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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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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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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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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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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

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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

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00:15:01,000 --> 00:15:05,000
over 4pi squared amount of
rotational energy.

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00:15:05,000 --> 00:15:11,000
For J equal three,

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00:15:11,000 --> 00:15:17,000
it has 3h squared over 2 pi
squared times the moment of

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inertia.

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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

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between those states?
Well, J equal one minus energy

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00:15:42,000 --> 00:15:46,000
J equal zero.
That is h squared over 4pi

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00:15:46,000 --> 00:15:51,000
squared times I minus zero.

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That is just h squared over 4pi
squared I.

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How do we make that transition?
Well, to make this transition

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00:16:03,000 --> 00:16:08,000
right here, we are going to need
a photon, and that photon is

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00:16:08,000 --> 00:16:13,000
going to have to have an energy
exactly equal to this.

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00:16:13,000 --> 00:16:17,000
Our photon E equal h nu,
that has got to be equal to h

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00:16:17,000 --> 00:16:22,000
squared 4pi squared times I.

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00:16:22,000 --> 00:16:27,000
I can solve for the frequency
of the photon that I need to

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00:16:27,000 --> 00:16:33,000
make that transition.
The frequency of that photon

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00:16:33,000 --> 00:16:36,000
then is h over 4pi squared times
I.

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00:16:36,000 --> 00:16:43,000
That is
the frequency of the photon that

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I need.
Now, in the case of HCl,

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we said that HCl has two
rotational modes.

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00:16:51,000 --> 00:16:58,000
It has a mode rotation around
this axis, and it also has a

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00:16:58,000 --> 00:17:06,000
mode rotation around this axis,
in the plane of the board.

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00:17:06,000 --> 00:17:09,000
These two rotations are
degenerate.

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00:17:09,000 --> 00:17:14,000
If you put in a photon with
this frequency,

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00:17:14,000 --> 00:17:21,000
here, it will excite either
this rotation or that rotation.

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00:17:21,000 --> 00:17:25,000
Now, let's look on the side
wall here.

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How do we do the experiment?
Again, just like we do it in

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00:17:32,000 --> 00:17:35,000
the infrared,
except that the energy of the

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00:17:35,000 --> 00:17:39,000
photon we now need is in the
microwave range.

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00:17:39,000 --> 00:17:43,000
Microwave spectra measure
rotational spectra of molecules.

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00:17:43,000 --> 00:17:47,000
And so, again we have a
microwave radiation,

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00:17:47,000 --> 00:17:51,000
monochromator,
coming out, going through the

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00:17:51,000 --> 00:17:55,000
sample, photo detector,
look at the intensity of the

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00:17:55,000 --> 00:18:00,000
photo detector as a function of
the frequency.

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00:18:00,000 --> 00:18:04,000
At some frequency here,
you see a dip in that

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00:18:04,000 --> 00:18:07,000
intensity.
Well, that is where the

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00:18:07,000 --> 00:18:12,000
molecule absorbs.
And, in the case of HCl here,

237
00:18:12,000 --> 00:18:17,000
the frequency of that
absorption for J equal zero to J

238
00:18:17,000 --> 00:18:21,000
equal one, well,
that frequency occurs at

239
00:18:21,000 --> 00:18:26,000
6.3x10^11 hertz.
That would be the frequency of

240
00:18:26,000 --> 00:18:32,000
the photon that you would need
for absorption.

241
00:18:32,000 --> 00:18:33,000
Yes?
I'm sorry?

242
00:18:33,000 --> 00:18:37,000
We got into the microwave
range, here, because the

243
00:18:37,000 --> 00:18:43,000
spacings between the rotational
states are much lower than the

244
00:18:43,000 --> 00:18:46,000
spacings between the vibrational
states.

245
00:18:46,000 --> 00:18:50,000
And I just actually wanted to
make that point,

246
00:18:50,000 --> 00:18:53,000
here.
We can calculate in terms of

247
00:18:53,000 --> 00:18:57,000
the energy, here,
what this spacing is.

248
00:18:57,000 --> 00:19:00,000
Let's do that.

249
00:19:05,000 --> 00:19:11,000
If we want to know the change
in energy from J equal zero to J

250
00:19:11,000 --> 00:19:19,000
equal one, that change in energy
is just h times the frequency of

251
00:19:19,000 --> 00:19:24,000
the photon, that photon,
there, that makes that

252
00:19:24,000 --> 00:19:30,000
transition happen.
You can plug that in,

253
00:19:30,000 --> 00:19:37,000
there, so we have 6.6261x10^-34
joule seconds,

254
00:19:37,000 --> 00:19:43,000
times the frequency
6.3479x10^11 hertz.

