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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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1855. --> 00:06:07
What Maxwell did was recognize

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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,

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but, and this is important,
the velocities of the molecules

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are not the same.
That is going to be important.

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Now, this expression here,
I want to simplify a little

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bit.
In particular,

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I want to simplify the sum of
the squares of the velocity

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components in the x direction
for each one of the molecules.

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To do that simplification,
I am going to introduce this

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quantity.
This quantity is the average of

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the square of the velocity in
the x direction.

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What this means is I take the
velocity in the x direction,

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I square it,
and then I take the average.

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This is not the square of the
average velocity in the x

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direction.
That is different.

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00:15:26 --> 00:15:31
This is the average of the
square of the velocity in the x

212
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direction.
How am I going to evaluate that

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quantity?
Well, I am going to take the

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velocity in the x direction for
molecule one and square it and

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00:15:43 --> 00:15:47
add to that the square of the
velocity in the x direction for

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molecule two and add to that the
square of the velocity in the x

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direction for molecule three,
all the way up to molecule n.

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And then, if I want the
average, I am going to divide by

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00:16:03 --> 00:16:06
n, the number of molecules there
are.

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00:16:06 --> 00:16:11
I am just going to bring n,
here, over to the other side.

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I have n times the average of
the velocity in the x direction

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squared as this sum,
and this is exactly the sum

223
00:16:21 --> 00:16:25
that I had in my expression for
the total force.

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I can simplify that now.
There is that same expression.

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Here is my total force.
What I am going to do is I am

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going to substitute n times the
average of the velocity in the x

227
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direction squared in for this
whole sum, and I have now

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something that is much tidier.
That is the total force exerted

229
00:16:52 --> 00:16:57
by all the collisions of the
molecules in the container on

230
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that front wall.
But now I want the pressure.

231
00:17:03 --> 00:17:08
And the pressure is just force
per unit area.

232
00:17:08 --> 00:17:15
And so, I am going to take my
expression for the force and

233
00:17:15 --> 00:17:21
divide it by the unit area.
The area is the area of this

234
00:17:21 --> 00:17:25
wall, here, in my initial
example.

235
00:17:25 --> 00:17:31
I am going to call that area A.
And if this is a Q,

236
00:17:31 --> 00:17:36
and I am going to make it a Q
because it is easier to do it

237
00:17:36 --> 00:17:42
that way, the area times the
length then of this box is just

238
00:17:42 --> 00:17:44
the volume.
Here is the volume.

239
00:17:44 --> 00:17:49
There is my expression for the
pressure due to all of the

240
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molecule colliding with the
front wall of that box.

241
00:17:54 --> 00:17:58
However, this is an expression
for the pressure in one

242
00:17:58 --> 00:18:03
dimension only,
the x dimension.

243
00:18:03 --> 00:18:05
And we know,
in real life,

244
00:18:05 --> 00:18:10
we have three dimensions.
We have to take care of that.

245
00:18:10 --> 00:18:14
Let's do that now.
Let's extend this problem to

246
00:18:14 --> 00:18:18
three dimensions.
To do so, I am just going to

247
00:18:18 --> 00:18:23
realize right here that the
square of the velocity is the

248
00:18:23 --> 00:18:26
sum of the squares of the
components.

249
00:18:26 --> 00:18:31
That, you understand.
That is okay.

250
00:18:31 --> 00:18:37
What is not so obvious is this.
The average of the square of

251
00:18:37 --> 00:18:43
the velocity is the sum of the
average of the squares of each

252
00:18:43 --> 00:18:47
one of the components.
That is true.

253
00:18:47 --> 00:18:52
You can prove that.
We are not going to prove that.

254
00:18:52 --> 00:18:57
I won't hold you responsible
for proving that,

255
00:18:57 --> 00:19:01
but that is true.
That is correct,

256
00:19:01 --> 00:19:05
but now here comes a critical
assumption in Boltzmann's

257
00:19:05 --> 00:19:08
treatment.
The critical assumption is that

258
00:19:08 --> 00:19:12
the motion of the molecules in
this gas, here,

259
00:19:12 --> 00:19:16
is random in the sense that the
molecules don't have a preferred

260
00:19:16 --> 00:19:19
direction.
They are going in the x

261
00:19:19 --> 00:19:23
direction as often as they are
going in the y direction as

262
00:19:23 --> 00:19:28
often as they are going in the z
direction, so the motion is

263
00:19:28 --> 00:19:32
random.
If that motion is random,

264
00:19:32 --> 00:19:36
then the average of the
velocity squared in the x

265
00:19:36 --> 00:19:42
direction is going to be equal
to that in the y direction.

