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PROFESSOR: OK, settle down.
Let's get started.
One announcement, yesterday we
had our first weekly I would
say minor celebration.
And by and large,
it went well.
Please make sure that you go to
your assigned recitation.
If you miss your recitation,
you need to get down to see
Hillary so that we make sure
that we have enough copies of
the weekly quizzes on hand.
If you've joined the class, you
have to check in with her.
She's down the hall here
in room 8-201.
And you'll be assigned
to a section.
What else do I have by
way of announcements?
Oh yes.
Just reminding you, this
was from 2003.
See it doesn't change.
It's the same s-block, p-block,
and d-block elements.
So that's coming up a week
from tomorrow, two
celebrations next.
And of course the contests, the
contests with hot prizes.
All right.
Let's get down to business.
Last day we looked at the
Rutherford-Geiger-Marsden
experiment.
Oh there's one other one.
If you look at the readings,
there's this one section
called the archives.
My predecessor, Professor Wit,
wrote a set of lecture notes.
And they look something
like this.
And some students have said that
they find these a little
more expository in certain
sections on certain topics
than the book to be.
And I have no preference.
But if you take a look at
this it'll say LN1,
lecture notes 1.
If you go to this, read this.
If you find that's
helpful, good.
If you don't find it helpful,
than stick with the book.
Just letting you know
what that is.
All right, so last day
we looked at the
Rutherford-Geiger-Marsden
experiment in which a high
energy beam of alpha particles
bombarded a thin, gold foil.
And on the basis of the
scattering results, namely
most of the particles went
through with minor scattering.
And a tiny fraction of
them were scattered
through large angles.
The Thomson plum pudding model
was rejected in favor of
Rutherford's nuclear
model of the atom.
And then subsequently, Bohr came
up with the quantitative
representation off the
Rutherford nuclear model.
And we were partway through the
treatment of Bohr last day
when we adjourned.
So let's right pick up the
thread from where we left off.
And so just to remind you, the
Bohr model is for a 1-electron
atom gas phase.
So this is either atomic
hydrogen, it could be helium
plus lithium 2 plus roentgenium
110 plus Its
doesn't matter how
many protons.
there's always only
want electron.
And it's a planetary model.
The positive charge concentrated
in the nucleus, Z
is the proton number.
And at a distance r from the
nucleus is a circular orbit in
which resides 1 electron.
It has a charge of minus E.
And I just designated
them q1 and q2.
You could have done
it the other way.
But I had to choose something.
So we went through
and looked at the
constitutive equations here.
So first of all, the energy
of the system--
and this is only going to be
the energy the electron.
Because we assume that the
nucleus is far more massive.
And so we don't have to get into
things like reduced mass
or anything like that.
So just measure the energy
of the electron.
1/2 mv squared is the
newtonian component.
And then coulombic energy that's
stored is z times e
squared over 4 pi epsilon zero
r, where epsilon zero is the
permittivity of vacuum.
And it's the factor, the 4 pi
epsilon zero, is the factor
that allows us to take
electrostatic energies and put
them on the same plane as
mechanical energies.
When we run through this, we
always end up in joules.
Then there's a force balanced to
make sure that the electron
neither falls into the
nucleus nor flees and
breaks free of the atom.
And the force balance is if you
put a ball at the end of a
string, and you whip it around
on a tether, you have a
centrifugal force that's trying
to get the ball to
break away.
And then the string
is pulling in.
So the pull in, in this case,
is the coulombic force.
And the force that makes the
ball want to flee is this mv
squared over r.
And that must be net zero.
Otherwise we're going to
have a shift in orbit.
And then lastly we have
the quantum condition.
And this was the breakthrough
of Bohr where he enunciated
that the quantum condition is
going to give us this energy
level quantization.
And this was a big departure
from what had been in the
past. The only antecedent idea
of this nature was the work by
Planck who said that
light is quantized.
But as I told you last
day, who knows what
light really is.
The Newtonian notion of
a ball orbiting was
very compelling here.
And the notion that the movement
of the electron could
in some way be discontinuous was
quite a major departure.
So I left you last
day with three
equations and three unknowns.
And what I'm not going
to do right now
is solve the equations.
Because I've been lecturing long
enough to know that's the
way to kill interest,
quench a lecture.
And so if you really want to go
through the algebra, be my
guest. You're smart
enough to do that.
Instead I'm going to show
you the results.
But those are the three
equations that you need.
So we have an equation in r.
We have an equation at v,
and an equation in e.
So let's go after them.
If you first look at
the solution for r.
This is the radius of the orbit
of the electron, the
orbit of the electron.
If you go through and solve,
you'll end up with this,
Epsilon zero times the square of
the Planck constant divided
by pi times m.
It's always the electron.
