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Last time we started talking
about the internal motions of

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molecules, the vibrations and
the rotations.

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And we talked about how the
different ways in which a

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molecule can store energy were
called modes.

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Or, sometimes called the
degrees of freedom.

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And so, in general,
if we have a molecule that has

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N atoms, it has 3N degrees of
freedom or 3N modes.

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And we said last time,
that three of those 3N modes

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are always translational modes.
We live in a three-dimensional

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universe, so there are three
translational modes,

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three directions in which the
molecule can travel,

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can move.
And, therefore,

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if we use three of these 3N
modes for translation,

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then we have 3 minus 3 modes
that are internal modes,

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internal degrees of freedom.
And that is what we really want

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to start looking at today.
On the diagram here,

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on the side,
here is our N molecule.

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There are 3N total modes.
I just said three of them were

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translational modes.
Now, how do we separate those

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remaining modes between rotation
and vibration?

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The way we do it this.
If you have a molecule that is

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linear, you always have two
rotational modes.

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I will show you that in a
moment.

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Linear molecules always have
two rotational modes.

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00:02:24 --> 00:02:30
Nonlinear molecules have three
rotational modes.

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Therefore, if we use,
in a linear molecule,

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three of them for translation,
two of them for rotation,

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the number of vibrational modes
are what is left over.

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For a linear molecule,
we have 3N minus 5 vibrational

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modes.
If we have a nonlinear

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molecule, since we have three
rotational modes in a nonlinear

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molecule, we have leftover 3N
minus 6 vibrational

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modes.

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00:03:16 --> 00:03:21
Now, let's take a look at this
a little bit more in detail.

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Let's start with nitrogen.
Two atoms.

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Six modes total.
Three translational.

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Two rotational modes of
nitrogen.

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What are those rotational
modes?

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00:03:35 --> 00:03:39
Well, here is our nitrogen.
And one of those rotational

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modes is going to be rotation
about an axis in the plane of

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this board, and its rotation
around this axis.

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That axis goes through the
center of mass of that molecule.

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That is one of the rotational
modes.

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Another rotational mode is
rotation around an axis,

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here, that is perpendicular to
the board.

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This is another rotational
mode.

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It is going to turn out that
these two rotational modes are

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degenerate, meaning they are
going to have the same frequency

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of rotation.
They are going to have the same

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energy.
We will talk about that in a

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moment.
What I want to point out,

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here, is that if you look at
the axis, here,

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in the plane of the board that
is along the bond axis,

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this, hey, you know what,
this is not a rotational mode

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because there is cylindrical
symmetry around that bond axis.

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It has no meaning,
rotation around that bond axis.

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The molecule looks the same for
all of the angles from zero to

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360 degrees.
That is not a rotational mode.

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Linear molecules,
diatomics, you have two

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rotational modes.
They are going to turn out to

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be degenerate,
as we are going to see.

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And then, our table over there
says for nitrogen,

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we ought to have one
vibrational mode.

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And we do.
That one vibrational mode,

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here, is the nitrogen-nitrogen
stretch.

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It is a stretch mode.
We said last time that these

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bonds function like a spring.
The molecule can stretch,

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the molecule can compress.
We are going to look at that in

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more detail in just a moment.
Next, in our table over here,

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let's look at CO two.
CO two has three atoms.

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Nine total modes.
Three of them are translation.

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CO two,
even though it has three atoms

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in it, is a linear molecule,
so it has only two rotational

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modes.
Let's look and see what those

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00:06:17 --> 00:06:22
two rotational modes are.
One of those rotational modes

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is, again, rotation around an
axis in the plane of the board,

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here, around the center of mass
of that molecule.

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That is one rotational mode.
The other rotational mode is

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around an axis perpendicular to
the board here.

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That is a second rotational
mode.

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These two modes are going to be
degenerate because they are

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actually the same motion,
just in a different plane.

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These are degenerate.
And then, again,

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just to emphasize,
if you think that you have a

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rotation here along that linear
bond axis, well,

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00:07:13 --> 00:07:18
the answer is no,
this is not a rotational mode.

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So, we have two rotational
modes for CO two.

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And now what does our table say
in terms of vibration?

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Well, in terms of vibration for
a linear molecule,

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3N minus 5,
we have used now five modes.

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We have four left,
3N minus 5.

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What are those modes?

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One of these modes is what we
call a symmetric stretch.

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00:08:03 --> 00:08:08


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A symmetric stretch means that
this oxygen and this oxygen are

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simultaneously stretching or are
simultaneously compressing.

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My body is the carbon,
and my arms are the oxygen.

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A symmetric stretch is the
oxygens both moving in or both

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moving out.
That is one vibrational mode,

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this symmetric stretch.
That is one vibrational mode.

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What is another vibrational
mode?

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Well, if you have a symmetric
stretch, it must mean you have

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00:08:49 --> 00:08:53
an anti-symmetric stretch.

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In this case,
we have one oxygen moving out

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00:09:01 --> 00:09:03
while the other oxygen is moving
in.

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00:09:03 --> 00:09:07
And this carbon is kind of
moving in like that,

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too.
Or, this one is moving in,

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00:09:09 --> 00:09:12
this one is moving in,
this one is moving out.

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00:09:12 --> 00:09:15
That is the anti-symmetric
stretch.

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00:09:15 --> 00:09:18
If my body is the carbon and my
arms are the oxygen,

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00:09:18 --> 00:09:23
this is the anti-symmetric
stretch with the carbon moving

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just a little bit to the side,
too.

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The anti-symmetric stretch,
another mode.

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These two modes are not
degenerate.

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00:09:32 --> 00:09:34
They have different
frequencies.

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They have different energies.
We will see that in a moment.

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What are the other modes?
Well, the other modes are a

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bending mode.
What can happen is that this

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oxygen and this oxygen can move
in, and this carbon will kind of

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move out.
Or, of course,

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the other way,
here.