266
00:19:42 --> 00:19:47
It is going to be equal to that
in the z direction if that

267
00:19:47 --> 00:19:52
motion is random.
And that is great because it is

268
00:19:52 --> 00:19:56
going to make things a little
simpler for us up here.

269
00:19:56 --> 00:20:00
If that is right,
then the average of the

270
00:20:00 --> 00:20:06
velocity squared is three times
the average of the velocity

271
00:20:06 --> 00:20:11
squared in any one of the
dimensions.

272
00:20:11 --> 00:20:16
That is going to make it easy
to extrapolate this to three

273
00:20:16 --> 00:20:22
dimensions because now I am
going to be able to substitute,

274
00:20:22 --> 00:20:28
which had the average of the
velocity squared only in the x

275
00:20:28 --> 00:20:34
direction, I am going to be able
to substitute in an expression

276
00:20:34 --> 00:20:40
for the average velocity in
three dimensions.

277
00:20:40 --> 00:20:43
That is just going to be
one-third that.

278
00:20:43 --> 00:20:48
This is going to be one-third
the average of the velocity

279
00:20:48 --> 00:20:50
squared.
That is great.

280
00:20:50 --> 00:20:56
Now, I am going to do that
substitution way up into there.

281
00:20:56 --> 00:21:00
And when I do that,
look at this.

282
00:21:00 --> 00:21:04
I have a result.
This is the kinetic theory

283
00:21:04 --> 00:21:07
result.
We just did exactly what

284
00:21:07 --> 00:21:11
Maxwell did.
We have an expression for the

285
00:21:11 --> 00:21:16
pressure times the volume,
which is written here in terms

286
00:21:16 --> 00:21:21
of the average of the velocity
squared of the molecules.

287
00:21:21 --> 00:21:28
It is written in terms of the
motion of the molecules.

288
00:21:28 --> 00:21:31
For the first time,
there is an understanding,

289
00:21:31 --> 00:21:36
here, of what gives rise to
pressure, and that is the

290
00:21:36 --> 00:21:41
velocity or the motion of these
molecules hitting the wall.

291
00:21:41 --> 00:21:45
That is great.
That is the kinetic theory

292
00:21:45 --> 00:21:48
result.
But now, Boltzmann also knew

293
00:21:48 --> 00:21:53
from experiment that P times V
is equal to nRT.

294
00:21:53 --> 00:21:57
That is the experimental
result, which had been known

295
00:21:57 --> 00:22:02
already for over a hundred
years.

296
00:22:02 --> 00:22:07
That is the experiment.
If his theory is correct,

297
00:22:07 --> 00:22:14
if PV is equal to N m average
velocity squared over three,

298
00:22:14 --> 00:22:20
it better be equal to nRT.

299
00:22:20 --> 00:22:26
That will give us,
here, a prediction for what the

300
00:22:26 --> 00:22:33
velocity of the molecules ought
to be in terms of something

301
00:22:33 --> 00:22:40
experimentally controllable.
We can see if this kinetic

302
00:22:40 --> 00:22:45
theory model is correct.
We can solve this for the

303
00:22:45 --> 00:22:52
average of the velocity squared.
It is equal to 3n RT N over m.

304
00:22:52 --> 00:22:57
We can go in the laboratory,

305
00:22:57 --> 00:23:02
vary T and see if,
in fact, the average of the

306
00:23:02 --> 00:23:07
velocity squared of the
molecules is equal to this

307
00:23:07 --> 00:23:11
expression here.
That is great.

308
00:23:11 --> 00:23:16
We have a way to experimentally
check this theory.

309
00:23:16 --> 00:23:21
And you also see,
here, now, a relationship

310
00:23:21 --> 00:23:26
between the velocity of the
molecules and this macroscopic

311
00:23:26 --> 00:23:32
quantity, temperature.
Temperature is related to the

312
00:23:32 --> 00:23:37
motion of these molecules.
But, before we go on,

313
00:23:37 --> 00:23:41
this is kind of a messy
expression here.

314
00:23:41 --> 00:23:44
It has too many n's and m's in
it.

315
00:23:44 --> 00:23:48
Let me simplify that a little
bit for you.

316
00:23:48 --> 00:23:54
I am going to simplify this so
that this is 3RT over capital M,

317
00:23:54 --> 00:24:00
where the following is true.

318
00:24:00 --> 00:24:04
Over here, n is the number of
moles in the gas.

