CHEMISTRY
The Molecular Nature of Matter and Change 3rd Edition
Chapter 7 Lecture Notes:
Quantum Theory and Atomic
Structure
Chem 150 - Ken Marr - Winter 2006
Welcome to Chem 150!!
Below are a few due dates and other useful
information
1.
2.
3.
Do the Prelab Preparation for tomorrow's lab activity,
Atomic Spectrum of Hydrogen. Turn in the prelab
questions at the start of lab and complete in your lab
notebook the following sections of the report for this lab
exercise: Title, Introduction, Materials/Methods and Data
Tables.
The completed report for lab 1 is due on Monday January
9, 2005.
Due Friday January 6, 2006: ALE 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
Section 7.1
The Nature of Light (Electromagnetic Radiation)
•
•
Light consists of waves with electrical and
magnetic components
Waves have a specific Frequency and Wavelength
» Symbol and Units of Each?
c = n l = 3.00 X 10
8
m/s
C = 2.99792 X 108 m/s
Figure 7.1
Frequency
and
Wavelength
c=ln
Amplitude (Intensity)
of a Wave
Figure 7.2
Figure 7.3
Regions of the Electromagnetic Spectrum
Increasing Wavelength
Increasing Frequency, S-1
Practice Problems:
Interconverting Frequency and Wavelength
1.
2.
Calculate the frequency in hertz of green light with
a wavelength of 550 nm.
Calculate the broadcast wavelength in meters of an
FM radio station that broadcasts at 104.3 MHz.
Answers:
1. 5.4 x 1014 hertz
2. 2.876 m
Wave-Particle Duality of Light: in some cases light
behaves as waves, in other times as photons (particles)
1.
Evidence for Wave Behavior of light
»
»
2.
Refraction of light
Diffraction of light
Evidence for Particle Behavior of light
»
»
Blackbody Radiation
Photoelectric Effect
Fig. 7.4
Different Behaviors of Waves and Particles
Refraction of
Light
Speed changes
when pebble
enters H2O
Diffraction of
Light
Evidence for
the wave
nature of light
Diffraction of
Light—
Diffraction of Light—
Figure 7.6
Blackbody Radiation
n
EE ==hhn
photon
Evidence for the
Blackbody
Particle Radiation
Behavior
of Light
~ 1000 K  emits a soft red glow
~ 1500 K  brighter & more
orange
7-10
~ 2000 K  brighter & white in
color
Blackbody Radiation
Evidence for Particle Behavior of Light
Only specific colors of light are emitted when
blackbodies (heated solids) are heated
1.
~ 1000 K  emits a soft red glow
~ 1500 K  brighter and more orange
~ 2000 K  brighter and white in color
Max Planck’s (1900): Atoms can only absorb or give off
specific packets or quanta of light energy.
2.
•
These packet of energy are called photons.
Particle Nature of Light
Max Planck (1900)
• EMR is emitted as weightless packets of energy called
photons
• Each photon has its own energy and frequency, n
Ephoton = hn
h = Planck’s constant = 6.626 x 10-34 J.s
Photoelectric Effect:
Evidence for Particle
Behavior of Light
• Light of a certain minimum
frequency (color) is needed to
dislodge electrons from a metal
plate.
•Wave theory predicts a wave
of a minimum amplitude.
Einstein’s Explanation of the
Photoelectric Effect (1905)
1.
2.
Light intensity is due to the number of
photons striking the metal per second, not
the amplitude
A photon of some minimum energy must be
absorbed by the metal
E photon= hn
Relationship between
Energy of Light and Wavelength
1.
Derive an equation that relates E and l from the
following equations: c = l n and E = hn
2.
Use this equation to Answer the following questions.....
a. Microwave ovens emit light of l = 3.00 mm. Calculate the
energy of each photon emitted from a microwave oven.
Ans. 6.63 x 10-23 J/photon
b. How many photons of light are needed for a microwave oven
to raise the temperature of a cup of water (236 g) from 20.0
oC to 100.0 oC?
Ans. 1.19 x 1027 photons
Section 7.2
Atomic Spectra
Continuous Spectrum
1.
•
Sunlight or from object heated to a very high
temperature (e.g. light filament)
Atomic Spectrum
2.
•
•
Also called line, bright line or emission spectrum
Due to an atom’s electron(s) excited by electricity or
heat falling from a higher to a lower energy level—
more about this later!!
Figure 7.8
The line spectra
of several
elements
Continuous
Spectrum
Line Spectra
7-13
Rydberg Equation
Predicts the Hydrogen Spectrum
Rydberg Equation
• Empirically derived to fit hydrogen’s atomic spectrum
• Predicts l’s of invisible line spectra
e.g. Hydrogen’s Ultraviolet line spectrum (nL = 1)
1   1  1 
R

l n n 
2
2
fL
iH
R = 1.096776 x 107 m-1
n = 1, 2, 3, 4, …
Using the Rydberg Equation
Practice Exercise:
Calculate the wavelength in nm and determine
the color of the line in the visible spectrum of
hydrogen for which nL = 2 and nH = 3.
