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Good afternoon.
I hope you had a nice weekend.

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Did you have a nice weekend?
Good.

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Today, we are going to start
talking about the motion of

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molecules.
We are first going to talk

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about the translational motion
of molecules today and Friday.

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And then we are going to talk
about the internal motion,

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in particular the vibrational
motion of molecules and their

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rotational motion.
That will be early next week.

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First, transitional motion.
Certainly, the quintessential

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equation that represents the
behavior of gases is,

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of course, the ideal gas law,
P equals n over V times RT.

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This equation accurately

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represents the behavior of gases
at low pressures.

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It is an empirical law,
of course.

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It is a law that Boyle and
Charles discovered by doing

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experiments.
They noted that as the

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temperature is raised,
the pressure went up.

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They noted as the amount of gas
or the number of moles of gas

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added goes up,
the pressure goes up.

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They noted that as the volume
of their gaseous container goes

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up, the pressure goes down.
Literally, this is an equation

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established by experiment and
just varying one variable at a

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time.
And since it looked like the

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pressure was directly
proportional to the temperature,

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they wrote an equation where it
was directly proportional to the

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temperature, etc.
And the proportionality

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constant here was always this
one constant R,

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no matter what gas you had.
But this ideal gas law

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describes the macroscopic
properties of a gas.

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And what I mean by macroscopic
properties are properties that

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describe a collection of
molecules.

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For example,
to really talk about a

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pressure, you have to have a
collection of molecules.

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We are going to talk about the
pressure due to one molecule,

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but that is really just a
model.

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If you want to talk about
pressure, you need to have a

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collection of molecules.
If you talk about temperature,

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it really only has meaning when
you have a collection of

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molecules.
But what we want to understand

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is what are the underlying
microscopic phenomenon that

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gives rise to this macroscopic
equation or these macroscopic

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properties.
We want to know what is going

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on in terms of the behavior of
the individual particles,

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the individual molecules that
make up this gas.

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Inquiring minds want to know
what the temperature means when

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we talk about individual
molecules.

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And that is also was Maxwell
and Boltzmann wanted to know.

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They wanted a microscopic
explanation for PV equal nRT.

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And, to do so,

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they put forth a theory called
the kinetic theory,

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or the kinetic theory for the
behavior of gases.

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And that is exactly what we are
going to take a look at here,

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this kinetic theory.
We are going to do what they

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did.
Basically, this kinetic theory

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allowed properties of gases at
low pressures to be predicted,

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and it allowed an understanding
of why some of the properties of

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real gases at higher pressures
deviated from this ideal gas

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law.
But, more importantly,

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what it did was it allowed the
quantity pressure times the

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volume **PV** to be understood
in terms of the motions of the

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molecules.
And it provided a means to

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understand this concept of a
temperature in terms of the

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motions of the molecules.
And that is what we are going

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to look at today.
We are going to see how we can

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describe pressure and volume in
terms of the motion of the

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molecules and temperature in
terms of the motion of the

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molecules.
That is our goal.

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In order to do that,
the first thing we have to do

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is to understand what we mean by
pressure.

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For example,
if you have some gas here in a

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container and you are measuring
the pressure of this gas in that

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container, --
what you are really measuring

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is the force that the gas exerts
on one of the walls of this

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container.
That is, pressure is the force

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exerted by the gas on one of
these walls.

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It is the force per unit area
that is exerted on the walls of

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that container.
That

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is what pressure is.
But what Maxwell did,

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--
-- and this is 1850,

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What Maxwell did was recognize
No surprise.

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that he could understand this
macroscopic pressure in terms of

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the individual forces of the
molecules when they hit the

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container.
In other words,

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he proposed that this gas was
composed of these molecules and

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that these molecules were
moving.

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That was his proposal.
And that when they moved and

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hit the walls of the container,
well, that was the force that

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was exerted by the gas.
In other words,

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the force was really the
individual forces,

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F sub i,
here, of the individual

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molecules.
That total force was the

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individual force of the
molecules hitting the walls of

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the container.
That is what led to this

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macroscopic concept,
the macroscopic quantity of

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pressure.
That was his idea.

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Well, if that was his idea,
he carried it through,

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now, to a prediction.
The idea is that this force

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arises from the individual
forces of these individual

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molecules hitting the walls of
the container.

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Let's look at what theory he
wrote down.

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His goal, here,
was to calculate these

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individual forces,
F sub i.

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And, of course,
go back to classical mechanics.

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Force is the mass times the
acceleration,

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the mass of the particle,
the molecule,

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times its acceleration.
And acceleration we can write

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just in terms of delta v over
delta t.

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That is just the mass times

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delta v over the change in time.
And if the mass is constant

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here, this numerator,
that is the change in momentum

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per unit change in time.
That is what this force is.

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What we have to do,
here, is calculate the change

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in the momentum of the wall when
the molecule hits the wall.

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That will be the force of the
individual molecule in the wall,

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that over the time between the
collisions of the molecule with

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the wall.
That is the individual forces.

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That is what we are trying to
calculate, right here.

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Let's do that and calculate
delta p, the change in the

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wall's momentum.
Let's do it in one dimension

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first.
Here is our box.

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This is just a cross-sectional
view of that box I drew over

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here.
And this is going to be the

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wall that we are going to be
interested in.

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The box has a length l.
And we have this molecule with

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some mass m.
It is coming into the wall and

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is going to collide with it,
and it has some velocity vector

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v.
But let's do it in one

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dimension because it is simpler
to do.

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And then we are going to extend
it to three dimensions in a few

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minutes.
We are just going to do it in

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one dimension now.
We are only going to be

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interested in then the
x-component of the velocity of

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this molecule coming into the
wall, and we want to know what

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is the force exerted by this
molecule when it collides with

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this wall.
First we have to get the change

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in the momentum.
This molecule comes in and hits

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the wall, and we are going to
consider it to be an elastic

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collision.
If the component of the

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velocity in the x direction
before the collision is v sub x,

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then after the collision,
it is minus v sub x.

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We have just changed the
direction of our velocity

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vector, not the magnitude.
The change in the atom's

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momentum, then,
is just the momentum after

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minus the moment before.
The momentum after was m times

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minus v sub x.
The momentum before,

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m sub v sub x.
Delta in the change in the

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atom's momentum,
is minus 2 m v sub x.

