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Chapter 7
The Quantum-Mechanical Model
of the Atom
The Electron (e-)
• Even though the electron is very small
(unmeasurable – LESS THAN A THRILLIONTH
OF A TRILLIONTH OF A GRAM!!)
they determine many of the chemical and
physical properties observed
• If we want to understand these properties, we
must understand the behaviors of the
electrons
Seeing
• Consider a baseball… you can see a baseball
because of the light bouncing off and entering
your eye
• An electron, on the other hand, would be
disturbed by the light itself… even if you could
see it with an instrument
What does this mean??
• Problem: we cannot observe an electron
without disturbing it
• It behaves differently when we observe it than
it would when we are not observing it (the
normal world)
Quantum-Mechanical Model
• A model that explains how electrons exist in
atoms and how those electrons determine the
chemical and physical properties of elements
• Properties that we expect from the table,
ionization, metals vs nonmetals, etc.
• This model explains why!! And how!!
Light
• Before we think about electrons, we must first
know a bit about light
• Light has many characteristics that are similar
to electrons – making it important to think
about first
Why is light relevant?
• Wave-particle duality
Light can be thought of as both a
wave AND as a particle – illustrates
properties of each
Radiation
• Light is electromagnetic radiation – a type of
energy embodies in oscillating electric and
magnetic fields
• An electric field is a region of space where an
electrically charged particle experiences a
force
• A magnetic field is a region of space where a
magnetic particle experiences a force
Magnetic vs Electric Field
How fast does light travel?
• In a vacuum, these waves move at a constant
speed of 3.00x108 m/s
– Fast enough to circle the Earth in one-seventh of a second
• This is the reason for the delay of seeing a
firework and hearing the blast
• SPEED OF LIGHT!!
• Sound only travels at
340 m/s
Electromagnetic Waves
• Electromagnetic Waves can be characterized
by its amplitude and wavelength (λ) -- lambda
Amplitude vs Wavelength
• The Amplitude is the vertical height (of depth
of a trough) AND determines its intensity or
brightness of the light
• The wavelength is the distance between
crests. It is measured in units of distance
(meters) – we use nanometers (nm) – 10-9
meters AND determines the color of the light
Energy of a Wave
• Wavelength and amplitude are both related to
the amount of energy carried by a wave
• Think of swimming in the crashing waves
– The higher the waves (the greater the amplitude)
the harder it is to swim
– The shorter wavelength (more closely spaced and
thus steeper) – getting pounded frequently – the
harder also
Different λ, Different Colors
• The different colors we see in the world
correlate to different colors
Different λ, Different Colors
Amplitude and λ are
independent of one another
• A wave can have a large amplitude and a long
wavelength, or a small amplitude and a short
wavelength
• The most energetic waves have large
amplitudes and short wavelengths
Frequency (v)
• Like all waves, light can also be characterized
by its frequency (v)
– The number of cycles that pass through a
stationary point in a given period of time
• Units are (cycles/s) of s-1
– An equivalent unit of frequency is the hertz (Hz),
defined as 1 cycle/sec
Relationships
• Frequency is directly proportional to the
speed at which the light is traveling
– The faster the wave, the more crests will pass
• It is inversely (indirectly) proportional to the
wavelength
– The farther apart the crests, the fewer will pass a
fixed location per unit time
• This inverse relationship allows us to derive
𝑐
𝑣=
λ
Where
“c” is the speed of light
“v” is the frequency
“λ” is the wavelength
**meaning (since you always know the speed of light) if you
are given one, you can find the other**
Practice
• Calculate the wavelength (in nm) of the light
emitting by a barcode scanner that has a
frequency of 4.62x1014 s-1
• What COLOR would this wavelength correlate
to?
Practice
• A laser used to dazzle the audience in a rock
concert emits light with a wavelength of 515
nm. Calculate the frequency of the light.
– What color is the laser??