So this is the radius of
the electron orbit.
This is the mass of the electron
times the square of
the elementary charge--
that whole thing I'm
going to group--
times the square of the quantum
number divided by z,
the proton number.
So what we see here?
Well, everything inside the
parentheses is constant.
These are all constant.
Pi obviously is geometric
constant.
And the rest of these are
constants you could look up in
your table of constants.
And we noticed that there is
a set of solutions to this.
The radius of the electron can
occupy various discrete values
defined by n.
So we say that the radius takes
on a plurality of values
a function of n.
And furthermore, the
functionality goes as the
square of n.
It's n squared times a constant,
where that constant
is inside those parentheses.
And we notice that because
the r goes as n
squared it's nonlinear.
It's nonlinear.
This is so important I'm
going to write it
down one more time.
So the radius of the electron
orbit takes multiple values.
It takes multiple values.
And they're discreet.
The physicists like to
use a different term.
When something is
discretized, the
physicists say it is quantized.
So these values are quantized.
You cannot continuously vary
the radius and nonlinear
values, multiple values.
So let's plug in.
Because I want to get
a sense of scale.
So let's look at the
simplest one.
The most primitive 1-electron
atom would be hydrogen.
In which case, Z equals 1.
So I've just got a proton
orbited by an electron.
So look at atomic hydrogen.
So in that case, Z equals 1.
And I'm going to look at n
equals 1, which is the lowest
number here, right?
R scales as n squared.
So the lowest value or r is
obtained when n equals 1.
And this is termed
the ground state.
The ground state.
So I want to ask what is the
radius of the electron orbit
ground state in atomic
hydrogen?
And if I plug in these values,
I'll call this r sub 1.
It turns out to be 5.29 times 10
to the minus 11 meters, or
0.529 angstroms. I love
the angstrom.
It's a great unit.
It's a great unit.
It's not an SI unit.
But I like the angstrom.
I'll show you why.
If you try to express this in
SI units, well there's 10 to
the minus 11 meters.
The Si units go in units
of clusters of 1,000.
So for example, you've
got the meter.
You've got the kilometer.
You've got the micrometer.
You've got 10 to the
minus 9 meters,
which is the nanometer.
This is a Goldilocks problem.
This one's too big.
And then the next one down here
is 10 to the minus 12
meters, which is
the picometer.
So this is either 52.9
picometers, or a 0.0529
nanometers.
And that's no good.
I want numbers like 3, 7,
simple to remember.
So 0.529, this is about
1/2 angstrom.
It's good to know.
But you try to publish, you
know what happens in a
scientific literature today?
The Literary Lions that control
the journals, they'll
circle that and say you have
to convert to SI units.
And so they have to
right some goofy
nanometer thing or something.
I know you think I'm crazy,
but I love the angstrom.
So there.
Anyway, so here it is.
Once you know that this is
0.529, this is on your table
of constants.
It's right on your table
of constants.
So you don't have to go and
calculate all this stuff.
Which means if you do your
homework with your table of
constants, you will know where
those numbers lie, as opposed
to opening this thing up for
the first time on the first
celebration of learning on
October the 7th, and with 47
entries and they're
tiny, tiny font.
And you're wondering where
is that thing.
Just a word to wise.
So now we know what this is.
We know this is 0.529 angstroms.
So now I can write
an equation for the radius
of a 1-electron
atom anywhere, anytime.
r of n is going to be equal
to a naught which
is this value here.
And it is termed the
Bohr radius.
So you can write it as a
function of the Bohr radius,
times the square of
n divided by Z.
So that's for all 1-electron
atoms, gas phase.
And you can see that as Z goes
up, the r goes down, which
makes sense.
So suppose instead of hydrogen,
we talk helium plus
What's the only difference?
Helium plus has 2 protons
in the nucleus.
Which means that the coulombic
force of attraction between
the same 1-electron and now a
doubly charged nucleus is
going to be stronger.
So the first orbit is going
to get pulled in.
And all the other orbits are
going to get pulled in.
By how much are they going
to get pulled in?
By that much.
So this is the functional
representation
of all of that physics.
All right, there's three
equations, three unknowns.
Let's look at energy.
So if you go through and solve
for energy, you get this one.
Minus this big monstrosity, mass
of the electron to the
fourth power of the elementary
charge times 8 times the
square of the permittivity of
vacuum times the square of the
Planck constant, all times the
square of the Proton number
divided by the square of
the quantum number.
And I just to make sure
everybody is with me here.
I always want to write n here.
n equals 1, 2, 3, takes on
integer values.
And we'll say it again here.
N equals 1, 2, 3, et
cetera, et cetera.
So again we say we see that
e is a function of n.
It's discretized.
It's quantized.
Why?
