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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The Wave Nature of Light
Wave properties are described by two
interdependent variables
Frequency n (nu) the number of cycles the
wave undergoes per second (units of s-1 or hertz
(Hz)) (cycles/s)
Wavelength l (lambda) the distance between
any point on a wave and a corresponding point on
the next wave (the distance the wave travels
during one cycle) (units of meters (m),
nanometers (10-9 m), picometers (10-12 m) or
angstroms (Å, 10-10 m) per cycle) (m/cycle)
Speed of a wave cycles/s x m/cycle m/s
c speed of light in a vacuum nl 3.00 x 108
m/s
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Figure 7.1
Frequency and Wavelength
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Amplitude (intensity) of a Wave
(a measure of the strength of the waves electric
and magnetic fields)
Figure 7.2
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Regions of the Electromagnetic Spectrum
Figure 7.3
Light waves all travel at the same speed through
a vacuum but differ in frequency and, therefore,
in wavelength.
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Sample Problem 7.1
Interconverting Wavelength and Frequency
PLAN
Use c ln
x
1.00 x 10-10 m
3.00 x 108 m/s
SOLUTION
n
3 x 1018 s-1
1.00 x 10-10 m
x
325 x 10-2 m
3.00 x 108 m/s
n
9.23 x 107 s-1
325 x 10-2 m
473 x 10-9 m
x
n
6.34 x 1014 s-1
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Distinguishing Between a Wave and a Particle
Refraction the change in the speed of a wave
when it passes from one transparent medium to
another
Diffraction the bending of a wave when it
strikes the edge of an object, as when it passes
through a slit an interference pattern develops
if the wave passes through two adjacent slits
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Different behaviors of waves and particles
Figure 7.4
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The diffraction pattern caused by light passing
through two adjacent slits
Figure 7.5
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Blackbody Radiation
Changes in the intensity and wavelength of
emitted light when an object is heated at
different temperatures
Figure 7.6
Plancks equation
E nhn
E energy of radiation h Plancks constant n
frequency n positive integer (a quantum
number)
h 6.626 x 10-34 J.s
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The Notion of Quantized Energy
If an atom can emit only certain quantities of
energy, then the atom can have only certain
quantities of energy. Thus, the energy of an
atom is quantized.
Each energy packet is called a quantum and
has energy equal to hn.
An atom changes its energy state by absorbing
or emitting one or more quanta of energy.
DEatom Eemitted (or absorbed) radiation
Dnhn
DE hn (Dn 1)energy change between adjacent
energy states
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Demonstration of the photoelectric effect
Wave model could not explain the (a) presence of
a threshold frequency, and (b) the absence of a
time lag.
Led to Einsteins photon theory of light
Ephoton hn DEatom
Figure 7.7
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Calculating the Energy of Radiation from its
Wavelength
Sample Problem 7.2
PLAN
After converting cm to m, we use the energy
equation, E hn combined with n c/l to find
the energy.
SOLUTION
E hc/l
(6.626 x 10-34J.s)
(3.00 x 108 m/s)
x
E
1.66 x 10-23 J
1.20 cm
x
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Atomic Spectra
Line spectra of several elements
Figure 7.8
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R
-
Rydberg equation
R is the Rydberg constant 1.096776 x 107 m-1
Three series of spectral lines of atomic hydrogen
for the visible series, n1 2 and n2 3, 4, 5,
...
Figure 7.9
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The Bohr Model of the Hydrogen Atom
Postulates
1. The H atom has only certain allowable energy
levels
2. The atom does not radiate energy while in one
of its stationary states
3. The atom changes to another stationary
state (i.e., the electron moves to another
orbit) only by absorbing or emitting a photon
whose energy equals the difference in energy
between the two states
The quantum number is associated with the radius
of an electron orbit the lower the n value, the
smaller the radius of the orbit and the lower the
energy level.
ground state and excited state
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Quantum staircase
Figure 7.10
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Limitations of the Bohr Model
Can only predict spectral lines for the hydrogen
atom (a one electron model)
For systems having gt1 electron, there are
additional nucleus-electron attractions and
electron-electron repulsions
Electrons do not travel in fixed orbits
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The Bohr explanation of the three series of
spectral lines for atomic hydrogen
Figure 7.11
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Figure B7.1
Flame tests
strontium 38Sr
copper 29Cu
Emission and absorption spectra of sodium atoms
Figure B7.2
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Fireworks emissions from compounds containing
specific elements
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Figure B7.3
The main components of a typical spectrophotometer
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The Absorption Spectrum of Chlorophyll a
Absorbs red and blue wavelengths but no green
and yellow wavelengths leaf appears green.
