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--fundamental frequency,
here, nu.

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Now, I should tell you that the
model that we used to get these

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vibrational energies for a
molecule is actually called a

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harmonic oscillator model.

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It is a model.
And it predicts that the

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spacings between these energies
are all equal.

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The reality is,
that the spacings are not all

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equal.
The reality is that this

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molecule is an anharmonic
oscillator.

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And that although way down here
in the well, the spacings are

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just about equal,
v equal zero,

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v equal one,
v equal two,

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as you get further up in the
well they do start to come

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together.
And they actually converge to

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the dissociation limit.
You don't have to know that,

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right now.
You will see that in later

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courses, but I just wanted to
make you aware of that,

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that the harmonic oscillator
model works pretty well down

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here.
But the harmonic oscillator

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potential function actually
looks like that.

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A real potential function looks
like that.

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This becomes anharmonic.
So those spacings do get closer

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together.
That is for the future.

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Now, you can also put enough
vibrational energy into the

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molecule to break a bond.
When you get up to here,

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you can put enough vibrational
energy.

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And this hydrogen and the
chlorine, as they oscillate out,

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they will just keep going,
If you put enough energy.

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You get up here,
and when they stretch,

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they will just keep on their
merry way.

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They won't come back.
There won't be a restoring

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force.
So, by putting a lot of

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vibrational energy into the
molecule, you can break your

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bond.
And in fact,

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that is what happens when you
break the bond.

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You have a lot of vibrational
energy in that bond,

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00:02:51 --> 00:02:54
and the two atoms just keep
flying apart.

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That was our diatomic molecule.
What I now want to just talk

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briefly about are polyatomic
molecules because we said in a

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polyatomic molecule,
such as water,

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we have several different
vibrational modes.

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And each of those vibrational
modes actually has a different

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fundamental frequency.
And each of those vibrational

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modes can be represented with
this sort of interaction

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potential.
Sometimes it is not as simple a

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coordinate system because you
have some bends so you have to

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plot this as a function of
angles, which is not easy to

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draw.
But each one of the vibrational

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modes in a polyatomic molecule
can, in effect,

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be represented by some kind of
energy of interaction,

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like we drew here for this
diatomic molecule.

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It is just a little bit more
complicated because your

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coordinate system is a little
more complicated.

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Particularly when you start
talking about bends.

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But let's take a look here on
the side walls.

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What I am showing you is a
vibrational spectrum of water.

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And, again, this is some
infrared radiation that is being

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directed at a sample of water
molecules.

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And we are measuring the
intensity of that radiation at a

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photo detector as a function of
the frequency of the radiation

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going through our sample.
And when the molecule absorbs

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radiation at that frequency,
that intensity at the detector

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goes down.
What you see,

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here, is just a hypothetical
spectrum for water.

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You see that the molecule is
absorbing some radiation,

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here, at 1,595 wave numbers.
Well, that 1,595 corresponds to

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the fundamental frequency of
vibration of this hydrogen

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bending mode.
That is 1,595.

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That is the frequency in
wavenumbers with which that

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hydrogen-oxygen-hydrogen bond is
bending or vibrating.

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This is also the energy between
the v equal zero and v equal one

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mode.
This represents excitation from

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v equal zero to v equal one.
And then, way up here,

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you see another transition at
3,652.

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Well that, we know to be the
symmetric stretch.

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That is, both hydrogens moving
in or out at the same time.

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3,652 wave numbers is the
fundamental frequency for that

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particular vibration.
It also represents a transition

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from v equal zero to v equal
one.

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And then, finally,
at 3,756, that is the

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anti-symmetric stretch,
where one of the hydrogens is

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moving out and one of the
hydrogens is moving in.

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Anti-symmetric stretch.
It also is the v equal zero to

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v equal one transition.
And so, that is what you would

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measure on an infrared spectrum
of water.

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Actually, infrared plus Raman,
but we will leave that out.

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And what is interesting is that
any molecule that has an OH

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stretch in it,
or any molecule that has an OH

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group in it --
If you take an infrared

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spectrum of it,
what you are going to see is

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that it has a transition
somewhere in between 3,400 wave

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numbers and about 3,800 wave
numbers.

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If you had an unknown compound
and you put it into your

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infrared spectrometer and saw a
transition somewhere in between

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3,400 and 3,800,
immediately that is going to

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clue you in that you have an OH
stretch.

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Because now other molecules
that are at all common are going

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to have a fundamental frequency
in that range.

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You can see how this infrared
spectroscopy can be used as an

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analytical tool to figure out
what molecule you have.

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And, of course,
the actual frequencies,

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then, are a fingerprint of the
molecule.

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But, just in general,
if you did not know anything

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about the molecule and you saw a
stretch in this range,

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you know you got an OH bond,
there, that is undergoing a

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symmetric or an anti-symmetric
stretch if you have two OH

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bonds.
Likewise, say you had a

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carbon-hydrogen bond,
what you would find is that all

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carbon-hydrogen bonds have
fundamental frequencies from

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about 2,800 wave numbers to
3,100 wave numbers.

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If you see a transition in that
range, you know that you have a

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hydrogen bonded to a carbon.
Because, again,

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there really isn't anything
else that is common that has a

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vibrational frequency in that
range.

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When you get to a little lower
frequencies, well,

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then it is a little more
difficult because there are lots

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of different other modes that
have vibrations,

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which are a little bit lower
frequency.