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These go like that,
this goes like that,

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00:10:14 --> 00:10:20
or that goes like that.
This is a bending vibration.

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00:10:20 --> 00:10:25


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Again, if my body is the carbon
and these are the oxygens,

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00:10:32 --> 00:10:37
the bending vibration looks
like this.

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00:10:37 --> 00:10:43
That is the bending vibration.
However, you could also imagine

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that you have a bend in another
plane.

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In other words,
you can imagine that these two

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oxygens here are coming out at
you or going in at you.

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00:11:00 --> 00:11:04
And so, in other words,
if I am the carbon,

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00:11:04 --> 00:11:10
these are the oxygens.
We also have a bend like this.

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These two bending vibrations
are degenerate.

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00:11:14 --> 00:11:18
These are four different
vibrations.

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00:11:18 --> 00:11:23
They are the four vibrational
modes, two bending,

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one symmetric stretch,
and the other,

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00:11:27 --> 00:11:34
the anti-symmetric stretch.
That is what we have in

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CO two.
What about another molecule?

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00:11:38 --> 00:11:41
How about water?
We have three atoms.

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00:11:41 --> 00:11:47
Again, nine total modes.
Three of them are translation,

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00:11:47 --> 00:11:50
but now water is not a linear
molecular.

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So, we are going to have three
rotational modes in the case of

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water.
Let's look at what those three

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00:12:00 --> 00:12:05
rotational modes ought to look
like.

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00:12:05 --> 00:12:12


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Here is our water molecule.
One of those rotations is,

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00:12:18 --> 00:12:24
again, a rotation around an
axis here in the plane of the

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board.
That is one rotational mode.

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00:12:29 --> 00:12:33
If I am the oxygen and my hands
are the hydrogen,

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this is a rotational mode.
You got that?

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00:12:37 --> 00:12:40
Okay.
And then another rotational

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00:12:40 --> 00:12:44
mode, here.
It can be a rotation around the

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center of the mass,
centered on the oxygen here.

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Axis perpendicular to the plane
of the board.

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This is that rotation.
And I am sorry,

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but I cannot do cartwheels.
That is one rotational mode.

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00:13:02 --> 00:13:07
And then a final rotational
mode involves rotation,

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00:13:07 --> 00:13:13
now, around this axis.
It is an axis perpendicular to

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this one, but in the plane of
the board.

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00:13:16 --> 00:13:23
And this is a rotational mode.
We have three rotational modes.

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This one requires me to do some
flips, which I am also not going

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to do.
But each one of these modes has

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a different frequency,
has a different energy.

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00:13:37 --> 00:13:42
And you can see we have three
of them now because we no longer

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have a linear molecule.
We have three rotational modes.

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Now, what about the vibrations
for water?

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00:13:50 --> 00:13:54
Well, you can see from our
graph up here that 3N minus 6,

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00:13:54 --> 00:13:59
in this case for
water, is going to be three

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vibrational modes.
Let's look at that.

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One of those vibrational modes
is a symmetric stretch,

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again.
The hydrogen is moving in or

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both moving out at the same
time.

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A picture of the symmetric
stretch is, again,

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I am the oxygen,
arms are the hydrogen.

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This is the symmetric stretch,
right?

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00:14:46 --> 00:14:50
And then, if there is a
symmetric stretch,

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00:14:50 --> 00:14:55
there is going to be an
anti-symmetric stretch.

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What that means is one of the
hydrogens is moving in and the

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00:15:00 --> 00:15:06
other one moving out.
Or, this way.

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00:15:06 --> 00:15:15
Of course, a picture for that
is this kind of a motion.

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00:15:15 --> 00:15:22
You get the idea.
Anti-symmetric stretch.

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00:15:22 --> 00:15:30
And then, finally,
we have a bending mode.

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A bending mode,
meaning these hydrogens moving

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00:15:35 --> 00:15:40
this way or those hydrogens
moving that way.

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00:15:40 --> 00:15:46
And, of course,
a picture for that is this.

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00:15:46 --> 00:15:50
That is a bending mode for the
hydrogens.

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00:15:50 --> 00:15:58
We have three vibrational modes
for the hydrogen.

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00:15:58 --> 00:16:02
Now we are going to look at
some of the principles behind

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00:16:02 --> 00:16:04
these internal degrees of
freedom.

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00:16:04 --> 00:16:08
But, of course,
the other reason for telling

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you about this-- Did you have a
question?

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00:16:12 --> 00:16:17


202
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It can, but it is not a
separate mode.

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00:16:20 --> 00:16:23
That is correct.
That is right.

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00:16:23 --> 00:16:26
It looks like a rotation,
actually.

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00:16:26 --> 00:16:32
That is absolutely correct.
The other reason for talking

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00:16:32 --> 00:16:37
about this is to tell you that
each one of these vibrations or

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00:16:37 --> 00:16:41
rotations occurs at a frequency
at an energy that is

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00:16:41 --> 00:16:45
characteristic of the molecule.
If we had a way to actually

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00:16:45 --> 00:16:50
measure the frequency of these
vibrations or the frequencies of

210
00:16:50 --> 00:16:54
the rotations,
well, then we have a great tool

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00:16:54 --> 00:16:59
for identifying what kind of
molecule we have.

212
00:16:59 --> 00:17:02
Analytically,
this is a wonderful technique

213
00:17:02 --> 00:17:06
for identifying,
ultimately, the molecule that

214
00:17:06 --> 00:17:11
you have, if you can measure
those frequencies of vibration

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00:17:11 --> 00:17:15
or rotation.
And we are going to do that in

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00:17:15 --> 00:17:18
just a moment.
But what we also have to

217
00:17:18 --> 00:17:23
understand, before we go and
measure these frequencies,

218
00:17:23 --> 00:17:28
is we have to understand that
the energies with which a

219
00:17:28 --> 00:17:33
molecule vibrates or rotates are
quantized.

220
00:17:33 --> 00:17:37
So, that is what we have to
spend some time thinking about

221
00:17:37 --> 00:17:40
right now.
Let's do that.

