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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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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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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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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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the anti-symmetric stretch.
That is what we have in

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CO two.
What about another molecule?

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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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If I am the oxygen and my hands
are the hydrogen,

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this is a rotational mode.
You got that?

00:12:37.000 --> 00:12:40.000
Okay.
And then another rotational

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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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And then a final rotational
mode involves rotation,

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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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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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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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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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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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other one moving out.
Or, this way.

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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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And then, finally,
we have a bending mode.

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A bending mode,
meaning these hydrogens moving

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this way or those hydrogens
moving that way.

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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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these internal degrees of
freedom.

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But, of course,
the other reason for telling

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you about this-- Did you have a
question?

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It can, but it is not a
separate mode.

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That is correct.
That is right.

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It looks like a rotation,
actually.

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That is absolutely correct.
The other reason for talking

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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rotations occurs at a frequency
at an energy that is

00:16:41.000 --> 00:16:45.000
characteristic of the molecule.
If we had a way to actually

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measure the frequency of these
vibrations or the frequencies of

00:16:50.000 --> 00:16:54.000
the rotations,
well, then we have a great tool

00:16:54.000 --> 00:16:59.000
for identifying what kind of
molecule we have.

00:16:59.000 --> 00:17:02.000
Analytically,
this is a wonderful technique

00:17:02.000 --> 00:17:06.000
for identifying,
ultimately, the molecule that

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you have, if you can measure
those frequencies of vibration

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or rotation.
And we are going to do that in

00:17:15.000 --> 00:17:18.000
just a moment.
But what we also have to

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understand, before we go and
measure these frequencies,

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is we have to understand that
the energies with which a

00:17:28.000 --> 00:17:33.000
molecule vibrates or rotates are
quantized.

00:17:33.000 --> 00:17:37.000
So, that is what we have to
spend some time thinking about

00:17:37.000 --> 00:17:40.000
right now.
Let's do that.

00:17:51.000 --> 00:17:54.000
And we are going to start with
vibration.

00:18:00.000 --> 00:18:04.000
To illustrate this,
I am going to draw one of these

00:18:04.000 --> 00:18:09.000
energies of interaction,
again, one of these curves that

00:18:09.000 --> 00:18:14.000
we have been talking about.
And I am going to set my zero

00:18:14.000 --> 00:18:18.000
of energy at the dissociated
atom limit.

00:18:18.000 --> 00:18:21.000
So, I am going to talk about H
plus Cl.

00:18:21.000 --> 00:18:25.000
And, of course,
right here, we talked about

00:18:25.000 --> 00:18:29.000
this being the equilibrium bond
length, r sub e,

00:18:29.000 --> 00:18:33.000
for the HCl molecule.

00:18:39.000 --> 00:18:42.000
That is the energy of
interaction.

00:18:42.000 --> 00:18:49.000
I am going to now draw in the
ground vibrational state of HCl,

00:18:49.000 --> 00:18:53.000
which I am going to represent
as a line.

00:18:53.000 --> 00:18:59.000
It is going to be the energy.
There is the ground vibrational

00:18:59.000 --> 00:19:04.000
state of HCl.
The ground vibrational state,

00:19:04.000 --> 00:19:09.000
that energy is characterized by
a quantum number.

00:19:09.000 --> 00:19:15.000
The quantum number is the
vibrational quantum number for

00:19:15.000 --> 00:19:20.000
the ground vibrational state.
The vibrational quantum number

00:19:20.000 --> 00:19:26.000
is v equal zero.
What this line represents is

00:19:26.000 --> 00:19:32.000
the ground vibrational energy.
It also represents the extent

00:19:32.000 --> 00:19:38.000
to which the molecule's bond
stretches and compresses.

00:19:38.000 --> 00:19:42.000
The bottom line,
here, is that the bond

00:19:42.000 --> 00:19:47.000
stretches up to here.
The inner section of this line

00:19:47.000 --> 00:19:52.000
with the interaction energy,
here, represents the maximum

00:19:52.000 --> 00:19:57.000
distance of the bond,
the maximum bond length,

00:19:57.000 --> 00:20:02.000
the most that the bond
stretches.

00:20:02.000 --> 00:20:07.000
r is going this way.
The intersection of this energy

00:20:07.000 --> 00:20:14.000
with this curve represents the
closest that the two nuclei get.

00:20:14.000 --> 00:20:21.000
This represents the most that
the molecule has compressed.

00:20:21.000 --> 00:20:24.000
This is the smallest value of
r.

00:20:24.000 --> 00:20:30.000
That is what that diagram
represents.

00:20:30.000 --> 00:20:34.000
If you start out,
here, with an HCl molecule at

00:20:34.000 --> 00:20:40.000
the equilibrium bond length,
right here, what happens is

00:20:40.000 --> 00:20:45.000
that the HCl will stretch,
and it will stretch to this

00:20:45.000 --> 00:20:48.000
position.
This is an exaggeration.