255
00:19:43,000 --> 00:19:50,000
When you do that,
you should get 4.2062x10^-22

256
00:19:50,000 --> 00:19:55,000
joules.
Or, if I convert that to

257
00:19:55,000 --> 00:20:03,000
kilojoules per mole,
it is 6.25330 kilojoules per

258
00:20:03,000 --> 00:20:08,000
mole.
In general, the difference in

259
00:20:08,000 --> 00:20:12,000
the spacings here,
the energy difference for

260
00:20:12,000 --> 00:20:18,000
vibration, we said is something
between three to 40 kilojoules

261
00:20:18,000 --> 00:20:22,000
per mole.
That is the difference between

262
00:20:22,000 --> 00:20:28,000
v equal zero and v equal one,
generally, for a large range of

263
00:20:28,000 --> 00:20:32,000
molecules.
The difference in the

264
00:20:32,000 --> 00:20:40,000
frequencies here for rotation,
delta E, that is more like

265
00:20:40,000 --> 00:20:47,000
something on the order of 0.01
to about 1.0 kilojoules per

266
00:20:47,000 --> 00:20:51,000
mole.
That is what is typical.

267
00:20:51,000 --> 00:20:58,000
So, rotations are much more
closely spaced in energy.

268
00:20:58,000 --> 00:21:02,000
Yes?
Well, in the case of the

269
00:21:02,000 --> 00:21:06,000
rotational spectra,
there is a little bit more

270
00:21:06,000 --> 00:21:09,000
diversity in what units are
used.

271
00:21:09,000 --> 00:21:13,000
And the reason is this.
They are more closely spaced.

272
00:21:13,000 --> 00:21:18,000
And the whole wavenumber came
into use in vibrational

273
00:21:18,000 --> 00:21:23,000
spectroscopy when the way people
would analyze the spectra would

274
00:21:23,000 --> 00:21:30,000
be to take a photographic plate
with the light coming in.

275
00:21:30,000 --> 00:21:33,000
And you would see the lines
separated in space.

276
00:21:33,000 --> 00:21:38,000
And people would take a ruler
in centimeters to measure the

277
00:21:38,000 --> 00:21:43,000
spacings between the lines.
That is historically how the

278
00:21:43,000 --> 00:21:48,000
wavenumber unit came to be.
The problem is that does not

279
00:21:48,000 --> 00:21:51,000
work so well in rotational
spectroscopy,

280
00:21:51,000 --> 00:21:56,000
often, because the spacings are
much closer together.

281
00:21:56,000 --> 00:22:01,000
And so, depending exactly on
what kind of diffraction grading

282
00:22:01,000 --> 00:22:06,000
used, the wavenumbers are not
always used.

283
00:22:06,000 --> 00:22:08,000
Sometimes they are the
frequency.

284
00:22:08,000 --> 00:22:13,000
We go back and forth.
And, in your homework problems,

285
00:22:13,000 --> 00:22:15,000
I go back and forth,
too.

286
00:22:15,000 --> 00:22:19,000
You really have to deal with
both kinds of units.

287
00:22:19,000 --> 00:22:23,000
But now, one of the
usefulnesses of this rotational

288
00:22:23,000 --> 00:22:28,000
spectroscopy in getting,
say, this frequency for the

289
00:22:28,000 --> 00:22:31,000
transition, one of the
usefulnesses,

290
00:22:31,000 --> 00:22:37,000
there, is to calculate the bond
length of the molecule.

291
00:22:37,000 --> 00:22:42,000
To determine the bond length of
the molecule really very

292
00:22:42,000 --> 00:22:46,000
accurately.
Because what we said over here

293
00:22:46,000 --> 00:22:51,000
is that the frequency for that
transition is this.

294
00:22:51,000 --> 00:22:57,000
The frequency of that
transition is h over 4pi squared

295
00:22:57,000 --> 00:23:02,000
times I.
And I is a moment of inertia.

296
00:23:02,000 --> 00:23:05,000
The moment of inertia is the
following.

297
00:23:05,000 --> 00:23:09,000
Maybe you have had this 8.01.
No, not yet?

298
00:23:09,000 --> 00:23:13,000
Oh, you will.
The moment of inertia is the

299
00:23:13,000 --> 00:23:19,000
reduced mass times the distance
between the two masses.

300
00:23:19,000 --> 00:23:22,000
In our case,
for the HCl molecule,

301
00:23:22,000 --> 00:23:28,000
it is equilibrium bond length,
r sub e squared.

302
00:23:28,000 --> 00:23:33,000
The reduced mass is what I gave
you before, m1 m2 over the sum

303
00:23:33,000 --> 00:23:36,000
of the two masses.

304
00:23:36,000 --> 00:23:41,000
Again, it is a way to reduce a
two-body problem to a one-body

305
00:23:41,000 --> 00:23:46,000
problem, where the one-body is
this fictitious body of reduced

306
00:23:46,000 --> 00:23:48,000
mass. But it is exact.