Because once you impose the
quantum condition here on
angular momentum, it propagates
through the entire model.
So radius this quantized, energy
is quantized, you're
going to see velocity
is quantized.
Because the quantum condition
is pervasive.
So e to the n, and I'm going to
take this whole quantity in
parentheses and just
call it giant K.
These are all positive
quantities.
Mass is positive.
And squares and fourth powers
of numbers must be positive.
So this is K times Z squared
over n squared That's good.
And we can go and evaluate K.
And when we evaluate L in SI
units, we get 2.18 times 10 to
the minus 18 joules.
This is joules per atom.
Or you can multiply this
by Avogadro's number.
If you multiply it by Avogadro's
number, then that
will give you 1.312 megajoules
per mol.
So that's the energy of the
electron in the ground state
of atomic hydrogen.
And then we can mediate that
with Z and n, and go to
electrons that are outside the
ground state, above the ground
state, or ground state electrons
in atoms that have
more than 1 proton, or both.
And so let's take a look
at the graphical
representation of that.
So instead of Cartesian
coordinates, because this is
spatial distribution, I'm
going to go to energy
coordinates and give you an
energy level diagram.
So again, not to scale.
Because this thing goes
is 1 over the square.
So that's going to be messy.
So let's start here.
And on the left side I'm going
to designate the energy.
And on the right side
I'm going to
designate the quantum number.
I'm going to start down here.
That's the lowest energy.
You see these are all negative
values, first of all.
They're all negative values.
Because Z is a square,
n is a square, and K
is a positive quantity.
So n equals 1 is the
lowest state.
It's the ground state.
It has a value of minus K.
So I'm going to do this one
just for atomic hydrogen.
So I'm going to write
atomic H.
So now Z equals 1.
You can do it later for Z
equals 2, 3, whatever.
So this is atomic hydrogen.
So ground state energy
is minus K.
What happens if we go to n
equals 2? n equals 2 it
becomes K divided
by 2 squared 4.
So it should be 3/4
of the way up.
I'm not going to go quite
3/4 of the way.
Because I want to leave room
for some fine structure.
That's why it's not to scale.
All right so this is
minus L over 4.
What if we go to n equals 3?
Well it's not symmetric here.
It's nonlinear.
But this should be
really what?
Minus K over 3 squared is 9.
You get the picture.
3, you can go 4, 5, and so on
until n equals infinity.
What happens when n
equals infinity?
I've got minus K over infinity,
which is vanishingly
small, zero.
Where is the electron when
n equals infinity?
r is n squared times
the Bohr radius.
That's a great, great
distance away.
What does it mean?
Physically it means that the
electron is so far away that
it is no longer bound.
It's no longer part
of the atom.
And when it's no longer part of
the atom, and the potential
energy that's stored is a result
of the charges coming
together from infinity.
I'm starting to talk like
someone out of 802.
What's the energy if I take 2
charged particles at infinite
separation?
I bring them into some
finite separation.
Voila.
There it is.
So when they're in infinite
separation
there's no energy stored.
Hence, you are at that point.
So n equals infinity means r
equals infinity, which means E
equal zero.
There's no stored energy.
So this means the electron
is no longer bound.
And therefore, if it's
no longer bound, we
have a term for it.
It's called free.
It's a free electron.
And if the electron
is free, then the
atom is electron deficient.
So if the electron is
free, that means the
atom is now an ion.
Because it's lost an electron.
It's no longer net neutral.
Or we say an atom hasn't
turned into an ion.
Or the electron has
been ionized.
What's the energy for that?
We can calculate what
that energy is.
It would be called the
ionization energy.
So if I started with an electron
down in here, and I
sent it all the way
to infinity.
See that's an energy space.
Which is the equivalent in
Cartesian space to go from
here to infinity, same idea.
Do you see the models?
This is Cartesian.
It's like a.
Map This is energy
coordinates.
It's different.
And you're going to be
able to think from
one model to another.
What's the energy
consequences?
What are the Cartesian
consequences?
Until we get to the point where
there is no Cartesian
representation.
Because the abstraction level
is too high, we'll have to
content ourselves with this.
So get comfortable moving
from there to there.
And then some day we're going
to say it's too complicated.
There's no Cartesian thing.
We'll be comfortable
by then with this.
OK so now we're taking an
electron from here up to here.
While we ask, what is the
ionization energy?
The ionization energy must
equal the delta E of the
transition.
So what's that?
The delta E of the transition
is going to equal always E
final minus E initial.
E final minus E initial.
Which is equal to E
at infinity minus
E1, the ground state.
Well E infinity, we just
said, is zero.
And the ground state energy
is equal to minus K.
So minus minus K is K.
So you also get the energy.
From infinity down to
ground state is
the ionization energy.
So we can define the ionization
energy in terms of
this transition.