319
00:24:04 --> 00:24:08
That is little n.
Big N was the number of

320
00:24:08 --> 00:24:12
molecules in the gas.
Little m was the mass per

321
00:24:12 --> 00:24:15
molecule.
All of this is equivalent to

322
00:24:15 --> 00:24:20
one over big M,
where big M was kilograms per

323
00:24:20 --> 00:24:23
mole.
You can convince yourselves of

324
00:24:23 --> 00:24:27
this equality.
I am taking all these N's and

325
00:24:27 --> 00:24:31
m's and making one big M.

326
00:24:31 --> 00:24:36


327
00:24:36 --> 00:24:38
That is my expression,
here.

328
00:24:38 --> 00:24:42
I have the average of the
velocity squared equal to 3RT

329
00:24:42 --> 00:24:47
over M.
But this quantity is the

330
00:24:47 --> 00:24:52
average of the velocity squared.
It is more convenient for us to

331
00:24:52 --> 00:24:57
talk about a quantity
proportional to the velocity and

332
00:24:57 --> 00:25:02
not the velocity squared.
What I am going to do is take

333
00:25:02 --> 00:25:04
the square root of it.
That is simple.

334
00:25:04 --> 00:25:09
I now have the square root of
the average of the velocity

335
00:25:09 --> 00:25:12
squared.
That is the square root of 3RT

336
00:25:12 --> 00:25:14
over M.
I am going to call that the

337
00:25:14 --> 00:25:18
root mean square velocity.
I am going to put an rms here

338
00:25:18 --> 00:25:23
as a subscript for the velocity.

339
00:25:23 --> 00:25:25
It is the root mean square
velocity.

340
00:25:25 --> 00:25:29
I wanted to talk about a
quantity proportional to the

341
00:25:29 --> 00:25:34
velocity, instead of the
velocity squared.

342
00:25:34 --> 00:25:38
That is all I did there.
That is the root mean square

343
00:25:38 --> 00:25:41
velocity.
But the other big thing about

344
00:25:41 --> 00:25:45
it is you can see,
for the first time,

345
00:25:45 --> 00:25:47
now we have got,
and Maxwell had,

346
00:25:47 --> 00:25:51
an understanding of what
temperature was.

347
00:25:51 --> 00:25:56
Temperature is related to the
motions of the molecules.

348
00:25:56 --> 00:25:59
Temperature is related,
in this way,

349
00:25:59 --> 00:26:06
to the speed of the molecules.
Those are the two important

350
00:26:06 --> 00:26:11
results.
And this kinetic theory makes a

351
00:26:11 --> 00:26:19
prediction for what those
velocities ought to be.

352
00:26:19 --> 00:26:26


353
00:26:26 --> 00:26:30
In addition,
the temperature is a measure of

354
00:26:30 --> 00:26:34
the kinetic energy of the
molecules.

355
00:26:34 --> 00:26:37
How is that?
Well, it is for this reason.

356
00:26:37 --> 00:26:43
Here is the expression we
derived from the kinetic theory.

357
00:26:43 --> 00:26:47
And then here is an expression
that I just wrote down,

358
00:26:47 --> 00:26:53
that says the average kinetic
energy of a molecule is one-half

359
00:26:53 --> 00:26:58
M, where M is kilograms per
mole, times the average of the

360
00:26:58 --> 00:27:04
velocity squared.

361
00:27:04 --> 00:27:09
If I substitute the average of
the velocity squared into here,

362
00:27:09 --> 00:27:13
I get three-halves RT.

363
00:27:13 --> 00:27:16
This is telling us,
right here, that the

364
00:27:16 --> 00:27:22
temperature is also a measure of
the kinetic energy of these

365
00:27:22 --> 00:27:25
molecules.
We are getting a microscopic

366
00:27:25 --> 00:27:30
view, here, of what temperature
is.

367
00:27:30 --> 00:27:34
It is related to the motions of
these molecules.

368
00:27:34 --> 00:27:40
Now, before I go on talking
about this, let me make one big

369
00:27:40 --> 00:27:43
point.
That is, this expression here,

370
00:27:43 --> 00:27:48
for the average energy,
notice that it is one-half M

371
00:27:48 --> 00:27:53
times the average of the
velocity squared.

372
00:27:53 --> 00:27:59
It is not one-half M times the
square of the average velocity

373
00:27:59 --> 00:28:03
This is important.

374
00:28:03 --> 00:28:09
The average energy is not the
square of the average velocity.

375
00:28:09 --> 00:28:14
Rather, the average energy is
the average of the velocity

376
00:28:14 --> 00:28:17
squared.
There is a big distinction.

377
00:28:17 --> 00:28:23
This is because the average
energy is the second moment of

378
00:28:23 --> 00:28:26
the velocity distribution
function.

