Download Section 7.5 Quantum Mechanics and the Atom

Chapter 7
The Quantum-Mechanical Model of the Atom
Chapter 7
Chapter 7
The Quantum-Mechanical Model of the Atom
“When you have
eliminated the
impossible, whatever
remains, no matter
how improbable,
must be the truth.”
Sherlock Holmes
Chapter 7
The Quantum-Mechanical Model of the Atom
7.1
7.2
7.3
7.4
7.5
7.6
Schrodinger’s Cat
The Nature of Light
Atoms, Spectroscopy and The Bohr Model
The Wave Nature of Matter: The de Broglie
Wavelength, the Uncertainty Principle and
Indeterminacy
Quantum Mechanics and the Atom
The Shapes of Atomic Orbitals
3
Section 7.1
Schrodinger’s Cat
Macroscopic vs Subatomic World
• The laws of physics that govern the macroscopic
world (apples and cars and trees and us) are
called Newton’s Laws or Classical Physics
– If something is travelling in a straight line – it will keep
travelling that way unless a force acts on it
– What goes up must come down
– Things like this
4
Section 7.1
Schrodinger’s Cat
Macroscopic vs Subatomic World
• The Quantum Mechanical Model
• The subatomic world is very strange, almost
unimaginably strange.
– Things don’t move in predictable ways
– Things appear and disappear
– Absolutely small particles (like electrons) can be in two
different states at the same time
• The scientists who developed the model were
actually kind of shocked and dismayed by what it
predicted.
5
Section 7.1
Schrodinger’s Cat
Macroscopic vs Subatomic World
• Schrödinger's Cat
• This is an absurd example of trying to apply the rules of the
subatomic world to the macroscopic world.
• A cat in a container with a poison that is released by the
emission of a radioactive particle
• If the container is closed we don’t know if the poison has
been released or not.
• So the cat is both dead and alive at the same time
• It is not til we open the container and make an observation
that we force the cat into one state or the other by the act of
observation.
6
Section 7.1
Schrodinger’s Cat
Macroscopic vs Subatomic World
• This chapter is about the Quantum Mechanical
Model
• We look at where the model came from
• This is a model so it explains things
• Like periodic behavior in the periodic table
7
Section 7.2
The Nature of Light
Properties of Light
• In the very beginning the experiments that would
lead to the QM model of the atom began with an
examination of the properties of light
• So we are going to start there too.
• First we will look at the wave nature of light.
• The way light was first understood
8
Section 7.2
The Nature of Light
The Wave Nature of Light
• Light is electromagnetic radiation.
• A type of energy characterized by oscillating
electric and magnetic fields
– Sounds awful doesn’t it.
– It not – both are actually familiar.
• Magnetic field is the space where a magnetic
particle feels a force (area around a magnet)
• Electric Field is the region of space where an
electrically charged particle feels a force
– Proton generates an electric field
9
Section 7.2
The Nature of Light
The Wave Nature of Light
• Light is electromagnetic radiation.
– Oscillating electric and magnetic fields
– Characterized by amplitude (intensity or brightness)
and wavelength (distance between peaks)
10
Section 7.2
The Nature of Light
The Wave Nature of Light
• Wavelength and amplitude are related to the
amount of energy in the wave
• Imagine swimming at the beach
– High waves (large amplitude)
– Close together (short wavelength)
• Very hard to swim against
• Because they have lots of energy
– Small waves (low amplitude)
– Far apart (long wavelength)
• Easy to swim against – low energy
11
Section 7.2
The Nature of Light
The Wave Nature of Light
• Light is also characterized by frequency (n) the
number of waves that pass a certain point in a given
period of time
• The relationship between n and l is
c
n
l
• Where c = a constant (speed of light 3.00 x 108 m/s)
• And
 l is the wavelength
• Notice the inverse relationship between n and l
12
Section 7.2
The Nature of Light
The Wave Nature of Light
• For visible light, wavelength (l) or frequency (n)
determines color.
13
Section 7.2
The Nature of Light
Concept Check
Determine the frequency of a type of
electromagnetic radiation with a wavelength of
2.12 x 10 – 10 m.
14
Section 7.2
The Nature of Light
Solution
Determine the frequency of a type of
electromagnetic radiation with a wavelength of
2.12 x 10 – 10 m.
nc/l
8
3.00 x 10
n
m
s  1.42 x 1018 s 1
10
2.12 x 10
m
15
Section 7.2
The Nature of Light
The Electromagnetic Spectrum
• Visible light is only a tiny portion of the
entire electromagnetic spectrum.
Fig 7-5
• Shortest wavelength have highest frequency (and energy)
• Longest wavelength have the lowest energy
16
Section 7.2
The Nature of Light
Interference and Diffraction
• Electromagnetic radiation (light) moves in
waves
• Waves can interact with each other
(interference)
• They can cancel each (destructive
interference) other or build each other up
(constructive interference)
17
Section 7.2
The Nature of Light
Interference and Diffraction
• Waves that are ‘in phase” align so that the
crests overlap
• Waves that are ‘out of phase” overlap so
that the crest of one overlaps with the
trough of another
18
Section 7.2
The Nature of Light
Interference and Diffraction
Fig 7-6
• Another
characteristic
of light is
diffraction.
Light bends
around the
slit. Particles
pass straight
through.
19
Section 7.2
The Nature of Light
Diffraction Patterns
• Diffraction patterns arise from constructive and destructive
interference of light passing through multiple slits. Fig 7-7
20
Section 7.2
The Nature of Light
Matter vs Energy
• At the end of the 19th century the structure of the atom
figured out.
• Matter and Energy considered distinct
• Matter
– Particles which have mass
– Position in space can be specified
• Energy
– Form of light described as a wave, which is massless
– Position in space cannot be localized
21
Section 7.2
The Nature of Light
The Particle Nature of Light
• So light is a wave and matter is a particle
• The classical view of light is that it was purely a
wave phenomenon
– This view was particularly supported by the
observation of the diffraction of light.
• So of course it can’t be that simple
• There was an observation that challenged this
view
• Called the Photoelectric Effect
22
Section 7.2
The Nature of Light
The Photoelectric Effect
• Many metals emit electrons when light shines on
them
Fig 7-8
23
Section 7.2
The Nature of Light
The Photoelectric Effect
• The explanation for the photoelectric effect
(according to classical physics) was that the
energy from the light was transferred to the
metal which dislodged the electrons.
• OK – so light is still a wave.
24
Section 7.2
The Nature of Light
The Photoelectric Effect
• According to this explanation the brighter the
light (higher intensity) the more electrons should
be dislodged
• If you used a dim light there should be a lag time
before you build up enough energy to dislodge
an electron
• But – that is not what the experimental results
showed
• Low intensity (dim) high frequency (energy) light
produced electrons without the predicted lag time
25
Section 7.2
The Nature of Light
The Photoelectric Effect
• In addition the photoelectric effect shows a
threshold frequency
• Below the threshold no electrons are ejected no
matter how long the light shines or how intense
(bright) it is.
Fig 7-9
26
Section 7.2
The Nature of Light
The Photoelectric Effect
• Low frequency (low energy) light does not eject
electrons from metal
– No matter how intense (bright)
– No matter how long it shines
• High frequency (high energy) light does eject
electrons
– Even if the light is dim
• How to explain this?
27
Section 7.2
The Nature of Light
The Photoelectric Effect
• Albert Einstein explained it using the reasoning
that light actually comes in packets (photons)
• And that the amount of energy in a photon is
given by the equation
Ephoton  hn
where h is a constant (Planck' s constant) with a
value of 6.626 x 10
sin ce n = c/l