Define ionization energy as the
minimum energy to remove
an electron from the
ground state of an
atom in a gas phase.
So that means there's no
solids, no liquids.
There's no work function here.
There's no lattice energy,
and so on.
So there's a definition of
the ionization energy.
And we can be a little
bit more elaborate.
Even though right now I'm
just going to do a
little break here.
I don't want to mislead
people.
But just an aside.
I'm nonlinear.
I can have multiple
conversations at once.
And you are capable
of stacking.
So we're going to break now.
We're not going to talk
about 1-electron atom.
We're going to follow the thread
of ionization energy.
I'm going to take lithium.
Lithium in its normal state has
3 protons, 3 electrons.
So I'm going to take
lithium gas.
And I'm going to ionize it
and make lithium plus.
So this is a lithium plus
ion in the gas phase.
It's still got 2 electrons.
So this isn't Bohr
model stuff.
But anyway, here we are.
So the energy for this action
would be called
the ionization energy.
Because I took a neutral atom,
and I pulled an electron out
of the ground state,
and so on.
So this is an ionization
energy.
But now I can continue
this process.
And I can take Lithium plus.
And I can lose an electron from
that, which will than
give me lithium 2 plus.
And this is called also
an ionization energy.
This is called the second
ionization energy.
So this is the first
ionization energy.
But just as when you write an
equation, when the coefficient
is 1, you don't write the 1.
I don't write 1 lithium here.
I know it's 1.
This is the second ionization
energy.
And then I can keep stripping
away electrons.
And I can take lithium 2 plus
in the gas phase, and take
away that electron leaving me
with just the lithium nucleus,
lithium 3 plus, plus electron.
And that's called the third
ionization energy.
OK, now what can we say?
What's the relationship here
between any of this except the
definition and the Bohr model?
well?
The Bohr model applies only to
1-electron atoms. Are there
any 1-electron atoms
on this board?
So we can calculate
the energy, the
third ionization energy.
We can get that from
the Bohr model.
You can do it in your head.
You can do it in your
head right?
It's just K times Z squared
right here.
It's going from 1.
So if this is 2.18, it's going
to be 9 times that trivially.
OK, so this is just
3 squared times K.
And this when you have to
get from the literature.
So you have to go to
primary sources.
Which is why you're going
to learn how to
use the proper database.
And this one here also you
get from the literature.
But this is on your
periodic table.
Your periodic table, one of the
data points it gives is
the first ionization energy
of all of the elements.
And so even though lithium
normally is a solid at room
temperature, the ionization
energy for lithium as given on
your periodic table is
for this reaction.
It's for the gas.
Anyway.
So that's little aside.
All right, the last quantity
that we could get from the
Bohr model is v, the velocity.
And I solve for the velocity.
We don't talk about
this very much.
But we're going to
do so once today.
And here it is.
So I went through the algebra.
And you get nh over 2 pi mr,
where r is the radius.
There's the quantum number.
This is quantized as well.
So I regrouped this.
And I already have a nice, cool
expression for r in terms
of the Bohr radius.
So I use that because
that's on the table.
So this is 2 pi times the mass
of the electron times the Bohr
radius, 1/2 angstrom, times Z
proton number divided by n,
where n equals 1,
2, 3, and so on.
So let's again get
a sense of scale.
So let's try for
sense of scale.
Let's do velocity of the ground
state electron in
atomic hydrogen.
So that means Z equals
1, n equals 1.
So I plug in the numbers.
And I get v1 for hydrogen,
atomic hydrogen, gives me 2.18
times 10 to the 6 meters
for second.
I don't know.
Is that fast?
Is that slow?
I don't know.
But I do know this much.
I know that the speed of light
is equal to 3 times 10 to the
8 meters per second.
So 10 to the 8 divided
by 10 to the 6.
2 and 3, that's roughly 1,
speaking as an engineer.
Who cares?
So this is about 1% of
the speed of light.
That's pretty good.
That gives me something
I can hang on to.
I would say that if this thing
is zipping around at 1% of the
speed of light, I would say
that's relatively fast. One
more time, 1% of the
speed of light.
That's relatively fast.
Remember last day I told you
that Bohr simply dismissed the
concept of the use of classical
electrodynamics down
to atomic dimensions.
Well here's another example of
why a lot of these assumptions
aren't going to work so well.
We're talking about the ground
state electron velocity in
this putative planetary model
of a 1-electron atom.
You're already getting into
relativistic effects.
So, just another example.
By the way, why do we use the
letter c for speed of light?
It comes from the latin word
celeritas, which means swiftness.
And we get the modern
word acceleration,
deceleration from that.
OK.
All right.
So the Bohr model, we've
now rolled it all out.
We have the energy portrait.
We have the radii, discrete.