00:20:48.000 --> 00:20:54.000
This is how much it stretches.
And then, it comes back and

00:20:54.000 --> 00:20:59.000
goes through the equilibrium
position.

00:20:59.000 --> 00:21:02.000
Over here, and this is another
exaggeration,

00:21:02.000 --> 00:21:06.000
this is how close they get to
each other.

00:21:06.000 --> 00:21:11.000
This is how much they compress.
This point is often called the

00:21:11.000 --> 00:21:15.000
inner turning point,
inner because it is the

00:21:15.000 --> 00:21:19.000
smallest bond distance.
This is often called the outer

00:21:19.000 --> 00:21:24.000
turning point because it is the
larger bond distance.

00:21:24.000 --> 00:21:28.000
It is outer because it is at
this distance,

00:21:28.000 --> 00:21:34.000
now, that the bond turns around
and begins to compress.

00:21:34.000 --> 00:21:39.000
This HCl molecule here is
vibrating, equilibrium position.

00:21:39.000 --> 00:21:44.000
Then it goes to the maximum
extension, comes back through

00:21:44.000 --> 00:21:49.000
the equilibrium position,
and then goes to the maximum

00:21:49.000 --> 00:21:54.000
compression, comes back to the
equilibrium position.

00:21:54.000 --> 00:21:59.000
That is what this line
literally represents.

00:21:59.000 --> 00:22:03.000
It is also the energy,
which I will explain a little

00:22:03.000 --> 00:22:08.000
bit more in just a moment.
So, this molecule is vibrating

00:22:08.000 --> 00:22:12.000
from here to here to here and
back and forth.

00:22:12.000 --> 00:22:18.000
Now, what I also want to point
out to you is something that I

00:22:18.000 --> 00:22:23.000
have been kind of misleading you
about for the last few weeks.

00:22:23.000 --> 00:22:28.000
That is that this molecule is
not sitting in the bottom of

00:22:28.000 --> 00:22:33.000
this well.
Remember that when we were

00:22:33.000 --> 00:22:39.000
drawing bond association
energies, I was drawing it from

00:22:39.000 --> 00:22:44.000
the bottom of this well to the
dissociated atom limit?

00:22:44.000 --> 00:22:49.000
I told you this was delta E sub
d.

00:22:49.000 --> 00:22:54.000
Well, the bottom line is that
it is not really correct.

00:22:54.000 --> 00:22:59.000
The reason is,
is because the molecule is

00:22:59.000 --> 00:23:05.000
never really sitting at the
bottom of this well.

00:23:05.000 --> 00:23:10.000
It is sitting this much above.
It is sitting at the v equal

00:23:10.000 --> 00:23:15.000
zero state.
It is not sitting at the bottom

00:23:15.000 --> 00:23:19.000
of the well.
If it were, it would not be

00:23:19.000 --> 00:23:22.000
vibrating.
If it is not vibration,

00:23:22.000 --> 00:23:28.000
it will violate the Uncertainty
Principle, something we didn't

00:23:28.000 --> 00:23:33.000
talk about.
But the bottom line is that it

00:23:33.000 --> 00:23:37.000
cannot sit at the bottom of the
well.

00:23:37.000 --> 00:23:42.000
It is always vibrating by about
this much energy.

00:23:42.000 --> 00:23:48.000
When you have an experimentally
determined dissociation energy

00:23:48.000 --> 00:23:54.000
for the bond energy,
the experimentally determined

00:23:54.000 --> 00:24:00.000
energy is actually the energy
from this level.

00:24:00.000 --> 00:24:05.000
From v equal zero up
to this dissociation limit.

00:24:05.000 --> 00:24:09.000
This is the experimental bond
energy.

00:24:09.000 --> 00:24:14.000
Because, if you are going to
measure a bond energy in the

00:24:14.000 --> 00:24:18.000
laboratory, well,
you can only measure it from

00:24:18.000 --> 00:24:24.000
the lowest level at which that
molecule can possibly be at.

00:24:24.000 --> 00:24:29.000
It is not at the bottom of the
well.

00:24:29.000 --> 00:24:34.000
The difference between where
this level is and the bottom of

00:24:34.000 --> 00:24:39.000
the well actually has a name.
That difference is called the

00:24:39.000 --> 00:24:42.000
zero point energy.

00:24:47.000 --> 00:24:52.000
And I will explain that in a
moment, when we look at the

00:24:52.000 --> 00:24:55.000
energies a little more
carefully.

00:24:55.000 --> 00:25:00.000
Now, just for completeness,
and this is not something that

00:25:00.000 --> 00:25:04.000
you have to know,
let me tell you what the

00:25:04.000 --> 00:25:10.000
nomenclature usually is here for
these well depths.

00:25:10.000 --> 00:25:14.000
The nomenclature from the
bottom of the well to here,

00:25:14.000 --> 00:25:18.000
which is not experimentally
measured, is usually D sub e,

00:25:18.000 --> 00:25:21.000
dissociation sub e.