Visible Light
• The sun or lights emit “white light”
– White light contains a variety of wavelengths
which determine what color we see
• When a substance absorbs some colors while
reflecting others, it appears colored
• Example: a red shirt reflects red and absorbs
the rest… meaning we see the red
The Electromagnetic Spectrum
• Visible light makes up only a small portion of
the entire electromagnetic spectrum which
includes all wavelengths of the spectrum
• The whole spectrum runs from 10-15 m
(gamma rays) to 105 m (radio waves)
Remember…
• Short-wavelength light has greater energy
• Gamma is the most dangerous and penetrates
the most (highest energy)
– The high energy causes damage to biological
molecules
X-Rays
• X-rays are also sufficiently energized enough
to damage cells
Ultraviolet (UV) Radiation
• Most familiar to us
• Produces burns
• Not as energized as gamma or x-rays but does
still contain enough energy to damage
biological molecules
Infrared (IR) Radiation
• Beyond visible light lies infrared (IR) radiation
– The heat you feel when you place your hand near
a hot object is IR radiation
Microwaves
• Beyond infrared are microwaves
– Used in radar and in microwave ovens
– Much longer wavelengths, and thus much less
energy… but is very efficiently absorbed by
water… which means it “excites” or heats things
that contain water
Radio Waves
• The longest wavelengths are radio waves
– Used to transmit signals responsible for AM and
FM radios, cellular telephones, televisions, and
other forms of communication
Page 286
• Read over the “chemistry and medicine”
scenario on page 286
• On a piece of (turn-in-able) paper, answer the
question presented AND
– Why can radiation be used to treat cancer??
What are some adverse affects you can think of?
Interference
• When waves interact with one another it is
called interference
– They can cancel each other out or build each
other up bigger
• There are two types:
– Constructive interference and
– Destructive interference
Diffraction
• When a wave encounters an obstacle or a slit
that is comparable to its wavelength, it bends
around it – a phenomenon called diffraction
Double-Slit Interference
• If there are two slits, the
resultant waves will
behave while interfering
with one another,
meaning there will be
constructive and
destructive interference
taking place
Now, Light as a Particle
• The photoelectric
effect
– The observation that
many metals emit
electrons when light
shines upon them –
light dislodges
electrons and pops
them off
A Threshold
• There is a minimum amount of energy needed
to pop off these electrons and see the light
– A threshold frequency
• This idea led to Einstein’s explanation:
– Light energy must come in packets
Energy
• Einstein said that the amount of energy (E) in
a light “packet” depends on its frequency (v)
according to the following equation:
𝐸 = ℎ𝑣
where “h” is Planck’s Constant (h=6.626x10-34J*s)
Photons
• A “packet” of light is called a photon or a
quantum of light
Since:
𝑣=
𝑐
λ
We can derive: 𝐸 =
• So… how many photons??
𝐸𝑝𝑢𝑙𝑠𝑒
# 𝑜𝑓 𝑝ℎ𝑜𝑡𝑜𝑛𝑠 =
𝐸𝑝ℎ𝑜𝑡𝑜𝑛
ℎ𝑐
λ
Practice
• A nitrogen gas laser with a wavelength of 337
nm contains 3.83 mJ of energy. How many
photons does it contain?
– Convert units… nm  m &
𝐸=
𝐸=
mJ to Joules
ℎ𝑐
λ
(6.626𝑥10−34𝐽∗𝑠)(3.00𝑥108𝑚∗𝑠−1)
3.37x10−7 m
= 5.90x10-19 J
Practice
• A nitrogen gas laser with a wavelength of 337 nm
contains 3.83 mJ of energy. How many photons
does it contain?
– Convert units… nm  m
𝐸=
ℎ𝑐
λ
𝐸=
&
mJ to Joules
(6.626𝑥10−34𝐽∗𝑠)(3.00𝑥108𝑚∗𝑠−1)
# 𝑜𝑓 𝑝ℎ𝑜𝑡𝑜𝑛𝑠 =
3.37x10−7 m
𝐸𝑝𝑢𝑙𝑠𝑒
𝐸𝑝ℎ𝑜𝑡𝑜𝑛
= 5.90x10-19 Joules
Practice
• A nitrogen gas laser with a wavelength of 337 nm
contains 3.83 mJ of energy. How many photons
does it contain?