We have quantization.
And we have velocities
if we ever want to
look at those again.
Now what's the next thing
we do in science?
We compare the predictions of
the Bohr model with data.
Are there any data
to support this?
Because remember, all Rutherford
said was plum
pudding doesn't make sense.
Instead I'm going to concentrate
the positive mass
in the center.
And then Bohr came along and
said, not only is it going to
be a planetary model, I'm going
to have circular orbits.
So now we've gone a long way
from Geiger-Marsden.
So is there any data?
Well there were data in 1853.
Remember, Bohr published
this in 1913.
In 1853 there was a
spectroscopist by the name--
I'm going to tell
you his name--
in Uppsala, Sweden.
And his name was Angstrom.
Angstrom was doing experiments
on hydrogen in
gas discharge tubes.
So he measured emissions from
gas discharge tube.
And it was filled with
various gases
including atomic hydrogen.
And in order to take his
data, what he used
was this device here--
there's the Bohr radius,
just an example.
He used the prism
spectrograph.
So here's a gas discharge
tube.
And I'm going to show you the
physics of that in a second.
Basically you've got a
pair of electrodes.
You've got gas in the tube.
And as this cartoon shows, you
apply a potential across the
electrodes.
And beyond a certain threshold
potential, the
tube begins to glow.
And the glow goes in
all directions.
And a blinds you when
you're in the lab.
So what you do, is you
cover this up a bit.
You have a narrow slit.
And then you force the light
to come through in a thin
ribbon, and then expose
it to a prism.
What the prism does, is it takes
the light and breaks it
into its components, sort of
rainbow-like, and magnifies
the difference.
As refraction goes, different
wavelengths will refract
different amounts.
And then you shoot this
across the room.
There's two counters.
One is a scintillation screen
and an army of graduate
students who sit there
in the dark.
But they're no good because you
can't stick them into the
publication.
You need to have data that
people will rely upon.
So instead you use a
photographic plate.
And if you put even a tiny,
tiny angle of separation
across a great enough distance,
you start to get
enough line splitting
that you can see.
And then you go backwards.
And you know the
geometry here.
And you can figure out what the
wavelength must have been
to go this distance, et
cetera, et cetera.
And these are all color coded,
not because they had color
film in those days, but just
to let you know that 656
nanometers, if you were the
graduate student sitting
there, you'd see a red line, a
green line, a blue line, and a
violent line.
So that's how he made
the measurements.
And he published those
measurements.
And so they lay.
And then the story gets
a little thicker.
In 1885, there's a Swiss high
school math teacher.
I mean, I can't make
this stuff.
This is true story.
There's a Swiss high school math
teacher by the name of J.
J.
Balmer.
We had J.
J.
Thomson.
Now we've got J.
J.
Balmer.
And J.
J.
Balmer, he loved to
play with numbers.
And he was studying
this set of lines.
And he was trying to come
up with a pattern.
Can you see a pattern there?
410 434 486, 656, do
they go squares?
Are they primes?
What's the pattern there?
So Balmer puzzled over
this for awhile.
And he finally came up with
the equation to represent
those lines.
So he studied Angstrom's
data found the pattern.
And here's the pattern
that he found.
He said that those
are wavelengths.
If I take instead wave number,
nu bar is called wave number,
which is the reciprocal
of the wavelength.
So if I take the reciprocal of
the wavelength, I end up with
those 4 lines conforming to a
series that goes like this.
1 over 2 squared minus 1
over n squared, where n
equals 3, 4, 5, 6.
And there's a constant here
which we're going
to designate r.
And the value of R--
I'm going to put that
on the next board--
the value of R as expressed in
SI units today, would be 1.1
times 10 to the 7 reciprocal
meters.
Wavelength is a meter.
Reciprocal wavelength
or wave number must
be reciprocal meters.
So how this all of this support
the Bohr model?
Well in order to explain it,
I've got a first tell you what
the physics of the gas
discharge tube are.
So let's go inside the gas
discharge tube and understand
those physics.
So here's the gas
discharge tube.
It's made are borosilicate
glass.
And we fill it with gas.
And in this case, the gas is
going to contain among other
things hydrogen.
So this is a hydrogen
gas phase atom.
And this is probably
at low pressure.
And then I said I
need electrode.
I'll get the electrodes
inside.
So I've got to have a really
good glass blower who can make
a glass to metal seal,
and have a
feedthrough to an electrode.
So this is still
a vacuum seal.
How do they get the gas
in the first place?
We don't show you this
in the books.
I'll tell you because I
did this in my Ph.D.
What you do is you have a
little side tube here.
You evacuate.
This goes to a vacuum pump.
And then over here you
have a gas source.
You evacuate.
Put in the gas to whatever
pressure you want.