379
00:28:26 --> 00:28:31
Variables don't always
correspond in a one-to-one

380
00:28:31 --> 00:28:36
manner.
You don't have to understand

381
00:28:36 --> 00:28:43
that, if this is foreign to you,
but I do want you to know this

382
00:28:43 --> 00:28:48
is correct.
Now, let me pick up back here.

383
00:28:48 --> 00:28:54
What I want you to notice is
that the root mean square

384
00:28:54 --> 00:29:00
velocity has a mass dependence
in it.

385
00:29:00 --> 00:29:04
What does that mean?
Well, it means the following.

386
00:29:04 --> 00:29:09
For some constant temperature,
say we pick 300 degrees Kelvin,

387
00:29:09 --> 00:29:14
the velocity of the molecule is
going to depend on its mass.

388
00:29:14 --> 00:29:18
And it is inversely
proportional to the mass,

389
00:29:18 --> 00:29:23
so heavier molecules move more
slowly, lighter molecules move

390
00:29:23 --> 00:29:25
more quickly.
For example,

391
00:29:25 --> 00:29:30
helium at 300 degrees Kelvin,
it is cruising along at 3

392
00:29:30 --> 00:29:36
miles per hour at
room temperature.

393
00:29:36 --> 00:29:40
Xenon, on the other hand,
which is much more massive,

394
00:29:40 --> 00:29:44
is moving at a measly 534 miles
per hour.

395
00:29:44 --> 00:29:47
There is a mass dependence
here.

396
00:29:47 --> 00:29:52
However, there is no mass
dependence to the kinetic

397
00:29:52 --> 00:29:54
energy.
The kinetic energy,

398
00:29:54 --> 00:29:58
we saw, was three-halves RT.

399
00:29:58 --> 00:30:04
You don't see a mass dependence
in here, do you?

400
00:30:04 --> 00:30:09
The kinetic energy is only
dependent on the temperature.

401
00:30:09 --> 00:30:15
Whether or not you have helium
or xenon, the kinetic energy of

402
00:30:15 --> 00:30:21
those atoms is 3.74 kilojoules
per mole at 300 degrees Kelvin.

403
00:30:21 --> 00:30:26
It does not matter that helium
is moving six times as fast as

404
00:30:26 --> 00:30:30
xenon.
They both have the same kinetic

405
00:30:30 --> 00:30:34
energy.
There is no mass dependence in

406
00:30:34 --> 00:30:36
kinetic energy.
Now you say,

407
00:30:36 --> 00:30:39
oh, but look at this,
here is a mass,

408
00:30:39 --> 00:30:43
there is a mass dependence.
No, because you have to

409
00:30:43 --> 00:30:46
remember that you substitute in
here.

410
00:30:46 --> 00:30:49
If you square this,
there is an M here,

411
00:30:49 --> 00:30:53
and that cancels.
There is no mass dependence in

412
00:30:53 --> 00:30:57
the kinetic energy,
but the velocity is dependent

413
00:30:57 --> 00:31:01
on the mass.
That is important.

414
00:31:01 --> 00:31:07
Well, I told you that this was
the kinetic theory result.

415
00:31:07 --> 00:31:12
That is, that the root mean
square velocity of these

416
00:31:12 --> 00:31:16
molecules was represented by
this equation.

417
00:31:16 --> 00:31:21
And, from this equation,
if you calculate at 300 degrees

418
00:31:21 --> 00:31:25
Kelvin, these are,
in fact, the velocities of

419
00:31:25 --> 00:31:30
those atoms.
But how do we know this is

420
00:31:30 --> 00:31:33
right?
How do we go and measure the

421
00:31:33 --> 00:31:38
velocities or the speeds of
molecules or atoms?

422
00:31:38 --> 00:31:44
Well, this is the way we do it.
It is called the time-of-flight

423
00:31:44 --> 00:31:47
technique.
Are we all on board,

424
00:31:47 --> 00:31:48
here?
Questions?

425
00:31:48 --> 00:31:51
Okay.
How are we going to do this

426
00:31:51 --> 00:31:56
time-of-flight technique?
What we are going to do is we

427
00:31:56 --> 00:32:02
are going to have a little
pinhole here that we can open

428
00:32:02 --> 00:32:08
and shut really quickly.
We are going to let out a

429
00:32:08 --> 00:32:12
little pulse of gas.
To measure the velocity of the

430
00:32:12 --> 00:32:16
molecules, we are literally
going to measure the time it

431
00:32:16 --> 00:32:21
takes the molecules to fly from
where we let them out to some