hc
Ephoton 
l
34
Js
28
Section 7.2
The Nature of Light
The Photoelectric Effect
• This was (of course) a revolutionary idea.
• Classical electromagnetic theory viewed light as
a wave whose intensity was continuously
variable
• Einstein is suggested that light is quantized into
particles and that a beam of light is not a wave
but rather a shower of particles each with a
discrete energy = hn

29
Section 7.2
The Nature of Light
• Dual nature of light:
 Electromagnetic radiation (and all matter)
exhibits wave properties and particulate
properties.
30
Section 7.2
The Nature of Light
The Photoelectric Effect
• So lets assume Einstein is correct.
• How does quantizing light explain the
experimental observations of the photoelectric
effect?
• The existence of the threshold frequency and
the lack of lag time with low intensity light
suggest that the light energy does not add up to
the point where the electron is ejected.
 • Rather you need a single event that provides the
appropriate amount of energy.
31
Section 7.2
The Nature of Light
How do Photons Explain The Photoelectric Effect
• Low frequency (low energy) light does not eject electrons
from metal
– No matter how intense (bright)
– No matter how long it shines
– These photons don’t have enough energy and they
don’t sum or add up
• High frequency (high energy) light does eject electrons
– Even if the light is dim
– These photons have enough energy – right frequency
– Remember amplitude (or intensity/brightness) is not
energy – frequency/color is energy
32
Section 7.2
The Nature of Light
The Photoelectric Effect
• The emission of electrons from the metal surface
depend on whether or not a single photon has
sufficient energy (hn) to dislodge a single
electron

33
Section 7.2
The Nature of Light
The Photoelectric Effect
• Think of photons like ping pong balls or
baseballs and the ejection of electrons like
breaking the pane of glass.
– ping pong balls are below the threshold frequency
– baseballs are above the threshold frequency
• Any photons above the threshold frequency
transfer the extra energy to the electron in the
form of kinetic energy