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However, we have to conserve

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momentum.
The momentum change in the atom

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plus the momentum change in the
wall has to equal zero.

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And what we want to know is the
momentum change in the wall,

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because we are after this
macroscopic quantity,

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pressure.
And so, if the change in the

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momentum of the atom is minus 2
m v sub x,

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then the change in the
momentum of the wall is 2 m v

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sub x .

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We have one quantity here.
We have delta p.

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But now, we have to calculate
how often that momentum in the

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wall changes due to this
molecule's collisions.

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We need the delta t.
And so, what happens here?

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The molecule comes in,
collides, and then reflects.

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And the molecule is now going
in this direction,

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where it hits the back wall,
and then it reflects.

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And ultimately it comes back
and hits the front wall.

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And what we want to know is
what is the time here between

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the collisions.
The time between when that

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molecule makes the momentum in
the wall change.

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What is this time,
delta t?

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Well, if we know the length of
the box and we know the value of

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v sub x, we can calculate that
time.

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That time is just two times the
length of the box because that

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molecule is traveling back and
then forth, --

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-- that is 2l divided by the
velocity component in the x

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direction, v sub x.
That

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is delta t.
Now we have delta p and delta

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t, and we can plug that in and
make it simple a little bit.

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And so, here is the force
exerted by the collision of one

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molecule on that one wall of the
container, m v sub x squared

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over l.

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But we said that Maxwell's idea
was that the macroscopic total

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force was the sum of these
individual forces.

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What we have to do is take the
force of each individual

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molecule and add them up over
all the molecules.

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There is that force for
molecule one,

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molecule two,
molecule three,

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all the way up to molecule n.
I am going to pull out an m

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over l out of this.
I just pull out an m over l,

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and I have left the sum of the
squares of the velocities in the

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x direction for each one of the
molecules.

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Notice here,
in this treatment,

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we have identical m's.
The particles are the same,

00:14:27.000 --> 00:14:33.000
but, and this is important,
the velocities of the molecules

00:14:33.000 --> 00:14:38.000
are not the same.
That is going to be important.

00:14:38.000 --> 00:14:44.000
Now, this expression here,
I want to simplify a little

00:14:44.000 --> 00:14:45.000
bit.
In particular,

00:14:45.000 --> 00:14:51.000
I want to simplify the sum of
the squares of the velocity

00:14:51.000 --> 00:14:56.000
components in the x direction
for each one of the molecules.

00:14:56.000 --> 00:15:02.000
To do that simplification,
I am going to introduce this

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quantity.
This quantity is the average of

00:15:06.000 --> 00:15:10.000
the square of the velocity in
the x direction.

00:15:10.000 --> 00:15:15.000
What this means is I take the
velocity in the x direction,

00:15:15.000 --> 00:15:18.000
I square it,
and then I take the average.

00:15:18.000 --> 00:15:23.000
This is not the square of the
average velocity in the x

00:15:23.000 --> 00:15:26.000
direction.
That is different.

00:15:26.000 --> 00:15:31.000
This is the average of the
square of the velocity in the x

00:15:31.000 --> 00:15:35.000
direction.
How am I going to evaluate that

00:15:35.000 --> 00:15:38.000
quantity?
Well, I am going to take the

00:15:38.000 --> 00:15:43.000
velocity in the x direction for
molecule one and square it and

00:15:43.000 --> 00:15:47.000
add to that the square of the
velocity in the x direction for

00:15:47.000 --> 00:15:52.000
molecule two and add to that the
square of the velocity in the x

00:15:52.000 --> 00:15:58.000
direction for molecule three,
all the way up to molecule n.

00:15:58.000 --> 00:16:03.000
And then, if I want the
average, I am going to divide by

00:16:03.000 --> 00:16:06.000
n, the number of molecules there
are.

00:16:06.000 --> 00:16:11.000
I am just going to bring n,
here, over to the other side.

00:16:11.000 --> 00:16:17.000
I have n times the average of
the velocity in the x direction

00:16:17.000 --> 00:16:21.000
squared as this sum,
and this is exactly the sum

00:16:21.000 --> 00:16:25.000
that I had in my expression for
the total force.

00:16:25.000 --> 00:16:31.000
I can simplify that now.
There is that same expression.

00:16:31.000 --> 00:16:36.000
Here is my total force.
What I am going to do is I am

00:16:36.000 --> 00:16:42.000
going to substitute n times the
average of the velocity in the x

00:16:42.000 --> 00:16:46.000
direction squared in for this
whole sum, and I have now

00:16:46.000 --> 00:16:52.000
something that is much tidier.
That is the total force exerted

00:16:52.000 --> 00:16:57.000
by all the collisions of the
molecules in the container on

00:16:57.000 --> 00:17:03.000
that front wall.
But now I want the pressure.

00:17:03.000 --> 00:17:08.000
And the pressure is just force
per unit area.

00:17:08.000 --> 00:17:15.000
And so, I am going to take my
expression for the force and

00:17:15.000 --> 00:17:21.000
divide it by the unit area.
The area is the area of this

00:17:21.000 --> 00:17:25.000
wall, here, in my initial
example.

00:17:25.000 --> 00:17:31.000
I am going to call that area A.
And if this is a Q,

00:17:31.000 --> 00:17:36.000
and I am going to make it a Q
because it is easier to do it

00:17:36.000 --> 00:17:42.000
that way, the area times the
length then of this box is just

00:17:42.000 --> 00:17:44.000
the volume.
Here is the volume.

00:17:44.000 --> 00:17:49.000
There is my expression for the
pressure due to all of the

00:17:49.000 --> 00:17:54.000
molecule colliding with the
front wall of that box.

00:17:54.000 --> 00:17:58.000
However, this is an expression
for the pressure in one

00:17:58.000 --> 00:18:03.000
dimension only,
the x dimension.

00:18:03.000 --> 00:18:05.000
And we know,
in real life,

00:18:05.000 --> 00:18:10.000
we have three dimensions.
We have to take care of that.

00:18:10.000 --> 00:18:14.000
Let's do that now.
Let's extend this problem to

00:18:14.000 --> 00:18:18.000
three dimensions.
To do so, I am just going to

00:18:18.000 --> 00:18:23.000
realize right here that the
square of the velocity is the

00:18:23.000 --> 00:18:26.000
sum of the squares of the
components.

00:18:26.000 --> 00:18:31.000
That, you understand.
That is okay.