432
00:32:21 --> 00:32:24
detector.
And since we know the distance,

433
00:32:24 --> 00:32:28
we are going to be able to
calculate the velocity from

434
00:32:28 --> 00:32:32
that.
The idea is at time t equals 0,

435
00:32:32 --> 00:32:38
we let out a little pulse of
gas, and then we start a clock

436
00:32:38 --> 00:32:41
running.
Then we just measure how long

437
00:32:41 --> 00:32:46
it takes the molecules to fly
from this origin here to this

438
00:32:46 --> 00:32:49
detector.
And since we built the

439
00:32:49 --> 00:32:54
apparatus, and we know what L
is, we can calculate the

440
00:32:54 --> 00:32:56
velocity.
Time-of-flight,

441
00:32:56 --> 00:33:02
that is what is done.
However, when we let this

442
00:33:02 --> 00:33:07
little pulse of gas out,
and now we let the molecules

443
00:33:07 --> 00:33:12
fly to that detector over here,
what happens as a function of

444
00:33:12 --> 00:33:15
time?
What will happen is that pulse

445
00:33:15 --> 00:33:21
of gas will spread out because
not all of the molecules or

446
00:33:21 --> 00:33:25
atoms in that pulse of gas have
the same velocity.

447
00:33:25 --> 00:33:32
Some of those atoms are moving
faster than the other atoms.

448
00:33:32 --> 00:33:36
And so, what is going to happen
is that the molecules or atoms

449
00:33:36 --> 00:33:41
that are moving faster are going
to hit the detector first.

450
00:33:41 --> 00:33:46
The molecules or atoms that are
moving more slowly are going to

451
00:33:46 --> 00:33:48
hit the detector at a later
time.

452
00:33:48 --> 00:33:53
And that is what we also want
to know, this distribution of

453
00:33:53 --> 00:33:55
velocities.
But in the measurement,

454
00:33:55 --> 00:34:01
what we are going to measure is
a distribution of times.

455
00:34:01 --> 00:34:04
Out of our detector,
we are going to have a plot

456
00:34:04 --> 00:34:08
that looks like this.
This is going to be f of t,

457
00:34:08 --> 00:34:12
essentially the number of
molecules hitting the detector

458
00:34:12 --> 00:34:15
at a certain time t versus the
time.

459
00:34:15 --> 00:34:20
When we first let our pulse of
gas out, that is time t equals

460
00:34:20 --> 00.


461
0. --> 00:34:22
Then, for a while,

462
00:34:22 --> 00:34:26
there are no molecules hitting
the detector because it takes a

463
00:34:26 --> 00:34:32
while for them to get over here
to this detector.

464
00:34:32 --> 00:34:35
But then, all of a sudden,
they start reaching the

465
00:34:35 --> 00:34:38
detector.
And this is essentially just a

466
00:34:38 --> 00:34:43
number of molecules that hit the
detector as a function of time.

467
00:34:43 --> 00:34:47
That number of molecules
increases and becomes a maximum

468
00:34:47 --> 00:34:50
here at some time,
and then it exponentially

469
00:34:50 --> 00:34:53
decays, here.
So, this is what we measure.

470
00:34:53 --> 00:34:58
This is a distribution here of
flight times of the molecules in

471
00:34:58 --> 00:35:02
this pulse of gas.
Well, that is nice,

472
00:35:02 --> 00:35:05
but this is a distribution of
flight times.

473
00:35:05 --> 00:35:10
It is not a distribution of
velocities, and we wanted a

474
00:35:10 --> 00:35:14
distribution of velocities.
We want to know the velocity,

475
00:35:14 --> 00:35:19
here, of these molecules.
You know how to convert time,

476
00:35:19 --> 00:35:22
given the path length,
to velocity,

477
00:35:22 --> 00:35:25
but it is not so
straightforward because we have

478
00:35:25 --> 00:35:31
a distribution function.
We have a distribution in time,

479
00:35:31 --> 00:35:36
and we want to convert that to
a distribution in velocity.

480
00:35:36 --> 00:35:41
We have to change the variable
here in a distribution function.

481
00:35:41 --> 00:35:45
How do we do that?
Well, we want this f of t to be

482
00:35:45 --> 00:35:50
an f of v.
We recognize here that this

483
00:35:50 --> 00:35:54
distribution in time,
the probability of finding a

484
00:35:54 --> 00:35:59
molecule between t and t plus
dt, has got to be equivalent to

485
00:35:59 --> 00:36:04
the probability of finding a
molecule with a velocity between

486
00:36:04 --> 00:36:09
v and v plus dv.
But to get from one

487
00:36:09 --> 00:36:13
distribution function to
another, for example,

488
00:36:13 --> 00:36:20
if we want f of v,
what we have to know is how one

489
00:36:20 --> 00:36:25
variable changes with respect to
another.

