Transcript Document 7509413

electron orbits
atomic spectra
the Bohr atom
“Never express yourself more clearly than you are able to think.”—Neils
Bohr
4.4 The Bohr Atom (1913)
The scenario that Beiser
presents in describing Bohr’s
model of hydrogen is not the
approach Bohr took in his
work, because de Broglie and
his matter waves didn't come
for another decade.
Nevertheless, let’s follow Beiser for a while, and at the end of
this section I’ll briefly mention Bohr’s approach.
“…if only sufficiently many physicists are placed at sufficiently big
machines, then everything will fall into place in the end.”—W. Heisenberg
(no, Heisenberg didn’t actually believe that)
Note that this is a model for the hydrogen atom.
Why choose the hydrogen atom...
Consider the wavelength of an orbiting electron.
An electron in orbit around a hydrogen nucleus has wavelength
 = h / mv and a velocity as given in section 4.2:
v=
e
.
4 0mr
You can solve these two equations for the wavelength:
h 4 ε0r
λ=
.
e
m
h 4 0r
λ=
.
e
m
You can plug in the electron mass, r = 5.3x10-11 m for the
radius of an electron orbit in hydrogen, e =1.6x10-19 C, and 0
= 8.85x10-12…
6.63 10-34 J  s
λ=
1.6 10-19 C
C2
-11
4  8.85 10

5.3

10
m
2
Nm
9.11 10-31 kg
-12
…and you get a wavelength  = 33x10-11 m, which
(coincidentally?) is the circumference of the electron orbit.
So what?
We have taken an electron and
calculated its wavelength. We
find that its wavelength exactly
corresponds to one orbit of the
hydrogen atom. See Figure 4.12.
Or try this visualization. (Click on the little “ball” to start.)
This gives us a clue as to how to construct electron orbits.
If an electron wave going around a
nucleus "meets itself" out of phase
after one revolution, destructive
interference will take place.
http://hyperphysics.phyastr.gsu.edu/hbase/ewav.html#c2
Actually, instead of "destroying" itself, such an electron simply
can't fit into an orbit. It must either gain or lose energy, so
as to have a velocity which gives it a wavelength which fits
into the orbit.
We thus arrive at the postulate that an electron can orbit a
nucleus only if its orbit contains an integral number of
de Broglie wavelengths.
Of course the idea of de Broglie waves was a decade in the
future when Bohr worked out his model for the hydrogen
atom, so he couldn't have made this postulate.
Expressed mathematically:
n λ = 2  rn
n = 1, 2, 3, ...
Combining this expression for  with the one we obtained
earlier in this section gives us an equation for rn, the orbital
radii in the Bohr atom.
n2h2ε0
rn =
 m2e
n = 1, 2, 3, ...
Like our particle in a box, an electron in hydrogen can “fit”
only if its wave can “fit.” The wavelength is quantized.
The orbital radii are quantized.
What else was quantized for our particle in box.
Energies!
I wonder if that will happen here…
(not a question, just
wondering out loud)
Here’s a visualization of the Bohr model.
If that visualization was supposed to mean something, there
are a couple of things we have to explain...
The integer n in the equations for  and r is called the
quantum number of the orbit.
When n=1,
2
r1 =
h ε0
 m2e
6.62×10   8.85×10 
=
= 5.292 × 10
  9.11×10 1.6×10 
-34 2
-31
-12
-19 2
-11
m
.
This value of r1 is the radius of the innermost orbit. It is called
the Bohr radius, a0, of the hydrogen atom, and a0=5.292x10-11
m. The other radii are given by rn = n2 a0.
If this wasn’t Bohr’s approach to his model of the atom, what
was?
As a student, Bohr hadn’t thought spectra were worth much.
However, when Bohr was trying to understand the hydrogen
atom, a colleague happened to show him some spectra and
Balmer’s formula.
1
1 1
= R  2 - 2  n = 3, 4,5,...