222
00:17:40 --> 00:17:51


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00:17:51 --> 00:17:54
And we are going to start with
vibration.

224
00:17:54 --> 00:18:00


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00:18:00 --> 00:18:04
To illustrate this,
I am going to draw one of these

226
00:18:04 --> 00:18:09
energies of interaction,
again, one of these curves that

227
00:18:09 --> 00:18:14
we have been talking about.
And I am going to set my zero

228
00:18:14 --> 00:18:18
of energy at the dissociated
atom limit.

229
00:18:18 --> 00:18:21
So, I am going to talk about H
plus Cl.

230
00:18:21 --> 00:18:25
And, of course,
right here, we talked about

231
00:18:25 --> 00:18:29
this being the equilibrium bond
length, r sub e,

232
00:18:29 --> 00:18:33
for the HCl molecule.

233
00:18:33 --> 00:18:39


234
00:18:39 --> 00:18:42
That is the energy of
interaction.

235
00:18:42 --> 00:18:49
I am going to now draw in the
ground vibrational state of HCl,

236
00:18:49 --> 00:18:53
which I am going to represent
as a line.

237
00:18:53 --> 00:18:59
It is going to be the energy.
There is the ground vibrational

238
00:18:59 --> 00:19:04
state of HCl.
The ground vibrational state,

239
00:19:04 --> 00:19:09
that energy is characterized by
a quantum number.

240
00:19:09 --> 00:19:15
The quantum number is the
vibrational quantum number for

241
00:19:15 --> 00:19:20
the ground vibrational state.
The vibrational quantum number

242
00:19:20 --> 00:19:26
is v equal zero.
What this line represents is

243
00:19:26 --> 00:19:32
the ground vibrational energy.
It also represents the extent

244
00:19:32 --> 00:19:38
to which the molecule's bond
stretches and compresses.

245
00:19:38 --> 00:19:42
The bottom line,
here, is that the bond

246
00:19:42 --> 00:19:47
stretches up to here.
The inner section of this line

247
00:19:47 --> 00:19:52
with the interaction energy,
here, represents the maximum

248
00:19:52 --> 00:19:57
distance of the bond,
the maximum bond length,

249
00:19:57 --> 00:20:02
the most that the bond
stretches.

250
00:20:02 --> 00:20:07
r is going this way.
The intersection of this energy

251
00:20:07 --> 00:20:14
with this curve represents the
closest that the two nuclei get.

252
00:20:14 --> 00:20:21
This represents the most that
the molecule has compressed.

253
00:20:21 --> 00:20:24
This is the smallest value of
r.

254
00:20:24 --> 00:20:30
That is what that diagram
represents.

255
00:20:30 --> 00:20:34
If you start out,
here, with an HCl molecule at

256
00:20:34 --> 00:20:40
the equilibrium bond length,
right here, what happens is

257
00:20:40 --> 00:20:45
that the HCl will stretch,
and it will stretch to this

258
00:20:45 --> 00:20:48
position.
This is an exaggeration.

259
00:20:48 --> 00:20:54
This is how much it stretches.
And then, it comes back and

260
00:20:54 --> 00:20:59
goes through the equilibrium
position.

261
00:20:59 --> 00:21:02
Over here, and this is another
exaggeration,

262
00:21:02 --> 00:21:06
this is how close they get to
each other.

263
00:21:06 --> 00:21:11
This is how much they compress.
This point is often called the

264
00:21:11 --> 00:21:15
inner turning point,
inner because it is the

265
00:21:15 --> 00:21:19
smallest bond distance.
This is often called the outer

266
00:21:19 --> 00:21:24
turning point because it is the
larger bond distance.

267
00:21:24 --> 00:21:28
It is outer because it is at
this distance,

268
00:21:28 --> 00:21:34
now, that the bond turns around
and begins to compress.

269
00:21:34 --> 00:21:39
This HCl molecule here is
vibrating, equilibrium position.

270
00:21:39 --> 00:21:44
Then it goes to the maximum
extension, comes back through

271
00:21:44 --> 00:21:49
the equilibrium position,
and then goes to the maximum

272
00:21:49 --> 00:21:54
compression, comes back to the
equilibrium position.

273
00:21:54 --> 00:21:59
That is what this line
literally represents.

274
00:21:59 --> 00:22:03
It is also the energy,
which I will explain a little

275
00:22:03 --> 00:22:08
bit more in just a moment.
So, this molecule is vibrating

276
00:22:08 --> 00:22:12
from here to here to here and
back and forth.

277
00:22:12 --> 00:22:18
Now, what I also want to point
out to you is something that I

278
00:22:18 --> 00:22:23
have been kind of misleading you
about for the last few weeks.

279
00:22:23 --> 00:22:28
That is that this molecule is
not sitting in the bottom of

280
00:22:28 --> 00:22:33
this well.
Remember that when we were

281
00:22:33 --> 00:22:39
drawing bond association
energies, I was drawing it from

282
00:22:39 --> 00:22:44
the bottom of this well to the
dissociated atom limit?

283
00:22:44 --> 00:22:49
I told you this was delta E sub
d.

284
00:22:49 --> 00:22:54
Well, the bottom line is that
it is not really correct.

285
00:22:54 --> 00:22:59
The reason is,
is because the molecule is

286
00:22:59 --> 00:23:05
never really sitting at the
bottom of this well.

287
00:23:05 --> 00:23:10
It is sitting this much above.
It is sitting at the v equal

288
00:23:10 --> 00:23:15
zero state.
It is not sitting at the bottom

289
00:23:15 --> 00:23:19
of the well.
If it were, it would not be

290
00:23:19 --> 00:23:22
vibrating.
If it is not vibration,

291
00:23:22 --> 00:23:28
it will violate the Uncertainty
Principle, something we didn't

292
00:23:28 --> 00:23:33
talk about.
But the bottom line is that it

293
00:23:33 --> 00:23:37
cannot sit at the bottom of the
well.

