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Chapter 12
Quantum Mechanics and Atomic Theory
5.1 Electromagnetic Radiation
5.2 The Nature of Matter
5.3 The Atomic Spectrum of Hydrogen
5.4 The Bohr Model
5.5 The Quantum Mechanical Description of the Atom
5.6 The Particle in a Box (skip)
5.7 The Wave Equation for the Hydrogen Atom
5.8 The Physical Meaning of a Wave Function
5.9 The Characteristics of Hydrogen Orbitals
5.10 Electron Spin and the Pauli Principle
5.11 Polyelectronic Atoms
5.12 The History of the Periodic Table
5.13 The Aufbau Principle and the Periodic Table
5.14 Further Development of the Polyelectronic Model
5.15 Periodic Trends in Atomic Properties
5.16 The Properties of Alkali Metals
Waves and Light
• Electromagnetic Radiation
– Energy travels through space as electromagnetic
radiation
– Examples: visible light, microwave radiation, radio
waves, X-rays, infra-red radiation, UV radiation
– Waves (characterized by λ, υ, amp, c)
– Travels at the speed of light (3x108 m/sec)
Electromagnetic Radiation
Light consists of waves of
oscillating electric (E) and
magnetic fields (H) that are
perpendicular to one another
and to the direction of
propagation of the light.
Electromagnetic Spectrum
400 nm (violet)
The visible spectrum
700 nm (red)
Important Equations
(that apply to EM radiation)
• c =  c=lambda nu)
c = 3 x 108 meters / second
 = wavelength [m, nm (10-9m), Å (10-10m)]
 = frequency (Hz = s-1)
[frequency and wavelength vary inversely]
• E = h (Energy = h nu)
h = Planck’s constant
(h = 6.62 x 10-34 J s = 6.62 x 10-34 kg m2 s-1)
[the energy of a wave increases with its frequency]
AM Radio Waves
• KJR Seattle, Channel 95 (AM)
950 kHz = 950,000 second-1
c = λν =>
λ = c/ν
λ = 3.0x108 m s-1/ 9.5 x 105 s-1 = 316 m
When the frequency (ν) of EM is 950 kHz, the
wavelength (λ) is 316 meters (about 1/5 mile).
FM Radio Waves
• WABE Atlanta: FM 90.1 MHz
c = λν =>
λ = c/ν
= 3.0x108 m s-1/ 90.1x106 s-1 = 3.33 m
FM radio waves are higher frequency, higher energy
and longer wavelength, than AM radio waves.
c = λν
E = hν
Problem
The X-ray generator in Loren Williams’ lab produces xradiation with wavelength of 1.54 Å (0.1 nm = 1 Å). What
is the frequency of the X-rays? What is the energy of each
X-ray photon?
X-rays
X-rays were discovered in 1895 by German scientist Wilhelm
Conrad Roentgen. He received a Nobel Prize in 1901. A week
after his discovery, Roentgen took an x-ray image of his wife’s
hand, visualizing the bones of her fingers and her wedding ring
- the world’s first x-ray image.
Roentgen ‘temporarily’ used the term “x”-ray to indicate the
unknown nature of this radiation. Max von Laue (Nobel Prize
1914) showed that x-rays are electromagnetic radiation, just
like visible light, but with higher frequency (and higher energy)
and smaller wavelength.
Within a few months of Roentgen’s discovery, doctors in New
York used x-rays to image broken bones.
c = λν
E = hν
Problem
The laser in an audio compact disc (CD) player produces
light with a wavelength of 780 nm. What is the frequency
of the light emitted from the laser?
Problem
The brilliant red color seen in fireworks displays is due
to 4.62 x 1014 s-1 strontium emission. Calculate the
wavelength of the light emitted.
Planck, Einstein, and Bohr
• 1901 Max Planck found that light (or energy) is quantized.
• In the microscopic world energy can be gained or lost only
in integer multiples of hν.
ΔE = n(hν)
n is an integer (1,2,3,…)
• h is Planck’s constant (h = 6.628X10-34 J s)
J: Joule, a unit of energy.