490
00:36:25 --> 00:36:30


491
00:36:30 --> 00:36:33
We have this distribution
function f of t,

492
00:36:33 --> 00:36:37
but we need to know how t
changes with v.

493
00:36:37 --> 00:36:40
We need dt by dv,
we need that,

494
00:36:40 --> 00:36:44
so let's get it.
I will tell you why in a

495
00:36:44 --> 00:36:47
moment.
We know how v changes with t.

496
00:36:47 --> 00:36:52
We are going to take the
derivative of v with respect to

497
00:36:52 --> 00:36:58
t and turn things around.
So, dt / dv is proportional to

498
00:36:58 --> 00:37:04
minus t squared over L.

499
00:37:04 --> 00:37:06
This is telling us,
essentially,

500
00:37:06 --> 00:37:09
how the variable t changes with
v.

501
00:37:09 --> 00:37:13
To every point in our
time-of-flight distribution,

502
00:37:13 --> 00:37:17
we are going to multiply this
by this, what is called the

503
00:37:17 --> 00:37:20
Jacobean.
We need this because the time

504
00:37:20 --> 00:37:24
and the velocity do not
correlate in a one-to-one

505
00:37:24 --> 00:37:27
manner.
That often happens with two

506
00:37:27 --> 00:37:33
distribution functions.
If you do not understand what I

507
00:37:33 --> 00:37:38
just said, it is okay.
This was just some extra.

508
00:37:38 --> 00:37:42
I do not hold you responsible
for it.

509
00:37:42 --> 00:37:48
I just changed my variable in
the distribution function.

510
00:37:48 --> 00:37:50
Yes?
I think in the notes,

511
00:37:50 --> 00:37:54
I might have had as a
proportionality.

512
00:37:54 --> 00:38:00
Up here I actually have the
equal sign.

513
00:38:00 --> 00:38:04
That actually won't matter in
this transformation.

514
00:38:04 --> 00:38:06
Pardon?
I understand that.

515
00:38:06 --> 00:38:10
That is fine.
I don't have the equal sign

516
00:38:10 --> 00:38:13
there.
That is why I left it out

517
00:38:13 --> 00:38:17
there, I think.
Anyway, this is the velocity

518
00:38:17 --> 00:38:20
distribution.
You don't have to understand

519
00:38:20 --> 00:38:24
how I got there.
This is what the velocity

520
00:38:24 --> 00:38:29
distribution looks like.
It is what is called the

521
00:38:29 --> 00:38:32
Maxwell-Boltzmann velocity
distribution.

522
00:38:32 --> 00:38:36
And the bottom line is that
Maxwell and Boltzmann predicted

523
00:38:36 --> 00:38:40
this, about 1855.
They actually predicted this

524
00:38:40 --> 00:38:43
distribution function.
We did not predict it.

525
00:38:43 --> 00:38:46
We did not go through that part
of kinetic theory,

526
00:38:46 --> 00:38:50
but they predicted it.
However, it was only until 

527
00:38:50 --> 00:38:54
that the technology existed,
fast enough timing and

528
00:38:54 --> 00:38:57
electronics existed,
to actually measure this

529
00:38:57 --> 00:39:02
experimentally.
This took a hundred years or so

530
00:39:02 --> 00:39:07
in order for this distribution
function to actually be

531
00:39:07 --> 00:39:10
measured, but here it is,
f of v.

532
00:39:10 --> 00:39:13
First of all,
there is all of this stuff,

533
00:39:13 --> 00:39:16
which is proportionality
constants.

534
00:39:16 --> 00:39:19
We will talk about that in a
moment.

535
00:39:19 --> 00:39:23
But the variable,
here, is v squared

536
00:39:23 --> 00:39:28
times an exponentially decaying
function with a v squared in

537
00:39:28 --> 00:39:31
there.
What does that mean?

538
00:39:31 --> 00:39:35
Well, if you look at the form
of f of v, this v squared is

539
00:39:35 --> 00:39:38
what gives rise to this increase
in f of v.

540
00:39:38 --> 00:39:42
Right here, at low velocities,
that is a quadratic,

541
00:39:42 --> 00:39:45
v squared.
But you are multiplying it by

542
00:39:45 --> 00:39:49
an exponentially decaying
function with this v squared in

543
00:39:49 --> 00:39:52
the argument.
And so that is what gives you

544
00:39:52 --> 00:39:54
this tail.
If you are multiplying a

545
00:39:54 --> 00:39:58
function that is going up and
one decreasing,

546
00:39:58 --> 00:40:03
you are going to get a maximum
at some value of v.