294
00:23:37 --> 00:23:42
It is always vibrating by about
this much energy.

295
00:23:42 --> 00:23:48
When you have an experimentally
determined dissociation energy

296
00:23:48 --> 00:23:54
for the bond energy,
the experimentally determined

297
00:23:54 --> 00:24:00
energy is actually the energy
from this level.

298
00:24:00 --> 00:24:05
From v equal zero up
to this dissociation limit.

299
00:24:05 --> 00:24:09
This is the experimental bond
energy.

300
00:24:09 --> 00:24:14
Because, if you are going to
measure a bond energy in the

301
00:24:14 --> 00:24:18
laboratory, well,
you can only measure it from

302
00:24:18 --> 00:24:24
the lowest level at which that
molecule can possibly be at.

303
00:24:24 --> 00:24:29
It is not at the bottom of the
well.

304
00:24:29 --> 00:24:34
The difference between where
this level is and the bottom of

305
00:24:34 --> 00:24:39
the well actually has a name.
That difference is called the

306
00:24:39 --> 00:24:42
zero point energy.

307
00:24:42 --> 00:24:47


308
00:24:47 --> 00:24:52
And I will explain that in a
moment, when we look at the

309
00:24:52 --> 00:24:55
energies a little more
carefully.

310
00:24:55 --> 00:25:00
Now, just for completeness,
and this is not something that

311
00:25:00 --> 00:25:04
you have to know,
let me tell you what the

312
00:25:04 --> 00:25:10
nomenclature usually is here for
these well depths.

313
00:25:10 --> 00:25:14
The nomenclature from the
bottom of the well to here,

314
00:25:14 --> 00:25:18
which is not experimentally
measured, is usually D sub e,

315
00:25:18 --> 00:25:21
dissociation sub e.

316
00:25:21 --> 00:25:28


317
00:25:28 --> 00:25:33
From the v equal zero
level to the dissociation limit,

318
00:25:33 --> 00:25:37
that energy here is D sub zero.

319
00:25:37 --> 00:25:42
You don't have to know these,
but I just want to tell you,

320
00:25:42 --> 00:25:46
when you see that.
All dissociation energies

321
00:25:46 --> 00:25:51
experimentally measured are
measured from here to here

322
00:25:51 --> 00:25:56
because that is the lowest
energy the molecule can have,

323
00:25:56 --> 00:26:01
is being in the v equal zero
ground vibrational

324
00:26:01 --> 00:26:07
state.
Well, now let's take a look at

325
00:26:07 --> 00:26:13
the energies a little more
carefully.

326
00:26:13 --> 00:26:31


327
00:26:31 --> 00:26:37
Here is our favorite diagram
again, our intermolecular

328
00:26:37 --> 00:26:41
interaction potential.
This is HCl,

329
00:26:41 --> 00:26:45
so here is the dissociated atom
limit.

330
00:26:45 --> 00:26:51
This is v equal zero,
right in there.

331
00:26:51 --> 00:26:57
Now, there is an excited state
for the HCl, or for any

332
00:26:57 --> 00:27:02
molecule.
v equal one is the first

333
00:27:02 --> 00:27:08
vibrationally excited state.
v equal two is the

334
00:27:08 --> 00:27:11
second vibrationally excited
state.

335
00:27:11 --> 00:27:15
v equal three is the
third.

336
00:27:15 --> 00:27:18
v equal four is the
fourth.

337
00:27:18 --> 00:27:22
v equal five is the
fifth.

338
00:27:22 --> 00:27:28
These are the allowed
vibrational energies.

339
00:27:28 --> 00:27:32
In other words,
the molecule can vibrate with

340
00:27:32 --> 00:27:38
this energy or this energy or
this energy, but not some

341
00:27:38 --> 00:27:43
arbitrary energy,
like right in here or like

342
00:27:43 --> 00:27:47
right in there.
These are the allowed

343
00:27:47 --> 00:27:52
vibrational states.
And we have a nice analytical

344
00:27:52 --> 00:27:56
expression to describe those
energies.

345
00:27:56 --> 00:28:02
That analytical expression is
right here.

346
00:28:02 --> 00:28:07
That energy is h times nu times
(that vibrational quantum number

347
00:28:07 --> 00:28:10
plus one-half).

348
00:28:10 --> 00:28:13
So, v is that vibrational
quantum number.

349
00:28:13 --> 00:28:17
This nu, here,
is the fundamental frequency,

350
00:28:17 --> 00:28:22
which is the frequency with
which the molecule vibrates.

351
00:28:22 --> 00:28:26
It is the frequency with which
that molecule stretches,

352
00:28:26 --> 00:28:31
comes back to the equilibrium
position, compresses,

353
00:28:31 --> 00:28:36
and comes back to the
equilibrium position.

354
00:28:36 --> 00:28:41
It is the number of cycles per
second that molecule makes.

355
00:28:41 --> 00:28:47
So, nu is the number of cycles,
here, that the molecule in v

356
00:28:47 --> 00:28:50
equal zero makes per
second.

357
00:28:50 --> 00:28:55
Stretches, compresses,
and back to the equilibrium

358
00:28:55 --> 00:28:58
position.
Notice it is also the

359
00:28:58 --> 00:29:04
vibrational frequency of v equal
one.

360
00:29:04 --> 00:29:09
The vibrational frequency does
not depend on the vibrational

361
00:29:09 --> 00:29:12
quantum number.
You see no dependence of the

362
00:29:12 --> 00:29:16
vibrational quantum number on
the frequency.

363
00:29:16 --> 00:29:21
v equal two vibrates
with the same frequency.

364
00:29:21 --> 00:29:23
v equal three.
v equal four.

365
00:29:23 --> 00:29:27
v equal five.
All of these states have the

366
00:29:27 --> 00:29:30
same nu here,
have the same fundamental

367
00:29:30 --> 00:29:34
frequency.
What does that mean?