00:25:28.000 --> 00:25:33.000
From the v equal zero
level to the dissociation limit,

00:25:33.000 --> 00:25:37.000
that energy here is D sub zero.

00:25:37.000 --> 00:25:42.000
You don't have to know these,
but I just want to tell you,

00:25:42.000 --> 00:25:46.000
when you see that.
All dissociation energies

00:25:46.000 --> 00:25:51.000
experimentally measured are
measured from here to here

00:25:51.000 --> 00:25:56.000
because that is the lowest
energy the molecule can have,

00:25:56.000 --> 00:26:01.000
is being in the v equal zero
ground vibrational

00:26:01.000 --> 00:26:07.000
state.
Well, now let's take a look at

00:26:07.000 --> 00:26:13.000
the energies a little more
carefully.

00:26:31.000 --> 00:26:37.000
Here is our favorite diagram
again, our intermolecular

00:26:37.000 --> 00:26:41.000
interaction potential.
This is HCl,

00:26:41.000 --> 00:26:45.000
so here is the dissociated atom
limit.

00:26:45.000 --> 00:26:51.000
This is v equal zero,
right in there.

00:26:51.000 --> 00:26:57.000
Now, there is an excited state
for the HCl, or for any

00:26:57.000 --> 00:27:02.000
molecule.
v equal one is the first

00:27:02.000 --> 00:27:08.000
vibrationally excited state.
v equal two is the

00:27:08.000 --> 00:27:11.000
second vibrationally excited
state.

00:27:11.000 --> 00:27:15.000
v equal three is the
third.

00:27:15.000 --> 00:27:18.000
v equal four is the
fourth.

00:27:18.000 --> 00:27:22.000
v equal five is the
fifth.

00:27:22.000 --> 00:27:28.000
These are the allowed
vibrational energies.

00:27:28.000 --> 00:27:32.000
In other words,
the molecule can vibrate with

00:27:32.000 --> 00:27:38.000
this energy or this energy or
this energy, but not some

00:27:38.000 --> 00:27:43.000
arbitrary energy,
like right in here or like

00:27:43.000 --> 00:27:47.000
right in there.
These are the allowed

00:27:47.000 --> 00:27:52.000
vibrational states.
And we have a nice analytical

00:27:52.000 --> 00:27:56.000
expression to describe those
energies.

00:27:56.000 --> 00:28:02.000
That analytical expression is
right here.

00:28:02.000 --> 00:28:07.000
That energy is h times nu times
(that vibrational quantum number

00:28:07.000 --> 00:28:10.000
plus one-half).

00:28:10.000 --> 00:28:13.000
So, v is that vibrational
quantum number.

00:28:13.000 --> 00:28:17.000
This nu, here,
is the fundamental frequency,

00:28:17.000 --> 00:28:22.000
which is the frequency with
which the molecule vibrates.

00:28:22.000 --> 00:28:26.000
It is the frequency with which
that molecule stretches,

00:28:26.000 --> 00:28:31.000
comes back to the equilibrium
position, compresses,

00:28:31.000 --> 00:28:36.000
and comes back to the
equilibrium position.

00:28:36.000 --> 00:28:41.000
It is the number of cycles per
second that molecule makes.

00:28:41.000 --> 00:28:47.000
So, nu is the number of cycles,
here, that the molecule in v

00:28:47.000 --> 00:28:50.000
equal zero makes per
second.

00:28:50.000 --> 00:28:55.000
Stretches, compresses,
and back to the equilibrium

00:28:55.000 --> 00:28:58.000
position.
Notice it is also the

00:28:58.000 --> 00:29:04.000
vibrational frequency of v equal
one.

00:29:04.000 --> 00:29:09.000
The vibrational frequency does
not depend on the vibrational

00:29:09.000 --> 00:29:12.000
quantum number.
You see no dependence of the

00:29:12.000 --> 00:29:16.000
vibrational quantum number on
the frequency.

00:29:16.000 --> 00:29:21.000
v equal two vibrates
with the same frequency.

00:29:21.000 --> 00:29:23.000
v equal three.
v equal four.

00:29:23.000 --> 00:29:27.000
v equal five.
All of these states have the

00:29:27.000 --> 00:29:30.000
same nu here,
have the same fundamental

00:29:30.000 --> 00:29:34.000
frequency.
What does that mean?

00:29:34.000 --> 00:29:39.000
That looks a little strange in
the sense that if you have a

00:29:39.000 --> 00:29:43.000
molecule in v equals zero,
the number of cycles per second

00:29:43.000 --> 00:29:48.000
that it makes is the same as a
molecule in v equal five.

00:29:48.000 --> 00:29:52.000
Well, you can see that in v
equal five, the molecule has to

00:29:52.000 --> 00:29:56.000
stretch further,
and it has to compress further.

