Ch 7

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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 (light) - Wave like
Wavelength, l, the distance from one crest to the
next in the wave. Measured in units of distance.
Frequency, n, the number of complete cycles per
sec., cps, Hz
Speed of Light, C, same for all EM
radiation.3.00 x 1010 cm/sec in vacuum
c l n
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Amplitude (Intensity) of a Wave
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Regions of the Electromagnetic Spectrum
c l n
n c / l
3 m
300 m
1000 kHz
100 MHz
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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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Different behaviors of waves and particles.
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The diffraction pattern caused by light passing
through two adjacent slits.
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Electromagnetic Radiation - Particle like
The view that EM was wavelike could not explain
certain phenomena like 1) Blackbody radiation
- when objects are heated, they give off shorter
and more intense radiation as the temperature
increases, e.g. dull red hot, to hotter orange,
to white hot....... But wavelike properties would
predict hotter temps would continue to give more
and more of shorter and shorter wavelengths. But
instead a bell shape curve of intensities is
obtained with the peak at the IR- Vis part of the
spectrum.
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Electromagnetic Radiation
E hn
E hc/l
Blackbody Radiation
Plancks constanth 6.626 x 10-34 J-s
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Electromagnetic Radiation - Particle like
The view that EM was wavelike could not explain
certain phenomena like 2) Photoelectron Effect
- light shinning on certain metal plates caused a
flow of electrons. However the the light had a
minimum frequency to cause the effect, i.e. not
any color would work. And although bright light
caused more electron flow than weak light,
electron flow started immediately with both
strong or weak light.
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Demonstration of the photoelectric effect
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Electromagnetic Radiation - Particle like
The better explanation for these experiments was
that EM consisted of packets of energy called
photons (particle-like) that had wave-like
properties as well. And that atoms could have
only certain quantities of energy, E nhn ,
where n is a positive integer, 1, 2, 3, etc.
This means energy is quantized. Ephoton hn
?Eatom
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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
6.626X10-34J-s
3x108m/s
x
E
1.66x10-23J
1.20cm
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Electromagnetic Radiation - Particle like
The view that EM was wavelike could not explain
certain phenomena like 3) Atomic Spectra -
Electrical discharges in tube of gaseous elements
produces light (EM).But not all wavelengths of
light were produced but just a few certain
wavelengths (or frequencies). And different
elements had different wavelengths associated
with them. Not just in the Visible but also IR
and UV regions.
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The line spectra of several elements
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Three Series of Spectral Lines of Atomic Hydrogen
Looking for an equation that would predict the
wavelength seen in H spectrum

R
Rydberg equation
-
for the visible series, n1 2 and n2 3, 4, 5,
...
R is the Rydberg constant 1.096776 x 107 m-1
But WHY does this equation work?
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Bohr Model of the Hydrogen Atom
Assumed the H atom has only certain allowable
energy levels for the electron orbits. (quantized
because it made the equations work)) When the
electron moves from one orbit to another, it has
to absorb or emit a photon whose energy equals
the difference in energy between the two
orbits. By assuming the electron traveled in
circular orbits, the energy level for each orbits
was E -2.18 x 10-18 J / n2 where n 1, 2,
3,....? Note because of negative sign, lowest
energy (most stable) when n 1 and highest
energy is E 0 when n ?. So energy released
when the electron moves from one n level to
another is ?E hn hc/l -2.18 x 10-18 J ( 1 /
n2final - 1 / n2initial )
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Quantum staircase
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The Bohr explanation of the three series of
spectral lines.
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But why must the electrons energy be quantized?
If EM can have particle-like properties in
addition to being wave-like, what if the electron
particles have wave-like properties? Quantization
is a natural consequence of having wave-like
properties.
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Wave motion in restricted systems
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de Broglie Wavelength - giving particles
wave-like properties
but E muc
E hc/l
so l hc/ E
The de Broglie Wavelengths of Several Objects
Substance
Mass (g)
Speed, u, (m/s)
l (m)
slow electron
9x10-28
1.0
7x10-4
fast electron
9x10-28
5.9x106
1x10-10
alpha particle
6.6x10-24
1.5x107
7x10-15
one-gram mass
1.0
0.01
7x10-29
baseball
142
25.0
2x10-34
Earth
6.0x1027
3.0x104
4x10-63
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Sample Problem 7.3
Calculating the de Broglie Wavelength of an
Electron
PLAN
Knowing the mass and the speed of the electron
allows to use the equation l h/mu to find the
wavelength.
SOLUTION
6.626x10-34kg-m2/s
l
7.27x10-10m
9.11x10-31kg
x
1.00x106m/s
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CLASSICAL THEORY
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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Observation
Theory
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The Heisenberg Uncertainty Principle
Heisenberg Uncertainty Principle expresses a
limitation on accuracy of simultaneous
measurement of observables such as the position
and the momentum of a particle.
.
D x m D u
?
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Sample Problem 7.4
Applying the Uncertainty Principle
PLAN
The uncertainty (D u) is given as 1 (0.01) of 6
x 106 m/s. Once we calculate this, plug it into
the uncertainty equation.
SOLUTION
D u (0.01) (6 x 106 m/s) 6 x 104 m/s
.
6.626 x 10-34 kg-m2/s
D x ?
? 10-9m
4p (9.11 x 10-31 kg) (6 x104 m/s)
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The Schrödinger Equation - Quantum Numbers and
Atomic Orbitals
A complicated equation with multiple solutions
which describes the probability of locating an
electron at the various allowed energy levels.
Solutions involve three interdependent variables
to describe an electron orbital.
i.e., an atomic orbital is specified by three
quantum numbers.
n the principal quantum number - a positive
integer
l the angular momentum quantum number - an
integer from 0 to n-1
ml the magnetic moment quantum number - an
integer from -l to l ms the spin quantum
number, 1/2 or - 1/2
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Electron probability in the ground-state H atom
Orbital showing 90 of electron probability
n 1 l 0 m 0
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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
sublevel namesl 0, called s l 1, p
l 2, d l 3, f
0
0
Magnetic, ml (orientation)
-l,,0,,l
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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
l
sublevel name
possible ml values
of orbitals
n
(a)
3d
-2, -1, 0, 1, 2
3 2
5
(b)
2s
0
2 0
1
(c)
5p
-1, 0, 1
5 1
3
(d)
4f
-3, -2, -1, 0, 1, 2, 3
4 3
7
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s orbitals
1s
2s
3s
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p orbitals - three of them
Combination
The 2p orbitals n 2, l 1
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d orbitals - five of them
The 3d orbitals n 3 l 2
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d orbitals - five of them
Combination
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f orbitals - seven of them
One of the seven possible 4f orbitals
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End of Chapter 7