Where do different theories
apply?
fast
The quantum theory was born in 1900, with the twentieth century, and future centuries
will list it among our own’s most remarkable achievements. Designed to account for the
puzzling behavior of matter at the submicroscopic scale of individual atoms, the theory
has enjoyed phenomenal success. It has accounted in a quantitative way for atomic
phenomena with numerical precision never before achieved in any field of science.
~ N. David Mermin
slow
Relativity
Newtonian
behavior
Wave-Particle
Duality
large
small
PS 110A Ch. 16 -3
PS 110A Ch. 16 -4
P1: What is the current understanding of what
“waves”
waves” when a particle acts like a wave?
Review of Models
a) The particle’s mass is
extended through space and
waves
b) The probability of finding a
particle in a given place is
spread out and waves
c) There is aluminiferous ether
spread throughout space that
waves
Continuous
Molecular
Thomson
Nuclear Solar System
Bohr Model
Today: Wave Model
PS 110A Ch. 16 -5
PS 110A Ch. 16 -6
P2:The uncertainty principle
states that
Standing Wave
Wavelength = h / (mass x speed)
a)
b)
c)
d)
.. a special kind of interference pattern produced
when 2 waves travel with same frequency
through the same medium.
E.g. -transverse (shear) wave sent along a string
fixed at both ends (wave 1)
-wave is reflected back (wave 2)
-for some special frequencies, constructive
interference: a standing wave is generated (wave 3)
We can’t know exactly
where a particle is
We can’t know exactly what
a particles velocity is
We can’t know exactly
where a particle is and what
it’s velocity is at the same
time
Scientists are kind of unsure
about what they are doing
3 (not traveling)
1
PS 110A Ch. 16 -7
2
PS 110A Ch. 16 -8
Nodes
Standing Waves
points along the standing wave
which have 0 amplitude at all times
-nodes are stationary; but energy travels
through them to either side
Hey,
there’s a
node,
eh!
And
there’s
an
antinode!
We can create one-dimensional standing
waves using a rope.
Only certain waves fit the constraints
antinodes
one antinode
three antinodes
nodes
No good. No
standing wave
will form.
Demo: jigsaw
PS 110A Ch. 16 -9
Standing waves also form for
other types of waves.
PS 110A Ch. 16 -10
Higher Dimensions
Standing waves are possible in higher dimension,
and can have complicated shapes. Different
standing wave shapes are called “Modes”.
In three dimensions you can also get standing
waves: for example when singing in the shower.
Demo: light covers
PS 110A Ch. 16 -11
Standing Wave Modes in 2
Dimensions
PS 110A Ch. 16 -12
So what do standing waves
have to do with the atom?
They help answer the problems
with the solar-system model of
the atom
Why are only certain orbits possible
How does the atom remember what its orbits are
to be like?
Why doesn’t an atom radiate energy
PS 110A Ch. 16 -13
PS 110A Ch. 16 -14
The Wave Model of the Atom
The wave-particle duality of the electron requires us
to treat electrons as waves as long as we aren’t
“looking” at them.
A 3-D electron standing probability wave surrounds
the nucleus.
We think of the waves as representing the
probability of finding an electron at each point in
space. We thus call their probability distribution
orbitals to reflect the idea that we cannot trace an
electron’s movement like we can in an orbit.
How do standing waves enter
into our picture of the atom?
The 3-D waves surround the nucleus. At certain radii,
or energies, the wave constructively interferes with
itself, at other radii or energies, it destructively
interferes.
n=4
Yes
n=5
Yes
n = 4 .3 3
THAT’S WHY THE ELECTRON ENERGIES
ARE QUANTIZED!
PS 110A Ch. 16 -15
Three Dimensional Atomic
Orbitals
The shape and energies of the actual
orbitals depends on the numbers of waves
surrounding the atom. They are given by
the Shrödinger Wave equation as
ih
∂Ψ ( x, t ) h 2 2
=
∇ Ψ ( x, t ) + V ( x , t ) Ψ ( x , t )
∂t
2m
Doesn’t that excite you?!
Never mind…
NO!
PS 110A Ch. 16 -16
Resolutions to
Bohr’s headaches
1. Why are only certain orbitals possible?
not orbits – not a path of particle movement, instead
orbital - standing wave of probability, allowed
shapes determined by Schrödinger equation
(wave equation).
2. How does atom remember?
the orbitals are the only ones allowed by nature
3. Why no radiation from undisturbed atom?
A standing probability wave does not change with
time and so the electron is not accelerating around an orbit.
PS 110A Ch. 16 -17
PS 110A Ch. 16 -18
Electron Energies
Important!
For multimulti-electron atoms, each set of
standing electron probability waves has a
different energy.
Like notes on a guitar string.
they govern spectra
If you can explain the spectra in detail, it
means you really know what’
what’s going
on.
™ all the wonders of modern chemistry
low note, low energy, few nodes
high note, high energy, many nodes
The music of the atoms.
PS 110A Ch. 16 -19
PS 110A Ch. 16 -20
Electron energies - analogy
with particle in a well
Classification of
Electron States
Major category: “Shells” or “Levels”
(corresponds to the principal # : 1, 2, 3, etc.)
