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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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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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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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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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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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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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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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an anti-symmetric stretch.

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In this case,
we have one oxygen moving out

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while the other oxygen is moving
in.

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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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this one is moving in,
this one is moving out.

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That is the anti-symmetric
stretch.

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If my body is the carbon and my
arms are the oxygen,

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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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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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or that goes like that.
This is a bending vibration.

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

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the bending vibration looks
like this.

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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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00:10:48,000 --> 00:10:53,000
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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00:11:00,000 --> 00:11:04,000
And so, in other words,
if I am the carbon,

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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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00:11:14,000 --> 00:11:18,000
These are four different
vibrations.

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They are the four vibrational
modes, two bending,

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one symmetric stretch,
and the other,

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00:11:27,000 --> 00:11:34,000
the anti-symmetric stretch.
That is what we have in

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CO two.
What about another molecule?

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00:11:38,000 --> 00:11:41,000
How about water?
We have three atoms.

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Again, nine total modes.
Three of them are translation,

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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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rotational modes ought to look
like.

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

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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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00:12:29,000 --> 00:12:33,000
If I am the oxygen and my hands
are the hydrogen,

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00:12:33,000 --> 00:12:37,000
this is a rotational mode.
You got that?

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00:12:37,000 --> 00:12:40,000
Okay.
And then another rotational

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00:12:40,000 --> 00:12:44,000
mode, here.
It can be a rotation around the

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00:12:44,000 --> 00:12:48,000
center of the mass,
centered on the oxygen here.

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00:12:48,000 --> 00:12:52,000
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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00:13:02,000 --> 00:13:07,000
And then a final rotational
mode involves rotation,

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00:13:07,000 --> 00:13:13,000
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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00:13:16,000 --> 00:13:23,000
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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00:13:37,000 --> 00:13:42,000
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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00:13:46,000 --> 00:13:50,000
Now, what about the vibrations
for water?

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00:13:50,000 --> 00:13:54,000
Well, you can see from our
graph up here that 3N minus 6,

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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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00:14:46,000 --> 00:14:50,000
And then, if there is a
symmetric stretch,

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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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00:15:00,000 --> 00:15:06,000
other one moving out.
Or, this way.

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00:15:06,000 --> 00:15:15,000
Of course, a picture for that
is this kind of a motion.

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You get the idea.
Anti-symmetric stretch.

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00:15:22,000 --> 00:15:30,000
And then, finally,
we have a bending mode.

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A bending mode,
meaning these hydrogens moving

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00:15:35,000 --> 00:15:40,000
this way or those hydrogens
moving that way.

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00:15:40,000 --> 00:15:46,000
And, of course,
a picture for that is this.

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That is a bending mode for the
hydrogens.

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We have three vibrational modes
for the hydrogen.

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Now we are going to look at
some of the principles behind

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00:16:02,000 --> 00:16:04,000
these internal degrees of
freedom.

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00:16:04,000 --> 00:16:08,000
But, of course,
the other reason for telling

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00:16:08,000 --> 00:16:12,000
you about this-- Did you have a
question?

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00:16:17,000 --> 00:16:20,000
It can, but it is not a
separate mode.

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00:16:20,000 --> 00:16:23,000
That is correct.
That is right.

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00:16:23,000 --> 00:16:26,000
It looks like a rotation,
actually.

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00:16:26,000 --> 00:16:32,000
That is absolutely correct.
The other reason for talking

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00:16:32,000 --> 00:16:37,000
about this is to tell you that
each one of these vibrations or

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00:16:37,000 --> 00:16:41,000
rotations occurs at a frequency
at an energy that is

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00:16:41,000 --> 00:16:45,000
characteristic of the molecule.
If we had a way to actually

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00:16:45,000 --> 00:16:50,000
measure the frequency of these
vibrations or the frequencies of

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00:16:50,000 --> 00:16:54,000
the rotations,
well, then we have a great tool

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00:16:54,000 --> 00:16:59,000
for identifying what kind of
molecule we have.

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00:16:59,000 --> 00:17:02,000
Analytically,
this is a wonderful technique

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00:17:02,000 --> 00:17:06,000
for identifying,
ultimately, the molecule that

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00:17:06,000 --> 00:17:11,000
you have, if you can measure
those frequencies of vibration

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00:17:11,000 --> 00:17:15,000
or rotation.
And we are going to do that in

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00:17:15,000 --> 00:17:18,000
just a moment.
But what we also have to

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00:17:18,000 --> 00:17:23,000
understand, before we go and
measure these frequencies,

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00:17:23,000 --> 00:17:28,000
is we have to understand that
the energies with which a

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00:17:28,000 --> 00:17:33,000
molecule vibrates or rotates are
quantized.

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00:17:33,000 --> 00:17:37,000
So, that is what we have to
spend some time thinking about

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00:17:37,000 --> 00:17:40,000
right now.
Let's do that.

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00:17:51,000 --> 00:17:54,000
And we are going to start with
vibration.

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00:18:00,000 --> 00:18:04,000
To illustrate this,
I am going to draw one of these

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00:18:04,000 --> 00:18:09,000
energies of interaction,
again, one of these curves that

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00:18:09,000 --> 00:18:14,000
we have been talking about.
And I am going to set my zero

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00:18:14,000 --> 00:18:18,000
of energy at the dissociated
atom limit.

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00:18:18,000 --> 00:18:21,000
So, I am going to talk about H
plus Cl.