00:29:56.000 --> 00:30:02.000
It has to travel more distance.
And so, if it has to travel

00:30:02.000 --> 00:30:06.000
more distance,
but it still carries out the

00:30:06.000 --> 00:30:11.000
number of cycles per unit time,
the same as v equal zero,

00:30:11.000 --> 00:30:17.000
well, then it must mean that
the molecule in v equal five is

00:30:17.000 --> 00:30:21.000
moving more quickly,
is moving faster.

00:30:21.000 --> 00:30:26.000
And that is what that means.
The energy of this state is

00:30:26.000 --> 00:30:30.000
higher.
It is moving faster.

00:30:30.000 --> 00:30:34.000
But it is moving with the same
frequency nu.

00:30:34.000 --> 00:30:37.000
That is the fundamental
frequency.

00:30:37.000 --> 00:30:42.000
That fundamental frequency,
of course, is in hertz,

00:30:42.000 --> 00:30:45.000
our unit for all frequencies
here.

00:30:45.000 --> 00:30:51.000
Now, I drew on the board and
called this energy from the

00:30:51.000 --> 00:30:56.000
bottom of the well to v equal
zero, the zero point energy.

00:30:56.000 --> 00:31:01.000
That is what it is.
But numerically what that

00:31:01.000 --> 00:31:05.000
energy is, is one-half h nu.
Stick in v equal zero,

00:31:05.000 --> 00:31:08.000
you get one-half h nu.

00:31:08.000 --> 00:31:11.000
This energy,
here, is measured from the

00:31:11.000 --> 00:31:15.000
bottom of this well.
It is important that you

00:31:15.000 --> 00:31:18.000
understand where this energy is
measured from.

00:31:18.000 --> 00:31:22.000
It is measured from the bottom
of this well.

00:31:22.000 --> 00:31:25.000
v equal one,
you put that in here and get

00:31:25.000 --> 00:31:29.000
three-halves h nu.

00:31:29.000 --> 00:31:33.000
v equal two,
you get five-halves h nu.

00:31:33.000 --> 00:31:35.000
v equal three,

00:31:35.000 --> 00:31:40.000
you get seven-halves h nu.

00:31:40.000 --> 00:31:45.000
The other thing to notice is
that these states are equally

00:31:45.000 --> 00:31:49.000
spaced.
The energy spacing between any

00:31:49.000 --> 00:31:54.000
two adjacent states is h times
nu.

00:32:02.000 --> 00:32:05.000
That is incorrect.
What I am drawing,

00:32:05.000 --> 00:32:09.000
here, is right.
I am sorry about that.

00:32:09.000 --> 00:32:16.000
One-half h nu is
from the bottom of the well to v

00:32:16.000 --> 00:32:20.000
equal zero.
I think I know what happened.

00:32:20.000 --> 00:32:24.000
What I have shown you here is
correct.

00:32:24.000 --> 00:32:30.000
What is in your notes is
printed incorrectly.

00:32:30.000 --> 00:32:35.000
You should make note of that
and change that.

00:32:45.000 --> 00:32:49.000
Suppose we have a molecule in
the v equal zero state,

00:32:49.000 --> 00:32:54.000
where the energy is one-half h
nu,

00:32:54.000 --> 00:33:00.000
that molecule can be excited to
the v equal one state.

00:33:00.000 --> 00:33:04.000
That molecule in v equal zero
is still vibrating,

00:33:04.000 --> 00:33:09.000
but we can make it vibrate even
faster, with more energy.

00:33:09.000 --> 00:33:13.000
The same cycles per second,
but with more energy,

00:33:13.000 --> 00:33:19.000
by promoting it to v equal one.
We can do that if the molecule

00:33:19.000 --> 00:33:23.000
absorbs a photon.
And the energy of that photon

00:33:23.000 --> 00:33:28.000
has to be equal to the
difference in the energies of

00:33:28.000 --> 00:33:33.000
these two states.
So, if the final energy,

00:33:33.000 --> 00:33:37.000
here, of the v equal one state,
that is the final state,

00:33:37.000 --> 00:33:43.000
if that energy is three-halves
h nu and the energy

00:33:43.000 --> 00:33:47.000
of the initial state was
one-half h nu,

00:33:47.000 --> 00:33:52.000
then the difference in energy
between the states is h nu.

00:33:52.000 --> 00:33:56.000
We have to have a photon that
has exactly this energy

00:33:56.000 --> 00:33:59.000
different, delta E.
That delta E,

00:33:59.000 --> 00:34:04.000
then, has to be gotten by a
photon with a frequency equal to

00:34:04.000 --> 00:34:07.000
nu.
And the frequency of that

00:34:07.000 --> 00:34:12.000
photon that is going to make
that transition is actually also

00:34:12.000 --> 00:34:15.000
the fundamental frequency of the
molecule.

00:34:15.000 --> 00:34:19.000
We are lucky on this one.
Nu is delta E over h.