And then the glass blower
disconnects.
This pulls this down
to a reduced
pressure, and seals it.
But the books don't
show you that.
That's a secret.
So we've got a gas
at low pressure.
And I've got an electrode
over here and an
electrode over here.
And they're connected to a
variable voltage power supply.
So I'm going to put an arrow
with the V meaning it's
variable voltage.
I can change the voltage.
And this convention, this
is the negative side.
So this means the electrons
leave the power
supply and go like this.
Which means this electrode
will be negative.
And this electrode
will be positive.
And thanks to Michael Faraday,
we will call
this one the cathode.
And this one we will
call the anode.
Now what happens?
We start turning up the
pressure, turning up the
voltage rather.
Low pressure here, but
electrical pressure is voltage.
The voltage gets high enough,
eventually the electrons will
boil off the cathode.
And this is a gas
at low pressure.
And they will accelerate from
rest and go all the way across
the tube and crash
into the anode.
And we complete the circuit.
But if there's a gas in here,
some of these electrons are
going to hit the
gas molecules.
And when they hit the gas
molecules, if they have enough
energy to do so, they will cause
electrons inside the gas
molecules to be excited.
And if they're excited enough,
the electrons will jump up to
a higher energy level.
But they can't be sustained.
Because this is a ballistic
collision.
It's a one of.
It's like a bowling alley.
One ball, one pin, one impact.
Now the pin is in the air.
What happens to the pin?
It falls back down.
Why?
Because gravity pulls it down.
In this case, you've got the
energetics pulling the
electron back down.
Now the electron goes from high
energy to low energy.
And when that happens, the
energy difference is given off
in the form of a photon.
So I get photon emission when
this falls back down.
And this photon has
a wavelength.
What I'm going to show you is
that the set of lines that you
get from exactly this
configuration using this
equation give you the
Balmer series.
So now you've got a model that
Bohr postulated for atomic
hydrogen on 1-electron atom that
exactly predicts that set
of 4 lines.
Which were measured
50 years before.
So let's go.
So first of all let's
get the energy here.
And I'm going to get the energy
of this electron.
This electron I'm going to call
a ballistic electron.
Why do I call it a ballistic
electron?
Because it's not bound.
It's free.
It boils off the cathode, flies
through free space, and
crashes into an anode.
Clearly it's not part
of an atom.
But there's a second electron
in this story.
And it's the ground state
electron in hydrogen.
And it lives here.
So what's the energy of the
ballistic electron?
Well that's just 1/2 mv squared
And where did it get
its energy from?
It got its energy from
the power supply.
And what's the electrostatic
energy?
It's the product of the charge
on the species times the
voltage through which
it was accelerated.
So away we go.
I know the charge on the
electron is minus E.
Whatever the voltage is
there, 1 volt, 10
volts, 100 volts, whatever.
Away we go.
By the way, I'm going to show
you just one other thing in
terms of order of magnitude.
The kinds of voltages you see
along here are 1 volt, 10
volts, that sort of thing.
So suppose, to get an order of
magnitude, suppose we had a
species of charge E.
So in other words, it's only 1
times the elementary charge.
A species of charge
E influenced by
voltage of 1 volt.
So this is 1 in the voltage
units, and 1 in the elementary
charge units.
How much energy would that be?
That's equivalent to making
1 volt and accelerate an
electron from rest
across this gap.
And the result would be, the
energy then would simply equal
1.6 times 10 to the minus 19
coulombs times 1 volt.
And what's the energy
going to be?
Well I've got coulombs
times volts.
And I don't know how I convert
one to the other.
I don't have to.
Why not?
Because that's an SI unit.
And that's an SI unit.
This is an energy.
So with impunity, I write
1.6 times 10 to
the minus 19 joules.
That's the beauty of SI units.
So that's a good news.
I know it's joules.
The bad news is I hate
this number.
It's a stupid number, 1.6 times
10 to the minus 19.
it's crazy.
Why don't I come up with
a number like 3, 7?
So what I could do,
is I could define.
I could define a unit such
that when the elementary
charge is accelerated across the
unit voltage, I would call
that unit 1 electron volt.
And so somebody thought
of this before me.
And hence, this is the unit
of the electron volt.
It takes these crazy things that
we've been spewing here
up until now, and rationalizes
them into numbers that people
can carry around
in their heads.
So what's K now?
K is 2.18 times 10 to
the minus 18 joules.
Yuck!
Let's convert that to
electron volts.
So I divide by 1.6 times
10 to the minus 19.
And I got 13.6 electron volts.
You'll remember that on your
death bed, ionization energy
of atomic hydrogen.
Maybe we don't have to give
out tables of constants.
You just know this stuff.
It's OK.
All right.
Now one last thing about this.
So this has got gas in it.