Ans. 656.4 nm Color????
1st The Good News….
Niels Bohr Planetary model of the atom explains
Hydrogen's Spectrum (1913)
An atom’s energy is quantized because electrons can
only move in fixed orbits (energy levels) around the
nucleus
Orbits are quantized
i.e. Each orbit can only have a certain radius
An electron can only move to another energy level
(orbit) when the energy absorbed or emitted equals
the difference in energy between the two energy levels
1.
2.
3.
•
Line spectra result as electrons emit light as they fall from a
higher to lower energy level
Bohr’s Explanation of the Three series of
Spectral Lines of the Hydrogen Spectrum
Figure 7.10
Quantum staircase
7-15
Animation of Bohr’s Planetary Model
1.
2.
Animation (Flash)
Animation (QuickTime)
Bohr’s Equation
Derived from the Ideas of Planck, Einstein & Classical Physics
1.
Eelectron = ELower - EHigher
2.
Eelectron = -2.18 x 10-19 J (1/n2Lower - 1/n2higher)
But…… E = hc/ l, substitution yields…
3.
1/l = 1.10 x107 m-1 (1/n2Lower - 1/n2higher)
•
•
or Eelectron = Efinal - Einitial
Bohr’s Constant is within 0.05 % of the Rydberg Constant
Equation provides a theoretical explanation of Hydrogen’s
Atomic Spectrum
Bohr’s Equation Accurately Predicts
the Ionization Energy of Hydrogen
Use Bohr’s equation to calculate the ionization
energy for
a.) one hydrogen atom
b.) one mole of hydrogen atoms
1/l = 1.10 x10
7
m-1 (1/n2Lower
Energy + H (g)
- 1/n2higher)
+
 H (g) + e
Answers: a.) 2.18 x 10-18J/atom ; b.) 1.31 x 103 kJ/mole
Now the Bad News…
Bohr’s Model is Incorrect!!
Closer inspection of spectral lines shows shows that
they are not all single lines
1.
•
Bohr’s model doesn’t account for the extra lines
Only works for atoms or ions with one electron
2.
•
Bohr’s model doesn’t account for presence of electronelectron repulsions and electron-nucleus attractions in
atoms with more than one electron.
Electrons do not orbit around the nucleus!!!
3.
•
•
A new model is needed
Would you believe that electrons behave as waves and as
particles????
Section 7.3
The Wave-Particle Duality of Matter
Electron Diffraction: Evidence that electrons behave as waves!
Davisson & Germer
(1927)
Electrons are
diffracted by
solids just like Xrays!
Hence, electrons
behave as waves!
X-Ray tube
Source of
electrons
Aluminum
X-Ray diffraction
pattern of Aluminum
Aluminum
Electron diffraction
pattern of Aluminum
Figure 7.14
Comparing the diffraction patterns of x-rays and electrons
7-23
Wave- Particle Duality of Matter and Energy
1.
Matter behaves as if it moves like a wave!!
2.
Only small, fast objects (e.g. e-, p+ , n0) have a measurable l
me = 9.11x10-31 kg; mp = mn = 1.67x10-27 kg
3.
Louis DeBroglie (1924) combined
2
E = mc
and
l matter = h/mu
4.
E = hc / l
to yield
m = mass; u = velocity
DeBroglie l too small to measure for heavy, slow objects
Locating an Electron....an uncertain affair!!
1. Orbital
•
Region in space where an electron wave is most likely to be
found
2. Exact location of an electron can’t be determined
3. Can only determine the probability of finding an
electron....why?
•
•
Electrons behave as waves!!
In order to “see” the position of an electron we must probe it
with radiation which changes its position and/or velocity
Heisenberg Uncertainty Principle
1.
Both the velocity and position of an electron
cannot be determined simultaneously
h
X  m  u 
4
x = uncertainty in position; u = uncertainty in velocity; m = mass of
object
2.
Can only determine the probability of finding an electron
»
3.
orbitals are regions in space where an electron will
most likely be found
See sample problem 7.4
Sample Problem 7.4
Applying the Uncertainty Principle
PROBLEM: An electron moving near an atomic nucleus has a speed 6x106 ± 1%
m/s. What is the uncertainty in its position (x)?
PLAN:
The uncertainty (x) is given as ±1%(0.01) of 6x106m/s. Once we
calculate this, plug it into the uncertainty equation.
SOLUTION:
u = (0.01)(6x106m/s) = 6x4m/s
x * mu ?
h
4
x ?
