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Chapter 7
Quantum Theory and Atomic Structure
7-1
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
7-2
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
7-3
Figure 7.1
Frequency
and
Wavelength
7-4
Amplitude (intensity)
of a Wave
(a measure of the
strength of the wave’s
electric and magnetic
fields)
Figure 7.2
7-5
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.
7-6
Sample Problem 7.1
Interconverting Wavelength and Frequency
PROBLEM: A dental hygienist uses x-rays (l = 1.00 Å) to take a series of dental
radiographs while the patient listens to a radio station (l = 325 cm)
and looks out the window at the blue sky (l = 473 nm). What is the
frequency (in s-1) of the electromagnetic radiation from each source?
Assume that the radiation travels at the speed of light, 3.00 x 108
m/s.
PLAN:
Use c = ln
-10
1.00 Å x 10 m
1Å
SOLUTION:
-2
325 cm x 10 m
1 cm
10-9 m
473 nm x
1 nm
= 1.00 x 10-10 m
3.00 x 108 m/s
n=
= 3 x 1018 s-1
-10
1.00 x 10 m
= 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
n=
7-7
3.00 x 108 m/s
473 x 10-9 m
= 6.34 x 1014 s-1
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
7-8
Different
behaviors
of waves
and
particles
Figure 7.4
7-9
The diffraction pattern caused by light passing
through two adjacent slits
Figure 7.5
7-10
Blackbody
Radiation
Changes in the intensity and
wavelength of emitted light when
an object is heated at different
temperatures
Planck’s equation
E = nhn
E = energy of radiation
h = Planck’s constant
n = frequency
n = positive integer (a
quantum number)
h = 6.626 x 10-34 J.s
7-11
Figure 7.6
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
7-12
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 Einstein’s photon theory of
light:
Ephoton = hn = DEatom
Figure 7.7
7-13
Sample Problem 7.2
Calculating the Energy of Radiation from its
Wavelength
PROBLEM: A cook uses a microwave oven to heat a meal. The wavelength of
the radiation is 1.20 cm. What is the energy of one photon of this
microwave radiation?
PLAN: After converting cm to m, we use the energy equation, E = hn
combined with n = c/l to find the energy.
SOLUTION:
E=
7-14
E = hc/l
(6.626 x 10-34J.s) x (3.00 x 108 m/s)
10 1.20 cm x 2
m
cm
= 1.66 x 10-23 J
Atomic Spectra
Line spectra of
several elements
Figure 7.8
7-15
Rydberg equation:
1
l
=
R
1
n12
1
n22
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
7-16
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
7-17
Quantum
staircase
Figure 7.10
7-18
Limitations of the Bohr Model
Can only predict spectral lines for the hydrogen atom
(a one electron model)
For systems having >1 electron, there are additional
nucleus-electron attractions and electron-electron
repulsions
Electrons do not travel in fixed orbits
7-19
The Bohr explanation of the three series of spectral
lines for atomic hydrogen
Figure 7.11
7-20
Figure B7.1
Flame tests
strontium
38Sr
Figure B7.2
7-21
copper 29Cu
Emission and absorption spectra of sodium atoms
Fireworks emissions
from compounds
containing specific
elements
7-22
Figure B7.3
The main components of a typical spectrophotometer
Lenses/slits/collimaters
narrow and align beam.
Source produces radiation
in region of interest. Must
be stable and reproducible.
In most cases, the source
emits many wavelengths.
7-23
Sample in compartment
absorbs characteristic
amount of each incoming
wavelength.
Monochromator
(wavelength selector)
disperses incoming
radiation into continuum
of component
wavelengths that are
scanned or individually
selected.
Computer converts
signal into displayed
data.
Detector converts
transmitted radiation
into amplified electrical
signal.
The Absorption
Spectrum of
Chlorophyll a
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
Absorbs red and
blue wavelengths
but no green and
yellow wavelengths;
leaf appears green.
7-24
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.