λ
2 n 
"As soon as I saw Balmer's formula, the whole thing
was immediately clear to me." –N. Bohr
Quoting from the class notes* of Taylor (U. of Virginia)…
*http://www.phys.virginia.edu/classes/252/Bohr_Atom/Bohr_Atom.html
“What he saw was that the set of allowed frequencies
(proportional to inverse wavelengths) emitted by the hydrogen
atom could all be expressed as differences.
This immediately suggested to him a generalization of his idea
of a "stationary state" lowest energy level, in which the
electron did not radiate.
There must be a whole sequence of these stationary states,
with radiation only taking place as the atom jumps from one to
another of lower energy, emitting a single quantum of
frequency f such that
h f = En – Em ,
the difference between the energies of the two states.
Did you spot the example of the misleading Physicsspeak* on
the previous slide?
Evidently, from the Balmer formula and its extension to general
integers m, n, these allowed non-radiating orbits, the
stationary states, could be labeled 1, 2, 3, ... , n, ... and had
energies -1, -1/4, -1/9, ..., -1/n2, ... in units of hcRH (using  f
= c and the Balmer equation above).”
The main thing is to realize that Bohr pictured a series of
“stationary states”* in which the electron could exist without
radiating, and that radiation takes place only when an electron
jumps between different states.
*How can an electron be orbiting if it is in a “stationary” state?
Furthermore, if, as we showed above, the speed is quantized,
then so is the electron’s angular momentum.
More specifically, it is not difficult to show that the angular
momentum is quantized in units of h/2, which we call ħ.
L = n h / 2 = n ħ .
I guess Beiser didn’t want to wrestle with angular momentum
in this chapter. I can’t say that I blame him.
Let’s think about Bohr’s model for a minute.
 It contains classical physics. Remember, we used our
“dynamically stable” orbit equations in the model.
e
v=
.
4 ε 0mr
 It doesn’t explain how an electron can exist in a “stationary
state” orbit without radiating. (Just giving it a name doesn’t
explain it.)
 It predicts Balmer’s (and all those others—see next section
in text) formula for the hydrogen spectral lines.
 But it doesn’t explain why the electron radiates when it
jumps between orbits.
Bohr’s model for hydrogen is a curious hybrid of classical and
quantum physics.
Some physics faculty refuse to teach it because it is wrong.
Several prominent physicists of Bohr’s day threatened to quit
the profession if Bohr was correct. “The prevailing impression
was one of scandal, or at least bewilderment, before the
undeserved success of such high-handed disregard of the
canons of formal logic.”*
*http://www.phys.virginia.edu/classes/252/Bohr_to_Waves/Bohr_to_Waves.html
"If you aren't confused by quantum physics, then you really haven't
understood it." –N. Bohr
Bohr himself admitted his model didn’t explain anything. But it
does tie together previously unexplained observations, and
tells us the direction we might go in looking for the “true”
model.
See here for another triumph* of Bohr’s model: singly ionized
helium (like hydrogen but with an extra neutron). However,
the model fails for helium. As we will see a few chapters from
now, that is nothing to be ashamed of.
There is a happy ending to this story: Bohr won the 1922
Nobel prize “for his services in the investigation of the
structure of atoms and of the radiation emanating from them.”
*"This is an enormous achievement. The theory of Bohr must then be
right." –A. Einstein
4.5 Energy Levels and Spectra
As suggested above, the Bohr model puts us in a position to
understand energy levels and atomic spectra.
Plugging rn into our expression for the energy of an electron in
an orbit gives
me 4  1  E 1
En = - 2 2  2  = 2
8ε0h  n  n
n = 1, 2, 3, ...
So electron energies are
quantized, as suggested
back on slide 5.
In this equation, E1 = -13.6 eV. These are the energy levels
of the hydrogen atom. Electrons in hydrogen atoms are
restricted to these energies. The negative sign indicates the
electrons are bound.
Electrons “fit into” discrete orbits…
…and have quantized energies.
The lowest energy level E1 is called the ground state of the
atom, and higher states are called excited states.
When n =  then E=0 and the electron is no longer bound.
Electrons with energies greater than E=0 are not restricted
to quantum states.
The ionization energy of hydrogen is E1 = -13.6 eV. The theory
from the Bohr model agrees with experiment!