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00:18:21,000 --> 00:18:25,000
And, of course,
right here, we talked about

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00:18:25,000 --> 00:18:29,000
this being the equilibrium bond
length, r sub e,

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00:18:29,000 --> 00:18:33,000
for the HCl molecule.

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00:18:39,000 --> 00:18:42,000
That is the energy of
interaction.

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00:18:42,000 --> 00:18:49,000
I am going to now draw in the
ground vibrational state of HCl,

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00:18:49,000 --> 00:18:53,000
which I am going to represent
as a line.

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00:18:53,000 --> 00:18:59,000
It is going to be the energy.
There is the ground vibrational

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00:18:59,000 --> 00:19:04,000
state of HCl.
The ground vibrational state,

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00:19:04,000 --> 00:19:09,000
that energy is characterized by
a quantum number.

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00:19:09,000 --> 00:19:15,000
The quantum number is the
vibrational quantum number for

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00:19:15,000 --> 00:19:20,000
the ground vibrational state.
The vibrational quantum number

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00:19:20,000 --> 00:19:26,000
is v equal zero.
What this line represents is

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00:19:26,000 --> 00:19:32,000
the ground vibrational energy.
It also represents the extent

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00:19:32,000 --> 00:19:38,000
to which the molecule's bond
stretches and compresses.

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00:19:38,000 --> 00:19:42,000
The bottom line,
here, is that the bond

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00:19:42,000 --> 00:19:47,000
stretches up to here.
The inner section of this line

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00:19:47,000 --> 00:19:52,000
with the interaction energy,
here, represents the maximum

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00:19:52,000 --> 00:19:57,000
distance of the bond,
the maximum bond length,

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00:19:57,000 --> 00:20:02,000
the most that the bond
stretches.

238
00:20:02,000 --> 00:20:07,000
r is going this way.
The intersection of this energy

239
00:20:07,000 --> 00:20:14,000
with this curve represents the
closest that the two nuclei get.

240
00:20:14,000 --> 00:20:21,000
This represents the most that
the molecule has compressed.

241
00:20:21,000 --> 00:20:24,000
This is the smallest value of
r.

242
00:20:24,000 --> 00:20:30,000
That is what that diagram
represents.

243
00:20:30,000 --> 00:20:34,000
If you start out,
here, with an HCl molecule at

244
00:20:34,000 --> 00:20:40,000
the equilibrium bond length,
right here, what happens is

245
00:20:40,000 --> 00:20:45,000
that the HCl will stretch,
and it will stretch to this

246
00:20:45,000 --> 00:20:48,000
position.
This is an exaggeration.

247
00:20:48,000 --> 00:20:54,000
This is how much it stretches.
And then, it comes back and

248
00:20:54,000 --> 00:20:59,000
goes through the equilibrium
position.

249
00:20:59,000 --> 00:21:02,000
Over here, and this is another
exaggeration,

250
00:21:02,000 --> 00:21:06,000
this is how close they get to
each other.

251
00:21:06,000 --> 00:21:11,000
This is how much they compress.
This point is often called the

252
00:21:11,000 --> 00:21:15,000
inner turning point,
inner because it is the

253
00:21:15,000 --> 00:21:19,000
smallest bond distance.
This is often called the outer

254
00:21:19,000 --> 00:21:24,000
turning point because it is the
larger bond distance.

255
00:21:24,000 --> 00:21:28,000
It is outer because it is at
this distance,

256
00:21:28,000 --> 00:21:34,000
now, that the bond turns around
and begins to compress.

257
00:21:34,000 --> 00:21:39,000
This HCl molecule here is
vibrating, equilibrium position.

258
00:21:39,000 --> 00:21:44,000
Then it goes to the maximum
extension, comes back through

259
00:21:44,000 --> 00:21:49,000
the equilibrium position,
and then goes to the maximum

260
00:21:49,000 --> 00:21:54,000
compression, comes back to the
equilibrium position.

261
00:21:54,000 --> 00:21:59,000
That is what this line
literally represents.

262
00:21:59,000 --> 00:22:03,000
It is also the energy,
which I will explain a little

263
00:22:03,000 --> 00:22:08,000
bit more in just a moment.
So, this molecule is vibrating

264
00:22:08,000 --> 00:22:12,000
from here to here to here and
back and forth.

265
00:22:12,000 --> 00:22:18,000
Now, what I also want to point
out to you is something that I

266
00:22:18,000 --> 00:22:23,000
have been kind of misleading you
about for the last few weeks.

267
00:22:23,000 --> 00:22:28,000
That is that this molecule is
not sitting in the bottom of

268
00:22:28,000 --> 00:22:33,000
this well.
Remember that when we were

269
00:22:33,000 --> 00:22:39,000
drawing bond association
energies, I was drawing it from

270
00:22:39,000 --> 00:22:44,000
the bottom of this well to the
dissociated atom limit?

271
00:22:44,000 --> 00:22:49,000
I told you this was delta E sub
d.

272
00:22:49,000 --> 00:22:54,000
Well, the bottom line is that
it is not really correct.

273
00:22:54,000 --> 00:22:59,000
The reason is,
is because the molecule is

274
00:22:59,000 --> 00:23:05,000
never really sitting at the
bottom of this well.

275
00:23:05,000 --> 00:23:10,000
It is sitting this much above.
It is sitting at the v equal

276
00:23:10,000 --> 00:23:15,000
zero state.
It is not sitting at the bottom

277
00:23:15,000 --> 00:23:19,000
of the well.
If it were, it would not be

278
00:23:19,000 --> 00:23:22,000
vibrating.
If it is not vibration,

279
00:23:22,000 --> 00:23:28,000
it will violate the Uncertainty
Principle, something we didn't

280
00:23:28,000 --> 00:23:33,000
talk about.
But the bottom line is that it

281
00:23:33,000 --> 00:23:37,000
cannot sit at the bottom of the
well.