00:18:31.000 --> 00:18:37.000
What is not so obvious is this.
The average of the square of

00:18:37.000 --> 00:18:43.000
the velocity is the sum of the
average of the squares of each

00:18:43.000 --> 00:18:47.000
one of the components.
That is true.

00:18:47.000 --> 00:18:52.000
You can prove that.
We are not going to prove that.

00:18:52.000 --> 00:18:57.000
I won't hold you responsible
for proving that,

00:18:57.000 --> 00:19:01.000
but that is true.
That is correct,

00:19:01.000 --> 00:19:05.000
but now here comes a critical
assumption in Boltzmann's

00:19:05.000 --> 00:19:08.000
treatment.
The critical assumption is that

00:19:08.000 --> 00:19:12.000
the motion of the molecules in
this gas, here,

00:19:12.000 --> 00:19:16.000
is random in the sense that the
molecules don't have a preferred

00:19:16.000 --> 00:19:19.000
direction.
They are going in the x

00:19:19.000 --> 00:19:23.000
direction as often as they are
going in the y direction as

00:19:23.000 --> 00:19:28.000
often as they are going in the z
direction, so the motion is

00:19:28.000 --> 00:19:32.000
random.
If that motion is random,

00:19:32.000 --> 00:19:36.000
then the average of the
velocity squared in the x

00:19:36.000 --> 00:19:42.000
direction is going to be equal
to that in the y direction.

00:19:42.000 --> 00:19:47.000
It is going to be equal to that
in the z direction if that

00:19:47.000 --> 00:19:52.000
motion is random.
And that is great because it is

00:19:52.000 --> 00:19:56.000
going to make things a little
simpler for us up here.

00:19:56.000 --> 00:20:00.000
If that is right,
then the average of the

00:20:00.000 --> 00:20:06.000
velocity squared is three times
the average of the velocity

00:20:06.000 --> 00:20:11.000
squared in any one of the
dimensions.

00:20:11.000 --> 00:20:16.000
That is going to make it easy
to extrapolate this to three

00:20:16.000 --> 00:20:22.000
dimensions because now I am
going to be able to substitute,

00:20:22.000 --> 00:20:28.000
which had the average of the
velocity squared only in the x

00:20:28.000 --> 00:20:34.000
direction, I am going to be able
to substitute in an expression

00:20:34.000 --> 00:20:40.000
for the average velocity in
three dimensions.

00:20:40.000 --> 00:20:43.000
That is just going to be
one-third that.

00:20:43.000 --> 00:20:48.000
This is going to be one-third
the average of the velocity

00:20:48.000 --> 00:20:50.000
squared.
That is great.

00:20:50.000 --> 00:20:56.000
Now, I am going to do that
substitution way up into there.

00:20:56.000 --> 00:21:00.000
And when I do that,
look at this.

00:21:00.000 --> 00:21:04.000
I have a result.
This is the kinetic theory

00:21:04.000 --> 00:21:07.000
result.
We just did exactly what

00:21:07.000 --> 00:21:11.000
Maxwell did.
We have an expression for the

00:21:11.000 --> 00:21:16.000
pressure times the volume,
which is written here in terms

00:21:16.000 --> 00:21:21.000
of the average of the velocity
squared of the molecules.

00:21:21.000 --> 00:21:28.000
It is written in terms of the
motion of the molecules.

00:21:28.000 --> 00:21:31.000
For the first time,
there is an understanding,

00:21:31.000 --> 00:21:36.000
here, of what gives rise to
pressure, and that is the

00:21:36.000 --> 00:21:41.000
velocity or the motion of these
molecules hitting the wall.

00:21:41.000 --> 00:21:45.000
That is great.
That is the kinetic theory

00:21:45.000 --> 00:21:48.000
result.
But now, Boltzmann also knew

00:21:48.000 --> 00:21:53.000
from experiment that P times V
is equal to nRT.

00:21:53.000 --> 00:21:57.000
That is the experimental
result, which had been known

00:21:57.000 --> 00:22:02.000
already for over a hundred
years.

00:22:02.000 --> 00:22:07.000
That is the experiment.
If his theory is correct,

00:22:07.000 --> 00:22:14.000
if PV is equal to N m average
velocity squared over three,

00:22:14.000 --> 00:22:20.000
it better be equal to nRT.

00:22:20.000 --> 00:22:26.000
That will give us,
here, a prediction for what the

00:22:26.000 --> 00:22:33.000
velocity of the molecules ought
to be in terms of something

00:22:33.000 --> 00:22:40.000
experimentally controllable.
We can see if this kinetic

00:22:40.000 --> 00:22:45.000
theory model is correct.
We can solve this for the

00:22:45.000 --> 00:22:52.000
average of the velocity squared.
It is equal to 3n RT N over m.

00:22:52.000 --> 00:22:57.000
We can go in the laboratory,

00:22:57.000 --> 00:23:02.000
vary T and see if,
in fact, the average of the

00:23:02.000 --> 00:23:07.000
velocity squared of the
molecules is equal to this

00:23:07.000 --> 00:23:11.000
expression here.
That is great.

00:23:11.000 --> 00:23:16.000
We have a way to experimentally
check this theory.

00:23:16.000 --> 00:23:21.000
And you also see,
here, now, a relationship

00:23:21.000 --> 00:23:26.000
between the velocity of the
molecules and this macroscopic

00:23:26.000 --> 00:23:32.000
quantity, temperature.
Temperature is related to the

00:23:32.000 --> 00:23:37.000
motion of these molecules.
But, before we go on,

00:23:37.000 --> 00:23:41.000
this is kind of a messy
expression here.

00:23:41.000 --> 00:23:44.000
It has too many n's and m's in
it.

00:23:44.000 --> 00:23:48.000
Let me simplify that a little
bit for you.

00:23:48.000 --> 00:23:54.000
I am going to simplify this so
that this is 3RT over capital M,

00:23:54.000 --> 00:24:00.000
where the following is true.

00:24:00.000 --> 00:24:04.000
Over here, n is the number of
moles in the gas.