34
Section 7.2
The Nature of Light
Concept Check
Determine the increment of energy (the energy of a
photon of light) that is emitted by light with a
frequency of 1.42 x 1018 /s. The value of Planck’s
constant is 6.626 x 10 – 34 Js.
35
Section 7.2
The Nature of Light
Solution
Determine the increment of energy (the energy of a
photon of light) that is emitted by light with a
frequency of 1.42 x 1018 s. The value of Planck’s
constant is 6.626 x 10 – 34 Js.
DE = hn= 6.626 x 10 – 34 Js x 1.42 x 1018 s = 9.41 x 10 – 16 J
36
Section 7.2
The Nature of Light
Conceptual Connection
Light of three different wavelengths 325 nm, 455 nm and 632
nm shines on a metal surface. Match the wavelength with the
observation.
A. No photoelectrons were observed.
B. Photoelectrons with a kinetic energy of 155 kJ/mole were
observed
C. Photoelectrons with a kinetic energy of 51 kJ/mole were
observed
37
Section 7.2
The Nature of Light
Solution
Light of three different wavelengths 325 nm, 455 nm and 632
nm shines on a metal surface. Match the wavelength with the
observation.
E = hn = hc/l so lower wavelength has higher E
A. No photoelectrons were observed. (632 nm – lowest E)
B. Photoelectrons with a kinetic energy of 155 kJ/mole were
observed (325 nm highest energy)
C. Photoelectrons with a kinetic energy of 51 kJ/mole were
observed (455 nm intermediate energy)
38
Section 7.3
Atomic Spectroscopy and the Bohr Model
Atomic Spectroscopy
• The discovery of the particle nature of light was
a breakthrough that began to challenge the
classical view that light was only a wave
• Similarly, certain observations about atoms
began to suggest a wave nature for particles
• The most significant of the observations was
atomic spectroscopy – the study of
electromagnetic radiation
39
Section 7.3
Atomic Spectroscopy and the Bohr Model
Atomic Spectroscopy
• When atoms absorb
energy (heat, light,
electricity) they can re-emit
that energy as light
Fig 7-10
– Think of a neon sign
• Different atoms emit light of
a characteristic color
Mercury, Helium and Hydrogen
40
Section 7.3
Atomic Spectroscopy and the Bohr Model
Atomic Spectroscopy
• If we pass the light emitted by an element
through a prism we see that it is actually
composed of several different wavelengths
• The color of light is determined by its wavelength
Fig 7-11
41
Section 7.3
Atomic Spectroscopy and the Bohr Model
Atomic Spectroscopy
• The series of bight lines is called an emission
spectrum
• Emission spectrum for an element is always the
same
• Can be used to identify an element
• Used to identify the composition of distant stars
Fig 7-11
42
Section 7.3
Atomic Spectroscopy and the Bohr Model
Atomic Spectroscopy
• Look at the difference between the white light
spectrum (continuous) and the barium spectrum
(discrete lines)
• Classical physics could not explain this
Fig 7-11
43
Section 7.3
Atomic Spectroscopy and the Bohr Model
Atomic Spectroscopy
• Classical physics actually predicts and
continuous spectrum where an electron orbiting
a nucleus would release light of every
wavelength
• An even bigger problem is that classical physics
predicts an electron orbiting the nucleus would
lose energy as it emits light and spiral into the
nucleus.
• According to classical physics the atom should
not even be stable! Hmmmmm!
44
Section 7.3
Atomic Spectroscopy and the Bohr Model
The Bohr Model
• So basically what we need is a new kind of
physics to explain the behavior of the atom.
• This process had already begun with the
breaking down of the barrier between light as a
wave and matter as a particle.
• Einstein showed that light behaves as a particle
• The Bohr model is the beginning to the process
of treating matter (electrons) as a wave.
45
Section 7.3
Atomic Spectroscopy and the Bohr Model
The Bohr Model
• There was a mathematician named Rydberg
who looked at lots and lots of atomic spectra and
he came up with an equation that predicted the
wavelengths of the lines in the hydrogen
emission spectra
1
1 
 1
= R 2  2 The Rydberg Equation
m
l
n 
• His equation worked but it didn’t explain
anything. He didn’t even know what m and n
were. (We are going to see it again later)
46
Section 7.3
Atomic Spectroscopy and the Bohr Model
The Bohr Model
• So Niels Bohr developed a model for the atom
that would explain atomic spectra.
• Electrons travel around the nucleus in circular
orbits
• These orbits exist only at specific fixed distances
– They are quantized
– So they obey classical physics but also posses an
unknown stability
– Bohr called these orbits stationary states
47
Section 7.3
Atomic Spectroscopy and the Bohr Model
The Bohr Model
• Bohr also stated that while the electron is circling
the nucleus in the stationary state, no radiation
is emitted
– Contradicts classical physics
• In the Bohr Model the only time radiation is
emitted or absorbed is when the electron
transitions from one stationary state to another.
48
Section 7.3
Atomic Spectroscopy and the Bohr Model
The Bohr Model
• Wavelengths for electronic transitions for
hydrogen
Fig 7-12
• Electron moving
from n= 5 orbit
to n = 1 orbit
released energy
with a wavelenth
of 434 nm
49
Section 7.3
Atomic Spectroscopy and the Bohr Model
The Bohr Model
• Transitions between stationary states in a
hydrogen atom are not like what you imagine in
the macroscopic world.
• The electron does not exactly travel from n = 5
to n = 1
• It is never observed between states
• It is only observed in one state or the other
• This stuff is weird.
50
Section 7.3
Atomic Spectroscopy and the Bohr Model
The Bohr Model
• The Bohr Model explained the line spectrum of
hydrogen
• But also left a lot of unanswered questions
• Was an intermediate between the classical view
of the electron and the fully quantum-mechanical
view.
• Which is what we will do next.
51
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
The Wave Nature of the Electron
• The heart of the quantum – mechanical model
(which replaced the Bohr Model) is the wave
nature of the electron.
• The wave nature of the electron was first
proposed by Louis de Broglie (we will look at
what he did next) but was not confirmed until
about 50 years later when scientists observed
diffraction of electrons.
52
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
• de Broglie figured if light has characteristic of
both waves and particles, what about matter?
• Took the equation for the energy of a photon
E
hc
l
hv
mv =
or
l
2
hv
h
m =