282
00:23:37,000 --> 00:23:42,000
It is always vibrating by about
this much energy.

283
00:23:42,000 --> 00:23:48,000
When you have an experimentally
determined dissociation energy

284
00:23:48,000 --> 00:23:54,000
for the bond energy,
the experimentally determined

285
00:23:54,000 --> 00:24:00,000
energy is actually the energy
from this level.

286
00:24:00,000 --> 00:24:05,000
From v equal zero up
to this dissociation limit.

287
00:24:05,000 --> 00:24:09,000
This is the experimental bond
energy.

288
00:24:09,000 --> 00:24:14,000
Because, if you are going to
measure a bond energy in the

289
00:24:14,000 --> 00:24:18,000
laboratory, well,
you can only measure it from

290
00:24:18,000 --> 00:24:24,000
the lowest level at which that
molecule can possibly be at.

291
00:24:24,000 --> 00:24:29,000
It is not at the bottom of the
well.

292
00:24:29,000 --> 00:24:34,000
The difference between where
this level is and the bottom of

293
00:24:34,000 --> 00:24:39,000
the well actually has a name.
That difference is called the

294
00:24:39,000 --> 00:24:42,000
zero point energy.

295
00:24:47,000 --> 00:24:52,000
And I will explain that in a
moment, when we look at the

296
00:24:52,000 --> 00:24:55,000
energies a little more
carefully.

297
00:24:55,000 --> 00:25:00,000
Now, just for completeness,
and this is not something that

298
00:25:00,000 --> 00:25:04,000
you have to know,
let me tell you what the

299
00:25:04,000 --> 00:25:10,000
nomenclature usually is here for
these well depths.

300
00:25:10,000 --> 00:25:14,000
The nomenclature from the
bottom of the well to here,

301
00:25:14,000 --> 00:25:18,000
which is not experimentally
measured, is usually D sub e,

302
00:25:18,000 --> 00:25:21,000
dissociation sub e.

303
00:25:28,000 --> 00:25:33,000
From the v equal zero
level to the dissociation limit,

304
00:25:33,000 --> 00:25:37,000
that energy here is D sub zero.

305
00:25:37,000 --> 00:25:42,000
You don't have to know these,
but I just want to tell you,

306
00:25:42,000 --> 00:25:46,000
when you see that.
All dissociation energies

307
00:25:46,000 --> 00:25:51,000
experimentally measured are
measured from here to here

308
00:25:51,000 --> 00:25:56,000
because that is the lowest
energy the molecule can have,

309
00:25:56,000 --> 00:26:01,000
is being in the v equal zero
ground vibrational

310
00:26:01,000 --> 00:26:07,000
state.
Well, now let's take a look at

311
00:26:07,000 --> 00:26:13,000
the energies a little more
carefully.

312
00:26:31,000 --> 00:26:37,000
Here is our favorite diagram
again, our intermolecular

313
00:26:37,000 --> 00:26:41,000
interaction potential.
This is HCl,

314
00:26:41,000 --> 00:26:45,000
so here is the dissociated atom
limit.

315
00:26:45,000 --> 00:26:51,000
This is v equal zero,
right in there.

316
00:26:51,000 --> 00:26:57,000
Now, there is an excited state
for the HCl, or for any

317
00:26:57,000 --> 00:27:02,000
molecule.
v equal one is the first

318
00:27:02,000 --> 00:27:08,000
vibrationally excited state.
v equal two is the

319
00:27:08,000 --> 00:27:11,000
second vibrationally excited
state.

320
00:27:11,000 --> 00:27:15,000
v equal three is the
third.

321
00:27:15,000 --> 00:27:18,000
v equal four is the
fourth.

322
00:27:18,000 --> 00:27:22,000
v equal five is the
fifth.

323
00:27:22,000 --> 00:27:28,000
These are the allowed
vibrational energies.

324
00:27:28,000 --> 00:27:32,000
In other words,
the molecule can vibrate with

325
00:27:32,000 --> 00:27:38,000
this energy or this energy or
this energy, but not some

326
00:27:38,000 --> 00:27:43,000
arbitrary energy,
like right in here or like

327
00:27:43,000 --> 00:27:47,000
right in there.
These are the allowed

328
00:27:47,000 --> 00:27:52,000
vibrational states.
And we have a nice analytical

329
00:27:52,000 --> 00:27:56,000
expression to describe those
energies.

330
00:27:56,000 --> 00:28:02,000
That analytical expression is
right here.

331
00:28:02,000 --> 00:28:07,000
That energy is h times nu times
(that vibrational quantum number

332
00:28:07,000 --> 00:28:10,000
plus one-half).

333
00:28:10,000 --> 00:28:13,000
So, v is that vibrational
quantum number.

334
00:28:13,000 --> 00:28:17,000
This nu, here,
is the fundamental frequency,

335
00:28:17,000 --> 00:28:22,000
which is the frequency with
which the molecule vibrates.

336
00:28:22,000 --> 00:28:26,000
It is the frequency with which
that molecule stretches,

337
00:28:26,000 --> 00:28:31,000
comes back to the equilibrium
position, compresses,

338
00:28:31,000 --> 00:28:36,000
and comes back to the
equilibrium position.

339
00:28:36,000 --> 00:28:41,000
It is the number of cycles per
second that molecule makes.