Electrons in hydrogen only exist in discrete energy levels, and
not in states in between. Electrons change energy levels by
absorbing a photon (and gaining energy) or emitting a photon
(and losing energy).
The difference between the initial and final electron energy is
equal to the photon energy:
Ei - Ef = hf .
Let ni and nf be the quantum numbers for the initial and final
electron states.
Then it is easy to show that the wavelength of the photon
emitted in this transition is
E1  1 1 
1
= 2 - 2 .
λ
hc  nf ni 
Wow! This is the generalization of Ballmer’s formula, as well
as all those other ones!
E1  1 1  Emission and absorption spectra for
1
= 2 - 2  . hydrogen will contain only the
λ
hc  nf ni 
wavelengths given by this equation.
If nf=1, the series of spectral lines known as the Lyman series
is observed. Please read your text and work the example
problems.
Best of all, the factor -E1/hc in our theory is equal to the
constant R in the experimentally observed spectral series. No
more adjustable parameters! Out with empiricism. In with
theory!
The Bohr model does an excellent job of explaining spectral
lines. We can’t claim to understand hydrogen, however,
because we still don’t understand the “stationary states.”
4.6 Correspondence Principle
It would be a good idea for you to read this section for ideas.
Quantum physics in the limit of large quantum
numbers should give the same results as classical
physics.
I won't test you on the material in this section.
4.7 Nuclear motion
Again, it would be a good idea to read this section for ideas,
but I won't test you on it.
The main idea is that electrons and nuclei orbit each other.
The much more massive nucleus moves very little, just as the
earth does most of the orbiting around the sun. However, on
the atomic scale, the corrections for nuclear motion are large
enough to be measurable.
4.8 Atomic Excitation
Electrons in atoms can be excited to energy levels above their
ground state by:
collisions with other atoms, ions, etc.,
which transfer kinetic energy,
or photons.
We are talking about electronic energy levels. The electrons
are absorbing the energy. When we say “atomic excitation”
we really mean “excitation of electrons in atoms.”
Images from http://library.thinkquest.org/16468/excit.htm.
Transitions back to the ground state occur via photon
emission. Typically this occurs within about 10-8 s of excitation
(a ballpark figure).
Suppose you have a pet electron that has too much kinetic
energy. How do you get rid of its* excess energy.
Door #1. Make it collide with something small?
*Being a responsible pet owner, you have had the electron neutered or
spayed, so it is really an “it,” not a “he” or “she.”
Door #2. Make it collide with something similar?
Door #3. Make it collide with something big?
Please vote now.
To get rid of excess energy…
?
?
?
Here is the answer.
Conclusion: electrons are most effective at transferring kinetic
energy to other electrons.
Examples of electron-electron collisions are neon lamps, the
fluorescent lamps in this room, and mercury vapor*
streetlamps.
Huh? This modern physics stuff really has some use in “real”
life?
*And you probably think these things are safe.
An electric field applied across a gas-filled tube* creates ions
and “free” electrons. The electric field accelerates the
electrons and ions. Electrons colliding with the ions excite
atomic energy levels.
Demo: spectrum tubes.
I’ve been discussing emission for a bit (“de-excitation”). An
example of excitation by a photon is hydrogen absorbing
photons of wavelength 121.7 nm and being excited from the
n=1 to n=2 state.
*Don’t try this at home.
The Franck-Hertz* Experiment (1914)
This famous experiment confirmed the existence of Bohr’s
atomic energy levels.
Accelerated electrons colliding
with atoms give up energy to
atomic electrons, provided their
energy is sufficient to promote
an atomic electron from one
energy level to a higher one.
*1925 Nobel Prize for Franck and Hertz. And no, not the Hertz of Hz fame.
Gustav Hertz, not Heinrich Hertz.
Hyperphysics shows the
experimental setup (previous
slide) and this actual data.
(Note: I purchased a
hyperphysics “license” to use
this material in my teaching.)
Here is an interactive Franck-Hertz experiment.
Things I might ask on a test: what did this experiment
demonstrate? Interpret the dips in the current versus voltage
curve.