00:34:19.000 --> 00:34:23.000
We have to have a photon with

00:34:23.000 --> 00:34:28.000
that frequency for the molecule
to be promoted from v equal zero

00:34:28.000 --> 00:34:33.000
to v equal one.
How do we know what that is?

00:34:33.000 --> 00:34:39.000
Well, we do an experiment.
We do an infrared spectroscopy

00:34:39.000 --> 00:34:44.000
type of experiment.
The difference in the energies

00:34:44.000 --> 00:34:51.000
between vibrational states is in
the infrared range of the

00:34:51.000 --> 00:34:56.000
electromagnetic spectrum.
And so we are going to use

00:34:56.000 --> 00:35:02.000
infrared radiation.
We have some source of infrared

00:35:02.000 --> 00:35:05.000
radiation.
It puts out all different

00:35:05.000 --> 00:35:09.000
frequencies in the infrared.
And then we send it through a

00:35:09.000 --> 00:35:13.000
monochromator.
The monochromator disperses

00:35:13.000 --> 00:35:17.000
that radiation in space and
allows only a certain frequency

00:35:17.000 --> 00:35:22.000
of radiation to come right
through the slits and out into

00:35:22.000 --> 00:35:24.000
our sample.
It is like you had the

00:35:24.000 --> 00:35:28.000
diffraction glasses,
where we disbursed the

00:35:28.000 --> 00:35:32.000
radiation in space.
Well, a monochromator is

00:35:32.000 --> 00:35:36.000
essentially the same thing.
There is a diffraction grading

00:35:36.000 --> 00:35:39.000
in it.
It allows, out of the front

00:35:39.000 --> 00:35:42.000
end, only radiation of a certain
frequency to come out.

00:35:42.000 --> 00:35:46.000
And we can adjust what that
radiation is or what frequency

00:35:46.000 --> 00:35:49.000
that radiation has.
We then allow it to pass

00:35:49.000 --> 00:35:51.000
through our sample,
say HCl.

00:35:51.000 --> 00:35:54.000
And then there is photo
detector over here,

00:35:54.000 --> 00:36:00.000
some kind of photomultiplier.
And what will happen is that if

00:36:00.000 --> 00:36:05.000
this molecule is going to absorb
frequency of that radiation,

00:36:05.000 --> 00:36:08.000
well, then the number of
photons reaching the photo

00:36:08.000 --> 00:36:13.000
detector is going to go down.
If the frequency does not match

00:36:13.000 --> 00:36:16.000
the frequency of the vibration
of HCl, well,

00:36:16.000 --> 00:36:20.000
the photons go right through,
onto the detector.

00:36:20.000 --> 00:36:24.000
But if it matches,
then the molecule absorbs those

00:36:24.000 --> 00:36:27.000
photons and they never make it
to the detector.

00:36:27.000 --> 00:36:33.000
So, a plot looks like this.
This is the intensity at the

00:36:33.000 --> 00:36:36.000
detector versus the frequency of
the radiation.

00:36:36.000 --> 00:36:40.000
The intensity will be high,
but now at some frequency,

00:36:40.000 --> 00:36:45.000
which will turn out to be the
vibrational frequency of the

00:36:45.000 --> 00:36:47.000
molecule, the intensity goes
down low.

00:36:47.000 --> 00:36:52.000
And then, as you go to higher
frequency, the intensity comes

00:36:52.000 --> 00:36:55.000
back up.
Right here, the molecule has

00:36:55.000 --> 00:36:59.000
made the transition from v equal
zero to v equal one because the

00:36:59.000 --> 00:37:04.000
frequency of that photon matches
the energy difference between v

00:37:04.000 --> 00:37:09.000
equal zero and v equal one
divided by h.

00:37:09.000 --> 00:37:13.000
It also happens to be the
fundamental frequency of the

00:37:13.000 --> 00:37:18.000
molecule for HCl.
So, that is how we know what

00:37:18.000 --> 00:37:21.000
those fundamental frequencies
are.

00:37:21.000 --> 00:37:24.000
Let's now look at what those
energies are.

00:37:24.000 --> 00:37:30.000
We have gone from v equal zero
to v equal one.

00:37:30.000 --> 00:37:34.000
Delta E equal h nu. We just

00:37:34.000 --> 00:37:39.000
measured nu as 8.6x10^13 inverse
seconds.

00:37:39.000 --> 00:37:44.000
Multiply it by h.
That is going to give us delta

00:37:44.000 --> 00:37:46.000
E.
The difference in energy

00:37:46.000 --> 00:37:51.000
between the two states is
5.7x10^-20 joules.

00:37:51.000 --> 00:37:57.000
If I turn that into kilojoules
per mole, that is 35 kilojoules

00:37:57.000 --> 00:38:02.000
per mole.
That is the energy difference

00:38:02.000 --> 00:38:06.000
from here to here,
about 35 kilojoules per mole.