340
00:28:41,000 --> 00:28:47,000
So, nu is the number of cycles,
here, that the molecule in v

341
00:28:47,000 --> 00:28:50,000
equal zero makes per
second.

342
00:28:50,000 --> 00:28:55,000
Stretches, compresses,
and back to the equilibrium

343
00:28:55,000 --> 00:28:58,000
position.
Notice it is also the

344
00:28:58,000 --> 00:29:04,000
vibrational frequency of v equal
one.

345
00:29:04,000 --> 00:29:09,000
The vibrational frequency does
not depend on the vibrational

346
00:29:09,000 --> 00:29:12,000
quantum number.
You see no dependence of the

347
00:29:12,000 --> 00:29:16,000
vibrational quantum number on
the frequency.

348
00:29:16,000 --> 00:29:21,000
v equal two vibrates
with the same frequency.

349
00:29:21,000 --> 00:29:23,000
v equal three.
v equal four.

350
00:29:23,000 --> 00:29:27,000
v equal five.
All of these states have the

351
00:29:27,000 --> 00:29:30,000
same nu here,
have the same fundamental

352
00:29:30,000 --> 00:29:34,000
frequency.
What does that mean?

353
00:29:34,000 --> 00:29:39,000
That looks a little strange in
the sense that if you have a

354
00:29:39,000 --> 00:29:43,000
molecule in v equals zero,
the number of cycles per second

355
00:29:43,000 --> 00:29:48,000
that it makes is the same as a
molecule in v equal five.

356
00:29:48,000 --> 00:29:52,000
Well, you can see that in v
equal five, the molecule has to

357
00:29:52,000 --> 00:29:56,000
stretch further,
and it has to compress further.

358
00:29:56,000 --> 00:30:02,000
It has to travel more distance.
And so, if it has to travel

359
00:30:02,000 --> 00:30:06,000
more distance,
but it still carries out the

360
00:30:06,000 --> 00:30:11,000
number of cycles per unit time,
the same as v equal zero,

361
00:30:11,000 --> 00:30:17,000
well, then it must mean that
the molecule in v equal five is

362
00:30:17,000 --> 00:30:21,000
moving more quickly,
is moving faster.

363
00:30:21,000 --> 00:30:26,000
And that is what that means.
The energy of this state is

364
00:30:26,000 --> 00:30:30,000
higher.
It is moving faster.

365
00:30:30,000 --> 00:30:34,000
But it is moving with the same
frequency nu.

366
00:30:34,000 --> 00:30:37,000
That is the fundamental
frequency.

367
00:30:37,000 --> 00:30:42,000
That fundamental frequency,
of course, is in hertz,

368
00:30:42,000 --> 00:30:45,000
our unit for all frequencies
here.

369
00:30:45,000 --> 00:30:51,000
Now, I drew on the board and
called this energy from the

370
00:30:51,000 --> 00:30:56,000
bottom of the well to v equal
zero, the zero point energy.

371
00:30:56,000 --> 00:31:01,000
That is what it is.
But numerically what that

372
00:31:01,000 --> 00:31:05,000
energy is, is one-half h nu.
Stick in v equal zero,

373
00:31:05,000 --> 00:31:08,000
you get one-half h nu.

374
00:31:08,000 --> 00:31:11,000
This energy,
here, is measured from the

375
00:31:11,000 --> 00:31:15,000
bottom of this well.
It is important that you

376
00:31:15,000 --> 00:31:18,000
understand where this energy is
measured from.

377
00:31:18,000 --> 00:31:22,000
It is measured from the bottom
of this well.

378
00:31:22,000 --> 00:31:25,000
v equal one,
you put that in here and get

379
00:31:25,000 --> 00:31:29,000
three-halves h nu.

380
00:31:29,000 --> 00:31:33,000
v equal two,
you get five-halves h nu.

381
00:31:33,000 --> 00:31:35,000
v equal three,

382
00:31:35,000 --> 00:31:40,000
you get seven-halves h nu.

383
00:31:40,000 --> 00:31:45,000
The other thing to notice is
that these states are equally

384
00:31:45,000 --> 00:31:49,000
spaced.
The energy spacing between any

385
00:31:49,000 --> 00:31:54,000
two adjacent states is h times
nu.

386
00:32:02,000 --> 00:32:05,000
That is incorrect.
What I am drawing,

387
00:32:05,000 --> 00:32:09,000
here, is right.
I am sorry about that.

388
00:32:09,000 --> 00:32:16,000
One-half h nu is
from the bottom of the well to v

389
00:32:16,000 --> 00:32:20,000
equal zero.
I think I know what happened.

390
00:32:20,000 --> 00:32:24,000
What I have shown you here is
correct.

391
00:32:24,000 --> 00:32:30,000
What is in your notes is
printed incorrectly.

392
00:32:30,000 --> 00:32:35,000
You should make note of that
and change that.

393
00:32:45,000 --> 00:32:49,000
Suppose we have a molecule in
the v equal zero state,

394
00:32:49,000 --> 00:32:54,000
where the energy is one-half h
nu,

395
00:32:54,000 --> 00:33:00,000
that molecule can be excited to
the v equal one state.

396
00:33:00,000 --> 00:33:04,000
That molecule in v equal zero
is still vibrating,

397
00:33:04,000 --> 00:33:09,000
but we can make it vibrate even
faster, with more energy.