307
00:23:48,000 --> 00:23:51,000
There are no approximations.
This is correct.

308
00:23:51,000 --> 00:23:55,000
That is the moment of inertia
of the molecule.

309
00:23:55,000 --> 00:24:00,000
It is a property of the
molecule, depending on the mass

310
00:24:00,000 --> 00:24:05,000
and the bond length.
You can see that if I

311
00:24:05,000 --> 00:24:10,000
substitute that in there,
h over 4pi squared times nu r

312
00:24:10,000 --> 00:24:15,000
sub e squared,
and then I go and

313
00:24:15,000 --> 00:24:20,000
solve for r sub e,
well, r sub e is (h over 4pi

314
00:24:20,000 --> 00:24:23,000
squared mu times nu) to the
one-half.

315
00:24:23,000 --> 00:24:28,000
In the case of HCl

316
00:24:28,000 --> 00:24:32,000
--
Actually, I think this is all

317
00:24:32,000 --> 00:24:37,000
in my slide, here.
If I go and stick in the value

318
00:24:37,000 --> 00:24:42,000
for nu, the equilibrium bond
length, here,

319
00:24:42,000 --> 00:24:48,000
is really 1.2748x10^-10 meters.
We can really measure these

320
00:24:48,000 --> 00:24:53,000
frequencies with high precision
and high accuracy.

321
00:24:53,000 --> 00:24:58,000
All bond lengths,
really, come from measurements,

322
00:24:58,000 --> 00:25:05,000
now, of rotational spectra.
All bond lengths in the gas

323
00:25:05,000 --> 00:25:12,000
phase, wherever we can measure
the rotational spectra of the

324
00:25:12,000 --> 00:25:16,000
molecule.
That is one of the main uses

325
00:25:16,000 --> 00:25:20,000
for rotational spectroscopy.
Questions?

326
00:25:20,000 --> 00:25:28,000
If not, what I am going to do
is leave the subject of internal

327
00:25:28,000 --> 00:25:32,000
motion.
I want to talk for the rest of

328
00:25:32,000 --> 00:25:38,000
the hour, and a little bit on
Friday about another topic,

329
00:25:38,000 --> 00:25:42,000
which is intermolecular
attraction.

330
00:25:47,000 --> 00:25:51,000
Interactions.
Or, I am going to write it here

331
00:25:51,000 --> 00:25:54,000
as attractions.
The bottom line is,

332
00:25:54,000 --> 00:25:58,000
I want to try to understand,
on a microscopic scale,

333
00:25:58,000 --> 00:26:05,000
deviations from the inner gas
law, PV equal nRT.

334
00:26:12,000 --> 00:26:17,000
You know PV equal nRT,
that if I made a plot of the

335
00:26:17,000 --> 00:26:23,000
volume versus the temperature
and kept the pressure constant,

336
00:26:23,000 --> 00:26:28,000
say the pressure is at one
atmosphere, and the number of

337
00:26:28,000 --> 00:26:34,000
moles in my gas is constant,
well, you know that what I

338
00:26:34,000 --> 00:26:40,000
should see from that equation is
a straight line.

339
00:26:40,000 --> 00:26:45,000
But suppose I took a balloon
filled with air and started to

340
00:26:45,000 --> 00:26:51,000
cool down that balloon--in that
case, the atmospheric pressure

341
00:26:51,000 --> 00:26:57,000
is essentially constant--in that
case, what would happen is that

342
00:26:57,000 --> 00:27:02,000
the volume would decrease.
It would decrease in a linear

343
00:27:02,000 --> 00:27:06,000
manner with temperature;
everything would be fine until

344
00:27:06,000 --> 00:27:11,000
at some low temperature,
this volume would start to

345
00:27:11,000 --> 00:27:16,000
deviate from the straight line
dependence, start to go down.

346
00:27:16,000 --> 00:27:20,000
And then, all of a sudden,
the volume would go very low

347
00:27:20,000 --> 00:27:24,000
because, of course,
at roughly 77 degrees Kelvin,

348
00:27:24,000 --> 00:27:29,000
which is the boiling point of
nitrogen, the liquid would

349
00:27:29,000 --> 00:27:34,000
condense.
So, the volume gets very small.

350
00:27:34,000 --> 00:27:37,000
I could also do that with
helium.

