Download The Quantum-Mechanical Model of the Atom

The Quantum-Mechanical Model
of the Atom
We shall be using the Atoms-First
Approach!
We shall first start with the
pHundamentals!
• Atoms are the fundamental particles of
chemistry.
• Atoms are composed of proton, neutrons, and
electrons.
– Electrons are where the “action” is in chemistry.
– But the other particles are important, too!
• Some of these particles, in turn, are composed
of others.
Now, before we go further...
• It is my job to
– explain what is in the
book AND
– to supplement your
knowledge!
• Thus, I want none of...
The Theory we Shall Dabble in is
Quantum Mechanics
• This is the theory of the “very small.”
• What do we mean by “small”?
– The smallest measurement that has physical
meaning is called the “Planck length” and is about
1.616199 × 10-35 meters.
– An atom is about 10-10 meters in diameter.
• So, what would “large” be?
– The universe is about 10+28 meters wide.
Particles within particles...
• Atoms are made of protons, neutrons, and
electrons.
• Electrons are fundamental in their own right
and are one type of lepton.
• But protons and neutrons are made up of
even smaller particles called quarks.
Quarks & Leptons
•
•
•
•
There are 6 basic quarks.
There are 6 basic leptons.
Each of these has an antiparticle.
Thus, there are 24 fundamental particles (for
mass) in nature.
• The next slide gives a list of all of them.
The List!
What holds these together?
• There is a set of force-carrying particles, too!
• These are listed according to the various
forces in nature:
– Gravity (gravitons)
– Weak force (W & Z bosons, the only fundamental
force particles with mass)
– Electromagnetism (photons)
– Strong force (gluons)
The photon as a force carrier...
For pHun, here is a weak force carrier
in action...
What goes on can be very complex (we
won’t discuss this one!)
Here is a summary for atomic
particles...
Ordinary matter is made of...
•
•
•
•
•
Protons,
Neutrons,
Electrons,
and the relevant force carriers.
All other particles serve no purpose that
anyone knows of...
There is even a song about this!
• Put in ear plugs if you can’t stand this...
• Also shut your eyes if you are too young to see
this...
http://www.youtube.com/watch?v=D3wBV1KN
6FQ
Of course, one could get even more
intellectual about this...
• One guy decided that a rap was needed.
• So...
http://www.youtube.com/watch?v=xYZkj2FPeoc
Here are the innards of a proton...
(Note the colors!)
Protons and neutrons...
• Belong to a class of particles called baryons.
• All baryons are composed of three quarks.
– Protons are made to two up quarks and one down
quark.
– Neutrons are made of one up quark to two down
quarks.
pHinally, the Nuclear Force!
• Gluons hold quarks together to make protons
and neutrons.
• Protons and neutrons, in turn are held
together by the nuclear force, which is derived
from the strong force.
• Protons and neutrons are held together by
pions; each pion composed of a quark and an
anti-quark!
Here are a proton and neutron being
attracted...
This is the last particle type we need to
know about!
• Let’s here it for quarks, leptons, and all those
happy force-carrying bosons!
http://www.youtube.com/watch?v=Ez9f5EQ84T
g
Here is a summary of what is in the
atom...
Of course, it is wise to read the pHine
print!
Some History!
• Quantum mechanics began as a basically a
classical description of problems with black
body radiation.
• Black body radiation explains the radiation
given off by warm objects (for instance, all red
hot metals are at about the same
temperature).
• But there was a fundamental problem!
The Ultraviolet CATastrophe!
What is that?
• According to classical physics, every object
with T > 0 K should radiate an infinite amount
of energy.
• This was very disturbing!
Here is what a BB curve is...
BB radiation explains the colors of hot objects!
And, below is what actually happens!
The problem was solved by Max Planck
in 1900...
What Planck did...
• He found an equation that fit the data.
• This equation forced vibrational frequencies to
be multiples of a fundamental frequency
instead of continuous. Energy is quantized!
• He put in an adjustable parameter, h.
• This parameter was later on found to be a
fundamental physical constant!
Let’s look at the nature of light...
• Light is a type of
electromagnetic
radiation.
• All radiation of this type
is composed of
oscillating electric and
magnetic fields
perpendicular to each
other.