398
00:33:09,000 --> 00:33:13,000
The same cycles per second,
but with more energy,

399
00:33:13,000 --> 00:33:19,000
by promoting it to v equal one.
We can do that if the molecule

400
00:33:19,000 --> 00:33:23,000
absorbs a photon.
And the energy of that photon

401
00:33:23,000 --> 00:33:28,000
has to be equal to the
difference in the energies of

402
00:33:28,000 --> 00:33:33,000
these two states.
So, if the final energy,

403
00:33:33,000 --> 00:33:37,000
here, of the v equal one state,
that is the final state,

404
00:33:37,000 --> 00:33:43,000
if that energy is three-halves
h nu and the energy

405
00:33:43,000 --> 00:33:47,000
of the initial state was
one-half h nu,

406
00:33:47,000 --> 00:33:52,000
then the difference in energy
between the states is h nu.

407
00:33:52,000 --> 00:33:56,000
We have to have a photon that
has exactly this energy

408
00:33:56,000 --> 00:33:59,000
different, delta E.
That delta E,

409
00:33:59,000 --> 00:34:04,000
then, has to be gotten by a
photon with a frequency equal to

410
00:34:04,000 --> 00:34:07,000
nu.
And the frequency of that

411
00:34:07,000 --> 00:34:12,000
photon that is going to make
that transition is actually also

412
00:34:12,000 --> 00:34:15,000
the fundamental frequency of the
molecule.

413
00:34:15,000 --> 00:34:19,000
We are lucky on this one.
Nu is delta E over h.

414
00:34:19,000 --> 00:34:23,000
We have to have a photon with

415
00:34:23,000 --> 00:34:28,000
that frequency for the molecule
to be promoted from v equal zero

416
00:34:28,000 --> 00:34:33,000
to v equal one.
How do we know what that is?

417
00:34:33,000 --> 00:34:39,000
Well, we do an experiment.
We do an infrared spectroscopy

418
00:34:39,000 --> 00:34:44,000
type of experiment.
The difference in the energies

419
00:34:44,000 --> 00:34:51,000
between vibrational states is in
the infrared range of the

420
00:34:51,000 --> 00:34:56,000
electromagnetic spectrum.
And so we are going to use

421
00:34:56,000 --> 00:35:02,000
infrared radiation.
We have some source of infrared

422
00:35:02,000 --> 00:35:05,000
radiation.
It puts out all different

423
00:35:05,000 --> 00:35:09,000
frequencies in the infrared.
And then we send it through a

424
00:35:09,000 --> 00:35:13,000
monochromator.
The monochromator disperses

425
00:35:13,000 --> 00:35:17,000
that radiation in space and
allows only a certain frequency

426
00:35:17,000 --> 00:35:22,000
of radiation to come right
through the slits and out into

427
00:35:22,000 --> 00:35:24,000
our sample.
It is like you had the

428
00:35:24,000 --> 00:35:28,000
diffraction glasses,
where we disbursed the

429
00:35:28,000 --> 00:35:32,000
radiation in space.
Well, a monochromator is

430
00:35:32,000 --> 00:35:36,000
essentially the same thing.
There is a diffraction grading

431
00:35:36,000 --> 00:35:39,000
in it.
It allows, out of the front

432
00:35:39,000 --> 00:35:42,000
end, only radiation of a certain
frequency to come out.

433
00:35:42,000 --> 00:35:46,000
And we can adjust what that
radiation is or what frequency

434
00:35:46,000 --> 00:35:49,000
that radiation has.
We then allow it to pass

435
00:35:49,000 --> 00:35:51,000
through our sample,
say HCl.

436
00:35:51,000 --> 00:35:54,000
And then there is photo
detector over here,

437
00:35:54,000 --> 00:36:00,000
some kind of photomultiplier.
And what will happen is that if

438
00:36:00,000 --> 00:36:05,000
this molecule is going to absorb
frequency of that radiation,

439
00:36:05,000 --> 00:36:08,000
well, then the number of
photons reaching the photo

440
00:36:08,000 --> 00:36:13,000
detector is going to go down.
If the frequency does not match

441
00:36:13,000 --> 00:36:16,000
the frequency of the vibration
of HCl, well,

442
00:36:16,000 --> 00:36:20,000
the photons go right through,
onto the detector.

443
00:36:20,000 --> 00:36:24,000
But if it matches,
then the molecule absorbs those

444
00:36:24,000 --> 00:36:27,000
photons and they never make it
to the detector.

445
00:36:27,000 --> 00:36:33,000
So, a plot looks like this.
This is the intensity at the

446
00:36:33,000 --> 00:36:36,000
detector versus the frequency of
the radiation.

447
00:36:36,000 --> 00:36:40,000
The intensity will be high,
but now at some frequency,

448
00:36:40,000 --> 00:36:45,000
which will turn out to be the
vibrational frequency of the

449
00:36:45,000 --> 00:36:47,000
molecule, the intensity goes
down low.

450
00:36:47,000 --> 00:36:52,000
And then, as you go to higher
frequency, the intensity comes

451
00:36:52,000 --> 00:36:55,000
back up.
Right here, the molecule has

452
00:36:55,000 --> 00:36:59,000
made the transition from v equal
zero to v equal one because the

453
00:36:59,000 --> 00:37:04,000
frequency of that photon matches
the energy difference between v

454
00:37:04,000 --> 00:37:09,000
equal zero and v equal one
divided by h.

455
00:37:09,000 --> 00:37:13,000
It also happens to be the
fundamental frequency of the

456
00:37:13,000 --> 00:37:18,000
molecule for HCl.
So, that is how we know what

457
00:37:18,000 --> 00:37:21,000
those fundamental frequencies
are.