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The Wave-Particle Duality of Matter and Energy
De Broglie If energy is particle-like, perhaps
matter is wavelike!
Electrons have wavelike motion and are restricted
to orbits of fixed radius explains why they will
have only certain possible frequencies and
energies.
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Wave motion in restricted systems
Figure 7.13
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The de Broglie Wavelength
Table 7.1 The de Broglie Wavelengths of
Several Objects
Substance
Mass (g)
Speed (m/s)
l (m)
slow electron
9 x 10-28
1.0
7 x 10-4
fast electron
9 x 10-28
5.9 x 106
1 x 10-10
alpha particle
6.6 x 10-24
1.5 x 107
7 x 10-15
one-gram mass
1.0
0.01
7 x 10-29
baseball
142
25.0
2 x 10-34
Earth
6.0 x 1027
3.0 x 104
4 x 10-63
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Calculating the de Broglie Wavelength of an
Electron
Sample Problem 7.3
PLAN
Knowing the mass and the speed of the electron
allows use of the equation, l h/mu, to find the
wavelength.
SOLUTION
6.626 x 10-34 kg.m2/s
7.27 x 10-10 m
l
9.11 x 10-31 kg
x
1.00 x 106 m/s
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Comparing the diffraction patterns of x-rays and
electrons
Figure 7.14
Electrons - particles with mass and charge -
create diffraction patterns in a manner similar
to electromagnetic waves!
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Figure 7.15
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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Figure 7.15 (continued)
Observation
Theory
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The Heisenberg Uncertainty Principle
It is impossible to know simultaneously the exact
position and momentum of a particle
Dx x m Du
Dx the uncertainty in position
Du the uncertainty in speed
A smaller Dx dictates a larger Du, and vice versa.
Implication cannot assign fixed paths for
electrons can know the probability of finding an
electron in a given region of space
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Sample Problem 7.4
Applying the Uncertainty Principle
PLAN
The uncertainty in the speed (Du) is given as 1
(0.01) of 6 x 106 m/s. Once we calculate this
value, the uncertainty equation is used to
calculate Dx.
SOLUTION
Du (0.01)(6 x 106 m/s) 6 x 104 m/s (the
uncertainty in speed)
6.626 x 10-34 kg.m2/s
1 x 10-9 m
Dx
4p (9.11 x 10-31 kg)(6 x 104 m/s)
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The Schrödinger Equation
HY EY
Each solution to this equation is associated with
a given wave function, also called an atomic
orbital
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Electron probability in the ground-state hydrogen
atom
Figure 7.16
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Quantum Numbers and Atomic Orbitals
An atomic orbital is specified by three quantum
numbers.
n the principal quantum number - a positive
integer (energy level)
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
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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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Anthony S. Serianni
Sample Problem 7.5
Determining Quantum Numbers for an Energy Level
PLAN
Follow the rules for allowable quantum numbers.
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 (s sublevel)
For l 1 ml -1, 0, or 1 (p sublevel)
For l 2 ml -2, -1, 0, 1, or 2 (d sublevel)
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, find ml and the number of
orbitals.
SOLUTION
n
l
sublevel name
possible ml values
no. 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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S orbitals
1s
3s
2s
spherical nodes
Figure 7.17
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2p orbitals
Figure 7.18
nodal planes
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3d orbitals
Figure 7.19
perpendicular nodal planes
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Figure 7.19 (continued)
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One of the seven possible 4f orbitals
Figure 7.20
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The energy levels in the hydrogen atom
The energy level depends only on the n value of
the orbital
Figure 7.21