00:24:04.000 --> 00:24:08.000
That is little n.
Big N was the number of

00:24:08.000 --> 00:24:12.000
molecules in the gas.
Little m was the mass per

00:24:12.000 --> 00:24:15.000
molecule.
All of this is equivalent to

00:24:15.000 --> 00:24:20.000
one over big M,
where big M was kilograms per

00:24:20.000 --> 00:24:23.000
mole.
You can convince yourselves of

00:24:23.000 --> 00:24:27.000
this equality.
I am taking all these N's and

00:24:27.000 --> 00:24:31.000
m's and making one big M.

00:24:36.000 --> 00:24:38.000
That is my expression,
here.

00:24:38.000 --> 00:24:42.000
I have the average of the
velocity squared equal to 3RT

00:24:42.000 --> 00:24:47.000
over M.
But this quantity is the

00:24:47.000 --> 00:24:52.000
average of the velocity squared.
It is more convenient for us to

00:24:52.000 --> 00:24:57.000
talk about a quantity
proportional to the velocity and

00:24:57.000 --> 00:25:02.000
not the velocity squared.
What I am going to do is take

00:25:02.000 --> 00:25:04.000
the square root of it.
That is simple.

00:25:04.000 --> 00:25:09.000
I now have the square root of
the average of the velocity

00:25:09.000 --> 00:25:12.000
squared.
That is the square root of 3RT

00:25:12.000 --> 00:25:14.000
over M.
I am going to call that the

00:25:14.000 --> 00:25:18.000
root mean square velocity.
I am going to put an rms here

00:25:18.000 --> 00:25:23.000
as a subscript for the velocity.

00:25:23.000 --> 00:25:25.000
It is the root mean square
velocity.

00:25:25.000 --> 00:25:29.000
I wanted to talk about a
quantity proportional to the

00:25:29.000 --> 00:25:34.000
velocity, instead of the
velocity squared.

00:25:34.000 --> 00:25:38.000
That is all I did there.
That is the root mean square

00:25:38.000 --> 00:25:41.000
velocity.
But the other big thing about

00:25:41.000 --> 00:25:45.000
it is you can see,
for the first time,

00:25:45.000 --> 00:25:47.000
now we have got,
and Maxwell had,

00:25:47.000 --> 00:25:51.000
an understanding of what
temperature was.

00:25:51.000 --> 00:25:56.000
Temperature is related to the
motions of the molecules.

00:25:56.000 --> 00:25:59.000
Temperature is related,
in this way,

00:25:59.000 --> 00:26:06.000
to the speed of the molecules.
Those are the two important

00:26:06.000 --> 00:26:11.000
results.
And this kinetic theory makes a

00:26:11.000 --> 00:26:19.000
prediction for what those
velocities ought to be.

00:26:26.000 --> 00:26:30.000
In addition,
the temperature is a measure of

00:26:30.000 --> 00:26:34.000
the kinetic energy of the
molecules.

00:26:34.000 --> 00:26:37.000
How is that?
Well, it is for this reason.

00:26:37.000 --> 00:26:43.000
Here is the expression we
derived from the kinetic theory.

00:26:43.000 --> 00:26:47.000
And then here is an expression
that I just wrote down,

00:26:47.000 --> 00:26:53.000
that says the average kinetic
energy of a molecule is one-half

00:26:53.000 --> 00:26:58.000
M, where M is kilograms per
mole, times the average of the

00:26:58.000 --> 00:27:04.000
velocity squared.

00:27:04.000 --> 00:27:09.000
If I substitute the average of
the velocity squared into here,

00:27:09.000 --> 00:27:13.000
I get three-halves RT.

00:27:13.000 --> 00:27:16.000
This is telling us,
right here, that the

00:27:16.000 --> 00:27:22.000
temperature is also a measure of
the kinetic energy of these

00:27:22.000 --> 00:27:25.000
molecules.
We are getting a microscopic

00:27:25.000 --> 00:27:30.000
view, here, of what temperature
is.

00:27:30.000 --> 00:27:34.000
It is related to the motions of
these molecules.

00:27:34.000 --> 00:27:40.000
Now, before I go on talking
about this, let me make one big

00:27:40.000 --> 00:27:43.000
point.
That is, this expression here,

00:27:43.000 --> 00:27:48.000
for the average energy,
notice that it is one-half M

00:27:48.000 --> 00:27:53.000
times the average of the
velocity squared.

00:27:53.000 --> 00:27:59.000
It is not one-half M times the
square of the average velocity

00:27:59.000 --> 00:28:03.000
This is important.

00:28:03.000 --> 00:28:09.000
The average energy is not the
square of the average velocity.

00:28:09.000 --> 00:28:14.000
Rather, the average energy is
the average of the velocity

00:28:14.000 --> 00:28:17.000
squared.
There is a big distinction.

00:28:17.000 --> 00:28:23.000
This is because the average
energy is the second moment of

00:28:23.000 --> 00:28:26.000
the velocity distribution
function.

00:28:26.000 --> 00:28:31.000
Variables don't always
correspond in a one-to-one

00:28:31.000 --> 00:28:36.000
manner.
You don't have to understand

00:28:36.000 --> 00:28:43.000
that, if this is foreign to you,
but I do want you to know this

00:28:43.000 --> 00:28:48.000
is correct.
Now, let me pick up back here.

00:28:48.000 --> 00:28:54.000
What I want you to notice is
that the root mean square

00:28:54.000 --> 00:29:00.000
velocity has a mass dependence
in it.

00:29:00.000 --> 00:29:04.000
What does that mean?
Well, it means the following.

00:29:04.000 --> 00:29:09.000
For some constant temperature,
say we pick 300 degrees Kelvin,

00:29:09.000 --> 00:29:14.000
the velocity of the molecule is
going to depend on its mass.

00:29:14.000 --> 00:29:18.000
And it is inversely
proportional to the mass,

00:29:18.000 --> 00:29:23.000
so heavier molecules move more
slowly, lighter molecules move

00:29:23.000 --> 00:29:25.000
more quickly.
For example,

00:29:25.000 --> 00:29:30.000
helium at 300 degrees Kelvin,
it is cruising along at 3

00:29:30.000 --> 00:29:36.000
miles per hour at
room temperature.

00:29:36.000 --> 00:29:40.000
Xenon, on the other hand,
which is much more massive,

00:29:40.000 --> 00:29:44.000
is moving at a measly 534 miles
per hour.

00:29:44.000 --> 00:29:47.000
There is a mass dependence
here.