458
00:37:21,000 --> 00:37:24,000
Let's now look at what those
energies are.

459
00:37:24,000 --> 00:37:30,000
We have gone from v equal zero
to v equal one.

460
00:37:30,000 --> 00:37:34,000
Delta E equal h nu. We just

461
00:37:34,000 --> 00:37:39,000
measured nu as 8.6x10^13 inverse
seconds.

462
00:37:39,000 --> 00:37:44,000
Multiply it by h.
That is going to give us delta

463
00:37:44,000 --> 00:37:46,000
E.
The difference in energy

464
00:37:46,000 --> 00:37:51,000
between the two states is
5.7x10^-20 joules.

465
00:37:51,000 --> 00:37:57,000
If I turn that into kilojoules
per mole, that is 35 kilojoules

466
00:37:57,000 --> 00:38:02,000
per mole.
That is the energy difference

467
00:38:02,000 --> 00:38:06,000
from here to here,
about 35 kilojoules per mole.

468
00:38:06,000 --> 00:38:12,000
How strong is the HCl bond?
Well, the HCl bond from here to

469
00:38:12,000 --> 00:38:15,000
here is about 420 kilojoules per
mole.

470
00:38:15,000 --> 00:38:20,000
This is about a tenth,
a little bit less than a tenth

471
00:38:20,000 --> 00:38:23,000
of the total bond strength for
HCl.

472
00:38:23,000 --> 00:38:27,000
If we make it vibrate,
we are not going to make that

473
00:38:27,000 --> 00:38:32,000
bond break.
It is an order of magnitude

474
00:38:32,000 --> 00:38:38,000
less energy here than what you
need to break that bond.

475
00:38:38,000 --> 00:38:44,000
Now, here is something that is
a little confusing.

476
00:38:44,000 --> 00:38:50,000
These fundamental frequencies,
it turns out that we actually

477
00:38:50,000 --> 00:38:56,000
use a different convention to
label them, different units.

478
00:38:56,000 --> 00:39:03,000
We do not use Hertz very often
to talk about the fundamental

479
00:39:03,000 --> 00:39:09,000
frequencies of molecules.
We use something called a

480
00:39:09,000 --> 00:39:15,000
wavenumber instead of hertz.
A wavenumber is an inverse

481
00:39:15,000 --> 00:39:19,000
centimeter, centimeter to the
minus one power.

482
00:39:19,000 --> 00:39:25,000
The symbol for a wavenumber is
a frequency sign with a bar on

483
00:39:25,000 --> 00:39:30,000
top of it.
That is a wave number.

484
00:39:30,000 --> 00:39:35,000
How do we get a wavenumber?
Well, we get a wavenumber by

485
00:39:35,000 --> 00:39:39,000
taking the frequency in Hertz,
inverse seconds,

486
00:39:39,000 --> 00:39:42,000
and dividing it by the speed of
light.

487
00:39:42,000 --> 00:39:46,000
But the speed of light,
of course, has to be in

488
00:39:46,000 --> 00:39:50,000
centimeters per second.
The seconds cancel,

489
00:39:50,000 --> 00:39:56,000
and what we have then left is
inverse centimeters.

490
00:39:56,000 --> 00:40:00,000
For example,
if the fundamental frequency of

491
00:40:00,000 --> 00:40:06,000
HCl was 8.6x10^13 hertz,
we divide that by the speed of

492
00:40:06,000 --> 00:40:12,000
light in centimeters per second
and the wavenumbers,

493
00:40:12,000 --> 00:40:17,000
or the fundamental frequency in
wavenumbers, then,

494
00:40:17,000 --> 00:40:24,000
is 2,886 wavenumbers for HCl.
These are the common units that

495
00:40:24,000 --> 00:40:29,000
are used to talk about the
fundamental vibrational

496
00:40:29,000 --> 00:40:35,000
frequencies of molecules.
And these are the common

497
00:40:35,000 --> 00:40:40,000
numbers used to discuss infrared
spectra of molecules.

498
00:40:40,000 --> 00:40:44,000
This unit of a wavenumber.
It is a non-SI unit.

499
00:40:44,000 --> 00:40:47,000
In other words,
what you will often see is an

500
00:40:47,000 --> 00:40:51,000
infrared spectrum intensity at
the photodiode,

501
00:40:51,000 --> 00:40:56,000
now, versus wave number.
And right at 2,886 wavenumbers,

502
00:40:56,000 --> 00:41:01,000
that is the fundamental
frequency in wavenumbers.

503
00:41:01,000 --> 00:41:06,000
That is where the molecule
makes the transition from v

504
00:41:06,000 --> 00:41:11,000
equal zero to v equal one.
Now, the question is,

505
00:41:11,000 --> 00:41:18,000
what determines the frequency
of the vibration of a molecule?

506
00:41:18,000 --> 00:41:23,000
Well, what determines the
frequency of that vibration,

507
00:41:23,000 --> 00:41:29,000
and here, I have now gone back
to frequency because I am

508
00:41:29,000 --> 00:41:34,000
writing it in terms of some
other parameters,

509
00:41:34,000 --> 00:41:40,000
are two parameters.
One is the force constant k,

510
00:41:40,000 --> 00:41:44,000
and the other is the mass.
What I have here is the reduced

511
00:41:44,000 --> 00:41:46,000
mass.
I will explain that to you in

512
00:41:46,000 --> 00:41:50,000
just a moment,
but this is essentially the

513
00:41:50,000 --> 00:41:52,000
mass.
The frequency is given by one

514
00:41:52,000 --> 00:41:57,000
over 2pi times the square root
of this k, the force constant

515
00:41:57,000 --> 00:42:00,000
over the mass.