7-25
Wave motion in
restricted systems
Figure 7.13
7-26
The de Broglie Wavelength
l =
Table 7.1
The de Broglie Wavelengths of Several Objects
Substance
slow electron
Mass (g)
9 x 10-28
Speed (m/s)
l (m)
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
3.0 x 104
4 x 10-63
Earth
7-27
h /mu
6.0 x 1027
Sample Problem 7.3
Calculating the de Broglie Wavelength of an
Electron
PROBLEM: Find the deBroglie wavelength of an electron with a speed of 1.00 x
106m/s (electron mass = 9.11 x 10-31 kg; h = 6.626 x 10-34 kg.m2/s).
PLAN:
Knowing the mass and the speed of the electron allows use of the
equation, l = h/mu, to find the wavelength.
SOLUTION:
l=
7-28
6.626 x 10-34 kg.m2/s
= 7.27 x 10-10
9.11 x 10-31 kg x 1.00 x 106 m/s m
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!
7-29
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
Since matter is discontinuous and particulate,
perhaps energy is discontinuous and particulate.
Observation
blackbody radiation
7-30
Theory
Planck:
photoelectric effect
Energy is quantized; only certain values
are allowed
Einstein: Light has particulate behavior (photons)
atomic line spectra
Bohr:
Energy of atoms is quantized; photon
emitted when electron changes orbit.
Figure 7.15 (continued)
Since energy is wavelike, perhaps matter is wavelike.
Observation
Davisson/Germer:
electron diffraction
by metal crystal
Theory
deBroglie: All matter travels in waves; energy of
atom is quantized due to wave motion of
electrons
Since matter has mass, perhaps energy has mass
Observation
Compton: photon
wavelength increases
(momentum decreases)
after colliding with
electron
Theory
Einstein/deBroglie: Mass and energy are
equivalent; particles have wavelength and
photons have momentum.
QUANTUM THEORY
Energy same as Matter:
particulate, massive, wavelike
7-31
The Heisenberg Uncertainty Principle
It is impossible to know simultaneously the exact
position and momentum of a particle
Dx x m Du ≥
h
4p
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
7-32
Sample Problem 7.4
Applying the Uncertainty Principle
PROBLEM: An electron moving near an atomic nucleus has a speed 6 x 106 ±
1% m/s. What is the uncertainty in its position (Dx)?
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)
Dx x mDu ≥
h
4p
Dx ≥
7-33
6.626 x 10-34 kg.m2/s
4p (9.11 x 10-31 kg)(6 x 104 m/s)
≥ 1 x 10-9 m
The Schrödinger Equation
HY = EY
wave function
d2Y
dx2
+
d2Y
dy2
mass of electron
+
d2Y
dz2
how y changes in space
potential energy at x,y,z
8p2mQ
+
(E-V(x,y,z)Y(x,y,z) = 0
2
h
total quantized energy of
the atomic system
Each solution to this equation is associated with a given
wave function, also called an atomic orbital
7-34
Electron probability in the ground-state hydrogen atom
Figure 7.16
7-35
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
7-36
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
Angular
momentum, l
(shape)
0 to n-1
0
0
0
0
Magnetic, ml
(orientation)
-l,…,0,…,+l
2
3
1
0
2
0
-1 0 +1
-1 0 +1
-2
7-37
1
-1
0
+1 +2
Anthony S.
Serianni:
Sample Problem 7.5
Determining Quantum Numbers for an Energy Level
PROBLEM: What values of the angular momentum (l) and magnetic (ml)
quantum numbers are allowed for a principal quantum number (n) of
3? How many orbitals are allowed for n = 3?
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
7-38
Sample Problem 7.6
Determining Sublevel Names and Orbital Quantum
Numbers
PROBLEM: Give the name, magnetic quantum numbers, and number of orbitals
for each sublevel with the following 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
(a)
3
2
3d
-2, -1, 0, 1, 2
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
7-39
sublevel name possible ml values
no. orbitals
S orbitals
1s
2s
spherical
nodes
Figure 7.17
7-40
3s
2p orbitals
Figure 7.18
nodal planes
7-41
3d orbitals
Figure 7.19
perpendicular nodal planes
7-42
Figure 7.19 (continued)
7-43
One of the seven
possible 4f orbitals
Figure 7.20
7-44
The energy levels
in the hydrogen atom
The energy level
depends only on the
n value of the orbital
Figure 7.21
7-45