• Each energy unit of size hν is called a packet or quantum
• 1905 Einstein suggested that electromagnetic
radiation can be viewed as a “stream of
particles” called photons
Ephoton = hυ = h(c/λ)
• About the same time, Einstein derived his
famous equation
E = mc2
• photons have mass.
5/24/2017
Zumdahl Chapter 12
14
Dual nature of light
E
m 2
c
c 
E  h  h 
 
Electrons and Atoms: The Atomic Spectrum of
Hydrogen (H.):
Put energy into a hydrogen atom (“excite it”),
what comes out?
ie., at what energies does excited Hydrogen emit light?
(1) A hydrogen atom consists of one
electron and one proton.
(2) A hydrogen atom has discrete
energy levels described by the primary
quantum number n (1,2,3…) which
gives the energy levels En (E1, E2, E3…)
E 4  E1
n=4
n=3
n=2
(3) Light is emitted from a hydrogen
atom when an electron changes from a
higher energy state (Ebig) to a lower
energy state (Esmall)
n=1
(4) The wavelengths emitted tell you
ΔE2-1, ΔE3-1, ΔE2-4… (where ΔE2-1 =
E2- E1).
E 3  E1



E 2  E1
(5) The observed emission spectrum of
a hydrogen atom (at specific λ) tells
you that the energy of a hydrogen atom
is quantized.
E 4  E1  E 4 1 
E 3  E1  E 31 

E 2  E1  E 21 


E 3  E 2  E 32 
hc
 656
E 4  E 2  E 4 2 
E 5  E 2  E 52 
hc
 486
hc
 434
hc
 97
hc
103
hc
121
The Bohr model of the hydrogen atom
1. The hydrogen atom is a small, positively charged nucleus
surrounded by a electron that travels in circular orbits. The atom
is analogous to the solar system, but with electrostatic forces
providing attraction, rather than gravity.
2. Unlike planets, electrons can occupy only certain orbits. Each
orbit represents a discrete energy state. In the Bohr model, the
energy of a hydrogen atom is quantized.
3. Light is emitted by a hydrogen atom when an electron falls
from a higher energy orbit to a lower energy orbit.
4. Since each orbit is of a definite fixed energy, the transition of
an electron from the higher energy orbit to the lower energy orbit
causes the emission of energy of a specific amount or size (a
quantum). The light emitted is at a specific frequency and
wavelength.
Electronic transitions in the Bohr model for
the hydrogen atom
E 3  E1  E 31 

hc
102
Bohr Model of the Atom (quantized energy)
Bohr calculated the angular momentum, radius and
energy of the electrons traveling in descrete orbits.
n2
rn  a 0  radius of each orbital
Z
a 0 called the bohr radius, a constant
Angular Momentum  mevr
h
 n
n  1,2,3,.....
2
n  orbitals, excited states
n  1,2,3,...
n  1 called ground state
Z is the postive charge on the nucleus
(1 of H, 2 for He, etc.)
2
Z
18
E n   2 (2.18x10 J)
n
Calculated ΔE’s match observed λ(emission)
Modern Quantum Mechanics (1)
• Bohr recognized that his model violates principles of
classical mechanics, which predict that electrons in
orbit would fall towards and collide with the nucleus.
Stable Bohr atoms are not possible.
• Modern quantum mechanics, with orbitals rather than
orbits, provides the only reasonable explanation for
the observed properties of the atoms
Modern Quantum Mechanics (2)
• Orbital Defn: Orbitals are the “quantum” states that
are available to electron. An orbital can be full (2 e-),
half full (1e-), or empty. An orbital is a wave
function, characterized by quantum numbers n
(energy), l (shape), and m (direction).
• An orbital is used to calculate the probability of
finding a electron at some location (Ψ2) – giving a
three-dimensional probability graph of an electron
position.
Orbitals
n=1
n=2
(like
Standing Waves)
n=3
5/24/2017Analogy: An electron in an orbital can be imagined to be a standing wave around
24the
nucleus. Electrons are not in the planet-like orbits.