– Convert units… nm  m
𝐸=
ℎ𝑐
λ
𝐸=
&
mJ to Joules
(6.626𝑥10−34𝐽∗𝑠)(3.00𝑥108𝑚∗𝑠−1)
# 𝑜𝑓 𝑝ℎ𝑜𝑡𝑜𝑛𝑠 =
3.37x10−7 m
𝐸𝑝𝑢𝑙𝑠𝑒
𝐸𝑝ℎ𝑜𝑡𝑜𝑛
=
= 5.90x10-19
3.83𝑥10−3𝐽
5.90𝑥10−19𝐽
= 6.49x1015 photons
More Practice
• A 100-watt lightbulb radiates energy at a rate of 100 J/s. (the
watt, a unit of power, or energy over time, is defined as 1 J/s).
If all the light emitted has a wavelength of 525 nm, how many
photons are emitted per second?
2.64x1020 photons
Review / Practice
• Arrange the following three types of
electromagnetic radiation – visible light, X-rays,
and microwaves – in order of increasing
A. Wavelength
B. Frequency
C. Energy per photon
Review / Practice
• Arrange the following three types of
electromagnetic radiation – visible light, X-rays,
and microwaves – in order of increasing
A. Wavelength
B. Frequency
C. Energy per photon
x rays < vis < micro
Review / Practice
• Arrange the following three types of
electromagnetic radiation – visible light, X-rays,
and microwaves – in order of increasing
A. Wavelength
x rays < vis < micro
B. Frequency
micro < vis < x rays
C. Energy per photon
Review / Practice
• Arrange the following three types of
electromagnetic radiation – visible light, X-rays,
and microwaves – in order of increasing
A. Wavelength
x rays < vis < micro
B. Frequency
micro < vis < x rays
C. Energy per photon
micro < vis < x rays
Back to the wave concept
• When an atom is excited (heat, light, elec) it
emits a spectrum of colors… unique to the
atom
• Emission spectrum is a series of bright lines
much like a “barcode” for each different atom
Bohr’s Model
• A model to explain the atomic spectra for each
atom
• In his model, electrons travel around the nucleus
in orbits
– Each orbit is a distance away from the nucleus and the
energy can be quantized..
• Electrons can also “jump” from orbital to orbital if
they are energized enough to make the leap
Bohr’s Concept
Wave Nature of Matter
• The wave nature of the electron replaced
Bohr’s Model and what first proposed by Louis
de Broglie in 1924 (confirmed in 1927)
• Electrons, which were thought of as particles
and have mass, also had a wave nature
– Hence why we focused a lot on the wave
behaviors of light
Electrons as a Wave
• Interference!!
• Single electrons interfere with themselves and
branch out, much like light in the double slit
interference
• They should (but do not) act like particles in
this experiment
The de Broglie Wavelength
• The wavelength of an electron of a mass (m)
moving at a velocity (v) is given by the de Broglie
Relation:
ℎ
𝜆=
𝑚𝑣
h = Plank’s Constant
-31 kg
????
9.11x10
m = mass of an electron
v = velocity of the electron Careful… its not “nu”!!
Practice
• Calculate the wavelength of an electron traveling with a
speed of 2.65x106 m/s
• Pay close attention to your units!!! (remember that 1 J =
1 kg*m2/s2
ℎ
6.626𝑥10 − 34 𝑘𝑔 ∗ 𝑚2 ∗ 𝑠 ∗ 𝑠 − 2
𝜆=
=
𝑚𝑣 9.11𝑥10 − 31 𝑘𝑔 ∗ 2.65𝑥106 𝑚/𝑠
= 2.74x10-10 m
Additional Practice
• What is the velocity of an electron have a de Broglie
wavelength that is approximately the length of a
chemical bond? Assume this length to be 1.2x10-10 m.
= 6.1x106 m/s
The Uncertainty Principle
• We just saw (de Broglie) that the velocity of an
electron is related to its wave nature
• The position (however) is related to its particle
nature
• These two concepts are known as complementary
properties
• We cannot observe the electron simultaneously as
both a particle and a wave (meaning we cannot
simultaneously measure its position and its velocity)
The Uncertainty Principle
• Heisenberg’s uncertainty Principle states the
more accurately you know the position of the
electron, the less accurately you can know its
velocity and vice versa
• Complementarity – an electron is observed as
either a particle or wave, but never both at once
Orbitals
• Orbitals describe an electron’s position within
the atom
• A distribution map that indicates the probable
location.