00:38:06.000 --> 00:38:12.000
How strong is the HCl bond?
Well, the HCl bond from here to

00:38:12.000 --> 00:38:15.000
here is about 420 kilojoules per
mole.

00:38:15.000 --> 00:38:20.000
This is about a tenth,
a little bit less than a tenth

00:38:20.000 --> 00:38:23.000
of the total bond strength for
HCl.

00:38:23.000 --> 00:38:27.000
If we make it vibrate,
we are not going to make that

00:38:27.000 --> 00:38:32.000
bond break.
It is an order of magnitude

00:38:32.000 --> 00:38:38.000
less energy here than what you
need to break that bond.

00:38:38.000 --> 00:38:44.000
Now, here is something that is
a little confusing.

00:38:44.000 --> 00:38:50.000
These fundamental frequencies,
it turns out that we actually

00:38:50.000 --> 00:38:56.000
use a different convention to
label them, different units.

00:38:56.000 --> 00:39:03.000
We do not use Hertz very often
to talk about the fundamental

00:39:03.000 --> 00:39:09.000
frequencies of molecules.
We use something called a

00:39:09.000 --> 00:39:15.000
wavenumber instead of hertz.
A wavenumber is an inverse

00:39:15.000 --> 00:39:19.000
centimeter, centimeter to the
minus one power.

00:39:19.000 --> 00:39:25.000
The symbol for a wavenumber is
a frequency sign with a bar on

00:39:25.000 --> 00:39:30.000
top of it.
That is a wave number.

00:39:30.000 --> 00:39:35.000
How do we get a wavenumber?
Well, we get a wavenumber by

00:39:35.000 --> 00:39:39.000
taking the frequency in Hertz,
inverse seconds,

00:39:39.000 --> 00:39:42.000
and dividing it by the speed of
light.

00:39:42.000 --> 00:39:46.000
But the speed of light,
of course, has to be in

00:39:46.000 --> 00:39:50.000
centimeters per second.
The seconds cancel,

00:39:50.000 --> 00:39:56.000
and what we have then left is
inverse centimeters.

00:39:56.000 --> 00:40:00.000
For example,
if the fundamental frequency of

00:40:00.000 --> 00:40:06.000
HCl was 8.6x10^13 hertz,
we divide that by the speed of

00:40:06.000 --> 00:40:12.000
light in centimeters per second
and the wavenumbers,

00:40:12.000 --> 00:40:17.000
or the fundamental frequency in
wavenumbers, then,

00:40:17.000 --> 00:40:24.000
is 2,886 wavenumbers for HCl.
These are the common units that

00:40:24.000 --> 00:40:29.000
are used to talk about the
fundamental vibrational

00:40:29.000 --> 00:40:35.000
frequencies of molecules.
And these are the common

00:40:35.000 --> 00:40:40.000
numbers used to discuss infrared
spectra of molecules.

00:40:40.000 --> 00:40:44.000
This unit of a wavenumber.
It is a non-SI unit.

00:40:44.000 --> 00:40:47.000
In other words,
what you will often see is an

00:40:47.000 --> 00:40:51.000
infrared spectrum intensity at
the photodiode,

00:40:51.000 --> 00:40:56.000
now, versus wave number.
And right at 2,886 wavenumbers,

00:40:56.000 --> 00:41:01.000
that is the fundamental
frequency in wavenumbers.

00:41:01.000 --> 00:41:06.000
That is where the molecule
makes the transition from v

00:41:06.000 --> 00:41:11.000
equal zero to v equal one.
Now, the question is,

00:41:11.000 --> 00:41:18.000
what determines the frequency
of the vibration of a molecule?

00:41:18.000 --> 00:41:23.000
Well, what determines the
frequency of that vibration,

00:41:23.000 --> 00:41:29.000
and here, I have now gone back
to frequency because I am

00:41:29.000 --> 00:41:34.000
writing it in terms of some
other parameters,

00:41:34.000 --> 00:41:40.000
are two parameters.
One is the force constant k,

00:41:40.000 --> 00:41:44.000
and the other is the mass.
What I have here is the reduced

00:41:44.000 --> 00:41:46.000
mass.
I will explain that to you in

00:41:46.000 --> 00:41:50.000
just a moment,
but this is essentially the

00:41:50.000 --> 00:41:52.000
mass.
The frequency is given by one

00:41:52.000 --> 00:41:57.000
over 2pi times the square root
of this k, the force constant

00:41:57.000 --> 00:42:00.000
over the mass.

00:42:00.000 --> 00:42:04.000
sqrt(k / m)**
That force constant,

00:42:04.000 --> 00:42:09.000
k, is a measure of the
stiffness of the spring.

00:42:09.000 --> 00:42:13.000
Another words,
if you have a very stiff

00:42:13.000 --> 00:42:19.000
spring, k is very large.
That is you need a lot of force

00:42:19.000 --> 00:42:25.000
to pull that spring apart,
a lot of force to compress that

00:42:25.000 --> 00:42:29.000
spring.
A large k means you have a very

00:42:29.000 --> 00:42:34.000
stiff spring.
If you have a small k,

00:42:34.000 --> 00:42:38.000
that means you have a very weak
spring.