This is the cathode.
And this beam of electrons, back
in the 1800s, there was a
popular term, it was
called the ray.
So instead of a beam of
light, people refer
to the ray of light.
So then when they got to
particle beams, they talk to
them as rays.
So this is now not an electron
beam, it's an electron ray.
And it comes off the cathode.
And it's in a vacuum tube.
So this could be called a
cathode ray tube, a CRT.
Now see, I could flatten this.
And I could spray it
with phosphors.
And then I could put some
charge plates here.
The electrons have a
negative charge.
So if I charge these plates, and
I was clever about how I
charge them and varied the
charge, I could raster the
electron beam like this about
30 times a second all up and
down the screen.
And then I could put some
program signal in there.
And I could sit here.
And I could watch TV.
It all started with the
gas discharge tube.
It says nothing about the
content unfortunately.
Very nice physics,
but no content.
All right.
So now we've got the electron.
The electron is moving, the
ballistic electron.
Now I want to look at what
happens when the ballistic
electron smashes into one of
those hydrogen atoms. So let's
go back over here.
So here's the incident
particle.
And it's going to be an
electron in this case.
So I'm going to designate
this electron.
This is my ballistic electron.
So here's the incident
electron.
And this is ballistic,
just to be clear.
It's the ballistic incident
electron.
Now this is a mixed
metaphor here.
Because I'm representing this
in Cartesian space.
But I've moved into
energy space.
So some people are going
to get really upset.
Because they're going to say,
well this is Cartesian, but
this isn't.
It doesn't matter.
It's my lecture.
It's my model.
It works.
I'm the professor.
So we've got this mixed
metaphor here.
But anyways, it helps a lot.
So what happens when this
thing comes in?
It depends on how much
energy it has.
Now if the incident energy,
if E incident is
tiny, nothing happens.
This thing just zooms
right on through.
But if the incident energy, if
E of the incident ballistic
electron is greater than
delta E for any
transition that's feasible--
and in this case I'm going to
assume I don't have thermal
distribution of electrons.
If I gave you Avogadro's number
of hydrogen atoms
because of the thermal
distribution of energies-- and
we'll come back to
this later--
there might actually be, at
any moment, some electrons
that are thermally excited
above the ground state.
We're going to forget
about that for now.
We're going to spiral up the
learning curve here.
So first time we're going to
assume all the electrons are
in the ground state.
If I don't enough energy to go
from n equals 1 to n equals 2,
nothing happens.
If I have more than enough
energy to go for n equals 1 to
n equals 2, I will
take that energy.
And the electron will jump,
steal that amount of energy,
and then this thing moves on--
and I'm purposely making this
vector shorter than the
incident vector--
with that amount
of energy raw.
And this is called the
scattered electron.
Go back to the bowling
ball analogy.
The bowling ball comes in with
a certain energy, hits the
pin, continues to roll.
But you know that there's a loss
of kinetic energy in the
bowling ball.
That's what we're seeing here.
It's purely ballistic.
Now let's say we do have more
than enough energy.
Suppose I have enough energy to
go from n equals 1 halfway
between n equals 2
and n equals 3.
There's only n equals
1, n equals 2.
I can't take the electron
up to n equals 2.3.
It's unallowed.
These are the only
allowed states.
So that differential amount of
energy then resides with the
electron that's ballistic.
And it moves on here.
So we've got conservation
of energy here.
We can say that E incident will
then equal the energy
that's lost in the transition
plus the energy that's still
left with the scattered
electron.
And we can calculate what that
transitional energy is.
That transitional energy
is going to equal
1/2 mv squared incident.
This is the velocity, the
incident electron.
What's the energy to go from
n equals 1 to n equals n?
Whatever it is, its minus K
times Z squared If it's
hydrogen, Z is 1.
It's going to be 1 over nf.
The final quantum number squared
minus 1 over the
square of the initial
quantum number.
And then, what's left over after
this has been robbed
from the incident ballistic
energy is 1/2 mv squared of
the scattered ballistic
electron.
And the quantization dictates
that only if E incident is
greater than delta E going 1
up to n-- here I'm assuming
everything is in its
ground state.
Later on we're going to
be more sophisticated.
But for first time
through, all are
ground state electrons.
I have to have enough energy to
go at least to n equals 2.
I can go to n equals
3, m equals 4.
In principle, if this thing had
more than 13.6 electron
volts, what would happen?
It would kick this
electron out.
Gone!
And you'd have 2
free electrons.
So if it's greater than this,
then the consequence is
electron promotion.
So we're moving along.
But this excited state
is unstable.
This excited state is unstable
because it's like the bowling
pin that got thrown up.
So what happens?
The electron standing up there
on n equals 2, for example,
looking down to n equals 1.
And it falls.