2
lv
lv
• Substituted mc2 for E and n for c because matter
does not move at the speed of light (yet).
• Allows us to calculate the wavelength of a
particle.
53
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
The de Broglie Wavelength
• So Louis de Broglie proposed that is was
possible to determine the wavelength of an
electron from its velocity
h
if m =
then
lv
h
l =
mv
• Where l = wavelength, h = Planck's constant
6.626 x 10 – 34 J s, m is the mass of the electron
(in kg) and v is its velocity
54
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
Concept Check
• Since quantum mechanical theory is universal that
means even macroscopic objects have a wavelength.
• Determine the wavelength of a car which has a mass of
1800 kg travelling at 19.4 m/s (appx 70 km/hour).
55
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
Solution
• Since quantum mechanical theory is universal that
means even macroscopic objects have a wavelength.
• Determine the wavelength of a car which has a mass of
1800 kg travelling at 19.4 m/s (appx 70 km/hour).


h
6.626x10 34 J s
l

 1.9 x 10 38 m
m
mv

1800 kg 19.4 
s

56
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
The Wave Nature of the Electron
• The wave nature of the electron which was
proposed by Louis de Broglie was not confirmed
until about 50 years later when scientists
observed diffraction of electrons.
57
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
The Wave Nature of the Electron
• We have seen diffraction before. Light diffracts
when it passes through a slit
58
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
The Wave Nature of the Electron
• We would not expect to see diffraction of
electrons if they were particles. We would
Fig 7-16b
expect something more like this.
59
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
The Wave Nature of the Electron
• Instead a beam of electrons gives this pattern.
Fig 7-16b
• This same pattern appears when single
electrons pass through the slit one at a time.
60
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
The Wave Nature of the Electron
• How can a single electron passing through the
slit produce a diffraction pattern
• Two waves are required to produce interference
and produce the diffraction pattern.
• This is where things get really weird.
• The single electron actually passes through both
slits at the same time.
• This is the proof of its wave nature.
61
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
The Uncertainty Principle
• So Louis de Broglie proposed that electrons
have a wave nature and we have seen the
diffraction pattern.
• But electrons also have a particle nature
because they have mass.
• How can the electron be both a particle and
wave at the same time.
• Lets return to the diffraction experiment.
62
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
The Uncertainty Principle
• The diffraction pattern from the single electrons
was assumed to arise from the electron passing
through both slits at the same time and
producing two new waves (thus interfering with
itself)
• Lets test this hypothesis
• Well it turns out we can’t
• The act of observing the electron forces it to go
through one slit or the other
• Diffraction pattern disappears.
63
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
The Uncertainty Principle
• If this laser beam detector is turned on the
diffraction pattern disappears.
64
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
The Uncertainty Principle
• We cannot simultaneously observe both the
particle and wave nature of the electron.
• When we try to observe which slit the electron
passes through (particle nature) we lose the
diffraction patter (wave nature)
• When we observe the diffraction pattern (wave
nature) we cannot determine which slit the
electron passes through (particle nature)
• Wave and particle nature are said to be
complementary properties
65
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
The Uncertainty Principle
• Complementary properties exclude one another.
• The more we know about one the less we know
about the other.
• Which property we observe depends on the
experiment
• In quantum mechanics the event affect the
outcome.
• Kind of like Schrödinger's Cat
66
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
The Uncertainty Principle
• The electron has other complementary properties
– The velocity of an electron is related to its wave nature
(de Broglie equation)
– The position of an electron is related to its particle nature
• So we cannot simultaneously know both velocity
and position
• Velocity and position are complementary properties
• Werner Heisenberg formalized this idea with an
equation
67
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
The Uncertainty Principle
• The Heisenberg Uncertainty Principle
h
Dx  mDv 
4
• Where Dx is uncertainty in position
 Dv is uncertainty in velocity
• h is Planck’s constant
68
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
Indeterminacy and the Probability Distribution Map