351
00:27:37,000 --> 00:27:41,000
If I took a helium balloon and
cooled it down,

352
00:27:41,000 --> 00:27:46,000
the volume would decrease.
But then, as I got pretty cold,

353
00:27:46,000 --> 00:27:52,000
the volume would start to
decrease faster than predicted

354
00:27:52,000 --> 00:27:57,000
by the inert gas law.
And right at 4 degrees Kelvin,

355
00:27:57,000 --> 00:28:02,000
the boiling point of liquid
helium, the volume would then

356
00:28:02,000 --> 00:28:09,000
just kind of plummet.
And so you can understand what

357
00:28:09,000 --> 00:28:14,000
happens right here,
but we also want to understand,

358
00:28:14,000 --> 00:28:20,000
why does the PV equal nRT start
to deviate before we get to a

359
00:28:20,000 --> 00:28:26,000
boiling point of the liquid?
The reason is because of these

360
00:28:26,000 --> 00:28:31,000
intermolecular attractions.
For example,

361
00:28:31,000 --> 00:28:35,000
if I had in my gas this
nitrogen molecule headed toward

362
00:28:35,000 --> 00:28:40,000
the wall of my container,
and it has some initial

363
00:28:40,000 --> 00:28:44,000
straight trajectory,
it is going to hit the wall,

364
00:28:44,000 --> 00:28:49,000
where it is going to exert this
force, which will lead to my

365
00:28:49,000 --> 00:28:53,000
macroscopic pressure.
But, if there is an oxygen

366
00:28:53,000 --> 00:28:57,000
molecule around,
that nitrogen molecule could

367
00:28:57,000 --> 00:29:02,000
indeed be deflected by these
attractive interactions,

368
00:29:02,000 --> 00:29:08,000
circle around it,
and then finally hit the wall.

369
00:29:08,000 --> 00:29:12,000
So the nitrogen would be
delayed in hitting the wall.

370
00:29:12,000 --> 00:29:16,000
If it is delayed,
then my force is going to be

371
00:29:16,000 --> 00:29:19,000
not as great,
because it is momentum change

372
00:29:19,000 --> 00:29:25,000
over the change in time between
collisions, my pressure is going

373
00:29:25,000 --> 00:29:28,000
to be lower if,
in fact, this nitrogen

374
00:29:28,000 --> 00:29:33,000
experiences some attractive
interaction that delays it from

375
00:29:33,000 --> 00:29:37,000
hitting the wall.
And, therefore,

376
00:29:37,000 --> 00:29:40,000
the pressure is lower,
or vice versa.

377
00:29:40,000 --> 00:29:45,000
In this case,
if I kept the outside pressure

378
00:29:45,000 --> 00:29:48,000
constant, then the volume would
go down.

379
00:29:48,000 --> 00:29:54,000
Now, why is that the case?
Why should nitrogen and oxygen

380
00:29:54,000 --> 00:30:00,000
actually attract each other?
In order to talk about that,

381
00:30:00,000 --> 00:30:05,000
we have to think a little bit
more carefully about what the

382
00:30:05,000 --> 00:30:10,000
electron distributions are
around nitrogen molecules,

383
00:30:10,000 --> 00:30:13,000
oxygen molecules.
We treated, in the sodium

384
00:30:13,000 --> 00:30:19,000
chloride, lithium chloride,
those things as point charges.

385
00:30:19,000 --> 00:30:24,000
We have to be a little more
sophisticated now in thinking

386
00:30:24,000 --> 00:30:28,000
about what the electron
distributions are in these kinds

387
00:30:28,000 --> 00:30:33,000
of molecules,
nitrogen or oxygen.

388
00:30:33,000 --> 00:30:37,000
And let me, for the ease of
just drawing this,

389
00:30:37,000 --> 00:30:40,000
talk about the inert gas,
here, argon.

390
00:30:40,000 --> 00:30:44,000
As you may or may not know,
on the average,

391
00:30:44,000 --> 00:30:48,000
the electron distribution
around argon is spherical.

392
00:30:48,000 --> 00:30:54,000
But, although quantum mechanics
does not allow us to see this,

393
00:30:54,000 --> 00:30:59,000
this electron distribution does
fluctuate.

394
00:30:59,000 --> 00:31:03,000
And, at some momentary time,
it could be that the electron

395
00:31:03,000 --> 00:31:08,000
distribution here is a little
bit larger on one side of this

396
00:31:08,000 --> 00:31:12,000
argon.
And then the argon nucleus here

397
00:31:12,000 --> 00:31:16,000
is a little bit deshielded,
so we kind of have a charge

398
00:31:16,000 --> 00:31:19,000
shift here, positive here,
minus.

399
00:31:19,000 --> 00:31:22,000
When we do that,
this is a dipole.

400
00:31:22,000 --> 00:31:25,000
We have separated charge in
space.

401
00:31:25,000 --> 00:31:30,000
That is what a definition of a
dipole is.