Sub-category: Orbital pattern or shape( ~ angular
The gravitational potential energy DECREASES as the ball
is located in positions nearer the earth.
Electrical potential energy decreases as e- “resides” in lower
energy orbitals.
Each electron state has an “energy” called an “energy level”.
Energy levels close together are called energy “shells”
momentum value and orientation, value is an s, or a p,
,..,orbital, orientation is which one of the p’s,…)
Sub-sub-category: spin up or down
Each electron state is defined by a unique combination
of these categories.
The Shell, Orbital Value and Orientation, and
Spin are like a unique “address” of the electron.
They tell you where the electron “lives”.
PS 110A Ch. 16 -21
PS 110A Ch. 16 -22
Orbitals are three dimensional
standing probability waves
Orbitals as addresses:
Shell Number (1, 2, 3, etc)
Town
Orbital Value & orientation designation = pattern
s, p, d, value the street
s
orientation
which
one on the street
p
= house
d
Spin state
upstairs or downstairs?
1s orbital
An f orbital
d orbitals
PS 110A Ch. 16 -23
Orbitals reside in “shells”
The first shell contains only an sorbital, electron can be in center)
PS 110A Ch. 16 -24
More on shells
Higher numbered shells are farther
from the nucleus (bigger radius).
Electrons in higher shells therefore
have more potential energy.
The orbitals in higher numbered
shells have more nodes.
The second shell contains an sorbital and 3 p-orbital patterns
The third shell contains an sorbital, 3 p-orbitals, and 5 dorbitals
Etc.
Similar in
size and energy
s have same shape
PS 110A Ch. 16 -25
PS 110A Ch. 16 -26
Electrons in an Atom
3
2
Energy
1
Shells
Spin
d
p
s
Notice order
p
s
s
Orbital sets
PS 110A Ch. 16 -27
The Pauli Exclusion Principle
No more than two
electrons can occupy
the same orbital
pattern (in a given
shell). If two
electrons are in the
same orbital pattern,
they must have
different spins. i.e.,
at most 2 electrons
can adopt the same
orbital orientation.
Electrons (and other particles) have
a property called “spin”.
It describes the magnetic properties
of the electron (in the electron it
is somewhat analogous to a tiny
bar magnet with a north and
south pole).
The spin of an electron can be
oriented one of two directions:
Spin up
Spin down
Although we call this spin, it does
not imply an actual spinning of
the electron.
PS 110A Ch. 16 -28
Electrons in an Atom
How many p electrons can
there be in a shell?
P3: Which orbitals are in the fourth shell?
PS 110A Ch. 16 -29
The Exclusion Principle (cont.)
At most 2 electrons can adopt the same wave
pattern (orientation). (2l +1 patterns for each
orbital)
• Each s orbital (l=0: 1 pattern) can have 2 states.
• Each p orbital (l=1: 3 patterns) can have 2 states
in each pattern.
• Each d orbital (l=2: 5 patterns) can have 2 states
in each pattern.
• Each f orbital (l=3: 7 patterns) can have 2 states
in each pattern.
PS 110A Ch. 16 -31
PS 110A Ch. 16 -30
Orbital Patterns
The pattern continues on as s, p, d, f, g, h, i, j, etc.
Each new orbital set has two more orbitals than the
previous one.
Orbital Type
s
Orbital shapes in set 1
Electrons
2
p
3
6
d f
g h i
j
5 7 9 11 13 15
10 14 18 22 26 30
PS 110A Ch. 16 -32
Understanding Atoms
Electrons in atoms
How do electrons fill the orbitals as we move along
the periodic table?
Fewer nodes in the standing waves correspond to lower
energies.
Electrons fill the lowest energy levels first.
High energy
3
21
Each combination of
shell
orbital orientation
spin
a “state”
Energies vary from element to element because of
positive charge of nucleus, different radii.
Low energy
Electrons fill lowest energies first, and then on up.
PS 110A Ch. 16 -33
PS 110A Ch. 16 -34
Hydrogen
Electron energies - analogy
For the neutral atom there are same number of electrons as protons.
The underground parking garage
free electron
Level 3
d
p
s
Shell 3
3d
3p
3s
Shell 2
2p
2s
Shell 1
1s
Level 2
p
s
Level 1
s
PS 110A Ch. 16 -35
Note that these energies are not to scale.
Helium
PS 110A Ch. 16 -36
Lithium
free electron
Shell 3
3d
3p
3s
Shell 2
2p
2s
Shell 1
1s
free electron
Shell 3
3d
3p
3s
Shell 2
2p
2s
Shell 1
1s
These energies are not to scale and they shift as you go to another element.
Note that these energies are not to scale.
PS 110A Ch. 16 -37
PS 110A Ch. 16 -38
An Energy “Well”
Beryllium
for a multimulti-electron atom
d
p
s
free electron
3
Shell 3
3d
3p
3s
Shell 2
2p
2s
Shell 1
1s
2
p
s
1
Shells
s
Orbital sets
P4: What atom is this?
Note that these energies are not to scale.
(look at periodic table)
PS 110A Ch. 16 -39
Filling sequence
Energy
PS 110A Ch. 16 -40
Re-emphisis
P5: How many electrons are in an atom that
has both the first and second shells filled?
PS 110A Ch. 16 -41
PS 110A Ch. 16 -42