368
00:29:34 --> 00:29:39
That looks a little strange in
the sense that if you have a

369
00:29:39 --> 00:29:43
molecule in v equals zero,
the number of cycles per second

370
00:29:43 --> 00:29:48
that it makes is the same as a
molecule in v equal five.

371
00:29:48 --> 00:29:52
Well, you can see that in v
equal five, the molecule has to

372
00:29:52 --> 00:29:56
stretch further,
and it has to compress further.

373
00:29:56 --> 00:30:02
It has to travel more distance.
And so, if it has to travel

374
00:30:02 --> 00:30:06
more distance,
but it still carries out the

375
00:30:06 --> 00:30:11
number of cycles per unit time,
the same as v equal zero,

376
00:30:11 --> 00:30:17
well, then it must mean that
the molecule in v equal five is

377
00:30:17 --> 00:30:21
moving more quickly,
is moving faster.

378
00:30:21 --> 00:30:26
And that is what that means.
The energy of this state is

379
00:30:26 --> 00:30:30
higher.
It is moving faster.

380
00:30:30 --> 00:30:34
But it is moving with the same
frequency nu.

381
00:30:34 --> 00:30:37
That is the fundamental
frequency.

382
00:30:37 --> 00:30:42
That fundamental frequency,
of course, is in hertz,

383
00:30:42 --> 00:30:45
our unit for all frequencies
here.

384
00:30:45 --> 00:30:51
Now, I drew on the board and
called this energy from the

385
00:30:51 --> 00:30:56
bottom of the well to v equal
zero, the zero point energy.

386
00:30:56 --> 00:31:01
That is what it is.
But numerically what that

387
00:31:01 --> 00:31:05
energy is, is one-half h nu.
Stick in v equal zero,

388
00:31:05 --> 00:31:08
you get one-half h nu.

389
00:31:08 --> 00:31:11
This energy,
here, is measured from the

390
00:31:11 --> 00:31:15
bottom of this well.
It is important that you

391
00:31:15 --> 00:31:18
understand where this energy is
measured from.

392
00:31:18 --> 00:31:22
It is measured from the bottom
of this well.

393
00:31:22 --> 00:31:25
v equal one,
you put that in here and get

394
00:31:25 --> 00:31:29
three-halves h nu.

395
00:31:29 --> 00:31:33
v equal two,
you get five-halves h nu.

396
00:31:33 --> 00:31:35
v equal three,

397
00:31:35 --> 00:31:40
you get seven-halves h nu.

398
00:31:40 --> 00:31:45
The other thing to notice is
that these states are equally

399
00:31:45 --> 00:31:49
spaced.
The energy spacing between any

400
00:31:49 --> 00:31:54
two adjacent states is h times
nu.

401
00:31:54 --> 00:32:02


402
00:32:02 --> 00:32:05
That is incorrect.
What I am drawing,

403
00:32:05 --> 00:32:09
here, is right.
I am sorry about that.

404
00:32:09 --> 00:32:16
One-half h nu is
from the bottom of the well to v

405
00:32:16 --> 00:32:20
equal zero.
I think I know what happened.

406
00:32:20 --> 00:32:24
What I have shown you here is
correct.

407
00:32:24 --> 00:32:30
What is in your notes is
printed incorrectly.

408
00:32:30 --> 00:32:35
You should make note of that
and change that.

409
00:32:35 --> 00:32:45


410
00:32:45 --> 00:32:49
Suppose we have a molecule in
the v equal zero state,

411
00:32:49 --> 00:32:54
where the energy is one-half h
nu,

412
00:32:54 --> 00:33:00
that molecule can be excited to
the v equal one state.

413
00:33:00 --> 00:33:04
That molecule in v equal zero
is still vibrating,

414
00:33:04 --> 00:33:09
but we can make it vibrate even
faster, with more energy.

415
00:33:09 --> 00:33:13
The same cycles per second,
but with more energy,

416
00:33:13 --> 00:33:19
by promoting it to v equal one.
We can do that if the molecule

417
00:33:19 --> 00:33:23
absorbs a photon.
And the energy of that photon

418
00:33:23 --> 00:33:28
has to be equal to the
difference in the energies of

419
00:33:28 --> 00:33:33
these two states.
So, if the final energy,

420
00:33:33 --> 00:33:37
here, of the v equal one state,
that is the final state,

421
00:33:37 --> 00:33:43
if that energy is three-halves
h nu and the energy

422
00:33:43 --> 00:33:47
of the initial state was
one-half h nu,

423
00:33:47 --> 00:33:52
then the difference in energy
between the states is h nu.

424
00:33:52 --> 00:33:56
We have to have a photon that
has exactly this energy

425
00:33:56 --> 00:33:59
different, delta E.
That delta E,

426
00:33:59 --> 00:34:04
then, has to be gotten by a
photon with a frequency equal to

427
00:34:04 --> 00:34:07
nu.
And the frequency of that

428
00:34:07 --> 00:34:12
photon that is going to make
that transition is actually also

429
00:34:12 --> 00:34:15
the fundamental frequency of the
molecule.

430
00:34:15 --> 00:34:19
We are lucky on this one.
Nu is delta E over h.

431
00:34:19 --> 00:34:23
We have to have a photon with

432
00:34:23 --> 00:34:28
that frequency for the molecule
to be promoted from v equal zero

433
00:34:28 --> 00:34:33
to v equal one.
How do we know what that is?