An orbital is a wavefunction (Ψ),
described by three quantum numbers [ψ (n, l, ml)]
1. n = principal quantum number
ψ (n, l, ml)
n = 1, 2, 3, …
n is related to the energy of the orbital
2. l = angular (azimuthal) quantum number
l = 0, 1, …. (n-1)
ψ (n, l, ml)
l gives the shape of the orbital
l = 0 is called an s orbital (these are spherical)
l = 1 is called a p orbital (these are orthogonal rabbit ears)
l = 2 is called a d orbital (these have strange shapes)
l = 3 is called an f orbital (these have stranger shapes)
l = 4 is called a g orbital (don’t even think about it)
An orbital is a wavefunction (Ψ),
described by three quantum numbers [ψ (n, l, ml)]
(continued)
3.
ml = magnetic quantum number Ψ (n, l, ml)
ml = -l, … , 0, ….+l
ml relates to the orientation of the orbital
Quantum NumbersΨ (n, l, ml)
Each orbital is specified by three quantum numbers (n, l, ml).
Each electron is specified by four quantum numbers (n, l, ml, ms).
ms = electron spin quantum number, indicated the electron’s
spin, which can be up or down.

ms = +1/2, -1/2 denoted by ,
•
Ψ (n, l , ml ) specifies an orbital.
•
Ψ (n, l , ml , ms) specifies an electron in an orbital.
Electrons and Orbitals
• Each orbital is specified by 3 quantum
numbers: (n,l,ml)
• Every orbital can hold two electrons
• Each electron is specified by 4 quantum
numbers: (n,l,ml,ms)
Summary
Ψ (n, l, ml)
– n: the primary quantum
number, controls size and energy,
and the possibilities for l.
n
l
orbital
designation
ml
# of
orbitals
1
0
0
1
0
1
2
0
1
2
3
1s
2s
2p
3s
3p
3d
4s
4p
4d
4f
0
0
-1, 0, +1
0
-1, 0, +1
-2, -1, 0, +1, +2
0
-1, 0, +1
-2, -1, 0, +1, +2
-3, -2, -1, 0, +1, +2, +3
1
1
3
1
3
5
1
3
5
7
2
– l: the angular quantum number,
controls orbital shape, and can
also effect energy. l controls the
possibilities for ml.
–ml: the orientation quantum
number
3
4
The First Three Orbitals Energy Levels (n=1,2 or 3)
Ψ (n, l, ml)
Ψ (1, 0, 0)
n
l
orbital
designation
ml
# of
orbitals
1
0
0
1
0
1
2
0
1
2
3
1s
2s
2p
3s
3p
3d
4s
4p
4d
4f
0
0
-1, 0, +1
0
-1, 0, +1
-2, -1, 0, +1, +2
0
-1, 0, +1
-2, -1, 0, +1, +2
-3, -2, -1, 0, +1, +2, +3
1
1
3
1
3
5
1
3
5
7
2
Ψ (3, 0, 0)
Ψ(3, 1, -1)
3
Ψ (3, 1, 0)
Ψ (3, 1, 1)
Ψ(3, 2,-2 )
Ψ (3, 2,-1 )
Ψ (3, 1, 0)
Ψ(3, 2, 1 )
Ψ (3, 1, 2)
4
Ψ(2, 0, 0)
Ψ(2, 1, -1)
Ψ(2, 1, 0)
Ψ(2, 1, +1)
Ψ (n, 0, 0)
Ψ (1, 0, 0)
Ψ (2, 0, 0)
l=0
s orbitals
Ψ (3, 0, 0)
Degeneracy
n2 = number of degenerate
orbitals with the same energy
(this applies to hydrogen only).
Ψ (2, 1, ml)
l=1
p orbitals
Ψ (2, 1, -1)
Ψ (2, 1, 0)
Ψ (2, 1, +1)
Ψ (3, 2, ml)
l=2
d orbitals
Ψ (3, 2, -2)
Ψ (3, 2, -1)
Ψ (3, 2, 1)
Ψ (3, 2, 0)
Ψ (3, 2, 2)
Ψ (4, 3, ml)
l=3
f orbitals
Ψ (4, 3, -3)
Ψ (4, 3, 0)
Ψ (4, 3, -2)
Ψ (4, 3, 1)
Ψ (4, 3, -1)
Ψ (4, 3, 2)
Ψ (4, 3, 3)
34
Orbital Energy Levels of Atoms
with many Electrons:
The degeneracy is lost.