Some Details
• EM radiation in
spectroscopy has simple
sinusoidal waves.
• Waves are characterized
by amplitude and
wavelength (λ).
• We see an example to
the right.
Some relationships...
• Light has a speed of
propagation, this is
c = 299792458 m/s.
• This is an exact number!
• The wavelength in
turn—along with c—
allows us to define the
frequency (ν) of the
wave.
The Range of Frequencies Leads to the
Electromagnetic Spectrum
Each type of radiation corresponds to a
type of atomic/molecular interaction
(Hence the importance of spectroscopy in
chemistry!)
Radiation Type
Atomic/Molecular Action
Radio waves
Nuclear Magnetic Resonance
Microwave
Molecular Rotation
Infrared
Molecular Vibrations
Visible/Ultraviolet
Outer electron excitations
X-ray
Inner electron excitations
γ-ray
Nuclear excitations
Two important aspects of the wave
nature of EM radiation...
• Interference (constructive and destructive)
• Diffraction
Look at interference first...
Constructive Interference
Destructive Interference
Diffraction occurs because of the wave
nature of light...
Diffraction and interference together
are more than twice the pHun!
The particle nature of light...
Isaac Newton thought that
light was a particle.
He called these particles,
“corpuscles.”
However...
• The wave model of
Thomas Young proved
to be much more
powerful.
• It explained diffraction
(which Newton knew
nothing about).
• Young looks very
pleased about this!
The truth is more complicated!
• Light acts both as a
wave and as a particle!
• It is composed of
massless particles called
photons.
• This came from the
photoelectric theory of
Einstein.
• He put Newton and
Young in their places!
This was a revolution!
• Young’s theory was
refined extensively by
James Clerk Maxwell.
• This seemed to solve all
problems dealing with
radiation.
• Some people even
thought that physics was
done.
• Maxwell probably
doubted this.
But we still admire his equations!
What is the photoelectric effect?
What was wrong...
• Maxwell’s theory said that the energy of
electrons coming off the metal surface should
be related to the amplitude of the wave.
• However, this was not true.
• Einstein found a different—and very simple—
relationship.
• We look at examples on the next slide.
Two examples...
Data for a single metal
Results for three metals (note
the identical slopes!)
And the slope was...
h
In general, the kinetic energy of the
ejected electrons is...
Photons
• Light comes in the form
of photons.
• Photons are massless
particles.
• Photons have definite
energy and
momentum!
• Photons are both a
particle and a wave!
And a little more to say...
• Remember that we
consider both
frequency and
wavelength.
• So, let’s combine our
rules and come up with
a more useful set of
equations.
h is a fundamental physical constant!
What exactly, then, is a photon?
• It is a particle.
• It is a wave.
• Sometimes, it is drawn
as shown to the right...
The depiction below may be more
“realistic”!
Some points about wave-particle
duality...
• Is this really a strange
concept?
• Why not just accept it
as the way things are?
• In some cultures such
duality may not be
strange at all; just look
to the right!
If you can’t understand, laugh!
We have seen h appear in two places!
• Black body radiation
• The photoelectric effect
• This gives us some suspicion that h is
universal.
• The next slide shows yet another appearance
of this now seemingly ubiquitous constant.
Atomic Spectra led to the next
appearance of h...
• Every element has a
unique atomic
spectrum.
• These are most easily
observed via gas
discharges.
• Some examples are to
the right.
A prism or grating can split the light
from an element into a SPECTRUM
Why is this happening?
The answer was found by
Niels Bohr.
The old bore is shown to
the right...
What Bohr showed...
• Electrons in an H-atom
are confined to specific
orbitals.
• Planck’s constant
figures into the orbital
energies.
• Only discrete
frequencies can be
absorbed or emitted.
More detail...
• Bohr got this (h and
other constants are
embedded in R).
• m and n are integers.
R = 10973731.568525m-1
(R is often labeled as RH
for reasons that will be
obvious later.)
Bohr’s Complete Equation for R
Various values for R...
Wow! Look at where h is...
•
•
•
•
•
•
Black body radiation.
The photoelectric effect.
The definition of the photon and its energy.
Rydberg’s constant.
Is there more?
Mais oui, mes amis!
Matter as a wave!