402
00:31:30,000 --> 00:31:35,000
And then, of course,
if there is another argon atom

403
00:31:35,000 --> 00:31:40,000
around, well,
this dipole then is going to

404
00:31:40,000 --> 00:31:46,000
induce the charge distribution
around another argon,

405
00:31:46,000 --> 00:31:51,000
so that now,
the positive end is going to

406
00:31:51,000 --> 00:31:57,000
attract or distort the electron
distribution around argon in

407
00:31:57,000 --> 00:32:02,000
this way.
And this will be the positive

408
00:32:02,000 --> 00:32:04,000
end.
And so, we have an

409
00:32:04,000 --> 00:32:09,000
instantaneous dipole which has
induced a dipole,

410
00:32:09,000 --> 00:32:14,000
an instantaneous dipole,
in a neighboring argon atom.

411
00:32:14,000 --> 00:32:20,000
Now we have these two dipoles
together, and they are oriented

412
00:32:20,000 --> 00:32:24,000
in opposite directions.
And that is an attractive

413
00:32:24,000 --> 00:32:27,000
interaction.
The next result is an

414
00:32:27,000 --> 00:32:32,000
attraction.
And, of course,

415
00:32:32,000 --> 00:32:38,000
we call that an induced
dipole-induced dipole

416
00:32:38,000 --> 00:32:42,000
interaction.
We also call that,

417
00:32:42,000 --> 00:32:49,000
sometimes, the London
dispersion force.

418
00:32:55,000 --> 00:33:02,000
This is not a permanent dipole.
This is a momentary dipole.

419
00:33:02,000 --> 00:33:06,000
This is an instantaneous
dipole, which induces,

420
00:33:06,000 --> 00:33:11,000
then, an instantaneous dipole
in the neighboring molecule or

421
00:33:11,000 --> 00:33:13,000
the atom.
The result is,

422
00:33:13,000 --> 00:33:18,000
because you have these two
dipoles now, align in opposite

423
00:33:18,000 --> 00:33:22,000
directions, a lowering of the
energy.

424
00:33:22,000 --> 00:33:26,000
There is a net attraction.
That is a reason why,

425
00:33:26,000 --> 00:33:33,000
in this nitrogen and oxygen,
there might be some attraction.

426
00:33:33,000 --> 00:33:37,000
The nitrogen may,
in fact, be deflected from its

427
00:33:37,000 --> 00:33:42,000
trajectory and hang around the
oxygen a little longer before it

428
00:33:42,000 --> 00:33:46,000
hits the wall.
Therefore, the pressure is

429
00:33:46,000 --> 00:33:51,000
lower than you would expect,
or the volume is lower than you

430
00:33:51,000 --> 00:33:54,000
would expect,
if you were keeping the

431
00:33:54,000 --> 00:33:57,000
pressure constant.
And we can draw that

432
00:33:57,000 --> 00:34:02,000
interaction energy for two
argons.

433
00:34:02,000 --> 00:34:06,000
Here are two argons separated.
Argon limit,

434
00:34:06,000 --> 00:34:12,000
we draw the energy of
interaction, there is some net

435
00:34:12,000 --> 00:34:15,000
attraction.
This is zero.

436
00:34:15,000 --> 00:34:20,000
And that net attraction,
here, as they come closer and

437
00:34:20,000 --> 00:34:26,000
closer together,
is a whopping 0.996 kilojoules

438
00:34:26,000 --> 00:34:30,000
per mole.
Not very large.

439
00:34:30,000 --> 00:34:36,000
But there is a net attraction.
You can also see that there is

440
00:34:36,000 --> 00:34:41,000
a value of r at which that
attraction is the maximum.

441
00:34:41,000 --> 00:34:46,000
And that is the equilibrium
bond length of the molecule

442
00:34:46,000 --> 00:34:51,000
argon two.
Can you form a molecule between

443
00:34:51,000 --> 00:34:54,000
inert gases?
You sure can.

444
00:34:54,000 --> 00:35:00,000
And sometimes we call this a
van der Waal's dimer.

445
00:35:00,000 --> 00:35:06,000
You can make these molecules.
There is the bond length.

446
00:35:06,000 --> 00:35:11,000
In the case of argon,
that bond length is 3.8

447
00:35:11,000 --> 00:35:15,000
angstroms.
But the origin of that

448
00:35:15,000 --> 00:35:20,000
attraction is this induced
dipole-induced dipole

449
00:35:20,000 --> 00:35:25,000
interaction.
You can make two argon atoms

450
00:35:25,000 --> 00:35:32,000
stick to each other.
But now, I do want to compare

451
00:35:32,000 --> 00:35:39,000
this energy of interaction right
here between two argons with the

452
00:35:39,000 --> 00:35:45,000
energy of interaction between
two hydrogen atoms that are

453
00:35:45,000 --> 00:35:50,000
covalently bonded.
The energy of interaction

454
00:35:50,000 --> 00:35:56,000
between two hydrogen atoms that
are covalently bonded,

455
00:35:56,000 --> 00:36:03,000
what does that look like?
Here are the two hydrogens

456
00:36:03,000 --> 00:36:12,000
separated, and the bond length,
here, is 432 kilojoules per

457
00:36:12,000 --> 00:36:17,000
mole.
Look at how much stronger the H

458
00:36:17,000 --> 00:36:26,000
two bond is in the case
of a covalently bound molecule,

459
00:36:26,000 --> 00:36:36,000
432 as compared to the 0.996 in
the case of the argon.