434
00:34:33 --> 00:34:39
Well, we do an experiment.
We do an infrared spectroscopy

435
00:34:39 --> 00:34:44
type of experiment.
The difference in the energies

436
00:34:44 --> 00:34:51
between vibrational states is in
the infrared range of the

437
00:34:51 --> 00:34:56
electromagnetic spectrum.
And so we are going to use

438
00:34:56 --> 00:35:02
infrared radiation.
We have some source of infrared

439
00:35:02 --> 00:35:05
radiation.
It puts out all different

440
00:35:05 --> 00:35:09
frequencies in the infrared.
And then we send it through a

441
00:35:09 --> 00:35:13
monochromator.
The monochromator disperses

442
00:35:13 --> 00:35:17
that radiation in space and
allows only a certain frequency

443
00:35:17 --> 00:35:22
of radiation to come right
through the slits and out into

444
00:35:22 --> 00:35:24
our sample.
It is like you had the

445
00:35:24 --> 00:35:28
diffraction glasses,
where we disbursed the

446
00:35:28 --> 00:35:32
radiation in space.
Well, a monochromator is

447
00:35:32 --> 00:35:36
essentially the same thing.
There is a diffraction grading

448
00:35:36 --> 00:35:39
in it.
It allows, out of the front

449
00:35:39 --> 00:35:42
end, only radiation of a certain
frequency to come out.

450
00:35:42 --> 00:35:46
And we can adjust what that
radiation is or what frequency

451
00:35:46 --> 00:35:49
that radiation has.
We then allow it to pass

452
00:35:49 --> 00:35:51
through our sample,
say HCl.

453
00:35:51 --> 00:35:54
And then there is photo
detector over here,

454
00:35:54 --> 00:36:00
some kind of photomultiplier.
And what will happen is that if

455
00:36:00 --> 00:36:05
this molecule is going to absorb
frequency of that radiation,

456
00:36:05 --> 00:36:08
well, then the number of
photons reaching the photo

457
00:36:08 --> 00:36:13
detector is going to go down.
If the frequency does not match

458
00:36:13 --> 00:36:16
the frequency of the vibration
of HCl, well,

459
00:36:16 --> 00:36:20
the photons go right through,
onto the detector.

460
00:36:20 --> 00:36:24
But if it matches,
then the molecule absorbs those

461
00:36:24 --> 00:36:27
photons and they never make it
to the detector.

462
00:36:27 --> 00:36:33
So, a plot looks like this.
This is the intensity at the

463
00:36:33 --> 00:36:36
detector versus the frequency of
the radiation.

464
00:36:36 --> 00:36:40
The intensity will be high,
but now at some frequency,

465
00:36:40 --> 00:36:45
which will turn out to be the
vibrational frequency of the

466
00:36:45 --> 00:36:47
molecule, the intensity goes
down low.

467
00:36:47 --> 00:36:52
And then, as you go to higher
frequency, the intensity comes

468
00:36:52 --> 00:36:55
back up.
Right here, the molecule has

469
00:36:55 --> 00:36:59
made the transition from v equal
zero to v equal one because the

470
00:36:59 --> 00:37:04
frequency of that photon matches
the energy difference between v

471
00:37:04 --> 00:37:09
equal zero and v equal one
divided by h.

472
00:37:09 --> 00:37:13
It also happens to be the
fundamental frequency of the

473
00:37:13 --> 00:37:18
molecule for HCl.
So, that is how we know what

474
00:37:18 --> 00:37:21
those fundamental frequencies
are.

475
00:37:21 --> 00:37:24
Let's now look at what those
energies are.

476
00:37:24 --> 00:37:30
We have gone from v equal zero
to v equal one.

477
00:37:30 --> 00:37:34
Delta E equal h nu. We just

478
00:37:34 --> 00:37:39
measured nu as 8.6x10^13 inverse
seconds.

479
00:37:39 --> 00:37:44
Multiply it by h.
That is going to give us delta

480
00:37:44 --> 00:37:46
E.
The difference in energy

481
00:37:46 --> 00:37:51
between the two states is
5.7x10^-20 joules.

482
00:37:51 --> 00:37:57
If I turn that into kilojoules
per mole, that is 35 kilojoules

483
00:37:57 --> 00:38:02
per mole.
That is the energy difference

484
00:38:02 --> 00:38:06
from here to here,
about 35 kilojoules per mole.

485
00:38:06 --> 00:38:12
How strong is the HCl bond?
Well, the HCl bond from here to

486
00:38:12 --> 00:38:15
here is about 420 kilojoules per
mole.

487
00:38:15 --> 00:38:20
This is about a tenth,
a little bit less than a tenth

488
00:38:20 --> 00:38:23
of the total bond strength for
HCl.

489
00:38:23 --> 00:38:27
If we make it vibrate,
we are not going to make that

490
00:38:27 --> 00:38:32
bond break.
It is an order of magnitude

491
00:38:32 --> 00:38:38
less energy here than what you
need to break that bond.

492
00:38:38 --> 00:38:44
Now, here is something that is
a little confusing.

493
00:38:44 --> 00:38:50
These fundamental frequencies,
it turns out that we actually

494
00:38:50 --> 00:38:56
use a different convention to
label them, different units.

495
00:38:56 --> 00:39:03
We do not use Hertz very often
to talk about the fundamental

496
00:39:03 --> 00:39:09
frequencies of molecules.
We use something called a

497
00:39:09 --> 00:39:15
wavenumber instead of hertz.
A wavenumber is an inverse

498
00:39:15 --> 00:39:19
centimeter, centimeter to the
minus one power.

499
00:39:19 --> 00:39:25
The symbol for a wavenumber is
a frequency sign with a bar on

500
00:39:25 --> 00:39:30
top of it.
That is a wave number.

501
00:39:30 --> 00:39:35
How do we get a wavenumber?
Well, we get a wavenumber by

502
00:39:35 --> 00:39:39
taking the frequency in Hertz,
inverse seconds,

503
00:39:39 --> 00:39:42
and dividing it by the speed of
light.

504
00:39:42 --> 00:39:46
But the speed of light,
of course, has to be in

505
00:39:46 --> 00:39:50
centimeters per second.
The seconds cancel,

506
00:39:50 --> 00:39:56
and what we have then left is
inverse centimeters.