00:29:47.000 --> 00:29:52.000
However, there is no mass
dependence to the kinetic

00:29:52.000 --> 00:29:54.000
energy.
The kinetic energy,

00:29:54.000 --> 00:29:58.000
we saw, was three-halves RT.

00:29:58.000 --> 00:30:04.000
You don't see a mass dependence
in here, do you?

00:30:04.000 --> 00:30:09.000
The kinetic energy is only
dependent on the temperature.

00:30:09.000 --> 00:30:15.000
Whether or not you have helium
or xenon, the kinetic energy of

00:30:15.000 --> 00:30:21.000
those atoms is 3.74 kilojoules
per mole at 300 degrees Kelvin.

00:30:21.000 --> 00:30:26.000
It does not matter that helium
is moving six times as fast as

00:30:26.000 --> 00:30:30.000
xenon.
They both have the same kinetic

00:30:30.000 --> 00:30:34.000
energy.
There is no mass dependence in

00:30:34.000 --> 00:30:36.000
kinetic energy.
Now you say,

00:30:36.000 --> 00:30:39.000
oh, but look at this,
here is a mass,

00:30:39.000 --> 00:30:43.000
there is a mass dependence.
No, because you have to

00:30:43.000 --> 00:30:46.000
remember that you substitute in
here.

00:30:46.000 --> 00:30:49.000
If you square this,
there is an M here,

00:30:49.000 --> 00:30:53.000
and that cancels.
There is no mass dependence in

00:30:53.000 --> 00:30:57.000
the kinetic energy,
but the velocity is dependent

00:30:57.000 --> 00:31:01.000
on the mass.
That is important.

00:31:01.000 --> 00:31:07.000
Well, I told you that this was
the kinetic theory result.

00:31:07.000 --> 00:31:12.000
That is, that the root mean
square velocity of these

00:31:12.000 --> 00:31:16.000
molecules was represented by
this equation.

00:31:16.000 --> 00:31:21.000
And, from this equation,
if you calculate at 300 degrees

00:31:21.000 --> 00:31:25.000
Kelvin, these are,
in fact, the velocities of

00:31:25.000 --> 00:31:30.000
those atoms.
But how do we know this is

00:31:30.000 --> 00:31:33.000
right?
How do we go and measure the

00:31:33.000 --> 00:31:38.000
velocities or the speeds of
molecules or atoms?

00:31:38.000 --> 00:31:44.000
Well, this is the way we do it.
It is called the time-of-flight

00:31:44.000 --> 00:31:47.000
technique.
Are we all on board,

00:31:47.000 --> 00:31:48.000
here?
Questions?

00:31:48.000 --> 00:31:51.000
Okay.
How are we going to do this

00:31:51.000 --> 00:31:56.000
time-of-flight technique?
What we are going to do is we

00:31:56.000 --> 00:32:02.000
are going to have a little
pinhole here that we can open

00:32:02.000 --> 00:32:08.000
and shut really quickly.
We are going to let out a

00:32:08.000 --> 00:32:12.000
little pulse of gas.
To measure the velocity of the

00:32:12.000 --> 00:32:16.000
molecules, we are literally
going to measure the time it

00:32:16.000 --> 00:32:21.000
takes the molecules to fly from
where we let them out to some

00:32:21.000 --> 00:32:24.000
detector.
And since we know the distance,

00:32:24.000 --> 00:32:28.000
we are going to be able to
calculate the velocity from

00:32:28.000 --> 00:32:32.000
that.
The idea is at time t equals 0,

00:32:32.000 --> 00:32:38.000
we let out a little pulse of
gas, and then we start a clock

00:32:38.000 --> 00:32:41.000
running.
Then we just measure how long

00:32:41.000 --> 00:32:46.000
it takes the molecules to fly
from this origin here to this

00:32:46.000 --> 00:32:49.000
detector.
And since we built the

00:32:49.000 --> 00:32:54.000
apparatus, and we know what L
is, we can calculate the

00:32:54.000 --> 00:32:56.000
velocity.
Time-of-flight,

00:32:56.000 --> 00:33:02.000
that is what is done.
However, when we let this

00:33:02.000 --> 00:33:07.000
little pulse of gas out,
and now we let the molecules

00:33:07.000 --> 00:33:12.000
fly to that detector over here,
what happens as a function of

00:33:12.000 --> 00:33:15.000
time?
What will happen is that pulse

00:33:15.000 --> 00:33:21.000
of gas will spread out because
not all of the molecules or

00:33:21.000 --> 00:33:25.000
atoms in that pulse of gas have
the same velocity.

00:33:25.000 --> 00:33:32.000
Some of those atoms are moving
faster than the other atoms.

00:33:32.000 --> 00:33:36.000
And so, what is going to happen
is that the molecules or atoms

00:33:36.000 --> 00:33:41.000
that are moving faster are going
to hit the detector first.

00:33:41.000 --> 00:33:46.000
The molecules or atoms that are
moving more slowly are going to

00:33:46.000 --> 00:33:48.000
hit the detector at a later
time.

00:33:48.000 --> 00:33:53.000
And that is what we also want
to know, this distribution of

00:33:53.000 --> 00:33:55.000
velocities.
But in the measurement,

00:33:55.000 --> 00:34:01.000
what we are going to measure is
a distribution of times.

00:34:01.000 --> 00:34:04.000
Out of our detector,
we are going to have a plot

00:34:04.000 --> 00:34:08.000
that looks like this.
This is going to be f of t,

00:34:08.000 --> 00:34:12.000
essentially the number of
molecules hitting the detector

00:34:12.000 --> 00:34:15.000
at a certain time t versus the
time.

00:34:15.000 --> 00:34:20.000
When we first let our pulse of
gas out, that is time t equals

00:00:00.000 --> 00:34:22.000
Then, for a while,

00:34:22.000 --> 00:34:26.000
there are no molecules hitting
the detector because it takes a

00:34:26.000 --> 00:34:32.000
while for them to get over here
to this detector.

00:34:32.000 --> 00:34:35.000
But then, all of a sudden,
they start reaching the

00:34:35.000 --> 00:34:38.000
detector.
And this is essentially just a

00:34:38.000 --> 00:34:43.000
number of molecules that hit the
detector as a function of time.

00:34:43.000 --> 00:34:47.000
That number of molecules
increases and becomes a maximum

00:34:47.000 --> 00:34:50.000
here at some time,
and then it exponentially

00:34:50.000 --> 00:34:53.000
decays, here.
So, this is what we measure.