460
00:36:36,000 --> 00:36:40,000
Look at what the equilibrium
distance is, here,

461
00:36:40,000 --> 00:36:45,000
in the case of H two.
It is 0.74 angstroms,

462
00:36:45,000 --> 00:36:49,000
compared to 3.8 angstroms in
the case of argon.

463
00:36:49,000 --> 00:36:54,000
This is not a covalent bond,
the induced dipole-induced

464
00:36:54,000 --> 00:37:01,000
dipole, but it is a bond.
But now, you might say that was

465
00:37:01,000 --> 00:37:06,000
not really a fair comparison
because hydrogen is much smaller

466
00:37:06,000 --> 00:37:11,000
than argon and,
of course, the bond length in

467
00:37:11,000 --> 00:37:17,000
hydrogen is going to be much
smaller than that in two argon

468
00:37:17,000 --> 00:37:20,000
atoms.
Therefore, if the two hydrogens

469
00:37:20,000 --> 00:37:26,000
are much closer together,
then the energy of interaction

470
00:37:26,000 --> 00:37:32,000
has got to be much stronger.
Well, to show you that isn't

471
00:37:32,000 --> 00:37:37,000
really the appropriate way to
think about it,

472
00:37:37,000 --> 00:37:42,000
look at this diagram,
here, on the side board,

473
00:37:42,000 --> 00:37:47,000
where what I am plotting for
you is the argon-argon

474
00:37:47,000 --> 00:37:51,000
interaction potential.
Here it is.

475
00:37:51,000 --> 00:37:55,000
It is this light kind of
reddish line,

476
00:37:55,000 --> 00:38:01,000
argon-argon.
Versus the chlorine-chlorine.

477
00:38:01,000 --> 00:38:05,000
Here is chlorine-chlorine.
Argon and chlorine are about

478
00:38:05,000 --> 00:38:08,000
the same mass.
They are about the same size.

479
00:38:08,000 --> 00:38:12,000
What do you see?
Well, you still see that the

480
00:38:12,000 --> 00:38:15,000
argon-argon interaction is much
weaker.

481
00:38:15,000 --> 00:38:17,000
You can hardly see,
in this drawing,

482
00:38:17,000 --> 00:38:22,000
the attractive part of the
interaction potential on this

483
00:38:22,000 --> 00:38:23,000
scale.
Chlorine-chlorine,

484
00:38:23,000 --> 00:38:26,000
on the other hand,
look at that,

485
00:38:26,000 --> 00:38:32,000
minus 200 kilojoules per mole.
And chlorine-chlorine is much

486
00:38:32,000 --> 00:38:35,000
closer in.
The equilibrium bond length,

487
00:38:35,000 --> 00:38:38,000
what is it?
1.9, or something like that.

488
00:38:38,000 --> 00:38:42,000
Whereas, we have 3.8 over here
for argon-argon.

489
00:38:42,000 --> 00:38:46,000
In that covalent bond between
two chlorine atoms,

490
00:38:46,000 --> 00:38:51,000
that is a different interaction
than this induced dipole-induced

491
00:38:51,000 --> 00:38:56,000
dipole where we have overlaps of
electrons, the wave functions

492
00:38:56,000 --> 00:39:01,000
constructively,
destructively interfering.

493
00:39:01,000 --> 00:39:07,000
That is different than the
induced dipole-induced dipole

494
00:39:07,000 --> 00:39:11,000
interaction.
Now, it turns out that we

495
00:39:11,000 --> 00:39:17,000
actually do have a nice
analytical form for the

496
00:39:17,000 --> 00:39:23,000
interaction potential due to
these induced dipole-induced

497
00:39:23,000 --> 00:39:27,000
dipole interactions.

498
00:39:32,000 --> 00:39:36,000
Let's take a look at that.
They are sometimes called

499
00:39:36,000 --> 00:39:41,000
dispersion interactions.
We leave off the name London.

500
00:39:41,000 --> 00:39:45,000
We have a nice analytical form
for these dispersion

501
00:39:45,000 --> 00:39:49,000
interactions.
And the name of that analytical

502
00:39:49,000 --> 00:39:53,000
form is the Lennard-Jones
potential.