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But, again, the specific number
will identify the molecule for

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you.
One other thing is that just in

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general, bending modes have
lower frequencies than stretch

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modes.
That is a general statement

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that is true.
Also, generally symmetric

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stretches have lower frequencies
than anti-symmetric stretches.

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That is true.
And you will see more of this

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infrared spectroscopy used as an
analytical tool,

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essentially,
when you take some organic

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chemistry.
Well, what I want to talk about

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now is the other internal degree
of freedom in molecules.

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That is molecular rotations.
Well, molecular rotations are

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quantized, just like molecular
vibrations are.

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Let me take my HCl again and
draw this intermolecular

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interaction potential.
And let me put my v equal zero

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level down here.
Then, just for ease of my

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diagram, I am going to put my v
equal one level up here.

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Rotational levels,
they are quantized.

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And the rotational quantum
number is given the symbol J.

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For example,
if you have a molecule in the

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first rotational state,
that first rotational state,

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then, will be right here.
We will call that J equal one.

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00:11:00 --> 00:11:06
And then, if you have it in the
second excited rotational state,

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that is going to be right
there.

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00:11:09 --> 00:11:14
We will call it J equal two.
And the third rotational state

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is going to be right there.
We will call it J equal three.

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And the fourth,
J equals four.

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Now, if you have a molecule in
the ground rotational state,

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that ground rotational state,
here, is J equal zero.

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And it is sitting here right on
top of the v equal zero level

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for the ground rotational state.
The bottom line is,

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you can have a molecule in the
ground vibrational state and the

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ground rotational state.
If that is the case,

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this is how much energy it has.
When you are in the ground

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rotational state,
that is zero energy,

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as we are going to see in a
moment.

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You can have a molecule in the
ground vibrational state in the

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first excited rotational state.
If that is the case,

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that is the energy.
You can have a molecule in the

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ground vibrational state and the
second excited rotational state.

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Then it has that energy.
You can have a molecule in the

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ground vibrational state in the
third excited rotational state.

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It has that energy.
But you can also have a

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molecule in the first excited
vibrational state,

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right here, and in the J equal
zero state.

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Well, if that is the case,
it has that energy.

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You could also have a molecule
in the first excited vibrational

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state and the first excited
rotational state.

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Well, then it has that energy.
Or, a molecule in the first

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excited vibrational state and
the second rotational state.

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Well, then it has that energy.
Each one of these vibrational

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states has, on top of it,
a manifold of rotational

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states.
The rotations and vibrations,

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for our purpose,
are not coupled.

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You can have so much in
vibration, so much in rotation.

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I want you to also notice,
that the difference in energies

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between rotational states is
much smaller than the difference

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in energies between vibrational
states.

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That is a general statement
that is correct.

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Now, we have a nice analytical
expression, again,

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for the allowed rotational
energies of molecules,

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and that analytical expression
is the following.

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E sub J is equal to h squared
times J, J plus one over 8 pi

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00:14:03 --> 00:14:08
squared times I.

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I is the moment of inertia.

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I will explain that in just a
moment.

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What does this say?
It says that when J is equal to

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zero, the rotational energy,
here, is equal to zero because

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it makes this all go away.
So, the molecule has no

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rotational energy in J equal
zero.

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When J is equal to one,
we put in J equal one,

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we calculate that,
and it comes out to be h

196
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squared over 4pi squared times
the moment of inertia.

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This is J equal one.

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For J equal two,
the molecule has 3h squared

199
00:15:01 --> 00:15:05
over 4pi squared amount of
rotational energy.

200
00:15:05 --> 00:15:11
For J equal three,

201
00:15:11 --> 00:15:17
it has 3h squared over 2 pi
squared times the moment of

202
00:15:17 --> 00:15:20
inertia.

203
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The energies go up.
Suppose I have a molecule in v

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equal zero, J equal zero,
and it makes the transition,

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here, to J equal one,
what is the energy difference

206
00:15:35 --> 00:15:42
between those states?
Well, J equal one minus energy

207
00:15:42 --> 00:15:46
J equal zero.
That is h squared over 4pi

208
00:15:46 --> 00:15:51
squared times I minus zero.

209
00:15:51 --> 00:15:59
That is just h squared over 4pi
squared I.

210
00:15:59 --> 00:16:03
How do we make that transition?
Well, to make this transition

211
00:16:03 --> 00:16:08
right here, we are going to need
a photon, and that photon is

212
00:16:08 --> 00:16:13
going to have to have an energy
exactly equal to this.

213
00:16:13 --> 00:16:17
Our photon E equal h nu,
that has got to be equal to h

214
00:16:17 --> 00:16:22
squared 4pi squared times I.

215
00:16:22 --> 00:16:27
I can solve for the frequency
of the photon that I need to

216
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make that transition.
The frequency of that photon

217
00:16:33 --> 00:16:36
then is h over 4pi squared times
I.

218
00:16:36 --> 00:16:43
That is
the frequency of the photon that

219
00:16:43 --> 00:16:47
I need.
Now, in the case of HCl,

220
00:16:47 --> 00:16:51
we said that HCl has two
rotational modes.

221
00:16:51 --> 00:16:58
It has a mode rotation around
this axis, and it also has a

222
00:16:58 --> 00:17:06
mode rotation around this axis,
in the plane of the board.

223
00:17:06 --> 00:17:09
These two rotations are
degenerate.

224
00:17:09 --> 00:17:14
If you put in a photon with
this frequency,

225
00:17:14 --> 00:17:21
here, it will excite either
this rotation or that rotation.