00:34:53.000 --> 00:34:58.000
This is a distribution here of
flight times of the molecules in

00:34:58.000 --> 00:35:02.000
this pulse of gas.
Well, that is nice,

00:35:02.000 --> 00:35:05.000
but this is a distribution of
flight times.

00:35:05.000 --> 00:35:10.000
It is not a distribution of
velocities, and we wanted a

00:35:10.000 --> 00:35:14.000
distribution of velocities.
We want to know the velocity,

00:35:14.000 --> 00:35:19.000
here, of these molecules.
You know how to convert time,

00:35:19.000 --> 00:35:22.000
given the path length,
to velocity,

00:35:22.000 --> 00:35:25.000
but it is not so
straightforward because we have

00:35:25.000 --> 00:35:31.000
a distribution function.
We have a distribution in time,

00:35:31.000 --> 00:35:36.000
and we want to convert that to
a distribution in velocity.

00:35:36.000 --> 00:35:41.000
We have to change the variable
here in a distribution function.

00:35:41.000 --> 00:35:45.000
How do we do that?
Well, we want this f of t to be

00:35:45.000 --> 00:35:50.000
an f of v.
We recognize here that this

00:35:50.000 --> 00:35:54.000
distribution in time,
the probability of finding a

00:35:54.000 --> 00:35:59.000
molecule between t and t plus
dt, has got to be equivalent to

00:35:59.000 --> 00:36:04.000
the probability of finding a
molecule with a velocity between

00:36:04.000 --> 00:36:09.000
v and v plus dv.
But to get from one

00:36:09.000 --> 00:36:13.000
distribution function to
another, for example,

00:36:13.000 --> 00:36:20.000
if we want f of v,
what we have to know is how one

00:36:20.000 --> 00:36:25.000
variable changes with respect to
another.

00:36:30.000 --> 00:36:33.000
We have this distribution
function f of t,

00:36:33.000 --> 00:36:37.000
but we need to know how t
changes with v.

00:36:37.000 --> 00:36:40.000
We need dt by dv,
we need that,

00:36:40.000 --> 00:36:44.000
so let's get it.
I will tell you why in a

00:36:44.000 --> 00:36:47.000
moment.
We know how v changes with t.

00:36:47.000 --> 00:36:52.000
We are going to take the
derivative of v with respect to

00:36:52.000 --> 00:36:58.000
t and turn things around.
So, dt / dv is proportional to

00:36:58.000 --> 00:37:04.000
minus t squared over L.

00:37:04.000 --> 00:37:06.000
This is telling us,
essentially,

00:37:06.000 --> 00:37:09.000
how the variable t changes with
v.

00:37:09.000 --> 00:37:13.000
To every point in our
time-of-flight distribution,

00:37:13.000 --> 00:37:17.000
we are going to multiply this
by this, what is called the

00:37:17.000 --> 00:37:20.000
Jacobean.
We need this because the time

00:37:20.000 --> 00:37:24.000
and the velocity do not
correlate in a one-to-one

00:37:24.000 --> 00:37:27.000
manner.
That often happens with two

00:37:27.000 --> 00:37:33.000
distribution functions.
If you do not understand what I

00:37:33.000 --> 00:37:38.000
just said, it is okay.
This was just some extra.

00:37:38.000 --> 00:37:42.000
I do not hold you responsible
for it.

00:37:42.000 --> 00:37:48.000
I just changed my variable in
the distribution function.

00:37:48.000 --> 00:37:50.000
Yes?
I think in the notes,

00:37:50.000 --> 00:37:54.000
I might have had as a
proportionality.

00:37:54.000 --> 00:38:00.000
Up here I actually have the
equal sign.

00:38:00.000 --> 00:38:04.000
That actually won't matter in
this transformation.

00:38:04.000 --> 00:38:06.000
Pardon?
I understand that.

00:38:06.000 --> 00:38:10.000
That is fine.
I don't have the equal sign

00:38:10.000 --> 00:38:13.000
there.
That is why I left it out

00:38:13.000 --> 00:38:17.000
there, I think.
Anyway, this is the velocity

00:38:17.000 --> 00:38:20.000
distribution.
You don't have to understand

00:38:20.000 --> 00:38:24.000
how I got there.
This is what the velocity

00:38:24.000 --> 00:38:29.000
distribution looks like.
It is what is called the

00:38:29.000 --> 00:38:32.000
Maxwell-Boltzmann velocity
distribution.

00:38:32.000 --> 00:38:36.000
And the bottom line is that
Maxwell and Boltzmann predicted

00:38:36.000 --> 00:38:40.000
this, about 1855.
They actually predicted this

00:38:40.000 --> 00:38:43.000
distribution function.
We did not predict it.

00:38:43.000 --> 00:38:46.000
We did not go through that part
of kinetic theory,

00:38:46.000 --> 00:38:50.000
but they predicted it.
However, it was only until

00:38:50.000 --> 00:38:54.000
that the technology existed,
fast enough timing and

00:38:54.000 --> 00:38:57.000
electronics existed,
to actually measure this

00:38:57.000 --> 00:39:02.000
experimentally.
This took a hundred years or so

00:39:02.000 --> 00:39:07.000
in order for this distribution
function to actually be

00:39:07.000 --> 00:39:10.000
measured, but here it is,
f of v.

00:39:10.000 --> 00:39:13.000
First of all,
there is all of this stuff,

00:39:13.000 --> 00:39:16.000
which is proportionality
constants.

00:39:16.000 --> 00:39:19.000
We will talk about that in a
moment.

00:39:19.000 --> 00:39:23.000
But the variable,
here, is v squared

00:39:23.000 --> 00:39:28.000
times an exponentially decaying
function with a v squared in

00:39:28.000 --> 00:39:31.000
there.
What does that mean?

00:39:31.000 --> 00:39:35.000
Well, if you look at the form
of f of v, this v squared is

00:39:35.000 --> 00:39:38.000
what gives rise to this increase
in f of v.

00:39:38.000 --> 00:39:42.000
Right here, at low velocities,
that is a quadratic,

00:39:42.000 --> 00:39:45.000
v squared.
But you are multiplying it by

00:39:45.000 --> 00:39:49.000
an exponentially decaying
function with this v squared in

00:39:49.000 --> 00:39:52.000
the argument.
And so that is what gives you

00:39:52.000 --> 00:39:54.000
this tail.
If you are multiplying a

00:39:54.000 --> 00:39:58.000
function that is going up and
one decreasing,

00:39:58.000 --> 00:40:03.000
you are going to get a maximum
at some value of v.