503
00:40:00,000 --> 00:40:06,000
Lennard-Jones was way ahead of
his time, a gentleman in England

504
00:40:06,000 --> 00:40:12,000
in the late 1800s who decided
when he got married that it was

505
00:40:12,000 --> 00:40:18,000
not really fair for his wife,
whose last name was Lennard,

506
00:40:18,000 --> 00:40:23,000
to take his name.
So, they both had hyphenated

507
00:40:23,000 --> 00:40:27,000
last names, Lennard-Jones.
That is Mr.

508
00:40:27,000 --> 00:40:32,000
Lennard-Jones.
And that potential function

509
00:40:32,000 --> 00:40:35,000
looks like this.
We are going to call it capital

510
00:40:35,000 --> 00:40:38,000
U, LJ, Lennard-Jones.

511
00:40:38,000 --> 00:40:40,000
It is going to be as a function
of r.

512
00:40:40,000 --> 00:40:45,000
R is the distance between,
I am going to use argon as the

513
00:40:45,000 --> 00:40:49,000
example, the two argon atoms.
That is going to be equal to 4

514
00:40:49,000 --> 00:40:52,000
times epsilon,
I will explain what epsilon is,

515
00:40:52,000 --> 00:40:57,000
times (sigma over r) to the 12
power, I will explain was sigma

516
00:40:57,000 --> 00:41:00,000
is in a moment,
--

517
00:41:00,000 --> 00:41:03,000
-- minus (sigma over r) to the
6 power.

518
00:41:07,000 --> 00:41:11,000
That is my potential form.
Now, when I plot it,

519
00:41:11,000 --> 00:41:16,000
it is going to look like every
other interaction potential that

520
00:41:16,000 --> 00:41:20,000
we have drawn here because I
cannot, on the board,

521
00:41:20,000 --> 00:41:24,000
draw things very accurately.
This is argon plus argon,

522
00:41:24,000 --> 00:41:29,000
way out here.
This is a function of r.

523
00:41:29,000 --> 00:41:33,000
We are going to start out at
zero, this is going to go down

524
00:41:33,000 --> 00:41:37,000
and then come back up.
That is the general form,

525
00:41:37,000 --> 00:41:42,000
and this is the actual
expression for that interaction.

526
00:41:42,000 --> 00:41:44,000
What are these parameters,
here?

527
00:41:44,000 --> 00:41:47,000
Well, the epsilon is this well
depth.

528
00:41:47,000 --> 00:41:52,000
It is measured from the bottom
of the well, although the argon

529
00:41:52,000 --> 00:41:56,000
atoms are never at the bottom of
the well.

530
00:41:56,000 --> 00:42:00,000
They have the zero point
energy.

531
00:42:00,000 --> 00:42:03,000
This epsilon,
here, is this energy,

532
00:42:03,000 --> 00:42:08,000
from the bottom to the
dissociated atom limit.

533
00:42:08,000 --> 00:42:13,000
What is this sigma?
Well, this sigma is related to

534
00:42:13,000 --> 00:42:19,000
the equilibrium bond length.
In the Lennard-Jones potential,

535
00:42:19,000 --> 00:42:25,000
here, the equilibrium bond
length is equal to 1.12 sigma.

536
00:42:25,000 --> 00:42:31,000
That is a parameter.
To get it, you would take the

537
00:42:31,000 --> 00:42:36,000
derivative of the potential
function, set it equal to zero,

538
00:42:36,000 --> 00:42:40,000
calculate the value of r,
and that would give you a zero

539
00:42:40,000 --> 00:42:44,000
in that derivative.
That is a maximum or a minimum.

540
00:42:44,000 --> 00:42:48,000
And it will turn out to be a
minimum in this case.

541
00:42:48,000 --> 00:42:52,000
That is what sigma is.
What are these two components,

542
00:42:52,000 --> 00:42:54,000
right here?
This component,

543
00:42:54,000 --> 00:43:00,000
this (sigma over r)
to the 12.

544
00:43:00,000 --> 00:43:06,000
What this describes are the
repulsions in this interaction.

545
00:43:06,000 --> 00:43:13,000
It is a good description of the
core electron-core electron

546
00:43:13,000 --> 00:43:18,000
repulsion.
Not the repulsion due to the

547
00:43:18,000 --> 00:43:25,000
outermost electrons in argon,
but the repulsion due to the

548
00:43:25,000 --> 00:43:28,000
core electrons,
the n equal one,

549
00:43:28,000 --> 00:43:35,000
the n equal two electrons.
And it describes the

550
00:43:35,000 --> 00:43:42,000
nuclear-nuclear repulsion.
It is a (one over r) to the 12

551
00:43:42,000 --> 00:43:47,000
dependence. If I plot that,

552
00:43:47,000 --> 00:43:51,000
it would look something like
this.

553
00:43:51,000 --> 00:43:57,000
It is repulsive everywhere.
It is what we call a

554
00:43:57,000 --> 00:44:04,000
short-range interaction.
And this is important.