226
00:17:21 --> 00:17:25
Now, let's look on the side
wall here.

227
00:17:25 --> 00:17:32
How do we do the experiment?
Again, just like we do it in

228
00:17:32 --> 00:17:35
the infrared,
except that the energy of the

229
00:17:35 --> 00:17:39
photon we now need is in the
microwave range.

230
00:17:39 --> 00:17:43
Microwave spectra measure
rotational spectra of molecules.

231
00:17:43 --> 00:17:47
And so, again we have a
microwave radiation,

232
00:17:47 --> 00:17:51
monochromator,
coming out, going through the

233
00:17:51 --> 00:17:55
sample, photo detector,
look at the intensity of the

234
00:17:55 --> 00:18:00
photo detector as a function of
the frequency.

235
00:18:00 --> 00:18:04
At some frequency here,
you see a dip in that

236
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intensity.
Well, that is where the

237
00:18:07 --> 00:18:12
molecule absorbs.
And, in the case of HCl here,

238
00:18:12 --> 00:18:17
the frequency of that
absorption for J equal zero to J

239
00:18:17 --> 00:18:21
equal one, well,
that frequency occurs at

240
00:18:21 --> 00:18:26
6.3x10^11 hertz.
That would be the frequency of

241
00:18:26 --> 00:18:32
the photon that you would need
for absorption.

242
00:18:32 --> 00:18:33
Yes?
I'm sorry?

243
00:18:33 --> 00:18:37
We got into the microwave
range, here, because the

244
00:18:37 --> 00:18:43
spacings between the rotational
states are much lower than the

245
00:18:43 --> 00:18:46
spacings between the vibrational
states.

246
00:18:46 --> 00:18:50
And I just actually wanted to
make that point,

247
00:18:50 --> 00:18:53
here.
We can calculate in terms of

248
00:18:53 --> 00:18:57
the energy, here,
what this spacing is.

249
00:18:57 --> 00:19:00
Let's do that.

250
00:19:00 --> 00:19:05


251
00:19:05 --> 00:19:11
If we want to know the change
in energy from J equal zero to J

252
00:19:11 --> 00:19:19
equal one, that change in energy
is just h times the frequency of

253
00:19:19 --> 00:19:24
the photon, that photon,
there, that makes that

254
00:19:24 --> 00:19:30
transition happen.
You can plug that in,

255
00:19:30 --> 00:19:37
there, so we have 6.6261x10^-34
joule seconds,

256
00:19:37 --> 00:19:43
times the frequency
6.3479x10^11 hertz.

257
00:19:43 --> 00:19:50
When you do that,
you should get 4.2062x10^-22

258
00:19:50 --> 00:19:55
joules.
Or, if I convert that to

259
00:19:55 --> 00:20:03
kilojoules per mole,
it is 6.25330 kilojoules per

260
00:20:03 --> 00:20:08
mole.
In general, the difference in

261
00:20:08 --> 00:20:12
the spacings here,
the energy difference for

262
00:20:12 --> 00:20:18
vibration, we said is something
between three to 40 kilojoules

263
00:20:18 --> 00:20:22
per mole.
That is the difference between

264
00:20:22 --> 00:20:28
v equal zero and v equal one,
generally, for a large range of

265
00:20:28 --> 00:20:32
molecules.
The difference in the

266
00:20:32 --> 00:20:40
frequencies here for rotation,
delta E, that is more like

267
00:20:40 --> 00:20:47
something on the order of 0.01
to about 1.0 kilojoules per

268
00:20:47 --> 00:20:51
mole.
That is what is typical.

269
00:20:51 --> 00:20:58
So, rotations are much more
closely spaced in energy.

270
00:20:58 --> 00:21:02
Yes?
Well, in the case of the

271
00:21:02 --> 00:21:06
rotational spectra,
there is a little bit more

272
00:21:06 --> 00:21:09
diversity in what units are
used.

273
00:21:09 --> 00:21:13
And the reason is this.
They are more closely spaced.

274
00:21:13 --> 00:21:18
And the whole wavenumber came
into use in vibrational

275
00:21:18 --> 00:21:23
spectroscopy when the way people
would analyze the spectra would

276
00:21:23 --> 00:21:30
be to take a photographic plate
with the light coming in.

277
00:21:30 --> 00:21:33
And you would see the lines
separated in space.

278
00:21:33 --> 00:21:38
And people would take a ruler
in centimeters to measure the

279
00:21:38 --> 00:21:43
spacings between the lines.
That is historically how the

280
00:21:43 --> 00:21:48
wavenumber unit came to be.
The problem is that does not

281
00:21:48 --> 00:21:51
work so well in rotational
spectroscopy,

282
00:21:51 --> 00:21:56
often, because the spacings are
much closer together.

283
00:21:56 --> 00:22:01
And so, depending exactly on
what kind of diffraction grading

284
00:22:01 --> 00:22:06
used, the wavenumbers are not
always used.

285
00:22:06 --> 00:22:08
Sometimes they are the
frequency.

286
00:22:08 --> 00:22:13
We go back and forth.
And, in your homework problems,

287
00:22:13 --> 00:22:15
I go back and forth,
too.

288
00:22:15 --> 00:22:19
You really have to deal with
both kinds of units.

289
00:22:19 --> 00:22:23
But now, one of the
usefulnesses of this rotational

290
00:22:23 --> 00:22:28
spectroscopy in getting,
say, this frequency for the

291
00:22:28 --> 00:22:31
transition, one of the
usefulnesses,

292
00:22:31 --> 00:22:37
there, is to calculate the bond
length of the molecule.