547
00:40:03 --> 00:40:09
That is where the shape of the
Maxwell-Boltzmann distribution

548
00:40:09 --> 00:40:14
function comes from.
And what this is telling you is

549
00:40:14 --> 00:40:21
the probability here of finding
a molecule in a gas with a speed

550
00:40:21 --> 00:40:26
between v and v plus dv.
That is what that is telling

551
00:40:26 --> 00:40:30
you.
We often characterize these

552
00:40:30 --> 00:40:35
distribution functions by some
quantities, and one of those

553
00:40:35 --> 00:40:39
quantities is what we call the
most probable speed.

554
00:40:39 --> 00:40:44
Here is the distribution
function, and I have the most

555
00:40:44 --> 00:40:48
probable speed labeled.
The most probable speed,

556
00:40:48 --> 00:40:53
(v)mp, is simply the value of v
at which the probability is the

557
00:40:53 --> 00:40:56
largest.
That was like our most probable

558
00:40:56 --> 00:41:02
value of r in the radial
distribution functions.

559
00:41:02 --> 00:41:04
That is what the most probable
speed is.

560
00:41:04 --> 00:41:07
If you wanted to get that
mathematically,

561
00:41:07 --> 00:41:11
what you would do is take this
distribution function,

562
00:41:11 --> 00:41:14
take the derivative,
set it equal to zero,

563
00:41:14 --> 00:41:18
and then solve for v.
That makes that derivative

564
00:41:18 --> 00:41:21
equal to zero.
The derivative is zero at

565
00:41:21 --> 00:41:24
maxima or minima.
And then, you would find that

566
00:41:24 --> 00:41:29
the value of the most probable
speed is the square root of 2RT

567
00:41:29 --> 00:41:33
over M.

568
00:41:33 --> 00:41:36
You are not responsible for
taking this derivative and

569
00:41:36 --> 00:41:41
setting it equal to zero.
You are responsible for knowing

570
00:41:41 --> 00:41:44
physically what the most
probable speed is.

571
00:41:44 --> 00:41:48
The fact that it is this value
here, where the probability is

572
00:41:48 --> 00:41:51
the largest.
It is the most probable value

573
00:41:51 --> 00:41:54
of v.
And you are responsible for

574
00:41:54 --> 00:41:57
recognizing this.
I don't ask you to memorize it

575
00:41:57 --> 00:42:01
or to write it down.
But, if you see it,

576
00:42:01 --> 00:42:07
you should know what it is.
That is one quantity that we

577
00:42:07 --> 00:42:11
use to characterize this
distribution function.

578
00:42:11 --> 00:42:16
Another quantity that we use is
the average speed,

579
00:42:16 --> 00:42:21
v bar, average speed.
And the first thing that you

580
00:42:21 --> 00:42:26
see is that the average speed is
a little bit higher than the

581
00:42:26 --> 00:42:31
most probable speed.
It is a little bit larger.

582
00:42:31 --> 00:42:35
Why is that?
Well, it is a little bit larger

583
00:42:35 --> 00:42:38
because on these
Maxwell-Boltzmann distribution

584
00:42:38 --> 00:42:43
functions, there are molecules
way out here that have very high

585
00:42:43 --> 00:42:45
speeds.
There are not a lot of

586
00:42:45 --> 00:42:50
molecules that have very high
speeds, but there are molecules

587
00:42:50 --> 00:42:54
with very high speeds.
And so, when you average over

588
00:42:54 --> 00:42:58
this distribution function,
because there is such a long

589
00:42:58 --> 00:43:01
Boltzmann tail here,
is what it is called,

590
00:43:01 --> 00:43:05
the average velocity is going
to be a little higher than the

591
00:43:05 --> 00:43:10
most probable velocity.
That is physically why the

592
00:43:10 --> 00:43:15
average velocity is a little bit
larger than the most probable

593
00:43:15 --> 00:43:17
velocity.
That is important.

594
00:43:17 --> 00:43:20
If I wanted to calculate what
the average velocity is,

595
00:43:20 --> 00:43:24
I would take the distribution
function, multiply it by v,

596
00:43:24 --> 00:43:28
and then integrate over the
range of v, which is zero to

597
00:43:28 --> 00:43:32
infinity.
You don't have to do that,

598
00:43:32 --> 00:43:36
but the quantity that you would
get, if you had done that,

599
00:43:36 --> 00:43:41
is the square root of 8RT over
pi M.