555
00:44:04,000 --> 00:44:10,000
What do we mean by short-range?
Well, short-range means that it

556
00:44:10,000 --> 00:44:16,000
only has a value when r is small
because it is a one over r

557
00:44:16,000 --> 00:44:19,000
raised to the 12 power.

558
00:44:19,000 --> 00:44:24,000
If r is large,
and you put a large number in

559
00:44:24,000 --> 00:44:28,000
the denominator and raise it to
the 12 power,

560
00:44:28,000 --> 00:44:34,000
the result is nothing.
The result is something close

561
00:44:34,000 --> 00:44:38,000
to zero.
This first term is zero when r

562
00:44:38,000 --> 00:44:42,000
is very large.
That is why we call it a

563
00:44:42,000 --> 00:44:47,000
short-range interaction.
And you can see that this

564
00:44:47,000 --> 00:44:52,000
really starts taking on value
when r is pretty small.

565
00:44:52,000 --> 00:44:58,000
But then there is this part,
the (sigma over r) to the 6

566
00:44:58,000 --> 00:45:04,000
term. That is the attractive

567
00:45:04,000 --> 00:45:07,000
interactions.
If I were to plot that,

568
00:45:07,000 --> 00:45:11,000
it would look like that.
That is the induced

569
00:45:11,000 --> 00:45:14,000
dipole-induced dipole
interaction.

570
00:45:14,000 --> 00:45:19,000
We could actually write down
the energy of interaction

571
00:45:19,000 --> 00:45:24,000
between these two induced
dipoles and find out that this

572
00:45:24,000 --> 00:45:30,000
is a one over the r to the 6
power.

573
00:45:30,000 --> 00:45:34,000
We are not going to do that,
but we could do that.

574
00:45:34,000 --> 00:45:38,000
These are always attractive.
The sum of the two,

575
00:45:38,000 --> 00:45:42,000
of course, gives us the actual
shape of that interaction

576
00:45:42,000 --> 00:45:45,000
potential.
But this interaction,

577
00:45:45,000 --> 00:45:49,000
here, is longer range,
or we call it longer range.

578
00:45:49,000 --> 00:45:54,000
It is longer range because it
is only one over r to the 6.

579
00:45:54,000 --> 00:45:57,000
And so r does not have to be

580
00:45:57,000 --> 00:46:04,000
that small in order for this
term to make the contribution.

581
00:46:04,000 --> 00:46:08,000
As r has a larger power here in
the denominator,

582
00:46:08,000 --> 00:46:12,000
that is shorter range
interaction.

583
00:46:12,000 --> 00:46:17,000
As it gets a smaller power in
the denominator,

584
00:46:17,000 --> 00:46:20,000
that is a shorter range
interaction.

585
00:46:20,000 --> 00:46:26,000
The other thing that is
interesting and important here

586
00:46:26,000 --> 00:46:33,000
is this value of sigma.
And I said that r sub e was

587
00:46:33,000 --> 00:46:37,000
1.12 times this sigma.

588
00:46:37,000 --> 00:46:44,000
The value of sigma is actually
the definition of what we call a

589
00:46:44,000 --> 00:46:50,000
van der Waal's radius.
That is, in the case of argon

590
00:46:50,000 --> 00:46:55,000
here, the equilibrium bond
length is 1.12 sigma.

591
00:46:55,000 --> 00:47:00,000
And so, r sub e,
1.12 sigma.

592
00:47:00,000 --> 00:47:08,000
For argon that parameter is 3.4
angstroms.

593
00:47:08,000 --> 00:47:16,000
And so the bond length is 3.8
angstroms.

594
00:47:16,000 --> 00:47:28,000
But the van der Waal's radius
is given by this 1.12 sigma over

595
00:00:02,000 --> 00:47:32,000
The van der Waal's radius,

596
00:47:32,000 --> 00:47:37,000
in this case 1.9 here,
is the radius that you use in

597
00:47:37,000 --> 00:47:42,000
these space filling models.
On the side wall,

598
00:47:42,000 --> 00:47:46,000
that top model is a space
filling model.

599
00:47:46,000 --> 00:47:51,000
And somehow you have to decide,
how large should those

600
00:47:51,000 --> 00:47:57,000
hydrogens be that are sticking
out in space that are not bonded

601
00:47:57,000 --> 00:48:03,000
to anything?
And the radii that are used are

602
00:48:03,000 --> 00:48:08,000
these van der Waal's radii,
because that is the radius at

603
00:48:08,000 --> 00:48:14,000
which you have this attractive
interaction due to the induced

604
00:48:14,000 --> 00:48:20,000
dipole-induced dipole.
And that is how those sizes are

605
00:48:20,115 --> 00:48:23,000
determined.