293
00:22:37 --> 00:22:42
To determine the bond length of
the molecule really very

294
00:22:42 --> 00:22:46
accurately.
Because what we said over here

295
00:22:46 --> 00:22:51
is that the frequency for that
transition is this.

296
00:22:51 --> 00:22:57
The frequency of that
transition is h over 4pi squared

297
00:22:57 --> 00:23:02
times I.
And I is a moment of inertia.

298
00:23:02 --> 00:23:05
The moment of inertia is the
following.

299
00:23:05 --> 00:23:09
Maybe you have had this 8.01.
No, not yet?

300
00:23:09 --> 00:23:13
Oh, you will.
The moment of inertia is the

301
00:23:13 --> 00:23:19
reduced mass times the distance
between the two masses.

302
00:23:19 --> 00:23:22
In our case,
for the HCl molecule,

303
00:23:22 --> 00:23:28
it is equilibrium bond length,
r sub e squared.

304
00:23:28 --> 00:23:33
The reduced mass is what I gave
you before, m1 m2 over the sum

305
00:23:33 --> 00:23:36
of the two masses.

306
00:23:36 --> 00:23:41
Again, it is a way to reduce a
two-body problem to a one-body

307
00:23:41 --> 00:23:46
problem, where the one-body is
this fictitious body of reduced

308
00:23:46 --> 00:23:48
mass. But it is exact.

309
00:23:48 --> 00:23:51
There are no approximations.
This is correct.

310
00:23:51 --> 00:23:55
That is the moment of inertia
of the molecule.

311
00:23:55 --> 00:24:00
It is a property of the
molecule, depending on the mass

312
00:24:00 --> 00:24:05
and the bond length.
You can see that if I

313
00:24:05 --> 00:24:10
substitute that in there,
h over 4pi squared times nu r

314
00:24:10 --> 00:24:15
sub e squared,
and then I go and

315
00:24:15 --> 00:24:20
solve for r sub e,
well, r sub e is (h over 4pi

316
00:24:20 --> 00:24:23
squared mu times nu) to the
one-half.

317
00:24:23 --> 00:24:28
In the case of HCl

318
00:24:28 --> 00:24:32
--
Actually, I think this is all

319
00:24:32 --> 00:24:37
in my slide, here.
If I go and stick in the value

320
00:24:37 --> 00:24:42
for nu, the equilibrium bond
length, here,

321
00:24:42 --> 00:24:48
is really 1.2748x10^-10 meters.
We can really measure these

322
00:24:48 --> 00:24:53
frequencies with high precision
and high accuracy.

323
00:24:53 --> 00:24:58
All bond lengths,
really, come from measurements,

324
00:24:58 --> 00:25:05
now, of rotational spectra.
All bond lengths in the gas

325
00:25:05 --> 00:25:12
phase, wherever we can measure
the rotational spectra of the

326
00:25:12 --> 00:25:16
molecule.
That is one of the main uses

327
00:25:16 --> 00:25:20
for rotational spectroscopy.
Questions?

328
00:25:20 --> 00:25:28
If not, what I am going to do
is leave the subject of internal

329
00:25:28 --> 00:25:32
motion.
I want to talk for the rest of

330
00:25:32 --> 00:25:38
the hour, and a little bit on
Friday about another topic,

331
00:25:38 --> 00:25:42
which is intermolecular
attraction.

332
00:25:42 --> 00:25:47


333
00:25:47 --> 00:25:51
Interactions.
Or, I am going to write it here

334
00:25:51 --> 00:25:54
as attractions.
The bottom line is,

335
00:25:54 --> 00:25:58
I want to try to understand,
on a microscopic scale,

336
00:25:58 --> 00:26:05
deviations from the inner gas
law, PV equal nRT.

337
00:26:05 --> 00:26:12


338
00:26:12 --> 00:26:17
You know PV equal nRT,
that if I made a plot of the

339
00:26:17 --> 00:26:23
volume versus the temperature
and kept the pressure constant,

340
00:26:23 --> 00:26:28
say the pressure is at one
atmosphere, and the number of

341
00:26:28 --> 00:26:34
moles in my gas is constant,
well, you know that what I

342
00:26:34 --> 00:26:40
should see from that equation is
a straight line.

343
00:26:40 --> 00:26:45
But suppose I took a balloon
filled with air and started to

344
00:26:45 --> 00:26:51
cool down that balloon--in that
case, the atmospheric pressure

345
00:26:51 --> 00:26:57
is essentially constant--in that
case, what would happen is that

346
00:26:57 --> 00:27:02
the volume would decrease.
It would decrease in a linear

347
00:27:02 --> 00:27:06
manner with temperature;
everything would be fine until

348
00:27:06 --> 00:27:11
at some low temperature,
this volume would start to

349
00:27:11 --> 00:27:16
deviate from the straight line
dependence, start to go down.

350
00:27:16 --> 00:27:20
And then, all of a sudden,
the volume would go very low

351
00:27:20 --> 00:27:24
because, of course,
at roughly 77 degrees Kelvin,

352
00:27:24 --> 00:27:29
which is the boiling point of
nitrogen, the liquid would

353
00:27:29 --> 00:27:34
condense.
So, the volume gets very small.

354
00:27:34 --> 00:27:37
I could also do that with
helium.