6.626x10-34kg*m2/s
4 (9.11x10-31kg)(6x104m/s)
7-27
= 10-9m
Section 7.4
Quantum Mechanical Model of the Atom:
Electron Waves in Atoms
Electrons are standing waves
1.
•
•
Peaks and troughs only move up and down
Similar to how guitar strings move
Orbitals
2.
•
•
Are areas in space where electron waves are most
likely to be found
Orbitals are made of electron waves
Quantum Mechanics and Atomic Orbitals
•
•
Erin Schrodinger (1926) developed a mathematical
equation called a wave function to describe the
energy of electrons
The square of the wave function gives the
probability of finding an electron at any point in
space, thus producing a map of an orbital
Atomic Orbital
An area in space where an electron wave is most likely to
be found outside of the nucleus
Quantum Numbers and Atomic Orbitals
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
7-30
Orbitals are Identified by 3 Quantum Numbers
Principle Quantum Number, n (n = 1,2,3…)
1.
•
•
Determines the orbital’s size and energy (I.e. which energy
level the electron occupies)
Relates to the average distance of the e- to the nucleus
Secondary Quantum Number, l
2.
•
•
•
Determines the orbital’s shape or sublevel : s, p, d or f
l = 0 to n-1
Orbitals with the same values for n and l are called sublevels
Orbitals are Identified by 3 Quantum Numbers
Magnetic Quantum Number, ml
3.
•
•
•
Determines the orbital’s orientation in space
ml = -l, …, 0 , …+l
ml represents the orbital within the sublevel.
S - sublevel has 1 orbital
p - sublevel has 3 orbitals
d - sublevel has 5 orbitals
F - sublevel has 7 orbitals
Table 7.2 The Hierarchy of Quantum Numbers for Atomic Orbitals
Name, Symbol
(Property)
Allowed Values
Quantum Numbers
n = Principal
Principal,
n
Positive integer
quantum
Number
(size,
energy)
(1, 2, 3, ...)
(size and
energy
of
orbital)
1
l =Angular
Angular
momentum,
l
0 to n-1
momentum
Q.N.
(shape)
(shape of orbital)
0
0
0
0
ml Magnetic,
= magneticmQ.N.
l
-l,…,0,…,+l
(orientation
of
(orientation)
orbital)
2
3
1
0
2
0
-1 0 +1
-1 0 +1
-2
7-31
1
-1
0
+1 +2
Relationship between Angular momentum Q.N. , l,
and sublevel names: s, p, d and f
Value of l
0
1
2
3
Sublevels only used
4
by electrons in the
5
excited state
Sublevel
s
p
d
f
g
h
f
Summary of Relationships
Between n, l and ml
ENERGY LEVEL
n
1
2
3
4
Sublevels
l
(0 to n-1)
Orbitals
ml
(-l to +l)
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 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.
7-32
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, we can find ml and the number of orbitals.
SOLUTION:
n
l
(a)
3
2
3d
-2, -1, 0, 1, 2
3
(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
7-33
sublevel name possible ml values # of orbitals
Practice Makes Perfect?
1.
What is the subshell (e.g. 1s, 2s, 2p, etc.) corresponding
to the following values for n and l?
a.
b.
c.
d.
e.
n = 2, l = 1
n = 4, l = 0
n = 3, l = 2
n = 5, l = 3
n = 3, l =3
Practice Makes Perfect?
2.
Which of the following sets of quantum numbers are
not possible?
a.
b.
c.
d.
e.
n=
n=
n=
n=
n=
2,
2,
2,
3,
0,
l=
l=
l=
l=
l=
1,
2,
1,
2,
0,
ml =0
ml =1
m l = -2
m l = -2
ml =0
The Relationship between the 4 Quantum
Numbers, Energy Levels, Sublevels and Orbitals
See figure 6.15, page 239 in Brady
(Transp.)
Practice Makes Perfect?
1.
2.
What subshells are found in the 4th shell?
Which subshell is higher in energy?
a. 3s or 3p
b. 4p or 4d
c. 3p or 4p
Shapes of
orbitals
As the value
for n
increases,
the electron
is more
likely to be
found
further
from the
nucleus
1s orbital
2s orbital
3s orbital
Fig. 7.18
Shapes of the three orbitals in the 2p sublevel: 2px 2py 2pz
Note that the three orbitals are mutually perpendicular to each other
(fig. D), thus contributing to an atoms overall spherical shape
An accurate
representation of
the 2pz orbital
Stylized shape
of 2pz used in
most texts
Fig. 7.19 c-g
Shapes of the five orbitals in the 3d sublevel
Note that the relative positions of the five orbitals in the 3d sublevel
contribute to the overall spherical shape of an atom (fig. H)
Fig. 7.20
One of the
possible seven
orbitals of the 4f
sublevel
Since only the s, p,
and d sublevels are
commonly involved
with bonding, we
will not be
concerned with the
shapes of the
orbitals of the fsublevel