• If light is both particles
and waves and
• quantum mechanics is
the theory of the very
small,
• then, maybe small
particles act as waves.
• This was shown by
Prince Louis de Broglie!
de Broglie put some things together in
a strange way...
For matter
For photons
Now, do some devious combining!
Bad math--but good science!
• de Broglie’s equation
came strictly from
analogy.
• But it is true!
• This was shown to be so
by Davisson & Germer
who first demonstrated
electron diffraction.
Now, since particles have wavelike
motion...
• How can we say exactly
where a particle is?
• That is, look at the
picture at the right;
where, exactly, is the
particle?
The Uncertainty Principle
• This was first proposed
by Werner Heisenberg.
• The old buzzard is to
the right.
• (Actually, he was a
spring chicken when
this picture was taken.)
Basic Idea of the U.P.
• The is an inherent limit
in how well we can
know the values of two
complementary physical
properties.
• Complementary
properties have
combined units of
action (energy × time).
Implications...
• The higher the precision
of one observable, the
more uncertain the
other.
• This tradeoff is shown
to the right!
Spectrum line widths...
• In passing, we also note
implications about
frequency and time.
• We shall look at a short
discussion of this to the
right.
• Note that width at halfheight of a spectrum
line is written as δν.
The main implications...
• In many cases, we cannot know a quantity
exactly.
• Rather, we have to live with statistics!
• The wave nature of matter thus dominates
things at the atomic/molecular level!
• We shall see that, in fact, we can visualize
things as standing waves.
Maybe this says it all...
The wave model turned out to be the
key to understanding atoms...
• The Bohr model was limited.
• What was needed was a more general model.
• This came about from the work of Erwin
Schrödinger.
• The next slide is the Schrödinger equation for
the hydrogen atom.
The whole thing...
Here is Schrödinger and a simplified
form of the equation...
How this equation solved many
problems...
• The wave characteristics of matter were
clearly included and defined.
• The energies were correctly determined;
these were the same as the Bohr model but
on a much firmer footing.
• Other things such as the angular momentum
of the electron orbits naturally emerged from
the solution.
All this devolved into 3 simple
quantum numbers!
• n: The principal quantum number
• l: The azimuthal quantum number
• ml: The magnetic quantum number.
n give the energy of the H-atom
This is easy to graph...
l defines the shape of the orbital
• l is determined by n.
• 0≤l≤n–1
n = 1 → l = 0
n = 2 → l = 0, 1
n = 3 → l = 0, 1, 2
Some special nomenclature...
•
•
•
•
l = 0: s orbitals
l = 1: p orbitals
l = 2: d orbitals
l = 3: f orbitals
(“sharp”)
(“principal”)
(“diffuse”)
(“fundamental”)
ml defines the direction of an orbital...
•
•
•
•
•
•
ml is defined by l.
-l ≤ ml ≤ +l
l = 0: ml = 0
l = 1: ml = -1, 0, +1
l = 2: ml = -2, -1, 0, +1, +2
ml takes on 2l + 1 values for a given l.
Atomic Spectroscopy Explained!
• n pretty much fills the
bill here (at least for
one-electron atoms).
• Looking to the right,
you see absorption and
emission.
Here is a more detailed look...
Orbitals and Wave Functions
• ψ2 corresponds to a probability density.
• We very often think of electron density in
terms of this probability density.
• There are various ways to envision this.
Probability Depictions (1s)
Dot Densities
Probability Density
Surface Descriptions (1s)
Orbital Surface (90%)
Radial Distribution Function
More complicated orbitals often have
nodes. (We see these first in a string.)
For pHun, there is an alternate way to
express this...
Nodes in the 2s and 3s Orbitals
Things get more interesting when we
get to p orbitals...
d orbitals are even more pHun!
f orbitals re really out there!
In math, these are the first few
spherical harmonics!
pHinal comments on interference and
waves...
• Sometimes, when we have constructive
interference, we say that waves are “in
phase.”
• Otherwise, we might say that they are “out of
phase.”
• The next slide shows a simple example of this.
Simple example of phase...
This applies to 3D waves, too
(Note the use of color)
Quick comment...
Do the ideas of constructive and destructive
interference and the idea of phase give you—
just maybe—and inkling of why atoms might
want to bond together?