516
00:42:00,000 --> 00:42:04,000
sqrt(k / m)**
That force constant,

517
00:42:04,000 --> 00:42:09,000
k, is a measure of the
stiffness of the spring.

518
00:42:09,000 --> 00:42:13,000
Another words,
if you have a very stiff

519
00:42:13,000 --> 00:42:19,000
spring, k is very large.
That is you need a lot of force

520
00:42:19,000 --> 00:42:25,000
to pull that spring apart,
a lot of force to compress that

521
00:42:25,000 --> 00:42:29,000
spring.
A large k means you have a very

522
00:42:29,000 --> 00:42:34,000
stiff spring.
If you have a small k,

523
00:42:34,000 --> 00:42:38,000
that means you have a very weak
spring.

524
00:42:38,000 --> 00:42:45,000
It does not take very much
force to stretch that spring or

525
00:42:45,000 --> 00:42:50,000
to compress that spring.
So k is a measure of the

526
00:42:50,000 --> 00:42:56,000
stiffness of the spring.
It also tells us something

527
00:42:56,000 --> 00:43:02,000
about the shape of the
interaction energy.

528
00:43:02,000 --> 00:43:06,000
Another words,
what k actually tells us here

529
00:43:06,000 --> 00:43:09,000
is kind of the width of this
well.

530
00:43:09,000 --> 00:43:14,000
Here is my interaction energy,
potential well.

531
00:43:14,000 --> 00:43:19,000
If you have a very narrow
width, this is a large k.

532
00:43:19,000 --> 00:43:25,000
If you have a very broad width,
here, this is a small k.

533
00:43:25,000 --> 00:43:30,000
So, k, the force constant,
tells us about the shape,

534
00:43:30,000 --> 00:43:36,000
here, of this well.
Very narrow well,

535
00:43:36,000 --> 00:43:40,000
large k.
Very broad well,

536
00:43:40,000 --> 00:43:46,000
small k.
Now, I think that maybe you

537
00:43:46,000 --> 00:43:55,000
have seen this before in 8.01.
Have you done the harmonic

538
00:43:55,000 --> 00:43:57,000
oscillator yet?
No?

539
00:43:57,000 --> 00:44:00,000
Yes?
Some yes.

540
00:44:00,000 --> 00:44:03,000
Some no.
Mostly no, it sounds to me.

541
00:44:03,000 --> 00:44:07,000
This expression,
here, for the frequency,

542
00:44:07,000 --> 00:44:10,000
you are going to see in 8.01
some time.

543
00:44:10,000 --> 00:44:14,000
It comes from the harmonic
oscillator model.

544
00:44:14,000 --> 00:44:18,000
You will remember that you have
seen this before.

545
00:44:18,000 --> 00:44:23,000
Those of you who have know what
I am talking about.

546
00:44:23,000 --> 00:44:28,000
Those of you who have not,
just remember you are going to

547
00:44:28,000 --> 00:44:35,000
see this very soon once again.
Now, the other parameter here

548
00:44:35,000 --> 00:44:42,000
that is important is this mu
here, this reduced mass.

549
00:44:42,000 --> 00:44:49,000
What the reduced mass is,
is essentially the mass.

550
00:44:49,000 --> 00:44:54,000
Now, if you had already done
this in 8.01,

551
00:44:54,000 --> 00:45:02,000
what you are going to look at
is a mass on the end of a spring

552
00:45:02,000 --> 00:45:08,000
attached to a wall.
There will be a wall,

553
00:45:08,000 --> 00:45:13,000
then you will see a spring,
and then there will be some

554
00:45:13,000 --> 00:45:16,000
mass m.
When you do this problem as a

555
00:45:16,000 --> 00:45:21,000
harmonic oscillator,
you are going to write nu here

556
00:45:21,000 --> 00:45:26,000
as k over this mass right there.
But, in this example,

557
00:45:26,000 --> 00:45:32,000
this mass here is the only body
that is moving.

558
00:45:32,000 --> 00:45:36,000
Your wall is still.
But, in the case of a molecule,

559
00:45:36,000 --> 00:45:40,000
both atoms are moving.
What mass do we use?

560
00:45:40,000 --> 00:45:46,000
Well, the fact that both are
moving means we have a two body

561
00:45:46,000 --> 00:45:49,000
problem.
And we are going to have to

562
00:45:49,000 --> 00:45:55,000
change the coordinate system to
take into account that both

563
00:45:55,000 --> 00:46:00,000
masses are moving.
We can do that easily.

564
00:46:00,000 --> 00:46:04,000
And doing that easily involves
using something called the

565
00:46:04,000 --> 00:46:08,000
reduced mass.
The reduced mass takes a two

566
00:46:08,000 --> 00:46:13,000
body problem and reduces it to a
single body, where the single

567
00:46:13,000 --> 00:46:18,000
body is this fictitious body
with a mass given by the reduced

568
00:46:18,000 --> 00:46:19,000
mass.
It is exact.

569
00:46:19,000 --> 00:46:22,000
I did not make any
approximations here.

570
00:46:22,000 --> 00:46:27,000
But the bottom line is that
this reduced mass is simply the

571
00:46:27,000 --> 00:46:31,000
mass of one atom,
times the mass of the other,

572
00:46:31,000 --> 00:46:36,000
divided by the sum of the two
masses.