355
00:27:37 --> 00:27:41
If I took a helium balloon and
cooled it down,

356
00:27:41 --> 00:27:46
the volume would decrease.
But then, as I got pretty cold,

357
00:27:46 --> 00:27:52
the volume would start to
decrease faster than predicted

358
00:27:52 --> 00:27:57
by the inert gas law.
And right at 4 degrees Kelvin,

359
00:27:57 --> 00:28:02
the boiling point of liquid
helium, the volume would then

360
00:28:02 --> 00:28:09
just kind of plummet.
And so you can understand what

361
00:28:09 --> 00:28:14
happens right here,
but we also want to understand,

362
00:28:14 --> 00:28:20
why does the PV equal nRT start
to deviate before we get to a

363
00:28:20 --> 00:28:26
boiling point of the liquid?
The reason is because of these

364
00:28:26 --> 00:28:31
intermolecular attractions.
For example,

365
00:28:31 --> 00:28:35
if I had in my gas this
nitrogen molecule headed toward

366
00:28:35 --> 00:28:40
the wall of my container,
and it has some initial

367
00:28:40 --> 00:28:44
straight trajectory,
it is going to hit the wall,

368
00:28:44 --> 00:28:49
where it is going to exert this
force, which will lead to my

369
00:28:49 --> 00:28:53
macroscopic pressure.
But, if there is an oxygen

370
00:28:53 --> 00:28:57
molecule around,
that nitrogen molecule could

371
00:28:57 --> 00:29:02
indeed be deflected by these
attractive interactions,

372
00:29:02 --> 00:29:08
circle around it,
and then finally hit the wall.

373
00:29:08 --> 00:29:12
So the nitrogen would be
delayed in hitting the wall.

374
00:29:12 --> 00:29:16
If it is delayed,
then my force is going to be

375
00:29:16 --> 00:29:19
not as great,
because it is momentum change

376
00:29:19 --> 00:29:25
over the change in time between
collisions, my pressure is going

377
00:29:25 --> 00:29:28
to be lower if,
in fact, this nitrogen

378
00:29:28 --> 00:29:33
experiences some attractive
interaction that delays it from

379
00:29:33 --> 00:29:37
hitting the wall.
And, therefore,

380
00:29:37 --> 00:29:40
the pressure is lower,
or vice versa.

381
00:29:40 --> 00:29:45
In this case,
if I kept the outside pressure

382
00:29:45 --> 00:29:48
constant, then the volume would
go down.

383
00:29:48 --> 00:29:54
Now, why is that the case?
Why should nitrogen and oxygen

384
00:29:54 --> 00:30:00
actually attract each other?
In order to talk about that,

385
00:30:00 --> 00:30:05
we have to think a little bit
more carefully about what the

386
00:30:05 --> 00:30:10
electron distributions are
around nitrogen molecules,

387
00:30:10 --> 00:30:13
oxygen molecules.
We treated, in the sodium

388
00:30:13 --> 00:30:19
chloride, lithium chloride,
those things as point charges.

389
00:30:19 --> 00:30:24
We have to be a little more
sophisticated now in thinking

390
00:30:24 --> 00:30:28
about what the electron
distributions are in these kinds

391
00:30:28 --> 00:30:33
of molecules,
nitrogen or oxygen.

392
00:30:33 --> 00:30:37
And let me, for the ease of
just drawing this,

393
00:30:37 --> 00:30:40
talk about the inert gas,
here, argon.

394
00:30:40 --> 00:30:44
As you may or may not know,
on the average,

395
00:30:44 --> 00:30:48
the electron distribution
around argon is spherical.

396
00:30:48 --> 00:30:54
But, although quantum mechanics
does not allow us to see this,

397
00:30:54 --> 00:30:59
this electron distribution does
fluctuate.

398
00:30:59 --> 00:31:03
And, at some momentary time,
it could be that the electron

399
00:31:03 --> 00:31:08
distribution here is a little
bit larger on one side of this

400
00:31:08 --> 00:31:12
argon.
And then the argon nucleus here

401
00:31:12 --> 00:31:16
is a little bit deshielded,
so we kind of have a charge

402
00:31:16 --> 00:31:19
shift here, positive here,
minus.

403
00:31:19 --> 00:31:22
When we do that,
this is a dipole.

404
00:31:22 --> 00:31:25
We have separated charge in
space.

405
00:31:25 --> 00:31:30
That is what a definition of a
dipole is.

406
00:31:30 --> 00:31:35
And then, of course,
if there is another argon atom

407
00:31:35 --> 00:31:40
around, well,
this dipole then is going to

408
00:31:40 --> 00:31:46
induce the charge distribution
around another argon,

409
00:31:46 --> 00:31:51
so that now,
the positive end is going to

410
00:31:51 --> 00:31:57
attract or distort the electron
distribution around argon in

411
00:31:57 --> 00:32:02
this way.
And this will be the positive

412
00:32:02 --> 00:32:04
end.
And so, we have an

413
00:32:04 --> 00:32:09
instantaneous dipole which has
induced a dipole,

414
00:32:09 --> 00:32:14
an instantaneous dipole,
in a neighboring argon atom.

415
00:32:14 --> 00:32:20
Now we have these two dipoles
together, and they are oriented

416
00:32:20 --> 00:32:24
in opposite directions.
And that is an attractive

417
00:32:24 --> 00:32:27
interaction.
The next result is an

418
00:32:27 --> 00:32:32
attraction.
And, of course,

419
00:32:32 --> 00:32:38
we call that an induced
dipole-induced dipole

420
00:32:38 --> 00:32:42
interaction.
We also call that,

421
00:32:42 --> 00:32:49
sometimes, the London
dispersion force.