507
00:39:56 --> 00:40:00
For example,
if the fundamental frequency of

508
00:40:00 --> 00:40:06
HCl was 8.6x10^13 hertz,
we divide that by the speed of

509
00:40:06 --> 00:40:12
light in centimeters per second
and the wavenumbers,

510
00:40:12 --> 00:40:17
or the fundamental frequency in
wavenumbers, then,

511
00:40:17 --> 00:40:24
is 2,886 wavenumbers for HCl.
These are the common units that

512
00:40:24 --> 00:40:29
are used to talk about the
fundamental vibrational

513
00:40:29 --> 00:40:35
frequencies of molecules.
And these are the common

514
00:40:35 --> 00:40:40
numbers used to discuss infrared
spectra of molecules.

515
00:40:40 --> 00:40:44
This unit of a wavenumber.
It is a non-SI unit.

516
00:40:44 --> 00:40:47
In other words,
what you will often see is an

517
00:40:47 --> 00:40:51
infrared spectrum intensity at
the photodiode,

518
00:40:51 --> 00:40:56
now, versus wave number.
And right at 2,886 wavenumbers,

519
00:40:56 --> 00:41:01
that is the fundamental
frequency in wavenumbers.

520
00:41:01 --> 00:41:06
That is where the molecule
makes the transition from v

521
00:41:06 --> 00:41:11
equal zero to v equal one.
Now, the question is,

522
00:41:11 --> 00:41:18
what determines the frequency
of the vibration of a molecule?

523
00:41:18 --> 00:41:23
Well, what determines the
frequency of that vibration,

524
00:41:23 --> 00:41:29
and here, I have now gone back
to frequency because I am

525
00:41:29 --> 00:41:34
writing it in terms of some
other parameters,

526
00:41:34 --> 00:41:40
are two parameters.
One is the force constant k,

527
00:41:40 --> 00:41:44
and the other is the mass.
What I have here is the reduced

528
00:41:44 --> 00:41:46
mass.
I will explain that to you in

529
00:41:46 --> 00:41:50
just a moment,
but this is essentially the

530
00:41:50 --> 00:41:52
mass.
The frequency is given by one

531
00:41:52 --> 00:41:57
over 2pi times the square root
of this k, the force constant

532
00:41:57 --> 00:42:00
over the mass.

533
00:42:00 --> 00:42:04
sqrt(k / m)**
That force constant,

534
00:42:04 --> 00:42:09
k, is a measure of the
stiffness of the spring.

535
00:42:09 --> 00:42:13
Another words,
if you have a very stiff

536
00:42:13 --> 00:42:19
spring, k is very large.
That is you need a lot of force

537
00:42:19 --> 00:42:25
to pull that spring apart,
a lot of force to compress that

538
00:42:25 --> 00:42:29
spring.
A large k means you have a very

539
00:42:29 --> 00:42:34
stiff spring.
If you have a small k,

540
00:42:34 --> 00:42:38
that means you have a very weak
spring.

541
00:42:38 --> 00:42:45
It does not take very much
force to stretch that spring or

542
00:42:45 --> 00:42:50
to compress that spring.
So k is a measure of the

543
00:42:50 --> 00:42:56
stiffness of the spring.
It also tells us something

544
00:42:56 --> 00:43:02
about the shape of the
interaction energy.

545
00:43:02 --> 00:43:06
Another words,
what k actually tells us here

546
00:43:06 --> 00:43:09
is kind of the width of this
well.

547
00:43:09 --> 00:43:14
Here is my interaction energy,
potential well.

548
00:43:14 --> 00:43:19
If you have a very narrow
width, this is a large k.

549
00:43:19 --> 00:43:25
If you have a very broad width,
here, this is a small k.

550
00:43:25 --> 00:43:30
So, k, the force constant,
tells us about the shape,

551
00:43:30 --> 00:43:36
here, of this well.
Very narrow well,

552
00:43:36 --> 00:43:40
large k.
Very broad well,

553
00:43:40 --> 00:43:46
small k.
Now, I think that maybe you

554
00:43:46 --> 00:43:55
have seen this before in 8.01.
Have you done the harmonic

555
00:43:55 --> 00:43:57
oscillator yet?
No?

556
00:43:57 --> 00:44:00
Yes?
Some yes.

557
00:44:00 --> 00:44:03
Some no.
Mostly no, it sounds to me.

558
00:44:03 --> 00:44:07
This expression,
here, for the frequency,

559
00:44:07 --> 00:44:10
you are going to see in 8.01
some time.

560
00:44:10 --> 00:44:14
It comes from the harmonic
oscillator model.

561
00:44:14 --> 00:44:18
You will remember that you have
seen this before.

562
00:44:18 --> 00:44:23
Those of you who have know what
I am talking about.

563
00:44:23 --> 00:44:28
Those of you who have not,
just remember you are going to

564
00:44:28 --> 00:44:35
see this very soon once again.
Now, the other parameter here

565
00:44:35 --> 00:44:42
that is important is this mu
here, this reduced mass.

566
00:44:42 --> 00:44:49
What the reduced mass is,
is essentially the mass.

567
00:44:49 --> 00:44:54
Now, if you had already done
this in 8.01,

568
00:44:54 --> 00:45:02
what you are going to look at
is a mass on the end of a spring

569
00:45:02 --> 00:45:08
attached to a wall.
There will be a wall,

570
00:45:08 --> 00:45:13
then you will see a spring,
and then there will be some

571
00:45:13 --> 00:45:16
mass m.
When you do this problem as a

572
00:45:16 --> 00:45:21
harmonic oscillator,
you are going to write nu here

573
00:45:21 --> 00:45:26
as k over this mass right there.
But, in this example,

574
00:45:26 --> 00:45:32
this mass here is the only body
that is moving.

575
00:45:32 --> 00:45:36
Your wall is still.
But, in the case of a molecule,

576
00:45:36 --> 00:45:40
both atoms are moving.
What mass do we use?