• In classical physics particles move in a trajectory
determined by position, velocity and forces
acting on the particle.
69
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
Indeterminacy and the Probability Distribution Map
• Classical physics is deterministic – the present
determines the future.
• Two baseballs hit the same way will travel the
same path and land in the same place.
• Electrons don’t behave this way.
• We can’t know both their position and velocity.
• In quantum mechanics trajectories are replaced
by probability distribution maps.
70
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
Indeterminacy and the Probability Distribution Map
• If a baseball behaved like an electron it would land
in a different place very time.
• This behavior of the electron is called
indeterminacy.
– If we were able to observe hundreds of electron baseballs
however we would be able to observe a statistical pattern
of where the electron-baseball was likely to land
71
Section 7.4
The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy
Indeterminacy and the Probability Distribution Map
• So how do we apply the probability distribution
map to actual electrons.
• In the next section we introduce the concept of
quantum-mechanical orbitals
• Probability distribution maps for electrons as
they exist in atoms.
72
Section 7.5
Quantum Mechanics and the Atom
Orbitals
• The electrons position is described in terms of an orbital
which is a probability distribution map where the electron
is likely to be found.
– A mathematically description of the probability of
finding an electron at a given point in space around
the nucleus
• Not a very satisfying definition
• Notice it says – likely.
• Why does an orbital describe a probability?
• Why isn’t an orbital a real physical thing?
73
Section 7.5
Quantum Mechanics and the Atom
Orbitals
• Remember the Heisenberg Uncertainty Principle.
• Says we can’t know both position and velocity of
an electron at the same time
• The orbital doesn’t define position very accurately
but it turns out we don’t need to know the position
very accurately.
• The next few slides explain why.
74
Section 7.5
Quantum Mechanics and the Atom
Position vs Energy
• Position and velocity are complementary
properties
– The more accurately we know one the less accurately
we know the other
• Velocity is directly related to energy (KE = ½
mv2),
• So position and energy are also complementary
properties.
• So we can only define one accurately
• Either Position or Energy.
75
Section 7.5
Quantum Mechanics and the Atom
Energies
• The properties of elements depend primarily on
the energies of its electrons.
– Whether and electron is transferred from one atom to
another to form an ionic bond depends on the relative
energies of the two atoms.
• Since the properties of elements depend on
energy this is the one we need to be well
defined
76
Section 7.5
Quantum Mechanics and the Atom
The Schrodinger Equation
• So a probability distribution map gives us an
idea of where the electron is (the orbital) and
each of these electrons has an very specific
energy.
• What are the values of these energies?
• The mathematical derivation of energies and
orbitals comes from solving the Schrodinger
Equation.
77
Section 7.5
Quantum Mechanics and the Atom
The Schrodinger Equation
• The Schrodinger Equation.
 HYEY
 H is a set of mathematical operations that represent
the total energy (kinetic and potential) of the electron
– E is the actual energy of the electron
 Y is called the wave function – describes the wavelike
nature of the electron
• A plot of Y2 represents an orbital – a position
probability distribution map of the electron
78
Section 7.5
Quantum Mechanics and the Atom
Plot of Y2 represents an orbital
• Imagine every one of
these dots
representing the
position of the
electron at a given
point in time.
• Notice how there are
more dots closer to
the origin (nucleus)
Fig 7-23a
79
Section 7.5
Quantum Mechanics and the Atom
Quantum numbers
• Plot of Y2 represents an orbital.
• The electrons in each orbital have very specific
energies.
• Each of these orbitals can be described by a series
of numbers (quantum numbers) which describe
various properties of the orbital.
• Kind of an orbital bookkeeping system.
80
Section 7.5
Quantum Mechanics and the Atom
Quantum numbers
• Principle quantum number (n) – overall size and
energy of the orbital
• Angular momentum quantum number (l) –
shape of orbital
• Magnetic quantum number (ml) – orientation of
orbital
• Spin quantum number (ms)– direction of spin of
the electron
81
Section 7.5
Quantum Mechanics and the Atom
Principle Quantum Number (n)
• Principle quantum number (n)
• Integer that determines the overall size and energy of
the orbital
• n = 1, 2, 3, ….
• The energy of the orbital is given by
En   2.178 x 10
–18
 1 
J 2
n 
82
Section 7.5
Quantum Mechanics and the Atom
Principle Quantum Number
• Why is this a negative value?
• An unbound electron would have an energy
of zero
En   2.178 x 10
–18
 1 
J 2 0
 