422
00:32:49 --> 00:32:55


423
00:32:55 --> 00:33:02
This is not a permanent dipole.
This is a momentary dipole.

424
00:33:02 --> 00:33:06
This is an instantaneous
dipole, which induces,

425
00:33:06 --> 00:33:11
then, an instantaneous dipole
in the neighboring molecule or

426
00:33:11 --> 00:33:13
the atom.
The result is,

427
00:33:13 --> 00:33:18
because you have these two
dipoles now, align in opposite

428
00:33:18 --> 00:33:22
directions, a lowering of the
energy.

429
00:33:22 --> 00:33:26
There is a net attraction.
That is a reason why,

430
00:33:26 --> 00:33:33
in this nitrogen and oxygen,
there might be some attraction.

431
00:33:33 --> 00:33:37
The nitrogen may,
in fact, be deflected from its

432
00:33:37 --> 00:33:42
trajectory and hang around the
oxygen a little longer before it

433
00:33:42 --> 00:33:46
hits the wall.
Therefore, the pressure is

434
00:33:46 --> 00:33:51
lower than you would expect,
or the volume is lower than you

435
00:33:51 --> 00:33:54
would expect,
if you were keeping the

436
00:33:54 --> 00:33:57
pressure constant.
And we can draw that

437
00:33:57 --> 00:34:02
interaction energy for two
argons.

438
00:34:02 --> 00:34:06
Here are two argons separated.
Argon limit,

439
00:34:06 --> 00:34:12
we draw the energy of
interaction, there is some net

440
00:34:12 --> 00:34:15
attraction.
This is zero.

441
00:34:15 --> 00:34:20
And that net attraction,
here, as they come closer and

442
00:34:20 --> 00:34:26
closer together,
is a whopping 0.996 kilojoules

443
00:34:26 --> 00:34:30
per mole.
Not very large.

444
00:34:30 --> 00:34:36
But there is a net attraction.
You can also see that there is

445
00:34:36 --> 00:34:41
a value of r at which that
attraction is the maximum.

446
00:34:41 --> 00:34:46
And that is the equilibrium
bond length of the molecule

447
00:34:46 --> 00:34:51
argon two.
Can you form a molecule between

448
00:34:51 --> 00:34:54
inert gases?
You sure can.

449
00:34:54 --> 00:35:00
And sometimes we call this a
van der Waal's dimer.

450
00:35:00 --> 00:35:06
You can make these molecules.
There is the bond length.

451
00:35:06 --> 00:35:11
In the case of argon,
that bond length is 3.8

452
00:35:11 --> 00:35:15
angstroms.
But the origin of that

453
00:35:15 --> 00:35:20
attraction is this induced
dipole-induced dipole

454
00:35:20 --> 00:35:25
interaction.
You can make two argon atoms

455
00:35:25 --> 00:35:32
stick to each other.
But now, I do want to compare

456
00:35:32 --> 00:35:39
this energy of interaction right
here between two argons with the

457
00:35:39 --> 00:35:45
energy of interaction between
two hydrogen atoms that are

458
00:35:45 --> 00:35:50
covalently bonded.
The energy of interaction

459
00:35:50 --> 00:35:56
between two hydrogen atoms that
are covalently bonded,

460
00:35:56 --> 00:36:03
what does that look like?
Here are the two hydrogens

461
00:36:03 --> 00:36:12
separated, and the bond length,
here, is 432 kilojoules per

462
00:36:12 --> 00:36:17
mole.
Look at how much stronger the H

463
00:36:17 --> 00:36:26
two bond is in the case
of a covalently bound molecule,

464
00:36:26 --> 00:36:36
432 as compared to the 0.996 in
the case of the argon.

465
00:36:36 --> 00:36:40
Look at what the equilibrium
distance is, here,

466
00:36:40 --> 00:36:45
in the case of H two.
It is 0.74 angstroms,

467
00:36:45 --> 00:36:49
compared to 3.8 angstroms in
the case of argon.

468
00:36:49 --> 00:36:54
This is not a covalent bond,
the induced dipole-induced

469
00:36:54 --> 00:37:01
dipole, but it is a bond.
But now, you might say that was

470
00:37:01 --> 00:37:06
not really a fair comparison
because hydrogen is much smaller

471
00:37:06 --> 00:37:11
than argon and,
of course, the bond length in

472
00:37:11 --> 00:37:17
hydrogen is going to be much
smaller than that in two argon

473
00:37:17 --> 00:37:20
atoms.
Therefore, if the two hydrogens

474
00:37:20 --> 00:37:26
are much closer together,
then the energy of interaction

475
00:37:26 --> 00:37:32
has got to be much stronger.
Well, to show you that isn't

476
00:37:32 --> 00:37:37
really the appropriate way to
think about it,

477
00:37:37 --> 00:37:42
look at this diagram,
here, on the side board,

478
00:37:42 --> 00:37:47
where what I am plotting for
you is the argon-argon

479
00:37:47 --> 00:37:51
interaction potential.
Here it is.

480
00:37:51 --> 00:37:55
It is this light kind of
reddish line,

481
00:37:55 --> 00:38:01
argon-argon.
Versus the chlorine-chlorine.

482
00:38:01 --> 00:38:05
Here is chlorine-chlorine.
Argon and chlorine are about

483
00:38:05 --> 00:38:08
the same mass.
They are about the same size.