When it falls it gives
off radiation.
And that radiation is
conservation of energy there.
And what we know is when it
gives off an energy of the
emitted photon, the energy of
the emitted photon must equal
delta E of the transition
falling from 2 to 1.
And we know how to
calculate that.
That's just that thing
flipped around.
And this thing is
equal to what?
This is equal to h nu.
According to Planck it's
hc over lambda is
equal to hc nu bar.
You know what this one is.
This is minus KZ squared
1 over-- in this case
it's going to be--
1 over nf squared 1 over 1
squared minus 1 over, in this
case, 1 over 2 squared And you
can generalize this 1 over nf.
So where am I going with this?
Well I'm going to flip
all of this around.
And when I flip it all around,
what I'm going to end up with
is this equation here.
And better than that,
I'm going to end
up with this equation.
And I'm going to end up with
this as the constant.
And when I get that, we're going
to say Bohr has done it.
The data support the theory.
So that's what we're
going to do.
But I think we're going to
stop at this point today.
So let me just jump here.
I mentioned to you that if you
go here on your periodic
table, there's the 13.6
electron volts.
In the case of lithium, this
is 5.4 electron volts.
So you can see all the
various values.
This by the way, people
no noise.
No noise.
It's 11:52.
I'm holding court until 11:55.
I'm simply changing topics.
It's not, oh this is
the part where I
can talk to my neighbor.
What are the rules?
No talking.
No food.
No horseplay until 11:55.
Then, still no horseplay,
gentle
talking, no food, no drink.
This is a sacred space.
I'm not kidding you.
Do you know why?
In this secular America, this
is sacred space because this
is where people learn.
The lecture hall is
sacred space.
Now, here's the 13.6
electron volts.
And there's the 1.6 times 10
to the minus 19 joules.
And if you multiply those 2,
you'll get the 2.18 over here.
All right.
Here's a cartoon showing the
photon, higher energy orbit,
lower energy orbit, electron
emission transition, right out
of your book.
And there's a postulate 6.
And we're going to finish this
up at the beginning of the
Friday lecture.
So here's the whole series from
n equals 3 to 2, n equals
4 to 2, and so on.
See how that works.
OK.
One of the things that we've
learned here, is that it
doesn't matter what the incident
energy is here.
The emission is characteristic
of the energy
levels inside the gas.
Instead of using an incident
electron, I could use an
incident proton.
I could use an incident
alpha particle.
I could use an incident
neutron.
Anything that has enough energy
to kick this up from n
equals 1 to n equals 2 will
result in photon emission of
this frequency.
That means the set of those
lines is unique to the target.
And this is the beginning
of chemical analysis.
I can use this to characterize
species, to it, stars.
How do we analyze the
composition a stars?
Well, do we send a NASA
spaceship out 25 light-years
and grab some gas and bring
it back to the lab?
No.
All we've got is the
spectrograph.
The star is hot.
That means thermal excitation
and cascading
down with photon emission.
And the lines we get are related
to the energy levels
within the stars.
So from a distance of a 100
light-years, I can tell you
what the composition is.
And if there's two gases
there, what if there's
hydrogen and helium?
the helium lines
will be there.
And they'll be superimposed on
the hydrogen lines unless they
lie directly on top
of one another.
I'm going to be able to figure
out what's there.
That's how it works.
Now here's a story about an
astronomer, Cecilia Payne.
She's the first woman
graduate student in
astronomy at Harvard.
She went on to chair the Faculty
of Arts and Sciences,
awarded tenure, but denied a
professorship for 18 years
because she was a woman.
And in her thesis, she was the
first person to figure out to
the sun is dominantly hydrogen,
not iron, which is
what most astronomers thought.
Why?
Well, because the earth
is made of iron.
Meteorites are made of iron.
The whole universe must
be made of iron.
Never mind the fact that
the sun is glowing.
So here's how spectroscopy
works.
Look at this.
See, what does this mean?
This is an analogy.
What could that mean?
Well, they said iron again.
I know this is misspelled.
But maybe it's just a glitch
in the instrumentation.
So most people would look
at that and say,
the message is iron.
But it's not about the word.
It's about the pattern.
It's about the pattern.
And that's what's
spectroscopy is.
By the way, if you somehow
didn't catch this lecture.
And you walked in, and all you
saw was those four lines.
You know those four lines.
That set of lines is
characteristic of atomic
hydrogen, and nothing else.
By the way, those lines
are very faint.
This is not to scale.
And they're so faint,
they're ghostlike.
And what is the latin
word for ghost?
Specter.
So what is a spectrum?
It is a set of ghostlike
lines.
To this day, the term
spectroscopy refers to the
ability to study data that are
so faint they're ghostlike.
All right, we'll see
you on Friday.