00:40:03.000 --> 00:40:09.000
That is where the shape of the
Maxwell-Boltzmann distribution

00:40:09.000 --> 00:40:14.000
function comes from.
And what this is telling you is

00:40:14.000 --> 00:40:21.000
the probability here of finding
a molecule in a gas with a speed

00:40:21.000 --> 00:40:26.000
between v and v plus dv.
That is what that is telling

00:40:26.000 --> 00:40:30.000
you.
We often characterize these

00:40:30.000 --> 00:40:35.000
distribution functions by some
quantities, and one of those

00:40:35.000 --> 00:40:39.000
quantities is what we call the
most probable speed.

00:40:39.000 --> 00:40:44.000
Here is the distribution
function, and I have the most

00:40:44.000 --> 00:40:48.000
probable speed labeled.
The most probable speed,

00:40:48.000 --> 00:40:53.000
(v)mp, is simply the value of v
at which the probability is the

00:40:53.000 --> 00:40:56.000
largest.
That was like our most probable

00:40:56.000 --> 00:41:02.000
value of r in the radial
distribution functions.

00:41:02.000 --> 00:41:04.000
That is what the most probable
speed is.

00:41:04.000 --> 00:41:07.000
If you wanted to get that
mathematically,

00:41:07.000 --> 00:41:11.000
what you would do is take this
distribution function,

00:41:11.000 --> 00:41:14.000
take the derivative,
set it equal to zero,

00:41:14.000 --> 00:41:18.000
and then solve for v.
That makes that derivative

00:41:18.000 --> 00:41:21.000
equal to zero.
The derivative is zero at

00:41:21.000 --> 00:41:24.000
maxima or minima.
And then, you would find that

00:41:24.000 --> 00:41:29.000
the value of the most probable
speed is the square root of 2RT

00:41:29.000 --> 00:41:33.000
over M.

00:41:33.000 --> 00:41:36.000
You are not responsible for
taking this derivative and

00:41:36.000 --> 00:41:41.000
setting it equal to zero.
You are responsible for knowing

00:41:41.000 --> 00:41:44.000
physically what the most
probable speed is.

00:41:44.000 --> 00:41:48.000
The fact that it is this value
here, where the probability is

00:41:48.000 --> 00:41:51.000
the largest.
It is the most probable value

00:41:51.000 --> 00:41:54.000
of v.
And you are responsible for

00:41:54.000 --> 00:41:57.000
recognizing this.
I don't ask you to memorize it

00:41:57.000 --> 00:42:01.000
or to write it down.
But, if you see it,

00:42:01.000 --> 00:42:07.000
you should know what it is.
That is one quantity that we

00:42:07.000 --> 00:42:11.000
use to characterize this
distribution function.

00:42:11.000 --> 00:42:16.000
Another quantity that we use is
the average speed,

00:42:16.000 --> 00:42:21.000
v bar, average speed.
And the first thing that you

00:42:21.000 --> 00:42:26.000
see is that the average speed is
a little bit higher than the

00:42:26.000 --> 00:42:31.000
most probable speed.
It is a little bit larger.

00:42:31.000 --> 00:42:35.000
Why is that?
Well, it is a little bit larger

00:42:35.000 --> 00:42:38.000
because on these
Maxwell-Boltzmann distribution

00:42:38.000 --> 00:42:43.000
functions, there are molecules
way out here that have very high

00:42:43.000 --> 00:42:45.000
speeds.
There are not a lot of

00:42:45.000 --> 00:42:50.000
molecules that have very high
speeds, but there are molecules

00:42:50.000 --> 00:42:54.000
with very high speeds.
And so, when you average over

00:42:54.000 --> 00:42:58.000
this distribution function,
because there is such a long

00:42:58.000 --> 00:43:01.000
Boltzmann tail here,
is what it is called,

00:43:01.000 --> 00:43:05.000
the average velocity is going
to be a little higher than the

00:43:05.000 --> 00:43:10.000
most probable velocity.
That is physically why the

00:43:10.000 --> 00:43:15.000
average velocity is a little bit
larger than the most probable

00:43:15.000 --> 00:43:17.000
velocity.
That is important.

00:43:17.000 --> 00:43:20.000
If I wanted to calculate what
the average velocity is,

00:43:20.000 --> 00:43:24.000
I would take the distribution
function, multiply it by v,

00:43:24.000 --> 00:43:28.000
and then integrate over the
range of v, which is zero to

00:43:28.000 --> 00:43:32.000
infinity.
You don't have to do that,

00:43:32.000 --> 00:43:36.000
but the quantity that you would
get, if you had done that,

00:43:36.000 --> 00:43:41.000
is the square root of 8RT over
pi M.

00:43:41.000 --> 00:43:43.000
It is larger than the most

00:43:43.000 --> 00:43:46.000
probable value.
And then, finally,

00:43:46.000 --> 00:43:50.000
the other way we characterize
the distribution function is by

00:43:50.000 --> 00:43:55.000
this root mean square speed that
we have already talked about a

00:43:55.000 --> 00:43:57.000
lot.
That root mean square speed,

00:43:57.000 --> 00:44:02.000
here, is even a larger value
than the average velocity or the

00:44:02.000 --> 00:44:07.000
average speed.
And the reason for that is,

00:44:07.000 --> 00:44:10.000
again, because of this
Maxwell-Boltzmann tail.

00:44:10.000 --> 00:44:14.000
We have molecules here that
have very high speeds.

00:44:14.000 --> 00:44:18.000
We don't have a lot of
molecules with high speeds,

00:44:18.000 --> 00:44:22.000
but we have very high speeds.
When we take that speed and we

00:44:22.000 --> 00:44:27.000
square it, and then take the
average, they make a big

00:44:27.000 --> 00:44:30.000
contribution,
and they push this root mean

00:44:30.000 --> 00:44:35.000
square speed even higher than
the average speed.

00:44:35.000 --> 00:44:38.000
Again, if you wanted to
calculate what that is,

00:44:38.000 --> 00:44:43.000
you would take v squared,
multiply it by the distribution

00:44:43.000 --> 00:44:49.000
function, integrate it over all
values of E, and you would get

00:44:49.000 --> 00:44:53.000
what we got before,
the square root of 3RT over M.