• The energy of an electron bound to a
nucleus would be lower (more stable)
compared to this reference state
– Thus – a negative sign in the equation
83
Section 7.5
Quantum Mechanics and the Atom
Angular Momentum Quantum Number
•
•
•
•
•
•
Angular momentum quantum number
Integer that determines the shape of the orbital
Indicated by the letter l
Possible values of l are from 0 to n-1
For example if n = 1 the only value of l = 0
If n = 2 the values of l are 0 and 1
84
Section 7.5
Quantum Mechanics and the Atom
Angular Momentum Quantum Number
• The different values of l have letter designations
• These are the familiar orbital designations.
85
Section 7.5
Quantum Mechanics and the Atom
Magnetic Quantum Number
•
•
•
•
•
•
Magnetic quantum number
Integer that determines the orientation of the orbital
Indicated by the letter ml
Possible values of ml are from –l to +l.
For example if n = 1 the only value of l = 0 so ml = 0
If l = 2 the values of l are 0 and 1
– For l = 0 ml = 0
– For l = 1 ml = –1, 0 and 1
86
Section 7.5
Quantum Mechanics and the Atom
Spin Quantum Number
•
•
•
•
Spin quantum number
Specifies the direction of spin of the electron
Indicated by the letter ms
Electrons either spin up + ½ or down – ½
87
Section 7.5
Quantum Mechanics and the Atom
Summary of Quantum Number
• Principle (n)
0, 1, 2, etc.
• Shape (l)
0 to n minus 1
• Orientation (ml)
– l to + l
• Electron Spin
+ ½ (up) – ½ (down)
88
Section 7.5
Quantum Mechanics and the Atom
Quantum Numbers Specify Orbitals
• n = principle
– Indicates energy
• l = sublevel or
subshell
– Indicates shape
• ml - indicates
orientation
89
Section 7.5
Quantum Mechanics and the Atom
Quantum Numbers Specify Orbitals
• n=1
• l=0
• ml = 0
90
Section 7.5
Quantum Mechanics and the Atom
Quantum Numbers Specify Orbitals
• n=2
• l=1
• ml = 0
91
Section 7.5
Quantum Mechanics and the Atom
Quantum Numbers Specify Orbitals
• n=3
• l=2
• ml = 1
92
Section 7.5
Quantum Mechanics and the Atom
Concept Check
For principal quantum level n = 3,
determine the number of allowed subshells
(different values of l), and give the
designation (number and letter) of each.
93
Section 7.5
Quantum Mechanics and the Atom
Solution
For principal quantum level n = 3, determine the
number of allowed subshells (different values of l ),
and give the designation (number and letter) of each.
The allowed values of l run from 0 to 2, so the number of allowed subshells
is 3.
Thus the subshells and their designations are:
l = 0, 3s
l = 1, 3p
l = 2, 3d
94
Section 7.5
Quantum Mechanics and the Atom
Concept Check
For l = 2, determine the magnetic quantum
numbers (ml) and the number of orbitals.
95
Section 7.5
Quantum Mechanics and the Atom
Solution
For l = 2, determine the magnetic quantum
numbers (ml) and the number of orbitals.
magnetic quantum numbers = –2, – 1, 0,
1, 2
number of orbitals = 5
96
Section 7.5
Quantum Mechanics and the Atom
Concept Check
Each set of quantum numbers below is
supposed to specify an orbital. However
each set contains one quantum number
that is not allowed. Replace the quantum
number that is not allowed with one that is
allowed
A. n = 3, l = 3, ml = +2
B. n= 2, l = 1, ml = –2
C. n = 1, l = 1, ml = 0
97
Section 7.5
Quantum Mechanics and the Atom
Solution
Each set of quantum numbers below is
supposed to specify an orbital. However each
set contains one quantum number that is not
allowed. Replace the quantum number that is
not allowed with one that is allowed
A. n = 3, l = 3, ml = +2
n = 3, l = 2, ml = +2
B. n= 2, l = 1, ml = –2
n= 2, l = 1, ml = –1,0 or 1
C. n = 1, l = 1, ml = 0
n = 1, l = 0, ml = 0
98
Section 7.5
Quantum Mechanics and the Atom
Atomic Spectroscopy Explained
• Quantum theory explains the observations made
from atomic spectra in Section 7.3
• Each line in the emission spectra is due the the
emission of a specific wavelength of light when
an electron transitions between quantum
mechanical orbitals.
99
Section 7.5
Quantum Mechanics and the Atom
Atomic Spectroscopy Explained
• When atom absorbs energy electron is excited to higher
energy orbital. But this atoms is unstable
• Electron quickly falls back (relaxes) to a lower energy
orbital – releases a photon of light. Energy of that photon
is exactly equal to the difference in energy between the
two levels
100
Section 7.5
Quantum Mechanics and the Atom
Atomic Spectroscopy Explained
• The energy of an orbital in a hydrogen atom with
a principal energy level of n is
En   2.178 x 10
–18
• The difference between the two energy levels
 DE = Efinal – Einitial