From the graph:
1. Are 2s and 2p degenerate (i.e., do they
have the same energy)?
2. Which is lower energy? 4s or 3d?
3. Which is lower energy? 6s or 4f?
4. Which is lower energy? 3d or 4p?
Energy Levels: Why is the 2s orbital higher
in energy than a 2p orbital?
Penetration: Electrons in
the 2s orbital are closer to
the nucleus (on average)
than electrons in a 2p
orbital.
So 2s electrons shield the
2p electrons from the
nucleus. This raises the
energy of the 2p electrons
(Coulomb’s law).
Many Electron Atoms (2)
Aufbau Principle
• The Aufbau principle assumes a process in
which an atom is "built up" by progressively
adding electrons and protons/neutrons. As
electrons are added, they enter the lowest
energy available orbital.
• Electrons fill orbitals of lowest available energy
before filling higher states. 1s fills before 2s,
which fills before 2p, which fills before 3s, which
fills before 3p.
Many Electron Atoms (3)
Filling Orbitals
with Electrons
1s (holds 2e-) then
2s (2e-) then
2p (6e-) then
3s (2e-) then
3p (6e-) then
4s (2e-) then
3d (10e-) then
4p (6e-) then
5s (2e-) …
Many Electron Atoms (4)
Pauli Exclusion Principle
No 2 electrons in an atom can have the same set
of quantum numbers
n, l, ml, ms
ms = electron spin quantum number
ms = +1/2, -1/2 denoted by


Hund’s Rules
Many Electron Atoms (5)
–
Every degenerate orbital is singly occupied (contains one
electron) before any orbital is doubly occupied (Electrons
distribute as much as possible within degenerate orbitals This is called the
"bus seat rule” It is analogous to the behavior of passengers who occupy
all double seats singly before occupying them doubly.
–
Multiple electrons in singly occupied orbitals have the same
spin.
Periodic Table
The Quantum Mechanical Periodic Table
Orbitals and the Periodic Table
PRS Question
The principle quantum number for the
outermost 2 electrons in Sr would be:
1) 3
2) 4
3) 5
4) 6
5) none of the above
PRS Question
The Angular quantum number (l) for the
outermost electron on K is:
1)
2)
3)
4)
5)
0
1
2
3
none of the above
PRS Question
An electron in which subshell will on
average be closer to the nucleus?
1)
2)
3)
4)
5)
3s
3p
3d
4d
none, they are all the same
distance from the nucleus
PRS Question
Which atom has a smaller 3s orbital?
1)
2)
3)
4)
5)
An atom with more protons
An atom with fewer protons
An atom with more neutrons
An atom with fewer neutrons
The size of the 3s orbital is the
same for all atoms.
Hund’s Rules
–
Every degenerate orbital is singly occupied (contains one
electron) before any orbital is doubly occupied (Electrons
distribute as much as possible within degenerate orbitals This is called
the "bus seat rule” It is analogous to the behavior of passengers who
occupy all double seats singly before occupying them doubly.
–
Multiple electrons in singly occupied orbitals have the
same spin.
5/24/2017
Zumdahl Chapter 12
50
“Aufbau” from Hydrogen to Boron
1s
1H:
2He:
1s1
1s2
22s1
Li:
1s
3
22s2
Be:
1s
4
5B:
1s22s22px1
2s
2px
2py
2pz
“Aufbau” from Carbon to Neon
1s
6C:
1s22s22px12py1
22s22p 12p 12p 1
N:
1s
7
x
y
z
8O:
1s22s22px22py12pz1
9F:
1s22s22px22py22pz1
10Ne:
1s22s22px22py22pz2
2s
2px
2py 2pz
• Valence Electrons
–
–
–
–
can become directly involved in chemical bonding
occupy the outermost (highest energy) shell of an atom
are beyond the immediately preceding noble-gas configuration
among the s-block and p-block elements, include electrons in s
and p subshells only
– among d-block and f-block elements, include electrons in s
orbitals plus electrons in unfilled d and f subshells
Why is Silver Ion Ag+
• Why not Ag0 or Ag2+ or Ag3+?