573
00:46:36,000 --> 00:46:42,000
And sometimes a useful
approximation is when one of the

574
00:46:42,000 --> 00:46:49,000
masses is much lower than the
other mass, so when m1 here is

575
00:46:49,000 --> 00:46:54,000
smaller than m2.
Well, you can see in that case,

576
00:46:54,000 --> 00:47:01,000
if m1 is very small compared to
m2, we are going to treat it

577
00:47:01,000 --> 00:47:06,000
like a zero.
And then these two m's are

578
00:47:06,000 --> 00:47:10,000
going to cancel.
You can see the reduced mass is

579
00:47:10,000 --> 00:47:14,000
roughly equivalent to the mass
of the smaller body.

580
00:47:14,000 --> 00:47:18,000
Sometimes I will ask you to use
that approximation.

581
00:47:18,000 --> 00:47:22,000
Other times,
I will ask you to calculate the

582
00:47:22,000 --> 00:47:25,000
reduced mass.
Let's look at some frequencies

583
00:47:25,000 --> 00:47:29,000
for some molecules,
here.

584
00:47:29,000 --> 00:47:33,000
I have an array of molecules.
All of them roughly have the

585
00:47:33,000 --> 00:47:38,000
same force constant.
I am going to look at the mass

586
00:47:38,000 --> 00:47:41,000
dependence on the fundamental
frequency.

587
00:47:41,000 --> 00:47:46,000
I have got the frequencies
written both in terms of wave

588
00:47:46,000 --> 00:47:50,000
numbers and Hertz.
The first thing I want you to

589
00:47:50,000 --> 00:47:54,000
notice is the fundamental
frequency for hydrogen,

590
00:47:54,000 --> 00:47:59,000
4,159 wave numbers.
That is the highest vibrational

591
00:47:59,000 --> 00:48:04,000
frequency that we know about.
Why is it so high?

592
00:48:04,000 --> 00:48:08,000
Well, because those two
hydrogens are so light,

593
00:48:08,000 --> 00:48:13,000
and the mass is in the
denominator of that fundamental

594
00:48:13,000 --> 00:48:17,000
frequency.
So if it is in the denominator,

595
00:48:17,000 --> 00:48:20,000
it is small,
and the fundamental frequency

596
00:48:20,000 --> 00:48:25,000
is going to be high.
Then look at these molecules,

597
00:48:25,000 --> 00:48:28,000
HCl, HBr, HI.
In this case here,

598
00:48:28,000 --> 00:48:33,000
it is all a hydrogen bonded to
a halogen.

599
00:48:33,000 --> 00:48:37,000
For the most part,
the reduced mass of these three

600
00:48:37,000 --> 00:48:41,000
molecules is that of hydrogen.
If that is of hydrogen,

601
00:48:41,000 --> 00:48:45,000
since that is in the
denominator for the expression

602
00:48:45,000 --> 00:48:50,000
for the frequency,
well, the frequencies are going

603
00:48:50,000 --> 00:48:53,000
to be high in general.
And they are,

604
00:48:53,000 --> 00:48:56,000
2,886, 2,559.
Not as high as molecular

605
00:48:56,000 --> 00:49:03,000
hydrogen, but still pretty high.
And then we look at these three

606
00:49:03,000 --> 00:49:06,000
molecules, chlorine,
bromine, iodine.

607
00:49:06,000 --> 00:49:12,000
Both particles very heavy.
The reduced mass is large.

608
00:49:12,000 --> 00:49:16,000
Look at what happened to the
frequency here.

609
00:49:16,000 --> 00:49:22,000
The frequency really has gone
down much lower than it was for

610
00:49:22,000 --> 00:49:28,000
HCl, HBr or HI.
So, that is one parameter.

611
00:49:28,000 --> 00:49:32,000
The other parameter,
of course, is the force

612
00:49:32,000 --> 00:49:35,000
constant.
Here are a set of molecules

613
00:49:35,000 --> 00:49:39,000
that have all roughly the same
reduced mass,

614
00:49:39,000 --> 00:49:41,000
fluorine, oxygen,
NO.

615
00:49:41,000 --> 00:49:46,000
And I have written down their
frequencies, both in wave

616
00:49:46,000 --> 00:49:51,000
numbers and Hertz.
For molecular fluorine,

617
00:49:51,000 --> 00:49:56,000
which has a single bond,
that frequency here is pretty

618
00:49:56,000 --> 00:50:01,000
low, 892 wave numbers.
It is pretty low.

619
00:50:01,000 --> 00:50:06,000
This is a single bond,
which is easy to stretch.

620
00:50:06,000 --> 00:50:10,000
It is not very stiff.
It is a loose bond,

621
00:50:10,000 --> 00:50:14,000
so to speak.
But now, look at oxygen and NO.

622
00:50:14,320 --> 00:50:18,000
They have double bonds,
which are stiffer.

623
00:50:18,000 --> 00:50:21,000
If they are stiffer,
k is larger,

624
00:50:21,000 --> 00:50:24,000
the fundamental frequency is
higher.

625
00:50:24,000 --> 00:50:27,000
Then, we look at CO and
nitrogen.

626
00:50:27,000 --> 00:50:34,000
They have triple bonds,
which are even stiffer.

627
00:50:34,000 --> 00:50:38,000
Stiffer bonds,
larger k, higher fundamental

628
00:50:38,000 --> 00:50:43,000
frequency, that are those two
parameters that dictate the

629
00:50:43,000 --> 00:50:47,000
frequency of vibration.
We will do rotation on

630
00:50:47,816 --> 00:50:50,000
Wednesday.
See you then.