00:42:38.000 --> 00:42:45.000
It does not take very much
force to stretch that spring or

00:42:45.000 --> 00:42:50.000
to compress that spring.
So k is a measure of the

00:42:50.000 --> 00:42:56.000
stiffness of the spring.
It also tells us something

00:42:56.000 --> 00:43:02.000
about the shape of the
interaction energy.

00:43:02.000 --> 00:43:06.000
Another words,
what k actually tells us here

00:43:06.000 --> 00:43:09.000
is kind of the width of this
well.

00:43:09.000 --> 00:43:14.000
Here is my interaction energy,
potential well.

00:43:14.000 --> 00:43:19.000
If you have a very narrow
width, this is a large k.

00:43:19.000 --> 00:43:25.000
If you have a very broad width,
here, this is a small k.

00:43:25.000 --> 00:43:30.000
So, k, the force constant,
tells us about the shape,

00:43:30.000 --> 00:43:36.000
here, of this well.
Very narrow well,

00:43:36.000 --> 00:43:40.000
large k.
Very broad well,

00:43:40.000 --> 00:43:46.000
small k.
Now, I think that maybe you

00:43:46.000 --> 00:43:55.000
have seen this before in 8.01.
Have you done the harmonic

00:43:55.000 --> 00:43:57.000
oscillator yet?
No?

00:43:57.000 --> 00:44:00.000
Yes?
Some yes.

00:44:00.000 --> 00:44:03.000
Some no.
Mostly no, it sounds to me.

00:44:03.000 --> 00:44:07.000
This expression,
here, for the frequency,

00:44:07.000 --> 00:44:10.000
you are going to see in 8.01
some time.

00:44:10.000 --> 00:44:14.000
It comes from the harmonic
oscillator model.

00:44:14.000 --> 00:44:18.000
You will remember that you have
seen this before.

00:44:18.000 --> 00:44:23.000
Those of you who have know what
I am talking about.

00:44:23.000 --> 00:44:28.000
Those of you who have not,
just remember you are going to

00:44:28.000 --> 00:44:35.000
see this very soon once again.
Now, the other parameter here

00:44:35.000 --> 00:44:42.000
that is important is this mu
here, this reduced mass.

00:44:42.000 --> 00:44:49.000
What the reduced mass is,
is essentially the mass.

00:44:49.000 --> 00:44:54.000
Now, if you had already done
this in 8.01,

00:44:54.000 --> 00:45:02.000
what you are going to look at
is a mass on the end of a spring

00:45:02.000 --> 00:45:08.000
attached to a wall.
There will be a wall,

00:45:08.000 --> 00:45:13.000
then you will see a spring,
and then there will be some

00:45:13.000 --> 00:45:16.000
mass m.
When you do this problem as a

00:45:16.000 --> 00:45:21.000
harmonic oscillator,
you are going to write nu here

00:45:21.000 --> 00:45:26.000
as k over this mass right there.
But, in this example,

00:45:26.000 --> 00:45:32.000
this mass here is the only body
that is moving.

00:45:32.000 --> 00:45:36.000
Your wall is still.
But, in the case of a molecule,

00:45:36.000 --> 00:45:40.000
both atoms are moving.
What mass do we use?

00:45:40.000 --> 00:45:46.000
Well, the fact that both are
moving means we have a two body

00:45:46.000 --> 00:45:49.000
problem.
And we are going to have to

00:45:49.000 --> 00:45:55.000
change the coordinate system to
take into account that both

00:45:55.000 --> 00:46:00.000
masses are moving.
We can do that easily.

00:46:00.000 --> 00:46:04.000
And doing that easily involves
using something called the

00:46:04.000 --> 00:46:08.000
reduced mass.
The reduced mass takes a two

00:46:08.000 --> 00:46:13.000
body problem and reduces it to a
single body, where the single

00:46:13.000 --> 00:46:18.000
body is this fictitious body
with a mass given by the reduced

00:46:18.000 --> 00:46:19.000
mass.
It is exact.

00:46:19.000 --> 00:46:22.000
I did not make any
approximations here.

00:46:22.000 --> 00:46:27.000
But the bottom line is that
this reduced mass is simply the

00:46:27.000 --> 00:46:31.000
mass of one atom,
times the mass of the other,

00:46:31.000 --> 00:46:36.000
divided by the sum of the two
masses.

00:46:36.000 --> 00:46:42.000
And sometimes a useful
approximation is when one of the

00:46:42.000 --> 00:46:49.000
masses is much lower than the
other mass, so when m1 here is

00:46:49.000 --> 00:46:54.000
smaller than m2.
Well, you can see in that case,

00:46:54.000 --> 00:47:01.000
if m1 is very small compared to
m2, we are going to treat it

00:47:01.000 --> 00:47:06.000
like a zero.
And then these two m's are

00:47:06.000 --> 00:47:10.000
going to cancel.
You can see the reduced mass is

00:47:10.000 --> 00:47:14.000
roughly equivalent to the mass
of the smaller body.