• What is the electron configuration for silver
(Ag0)?
• What happens to the configuration if we
remove one electron from Ag0?
Can you identify this element?
• 1s22s22p65p1
• Why is the electron configuration written
as such? (why not 1s22s22p63s1)
• Is 1s22s22p6 a different element?
PRS Question
What is the maximum number of
electrons that can occupy the orbitals
with principal quantum number = 4?
1)
2)
3)
4)
5)
2
8
18
32
none of the above
PRS question
Which of the following have 4 valance
electrons?
1)
2)
3)
4)
5)
Al
Si
P
As
Be
PRS question
What is the maximum number of
electrons that can occupy the orbitals
with principal quantum number = 3?
1)
2)
3)
4)
5)
2
8
18
32
none of the above
PRS question
Which of the following is the electron
configuration of a ground state Se atom?
1)
2)
3)
4)
5)
[Ar]4s24d104p4
[Ar]3s23d103p3
[Ar]4s23d104p3
[Ar]4s23d104p4
none of the above
PRS question
What is the electron configuration
of a phosphorous atom?
1)
2)
3)
4)
5)
1s22s23s22p63p2
1s22s22p63s23p3
1s22s22p63s23p2
1s22s22p63p4
1s22s22p63s4
PRS question
How many unpaired electrons are on
a phosphorous atom?
1)
2)
3)
4)
5)
2
3
4
5
6
PRS question
How many valence electrons are there
in a Cl atom?
1)
2)
3)
4)
5)
4
5
6
7
8
Problem:
Write the valance-electron configuration and state the
number of valence electrons in each of the following
atoms and ions: (a) Y, (b) Lu, (c) Mg2+
(a) Y (Yttrium): atomic number Z = 39
[Kr] 5s2 4d 1
3 valence electrons
(b) Lu (Lutetium): Z = 71
[Xe] 6s 2 4f 14 5d 1
3 valence electrons
Note filled 4f sub shell
(c) Mg2+ (Magnesium (II) ion): Z = 12
This is the 2+ ion, thus 10 electrons
[Ne] configuration or 1s2 2s2 2p6
0 valence electrons
Periodic Trends in Atomic Properties
•
•
•
•
Ionization Energy
Electron Affinity
Atomic Radius
Electronegativity
Ionization energy of an atom is the minimum amount
of energy necessary to detach an electron form an atom
that is in its ground state.
X → X+ + e - ΔE = IE1
X+ → X2+ + e - ΔE = IE2
The first ionization energy
values decrease in going
down a group
Electron Affinity
X + e─ → X─
ΔE = electron attachment energy
EA tends to parallel IE, but shifted one
atomic number lower
e.g. Halogens have a much higher EA
than noble gases
Electron Affinity
PRS
Which of the following has the greatest
magnitude?
1) The first ionization energy of strontium (Sr)
2) The first electron affinity of fluorine (F)
3) The second ionization energy of magnesium
(Mg)
4) The first ionization energy of oxygen (O)
5) The third ionization energy of magnesium
(Mg)
PRS
Which of the following has the greatest
magnitude?
1) The first ionization energy of strontium
2) The first electron affinity of fluorine
3) The second ionization energy of
magnesium
4) The first ionization energy of oxygen
5) The third ionization energy of
magnesium
Electron Affinity vs Electronegativity
1. ELECTRON AFFINITY is the ENERGY RELEASED when an atom in the gas
phase adds an electron to form a negative ion: E + e(-1) ---> E(-1). This quantity can be
measured experimentally.
Unfortunately, even though most electron affinities tend to be EXOTHERMIC,
They are given as positive quantities, which is the opposite the normal sign convention.