484
00:38:08 --> 00:38:12
What do you see?
Well, you still see that the

485
00:38:12 --> 00:38:15
argon-argon interaction is much
weaker.

486
00:38:15 --> 00:38:17
You can hardly see,
in this drawing,

487
00:38:17 --> 00:38:22
the attractive part of the
interaction potential on this

488
00:38:22 --> 00:38:23
scale.
Chlorine-chlorine,

489
00:38:23 --> 00:38:26
on the other hand,
look at that,

490
00:38:26 --> 00:38:32
minus 200 kilojoules per mole.
And chlorine-chlorine is much

491
00:38:32 --> 00:38:35
closer in.
The equilibrium bond length,

492
00:38:35 --> 00:38:38
what is it?
1.9, or something like that.

493
00:38:38 --> 00:38:42
Whereas, we have 3.8 over here
for argon-argon.

494
00:38:42 --> 00:38:46
In that covalent bond between
two chlorine atoms,

495
00:38:46 --> 00:38:51
that is a different interaction
than this induced dipole-induced

496
00:38:51 --> 00:38:56
dipole where we have overlaps of
electrons, the wave functions

497
00:38:56 --> 00:39:01
constructively,
destructively interfering.

498
00:39:01 --> 00:39:07
That is different than the
induced dipole-induced dipole

499
00:39:07 --> 00:39:11
interaction.
Now, it turns out that we

500
00:39:11 --> 00:39:17
actually do have a nice
analytical form for the

501
00:39:17 --> 00:39:23
interaction potential due to
these induced dipole-induced

502
00:39:23 --> 00:39:27
dipole interactions.

503
00:39:27 --> 00:39:32


504
00:39:32 --> 00:39:36
Let's take a look at that.
They are sometimes called

505
00:39:36 --> 00:39:41
dispersion interactions.
We leave off the name London.

506
00:39:41 --> 00:39:45
We have a nice analytical form
for these dispersion

507
00:39:45 --> 00:39:49
interactions.
And the name of that analytical

508
00:39:49 --> 00:39:53
form is the Lennard-Jones
potential.

509
00:39:53 --> 00:40:00


510
00:40:00 --> 00:40:06
Lennard-Jones was way ahead of
his time, a gentleman in England

511
00:40:06 --> 00:40:12
in the late 1800s who decided
when he got married that it was

512
00:40:12 --> 00:40:18
not really fair for his wife,
whose last name was Lennard,

513
00:40:18 --> 00:40:23
to take his name.
So, they both had hyphenated

514
00:40:23 --> 00:40:27
last names, Lennard-Jones.
That is Mr.

515
00:40:27 --> 00:40:32
Lennard-Jones.
And that potential function

516
00:40:32 --> 00:40:35
looks like this.
We are going to call it capital

517
00:40:35 --> 00:40:38
U, LJ, Lennard-Jones.

518
00:40:38 --> 00:40:40
It is going to be as a function
of r.

519
00:40:40 --> 00:40:45
R is the distance between,
I am going to use argon as the

520
00:40:45 --> 00:40:49
example, the two argon atoms.
That is going to be equal to 4

521
00:40:49 --> 00:40:52
times epsilon,
I will explain what epsilon is,

522
00:40:52 --> 00:40:57
times (sigma over r) to the 12
power, I will explain was sigma

523
00:40:57 --> 00:41:00
is in a moment,
--

524
00:41:00 --> 00:41:03
-- minus (sigma over r) to the
6 power.

525
00:41:03 --> 00:41:07


526
00:41:07 --> 00:41:11
That is my potential form.
Now, when I plot it,

527
00:41:11 --> 00:41:16
it is going to look like every
other interaction potential that

528
00:41:16 --> 00:41:20
we have drawn here because I
cannot, on the board,

529
00:41:20 --> 00:41:24
draw things very accurately.
This is argon plus argon,

530
00:41:24 --> 00:41:29
way out here.
This is a function of r.

531
00:41:29 --> 00:41:33
We are going to start out at
zero, this is going to go down

532
00:41:33 --> 00:41:37
and then come back up.
That is the general form,

533
00:41:37 --> 00:41:42
and this is the actual
expression for that interaction.

534
00:41:42 --> 00:41:44
What are these parameters,
here?

535
00:41:44 --> 00:41:47
Well, the epsilon is this well
depth.

536
00:41:47 --> 00:41:52
It is measured from the bottom
of the well, although the argon

537
00:41:52 --> 00:41:56
atoms are never at the bottom of
the well.

538
00:41:56 --> 00:42:00
They have the zero point
energy.

539
00:42:00 --> 00:42:03
This epsilon,
here, is this energy,

540
00:42:03 --> 00:42:08
from the bottom to the
dissociated atom limit.

541
00:42:08 --> 00:42:13
What is this sigma?
Well, this sigma is related to

542
00:42:13 --> 00:42:19
the equilibrium bond length.
In the Lennard-Jones potential,

543
00:42:19 --> 00:42:25
here, the equilibrium bond
length is equal to 1.12 sigma.

544
00:42:25 --> 00:42:31
That is a parameter.
To get it, you would take the

545
00:42:31 --> 00:42:36
derivative of the potential
function, set it equal to zero,

546
00:42:36 --> 00:42:40
calculate the value of r,
and that would give you a zero

547
00:42:40 --> 00:42:44
in that derivative.
That is a maximum or a minimum.