DE   2.178 x 10
–18
DE   2.178 x 10
–18
 1 
J 2   2.178 x 10 –18
n f 
 1 
J 2
n 
 1 
J 2 
n i 
 1
1 
J 2 – 2  look - its the Rydberg Equation
n f
n i 
101
Section 7.5
Quantum Mechanics and the Atom
Atomic Spectroscopy Explained
• This equation allows us to calculate changes
in energy of an electron when the electron
changes orbits. From n = 6 to n = 1
DE   2.178 x 10
–18
 1
1 
J 2 – 2 
n f
n i 
DE   2.178 x 10
–18
1 
1
J 2 – 2
1
6 
DE   2.178 x 10 –18 J1 – .028 DE   2.178 x 10 –18 J x 0.972
DE   2.117 x 10
18
J
102
Section 7.5
Quantum Mechanics and the Atom
Atomic Spectroscopy Explained
• The DE is negative because the atom emits
the energy
 DEatom = – DEphoton
• Once we have the change in energy we can
determine the wavelength of light
l 
hc
DE
103
Section 7.5
Quantum Mechanics and the Atom
Atomic Spectroscopy Explained
l 
l 
hc
DE
l 
hc
 2.118 x 10 18 J
6.626x1034 J  s3.00x108 m




s 
2.117 x 10 18 J
 9.390x10
8
m
• Notice DE in this calculation is not negative. Not relative to the atom
anymore.
104
Section 7.5
Quantum Mechanics and the Atom
Exercise
What color of light is emitted when an excited
electron in the hydrogen atom falls from:
a) n = 5 to n = 2
b) n = 4 to n = 2
c) n = 3 to n = 2
105
Section 7.5
Quantum Mechanics and the Atom
Solution
What color of light is emitted when an excited
electron in the hydrogen atom falls from:
a) n = 5 to n = 2 blue l = 434.6 nm
DE 



DE  



 2.178 x 10-18 



1    1 
  n 2   n 2 
 f   i 

4.574x10 -19
hc
l
4.574x10-19

l 



DE  2.178 x

hc
-19

 4.574x10
l

 

10-18  12   12 
2  5 
l  4.574x10 -19   hc
6.626x1034 J s3.00x108 m




s 
4.574x10-19
 4.346x10
7
m
106
Section 7.5
Quantum Mechanics and the Atom
Solution
What color of light is emitted when an excited
electron in the hydrogen atom falls from:
a) n = 5 to n = 2
blue l = 434.4 nm
a) n = 4 to n = 2 green l= 486.7 nm
DE 
- 4.084x10
-19
c) n = 3 to n = 2 orange/red l = 657.1 nm