00:44:53.000 --> 00:44:57.000
Bottom line here,

00:44:57.000 --> 00:45:01.000
for argon at 300 degrees
Kelvin, the most probable speed

00:45:01.000 --> 00:45:07.000
is 353 meters per second.
For argon at 300 degrees

00:45:07.000 --> 00:45:11.000
Kelvin, the average speed is
meters per second.

00:45:11.000 --> 00:45:17.000
And the root mean square speed
is 433 meters per second.

00:45:17.000 --> 00:45:22.000
Here, you can see how these
three quantities increase as you

00:45:22.000 --> 00:45:28.000
go from most probable to the
root mean square speed.

00:45:28.000 --> 00:45:31.000
And, in fact,
no surprise,

00:45:31.000 --> 00:45:38.000
if you actually go and measure
the argon speeds distribution

00:45:38.000 --> 00:45:44.000
function and then evaluate these
characteristics,

00:45:44.000 --> 00:45:51.000
you find that indeed those
measurements agree with what Mr.

00:45:51.000 --> 00:45:56.000
Maxwell and Mr.
Boltzmann predicted in 1850,

00:46:00.000 --> 00:46:02.000
Back to this distribution
function.

00:46:02.000 --> 00:46:06.000
We saw how to characterize it,
but I want to talk about a

00:46:06.000 --> 00:46:10.000
couple of other parameters that
it has in it.

00:46:10.000 --> 00:46:13.000
It has in it the mass,
and it has it in the

00:46:13.000 --> 00:46:16.000
temperature.
Let's talk about what this

00:46:16.000 --> 00:46:20.000
distribution function looks like
for different masses and

00:46:20.000 --> 00:46:24.000
different temperatures.
That is the general form,

00:46:24.000 --> 00:46:29.000
but different masses and
different temperatures.

00:46:29.000 --> 00:46:32.000
Let's start by keeping the
temperature constant,

00:46:32.000 --> 00:46:35.000
300 degrees Kelvin.
And now, we are going to look

00:46:35.000 --> 00:46:39.000
at the distribution functions
for three different masses,

00:46:39.000 --> 00:46:43.000
and I am going to plot that.
Here it is for xenon.

00:46:43.000 --> 00:46:46.000
What you can see is xenon,
very narrow distribution.

00:46:46.000 --> 00:46:49.000
Here it is for argon,
lighter mass,

00:46:49.000 --> 00:46:52.000
broader distribution.
Here it is for helium,

00:46:52.000 --> 00:46:55.000
really broad distribution.
Well, first of all,

00:46:55.000 --> 00:46:59.000
you can see that the average
speed of helium is much greater

00:46:59.000 --> 00:47:04.000
than it is for argon,
than it is for xenon.

00:47:04.000 --> 00:47:09.000
Because we already saw that it
was inversely proportional to

00:47:09.000 --> 00:47:15.000
the square root of the mass.
But the other thing to note is

00:47:15.000 --> 00:47:21.000
how broad the distribution is
for helium compared to xenon.

00:47:21.000 --> 00:47:26.000
And that is the general case.
The lighter masses have broader

00:47:26.000 --> 00:47:30.000
distributions.
Helium is so broad,

00:47:30.000 --> 00:47:36.000
meaning that there are helium
atoms here that are so fast that

00:47:36.000 --> 00:47:41.000
this is the reason why there is
relatively little helium and

00:47:41.000 --> 00:47:45.000
hydrogen in our atmosphere.
That is because,

00:47:45.000 --> 00:47:49.000
at our temperatures,
there are enough molecules with

00:47:49.000 --> 00:47:55.000
high enough velocities to escape
the earth's gravitational pull,

00:47:55.000 --> 00:48:00.000
because they are here at the
tail end.

00:48:00.000 --> 00:48:05.000
And so, the helium and the
hydrogen leave our atmosphere.

00:48:05.000 --> 00:48:09.000
Unlike Jupiter,
which is 300 times more massive

00:48:09.000 --> 00:48:14.000
than the earth,
where the gravitational pull is

00:48:14.000 --> 00:48:16.000
greater.
And, in that case,

00:48:16.000 --> 00:48:22.000
then those helium atoms don't
have enough velocity to escape

00:48:22.000 --> 00:48:26.000
the earth's gravitational pull.
So, that is mass.

00:48:26.000 --> 00:48:32.000
What about temperature?
Now we are going to keep the

00:48:32.000 --> 00:48:35.000
mass constant and we are going
to look at temperature.

00:48:35.000 --> 00:48:39.000
Here is the distribution
function at 100 degrees Kelvin,

00:48:39.000 --> 00:48:42.000
pretty narrow.
Here is the distribution

00:48:42.000 --> 00:48:45.000
function for argon at
degrees Kelvin,

00:48:45.000 --> 00:48:47.000
broader.
Here is the distribution

00:48:47.000 --> 00:48:50.000
function at 1000 degrees kelvin,
broader again.

00:48:50.000 --> 00:48:55.000
You see that as we increase the
temperature, the average speed

00:48:55.000 --> 00:48:58.000
increases.
As we increase the temperature,

00:48:58.000 --> 00:49:03.000
the width of that distribution
function increases.

00:49:03.000 --> 00:49:07.000
Now, one thing to note here is
this is a probability.

00:49:07.000 --> 00:49:12.000
The areas under these curves
have to all add up to one.

00:49:12.000 --> 00:49:17.000
If this distribution function
is going to move to higher

00:49:17.000 --> 00:49:21.000
velocities as we increase the
temperature, well,

00:49:21.000 --> 00:49:26.000
of course this maximum
probability is going to have to

00:49:26.000 --> 00:49:31.000
go down because we cannot lose
any molecules.

00:49:31.000 --> 00:49:35.000
All the area under this curve,
at 100 degrees Kelvin,

00:49:35.000 --> 00:49:40.000
has to equal the area under
this curve, at 1000 degrees

00:49:40.000 --> 00:49:45.000
Kelvin, because this is a
probability that we are plotting

00:49:45.000 --> 00:49:48.000
here.
That is our description of the

00:49:48.582 --> 00:49:51.000
Maxwell-Boltzmann distribution.
See you on Friday.