Orbitals
• s
spherical, holds 2 electrons
• p
contains 6 electrons
• d
contains 10 electrons
• f
contains 14 electrons
Orbitals
• s
spherical, holds 2 electrons
Orbitals
• p
contains 6 electrons
Orbitals
• d
contains 10 electrons
Orbitals
• f
contains 14 electrons
Quantum Numbers
• The Principal QN (n):
• The Angular Momentum QN (ɻ):
• The Magnetic QN (ml):
• The Spin QN (ms):
Quantum Numbers
• The Principal QN (n):
– The number that determines the overall size and
energy of the orbital
– Possible values are n=1, 2, 3, …
– Assign by going down the groups on the periodic table
Quantum Numbers
• The Angular Momentum QN (ɻ):
– Determines the shape of the orbital
– Possible values are 0, 1, 2, 3, … , (n-1)
Easy typical pattern:
– s orbital, ɻ = 0
– p orbital, ɻ = 1
– d orbital, ɻ = 2
– f orbital, ɻ = 3
Lowest energy
Highest energy
Compare n and ɻ
• Orbitals with the same value of n are said to
be in the same principal level (or principal
shell)
• Orbitals with the same value for n and ɻ are
said to be in the same sublevel (or subshell)
Quantum Numbers
• The Magnetic QN (ml):
– Specifies the orientation of the orbital
– Possible numbers range for +ɻ to -ɻ (including 0)
So, within the n=2 level, there are ɻ =0 and ɻ =1
If ɻ = 1 then mɻ can be -1, 0, 1 (a total of three orbitals)
Quantum Numbers
• The Spin QN (ms):
– Describes the spin… simply put: clockwise (+1/2)
or counter clockwise (-1/2)
– Does not matter… as long as you have opposites
Orbitals, revisited
• The total number of orbitals in any sublevel is
equal to 2ɻ+1
• The total number of orbitals in a level is equal
to n2
Practice
• What are the quantum numbers and names (for
example, 2s, 2p) of the orbitals in the n=4
principal level? How many n=4 orbitals exist?
n = 4, therefore ɻ = 0, 1, 2, and 3
ɻ values
mɻ values
orbital names
ɻ=0
0
4s (1 orbital)
ɻ=1
-1, 0 , +1
4p (3 orbitals)
ɻ=2
-2, -1, 0, +1, +2
4d (5 orbitals)
ɻ=3
-3, -2, -1, 0, +1, +2, +3 4f (7 orbitals)
total number of orbitals = 42 = 16
More Practice
• List the quantum numbers associated with all of
the 5d orbitals. How many 5d orbitals exist?
n=5
“d” means ɻ = 2
Therefore, mɻ (can) = -2, -1, 0, 1, 2
Meaning…. 5 orbitals
Quantum Numbers Practice
• The following sets of QN’s are each supposed to
specify an orbital. One set, however, is
erroneous. Which one and why?
(a) n=3; ɻ = 0, mɻ = 0
(b) n=2; ɻ = 1, mɻ = -1
(c) n=1; ɻ = 0, mɻ = 0
(d) n=4; ɻ = 1, mɻ = -2
(d) is wrong because, for ɻ = 1, the possible values
of mɻ are only -1, 0, and +1
Shapes of Atomic Orbitals
• Nodes – a point where the wave function and
electron distribution is zero
– In layperson terms: A node is a point where the
electron probability is zero.
Shapes of Orbitals
Great Concept… hint
• If some orbitals are shaped like dumbbells and 3-D
cloverleaves, and if most of the volume of an atom is
empty space diffusely occupied by electrons in these
orbitals, then why do we often depict atoms as spheres?
• The shape of the atom (as spherical) is obtained by
superimposing all of the orbitals it contains. Meaning
laying each over top of the others.