0 to n-1
0
0
1
0
0
Magnetic, ml (orientation)
-l,,0,,l
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Orbital Notation

• Energy levels were assigned a principal quantum
  number, n, which could equal 1, 2, 3 This
  quantum number, n designates the energy and size
  of the region in space the electrons might be
  found.
• In this model each energy level or shell is
  broken into sublevels or subshells. The sublevel
  designations tell the shape of the region in
  space the electrons might be found.

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Orbital Notation

• These sublevels are designated s, p, d, f, etc
• the 1st energy level has one type of sublevel, s
• From 0 to n-1 only value is 0(s)
• the 2nd level has 2 types of sublevels, s and p
• From 0 to n-1 value is 0(s), and 1(p)
• the 3rd has 3, s, p, and d
• sublevels differ in the shape of their
  probability distribution
• each type of sublevel has one or more orbitals
• the orbitals of a sublevel differ by their
  orientation in space

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Orbital Notation

• s sublevel has 1 orbital, spherical so only 1
  orientation
• From l to l is 0
• p sublevel has 3 orbitals, 1 type with 3
  orientations px,py,pz
• From l to l, is 1, 0, 1
• d has 5 orbitals
• f has 7 orbitals
• g has 2 more, etc
• s sublevels are spherical
• p sublevels are shaped like dumbbells
• be familiar with the shapes

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Sample Problem 7.5
Determining Quantum Numbers for an Energy Level
PLAN
Follow the rules for allowable quantum numbers
found in the text.
l values can be integers from 0 to n-1 ml can
be integers from -l through 0 to l.
SOLUTION
For n 3, l 0, 1, 2
For l 0 ml 0
For l 1 ml -1, 0, or 1
For l 2 ml -2, -1, 0, 1, or 2
There are 9 ml values and therefore 9 orbitals
with n 3.
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Sample Problem 7.6
Determining Sublevel Names and Orbital Quantum
Numbers
(a) n 3, l 2
(b) n 2, l 0
(c) n 5, l 1
(d) n 4, l 3
PLAN
Combine the n value and l designation to name the
sublevel. Knowing l, we can find ml and the
number of orbitals.
SOLUTION
n
l
sublevel name
possible ml values
of orbitals
(a)
2
3d
-2, -1, 0, 1, 2
3
5
(b)
2
0
2s
0
1
(c)
5
1
5p
-1, 0, 1
3
(d)
4
3
4f
-3, -2, -1, 0, 1, 2, 3
7
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Figure 7.17
1s
2s
3s
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Figure 7.18
The 2p orbitals
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Figure 7.19
The 3d orbitals
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Figure 7.19 continued
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Figure 7.20
One of the seven possible 4f orbitals