548
00:42:44 --> 00:42:48
And it will turn out to be a
minimum in this case.

549
00:42:48 --> 00:42:52
That is what sigma is.
What are these two components,

550
00:42:52 --> 00:42:54
right here?
This component,

551
00:42:54 --> 00:43:00
this (sigma over r)
to the 12.

552
00:43:00 --> 00:43:06
What this describes are the
repulsions in this interaction.

553
00:43:06 --> 00:43:13
It is a good description of the
core electron-core electron

554
00:43:13 --> 00:43:18
repulsion.
Not the repulsion due to the

555
00:43:18 --> 00:43:25
outermost electrons in argon,
but the repulsion due to the

556
00:43:25 --> 00:43:28
core electrons,
the n equal one,

557
00:43:28 --> 00:43:35
the n equal two electrons.
And it describes the

558
00:43:35 --> 00:43:42
nuclear-nuclear repulsion.
It is a (one over r) to the 12

559
00:43:42 --> 00:43:47
dependence. If I plot that,

560
00:43:47 --> 00:43:51
it would look something like
this.

561
00:43:51 --> 00:43:57
It is repulsive everywhere.
It is what we call a

562
00:43:57 --> 00:44:04
short-range interaction.
And this is important.

563
00:44:04 --> 00:44:10
What do we mean by short-range?
Well, short-range means that it

564
00:44:10 --> 00:44:16
only has a value when r is small
because it is a one over r

565
00:44:16 --> 00:44:19
raised to the 12 power.

566
00:44:19 --> 00:44:24
If r is large,
and you put a large number in

567
00:44:24 --> 00:44:28
the denominator and raise it to
the 12 power,

568
00:44:28 --> 00:44:34
the result is nothing.
The result is something close

569
00:44:34 --> 00:44:38
to zero.
This first term is zero when r

570
00:44:38 --> 00:44:42
is very large.
That is why we call it a

571
00:44:42 --> 00:44:47
short-range interaction.
And you can see that this

572
00:44:47 --> 00:44:52
really starts taking on value
when r is pretty small.

573
00:44:52 --> 00:44:58
But then there is this part,
the (sigma over r) to the 6

574
00:44:58 --> 00:45:04
term. That is the attractive

575
00:45:04 --> 00:45:07
interactions.
If I were to plot that,

576
00:45:07 --> 00:45:11
it would look like that.
That is the induced

577
00:45:11 --> 00:45:14
dipole-induced dipole
interaction.

578
00:45:14 --> 00:45:19
We could actually write down
the energy of interaction

579
00:45:19 --> 00:45:24
between these two induced
dipoles and find out that this

580
00:45:24 --> 00:45:30
is a one over the r to the 6
power.

581
00:45:30 --> 00:45:34
We are not going to do that,
but we could do that.

582
00:45:34 --> 00:45:38
These are always attractive.
The sum of the two,

583
00:45:38 --> 00:45:42
of course, gives us the actual
shape of that interaction

584
00:45:42 --> 00:45:45
potential.
But this interaction,

585
00:45:45 --> 00:45:49
here, is longer range,
or we call it longer range.

586
00:45:49 --> 00:45:54
It is longer range because it
is only one over r to the 6.

587
00:45:54 --> 00:45:57
And so r does not have to be

588
00:45:57 --> 00:46:04
that small in order for this
term to make the contribution.

589
00:46:04 --> 00:46:08
As r has a larger power here in
the denominator,

590
00:46:08 --> 00:46:12
that is shorter range
interaction.

591
00:46:12 --> 00:46:17
As it gets a smaller power in
the denominator,

592
00:46:17 --> 00:46:20
that is a shorter range
interaction.

593
00:46:20 --> 00:46:26
The other thing that is
interesting and important here

594
00:46:26 --> 00:46:33
is this value of sigma.
And I said that r sub e was

595
00:46:33 --> 00:46:37
1.12 times this sigma.

596
00:46:37 --> 00:46:44
The value of sigma is actually
the definition of what we call a

597
00:46:44 --> 00:46:50
van der Waal's radius.
That is, in the case of argon

598
00:46:50 --> 00:46:55
here, the equilibrium bond
length is 1.12 sigma.

599
00:46:55 --> 00:47:00
And so, r sub e,
1.12 sigma.

600
00:47:00 --> 00:47:08
For argon that parameter is 3.4
angstroms.

601
00:47:08 --> 00:47:16
And so the bond length is 3.8
angstroms.

602
00:47:16 --> 00:47:28
But the van der Waal's radius
is given by this 1.12 sigma over

603
00:47:28.389 --> 2.


604
2. --> 00:47:32
The van der Waal's radius,

605
00:47:32 --> 00:47:37
in this case 1.9 here,
is the radius that you use in

606
00:47:37 --> 00:47:42
these space filling models.
On the side wall,

607
00:47:42 --> 00:47:46
that top model is a space
filling model.

608
00:47:46 --> 00:47:51
And somehow you have to decide,
how large should those

609
00:47:51 --> 00:47:57
hydrogens be that are sticking
out in space that are not bonded

610
00:47:57 --> 00:48:03
to anything?
And the radii that are used are

611
00:48:03 --> 00:48:08
these van der Waal's radii,
because that is the radius at

612
00:48:08 --> 00:48:14
which you have this attractive
interaction due to the induced

613
00:48:14 --> 00:48:20
dipole-induced dipole.
And that is how those sizes are

614
00:48:20.115 --> 48:23
determined.