2. ELECTRONEGATIVITY is an empirical scale of the ability of an atom IN A
COVALENTLY BONDED MOLECULE to attract electrons from other atoms in the
molecule.
Electronegativity is related to but is no the same as electron affinity.
Atomic Radius
The radius of an atom (r) is defined
as half the distance between the nuclei in a
molecule consisting of identical atoms.
Atomic radii (in
picometers) for
selected atoms.
Nuclear Charge
What does increasing
the nuclear charge do to
the orbital energy?
A. More Positive
B. Closer to Zero
C. More Negative
What does this mean?
Shielding
Sizes of Atoms and Ions
Atomic size generally increases moving down a group
Among s-block and p-block elements, atomic size generally
decreases moving from left to right
The Trend in Atomic Size
Ions
Observations:
How does this
trend differ
from atoms?
Explain.
Think-Pair-Share
PRS Question
What is the correct order of decreasing size of the
following ions?
A.
P3- > Cl- > K+ > Ca2+
B.
Ca2+ > K+ > Cl- > P3-
C.
K+ > Cl- > Ca2+ > P3-
D.
K+ > Cl- > P3- > Ca2+
The “Trends”
Iron Compounds
Fe
4s
3d
3d
3d
3d
3d
4p
4p
4p
4s
3d
3d
3d
3d
3d
4p
4p
4p
Fe3+
PRS Question
47. Select the diamagnetic io n.
2+
A.
Cu
2+
B.
Ni
3+
C.
Cr
3+
D.
Sc
2+
E.
Cr
Niels Bohr
(1885-1962 )
Highlights
– Worked with J.J. Thomson (1911) who
discovered the electron in 1896
– 1913 developed a quantum model for
the hydrogen atom
– During the Nazi occupation of Denmark
in World War II, escaped England and
America
– Associated with the Atomic Energy
Project.
– Open Letter to the United Nations in
1950 peaceful application of atomic
physics
Moments in a Life
– Nobel Prize in Physics 1922
Quantum Mechanics and Atomic
Structure (Part 1)
The Uncertainty Principle. In 1927, Werner Heisenberg
established that it is impossible to know (or measure),
with arbitrary precision, both the position and the
momentum of an object
h
x  mv 
4
imprecision of
position
imprecision of momentum
The better position is
known, the less well
known is momentum
(and vice versa).
Heisenberg’s Uncertainty Principle:
You cannot measure/observe something without changing
that which you are observing/measuring.
Determine the position of an electron with the
precision on the order of the size of an atom.
what is the uncertainty of the velocity,
∆v?
Determine the velocity of an apple to be zero, but
with uncertainty of 10-5 m/s
what is the uncertainty of the position,
∆x?
PRS Quiz
The Heisenberg uncertainty principle:
1)
2)
3)
4)
5)
places limits on the accuracy of
measuring both position and motion
is most important for microscopic
objects
makes the idea of “orbits” for
electrons meaningless
none of the above
a, b and c
The Heisenberg uncertainty principle:
1)
2)
3)
4)
5)
places limits on the accuracy of
measuring both position and motion
is most important for microscopic
objects
makes the idea of “orbits” for
electrons meaningless
none of the above
a, b and c
Quantum Mechanics and Atomic
Structure (Part 2)
DeBroglie: Even baseballs are waves. Particles move
with linear momentum (p) and have wave like properties
and a wavelength
(λ = h/p = h/mev)
For macroscopic objects, we can ignore the wave
properties since m is large.
Quantum Mechanics and Atomic
Structure (Part 3)
Schrödinger Equation
Ĥ = E
Schrödinger Equation
Ĥ = E
Results in many solutions, each solution consists
of a wave function,  (n, l, ml) that is a function
of quantum numbers.
(x, y, z) is a complex function defined over
three dimensional space. Its complex square is
a three dimensional probability function, i.e 2 =
the probability that an electron is in a certain
region of space. 2 defines the shapes of
orbitals.
The wave function provides a complete
description of how electrons behave. Each n, l,
ml describes one atomic orbital.
Schrodinger Eqn
Solutions