577
00:45:40 --> 00:45:46
Well, the fact that both are
moving means we have a two body

578
00:45:46 --> 00:45:49
problem.
And we are going to have to

579
00:45:49 --> 00:45:55
change the coordinate system to
take into account that both

580
00:45:55 --> 00:46:00
masses are moving.
We can do that easily.

581
00:46:00 --> 00:46:04
And doing that easily involves
using something called the

582
00:46:04 --> 00:46:08
reduced mass.
The reduced mass takes a two

583
00:46:08 --> 00:46:13
body problem and reduces it to a
single body, where the single

584
00:46:13 --> 00:46:18
body is this fictitious body
with a mass given by the reduced

585
00:46:18 --> 00:46:19
mass.
It is exact.

586
00:46:19 --> 00:46:22
I did not make any
approximations here.

587
00:46:22 --> 00:46:27
But the bottom line is that
this reduced mass is simply the

588
00:46:27 --> 00:46:31
mass of one atom,
times the mass of the other,

589
00:46:31 --> 00:46:36
divided by the sum of the two
masses.

590
00:46:36 --> 00:46:42
And sometimes a useful
approximation is when one of the

591
00:46:42 --> 00:46:49
masses is much lower than the
other mass, so when m1 here is

592
00:46:49 --> 00:46:54
smaller than m2.
Well, you can see in that case,

593
00:46:54 --> 00:47:01
if m1 is very small compared to
m2, we are going to treat it

594
00:47:01 --> 00:47:06
like a zero.
And then these two m's are

595
00:47:06 --> 00:47:10
going to cancel.
You can see the reduced mass is

596
00:47:10 --> 00:47:14
roughly equivalent to the mass
of the smaller body.

597
00:47:14 --> 00:47:18
Sometimes I will ask you to use
that approximation.

598
00:47:18 --> 00:47:22
Other times,
I will ask you to calculate the

599
00:47:22 --> 00:47:25
reduced mass.
Let's look at some frequencies

600
00:47:25 --> 00:47:29
for some molecules,
here.

601
00:47:29 --> 00:47:33
I have an array of molecules.
All of them roughly have the

602
00:47:33 --> 00:47:38
same force constant.
I am going to look at the mass

603
00:47:38 --> 00:47:41
dependence on the fundamental
frequency.

604
00:47:41 --> 00:47:46
I have got the frequencies
written both in terms of wave

605
00:47:46 --> 00:47:50
numbers and Hertz.
The first thing I want you to

606
00:47:50 --> 00:47:54
notice is the fundamental
frequency for hydrogen,

607
00:47:54 --> 00:47:59
4,159 wave numbers.
That is the highest vibrational

608
00:47:59 --> 00:48:04
frequency that we know about.
Why is it so high?

609
00:48:04 --> 00:48:08
Well, because those two
hydrogens are so light,

610
00:48:08 --> 00:48:13
and the mass is in the
denominator of that fundamental

611
00:48:13 --> 00:48:17
frequency.
So if it is in the denominator,

612
00:48:17 --> 00:48:20
it is small,
and the fundamental frequency

613
00:48:20 --> 00:48:25
is going to be high.
Then look at these molecules,

614
00:48:25 --> 00:48:28
HCl, HBr, HI.
In this case here,

615
00:48:28 --> 00:48:33
it is all a hydrogen bonded to
a halogen.

616
00:48:33 --> 00:48:37
For the most part,
the reduced mass of these three

617
00:48:37 --> 00:48:41
molecules is that of hydrogen.
If that is of hydrogen,

618
00:48:41 --> 00:48:45
since that is in the
denominator for the expression

619
00:48:45 --> 00:48:50
for the frequency,
well, the frequencies are going

620
00:48:50 --> 00:48:53
to be high in general.
And they are,

621
00:48:53 --> 00:48:56
2,886, 2,559.
Not as high as molecular

622
00:48:56 --> 00:49:03
hydrogen, but still pretty high.
And then we look at these three

623
00:49:03 --> 00:49:06
molecules, chlorine,
bromine, iodine.

624
00:49:06 --> 00:49:12
Both particles very heavy.
The reduced mass is large.

625
00:49:12 --> 00:49:16
Look at what happened to the
frequency here.

626
00:49:16 --> 00:49:22
The frequency really has gone
down much lower than it was for

627
00:49:22 --> 00:49:28
HCl, HBr or HI.
So, that is one parameter.

628
00:49:28 --> 00:49:32
The other parameter,
of course, is the force

629
00:49:32 --> 00:49:35
constant.
Here are a set of molecules

630
00:49:35 --> 00:49:39
that have all roughly the same
reduced mass,

631
00:49:39 --> 00:49:41
fluorine, oxygen,
NO.

632
00:49:41 --> 00:49:46
And I have written down their
frequencies, both in wave

633
00:49:46 --> 00:49:51
numbers and Hertz.
For molecular fluorine,

634
00:49:51 --> 00:49:56
which has a single bond,
that frequency here is pretty

635
00:49:56 --> 00:50:01
low, 892 wave numbers.
It is pretty low.

636
0:50:01 --> 00:50:06
This is a single bond,
which is easy to stretch.

637
0:50:06 --> 00:50:10
It is not very stiff.
It is a loose bond,

638
0:50:10 --> 00:50:14
so to speak.
But now, look at oxygen and NO.

639
0:50:14.32 --> 00:50:18
They have double bonds,
which are stiffer.

640
0:50:18 --> 00:50:21
If they are stiffer,
k is larger,

641
0:50:21 --> 00:50:24
the fundamental frequency is
higher.

642
0:50:24 --> 00:50:27
Then, we look at CO and
nitrogen.

643
0:50:27 --> 00:50:34
They have triple bonds,
which are even stiffer.

644
0:50:34 --> 00:50:38
Stiffer bonds,
larger k, higher fundamental

645
0:50:38 --> 00:50:43
frequency, that are those two
parameters that dictate the

646
0:50:43 --> 00:50:47
frequency of vibration.
We will do rotation on

647
0:50:47.816 --> 50:50
Wednesday.
See you then.