DE 
- 3.025 x 10
-19
107
Section 7.5
Quantum Mechanics and the Atom
The Shapes of Atomic Orbitals
• The shapes of orbitals are important because
covalent bonding involves the sharing of
electrons that occupy these orbitals.
• In one model of bonding a bond consists of the
overlap of atomic orbitals on adjacent atoms.
• The shapes of atomic orbitals are determined
primarily by the angular momentum quantum
number (l).
108
Section 7.5
Quantum Mechanics and the Atom
The Shapes of Atomic Orbitals
•
•
•
•
Orbitals with l = 0 are called s orbitals
Orbitals with l = 1 are called p orbitals
Orbitals with l = 2 are called d orbitals
Orbitals with l = 3 are called f orbitals
109
Section 7.5
Quantum Mechanics and the Atom
s Orbitals (l = 0)
• The lowest energy orbital is the spherically
symmetrical 1s orbital. s orbital in 1st principal
energy level.
• There are a couple of different ways to visualize
an orbital.
• But always remember – an orbital is a region of
space where there is a probability of finding an
electron.
• It is not a real physical thing.
110
Section 7.5
Quantum Mechanics and the Atom
Physical Meaning of a Wave Function
• The wave function (ψ itself has no physical
meaning
• The square of the wave function (ψ2 indicates
the probability of finding an electron near a
particular point in space (this is what we call an
orbital).
– Most commonly represented as a probability
density – intensity of color is used to indicate
the probability value near a given point in
space.
111
Section 7.5
Quantum Mechanics and the Atom
Visualizing Orbitals
• Picture an orbital as a three-dimensional
probability density map.
112
Section 7.5
Quantum Mechanics and the Atom
s Orbitals (l = 0)
• Three dimensional
plot of the wave
function squared y2.
This represents a
probability density.
Probability (per unit
volume) of finding the
electron at a point in
space
Fig 7-23a
113
Section 7.5
Quantum Mechanics and the Atom
s Orbitals (l = 0)
• Imagine every one of
these dots
representing the
position of the
electron at a given
point in time.
• Notice how there are
more dots closer to
the origin (nucleus)
Fig 7-23a
114
Section 7.5
Quantum Mechanics and the Atom
s Orbitals (l = 0)
Fig 7-24
• This representation is
the surface of a
sphere that
encompasses the
volume where the
electrons is found
90% of the time.
115
Section 7.5
Quantum Mechanics and the Atom
Radial Probability Distribution for 1s Orbital
Notice now the
probability of finding
an electron actually
increases a little bit
out from the nucleus
because the volume of
the cross section is
larger.
Fig B
116
Section 7.5
Quantum Mechanics and the Atom
2s Orbital
• Notice in the s orbital in the 2nd
principal energy level (n= 2) there
is a place on both the radial
probability distribution where
probability is 0.
• Electron cannot exist here.
• Called a node
• The probability of finding the
electron here is zero.
117
Section 7.5
Quantum Mechanics and the Atom
The 3s Orbital
• The s orbital in the 3rd
principal energy level (n = 3)
has 2 nodes.
118
Section 7.5
Quantum Mechanics and the Atom
Two Representations of the
Hydrogen 1s, 2s, and 3s
Orbitals
a = electron probability
distribution
b = surface that contains 90% of
the total electron probability
(size of orbital)
As n increases the number
of nodes increases
119
Section 7.5
Quantum Mechanics and the Atom
p orbitals (l = 1)
• Each principle energy level with n = 2 or greater
has three p orbitals.
• l = 1 so ml = –1, 0 and 1
• The three p orbitals are not spherical but have a
dumbbell shape with a node in the center.
• The three orbitals are arranged on the x, y and z
axis.
120
Section 7.5
Quantum Mechanics and the Atom
p orbitals (l = 1)
• The 2p orbitals and their radial distribution
function. 3p, 4p and higher p orbitals are larger
and contain more nodes.
Fig 7-27
121
Section 7.5
Quantum Mechanics and the Atom
d orbitals (l = 2)
• Each principle energy level with n = 3 or greater
has five d orbitals.
• l = 2 so ml = –2, –1, 0, 1 and 2
• These orbitals have more complicated cloverleaf
shapes
122
Section 7.5
Quantum Mechanics and the Atom
d orbitals (l = 2)
• The 3d orbitals. The higher d orbitals are larger
and contain more nodes.
Fig 7-28
123
Section 7.5
Quantum Mechanics and the Atom
f orbitals (l = 3)
• Each principle energy level with n = 3 or greater
has seven f orbitals.
• l = 3 so ml = –3,–2, –1, 0, 1, 2 and 3
• These orbitals are very complex with many more
lobes and nodes than d orbitals.
124
Section 7.5
Quantum Mechanics and the Atom
f orbitals (l = 3)
• The 4f orbitals
Fig 7-29
125
Section 7.5
Quantum Mechanics and the Atom
The Phases of Orbitals
• The orbitals we have been looking at are
actually 3 dimensional waves.
• If we look at a 1 dimensional wave.
• We see the wave on the left has positive
amplitude over the entire length where the wave
on the right has positive amplitude over the first
half and negative amplitude over the second
half.
126
Section 7.5
Quantum Mechanics and the Atom
The Phases of Orbitals
• The sign of the amplitude of a wave is known as
its phase
• Blue is positive phase and red is negative.
127
Section 7.5
Quantum Mechanics and the Atom
The Phases of Orbitals
• We can also
represent phase of
a three dimensional
quantummechanical orbital.
• Blue is positive
phase and red is
negative.
• Phase is important
in bonding.
128
Section 7.5
Quantum Mechanics and the Atom
The Shapes of Atoms
Fig 7-30
• Atoms are
represented as
spheres because
when we
superimpose all
the different
orbitals we get a
roughly spherical
shape.
129