00:47:14.000 --> 00:47:18.000
Sometimes I will ask you to use
that approximation.

00:47:18.000 --> 00:47:22.000
Other times,
I will ask you to calculate the

00:47:22.000 --> 00:47:25.000
reduced mass.
Let's look at some frequencies

00:47:25.000 --> 00:47:29.000
for some molecules,
here.

00:47:29.000 --> 00:47:33.000
I have an array of molecules.
All of them roughly have the

00:47:33.000 --> 00:47:38.000
same force constant.
I am going to look at the mass

00:47:38.000 --> 00:47:41.000
dependence on the fundamental
frequency.

00:47:41.000 --> 00:47:46.000
I have got the frequencies
written both in terms of wave

00:47:46.000 --> 00:47:50.000
numbers and Hertz.
The first thing I want you to

00:47:50.000 --> 00:47:54.000
notice is the fundamental
frequency for hydrogen,

00:47:54.000 --> 00:47:59.000
4,159 wave numbers.
That is the highest vibrational

00:47:59.000 --> 00:48:04.000
frequency that we know about.
Why is it so high?

00:48:04.000 --> 00:48:08.000
Well, because those two
hydrogens are so light,

00:48:08.000 --> 00:48:13.000
and the mass is in the
denominator of that fundamental

00:48:13.000 --> 00:48:17.000
frequency.
So if it is in the denominator,

00:48:17.000 --> 00:48:20.000
it is small,
and the fundamental frequency

00:48:20.000 --> 00:48:25.000
is going to be high.
Then look at these molecules,

00:48:25.000 --> 00:48:28.000
HCl, HBr, HI.
In this case here,

00:48:28.000 --> 00:48:33.000
it is all a hydrogen bonded to
a halogen.

00:48:33.000 --> 00:48:37.000
For the most part,
the reduced mass of these three

00:48:37.000 --> 00:48:41.000
molecules is that of hydrogen.
If that is of hydrogen,

00:48:41.000 --> 00:48:45.000
since that is in the
denominator for the expression

00:48:45.000 --> 00:48:50.000
for the frequency,
well, the frequencies are going

00:48:50.000 --> 00:48:53.000
to be high in general.
And they are,

00:48:53.000 --> 00:48:56.000
2,886, 2,559.
Not as high as molecular

00:48:56.000 --> 00:49:03.000
hydrogen, but still pretty high.
And then we look at these three

00:49:03.000 --> 00:49:06.000
molecules, chlorine,
bromine, iodine.

00:49:06.000 --> 00:49:12.000
Both particles very heavy.
The reduced mass is large.

00:49:12.000 --> 00:49:16.000
Look at what happened to the
frequency here.

00:49:16.000 --> 00:49:22.000
The frequency really has gone
down much lower than it was for

00:49:22.000 --> 00:49:28.000
HCl, HBr or HI.
So, that is one parameter.

00:49:28.000 --> 00:49:32.000
The other parameter,
of course, is the force

00:49:32.000 --> 00:49:35.000
constant.
Here are a set of molecules

00:49:35.000 --> 00:49:39.000
that have all roughly the same
reduced mass,

00:49:39.000 --> 00:49:41.000
fluorine, oxygen,
NO.

00:49:41.000 --> 00:49:46.000
And I have written down their
frequencies, both in wave

00:49:46.000 --> 00:49:51.000
numbers and Hertz.
For molecular fluorine,

00:49:51.000 --> 00:49:56.000
which has a single bond,
that frequency here is pretty

00:49:56.000 --> 00:50:01.000
low, 892 wave numbers.
It is pretty low.

00:50:01.000 --> 00:50:06.000
This is a single bond,
which is easy to stretch.

00:50:06.000 --> 00:50:10.000
It is not very stiff.
It is a loose bond,

00:50:10.000 --> 00:50:14.000
so to speak.
But now, look at oxygen and NO.

00:50:14.320 --> 00:50:18.000
They have double bonds,
which are stiffer.

00:50:18.000 --> 00:50:21.000
If they are stiffer,
k is larger,

00:50:21.000 --> 00:50:24.000
the fundamental frequency is
higher.

00:50:24.000 --> 00:50:27.000
Then, we look at CO and
nitrogen.

00:50:27.000 --> 00:50:34.000
They have triple bonds,
which are even stiffer.

00:50:34.000 --> 00:50:38.000
Stiffer bonds,
larger k, higher fundamental

00:50:38.000 --> 00:50:43.000
frequency, that are those two
parameters that dictate the

00:50:43.000 --> 00:50:47.000
frequency of vibration.
We will do rotation on

00:50:47.816 --> 00:50:50.000
Wednesday.
See you then.