600
00:43:41 --> 00:43:43
It is larger than the most

601
00:43:43 --> 00:43:46
probable value.
And then, finally,

602
00:43:46 --> 00:43:50
the other way we characterize
the distribution function is by

603
00:43:50 --> 00:43:55
this root mean square speed that
we have already talked about a

604
00:43:55 --> 00:43:57
lot.
That root mean square speed,

605
00:43:57 --> 00:44:02
here, is even a larger value
than the average velocity or the

606
00:44:02 --> 00:44:07
average speed.
And the reason for that is,

607
00:44:07 --> 00:44:10
again, because of this
Maxwell-Boltzmann tail.

608
00:44:10 --> 00:44:14
We have molecules here that
have very high speeds.

609
00:44:14 --> 00:44:18
We don't have a lot of
molecules with high speeds,

610
00:44:18 --> 00:44:22
but we have very high speeds.
When we take that speed and we

611
00:44:22 --> 00:44:27
square it, and then take the
average, they make a big

612
00:44:27 --> 00:44:30
contribution,
and they push this root mean

613
00:44:30 --> 00:44:35
square speed even higher than
the average speed.

614
00:44:35 --> 00:44:38
Again, if you wanted to
calculate what that is,

615
00:44:38 --> 00:44:43
you would take v squared,
multiply it by the distribution

616
00:44:43 --> 00:44:49
function, integrate it over all
values of E, and you would get

617
00:44:49 --> 00:44:53
what we got before,
the square root of 3RT over M.

618
00:44:53 --> 00:44:57
Bottom line here,

619
00:44:57 --> 00:45:01
for argon at 300 degrees
Kelvin, the most probable speed

620
00:45:01 --> 00:45:07
is 353 meters per second.
For argon at 300 degrees

621
00:45:07 --> 00:45:11
Kelvin, the average speed is
meters per second.

622
00:45:11 --> 00:45:17
And the root mean square speed
is 433 meters per second.

623
00:45:17 --> 00:45:22
Here, you can see how these
three quantities increase as you

624
00:45:22 --> 00:45:28
go from most probable to the
root mean square speed.

625
00:45:28 --> 00:45:31
And, in fact,
no surprise,

626
00:45:31 --> 00:45:38
if you actually go and measure
the argon speeds distribution

627
00:45:38 --> 00:45:44
function and then evaluate these
characteristics,

628
00:45:44 --> 00:45:51
you find that indeed those
measurements agree with what Mr.

629
00:45:51 --> 00:45:56
Maxwell and Mr.
Boltzmann predicted in 1850,

630
00:45:56.567 --> 1855.


631
1855. --> 00:46:00
No surprise.

632
00:46:00 --> 00:46:02
Back to this distribution
function.

633
00:46:02 --> 00:46:06
We saw how to characterize it,
but I want to talk about a

634
00:46:06 --> 00:46:10
couple of other parameters that
it has in it.

635
00:46:10 --> 00:46:13
It has in it the mass,
and it has it in the

636
00:46:13 --> 00:46:16
temperature.
Let's talk about what this

637
00:46:16 --> 00:46:20
distribution function looks like
for different masses and

638
00:46:20 --> 00:46:24
different temperatures.
That is the general form,

639
00:46:24 --> 00:46:29
but different masses and
different temperatures.

640
00:46:29 --> 00:46:32
Let's start by keeping the
temperature constant,

641
00:46:32 --> 00:46:35
300 degrees Kelvin.
And now, we are going to look

642
00:46:35 --> 00:46:39
at the distribution functions
for three different masses,

643
00:46:39 --> 00:46:43
and I am going to plot that.
Here it is for xenon.

644
00:46:43 --> 00:46:46
What you can see is xenon,
very narrow distribution.

645
00:46:46 --> 00:46:49
Here it is for argon,
lighter mass,

646
00:46:49 --> 00:46:52
broader distribution.
Here it is for helium,

647
00:46:52 --> 00:46:55
really broad distribution.
Well, first of all,

648
00:46:55 --> 00:46:59
you can see that the average
speed of helium is much greater

649
00:46:59 --> 00:47:04
than it is for argon,
than it is for xenon.

650
00:47:04 --> 00:47:09
Because we already saw that it
was inversely proportional to

651
00:47:09 --> 00:47:15
the square root of the mass.
But the other thing to note is

652
00:47:15 --> 00:47:21
how broad the distribution is
for helium compared to xenon.

653
00:47:21 --> 00:47:26
And that is the general case.
The lighter masses have broader

654
00:47:26 --> 00:47:30
distributions.
Helium is so broad,

655
00:47:30 --> 00:47:36
meaning that there are helium
atoms here that are so fast that

656
00:47:36 --> 00:47:41
this is the reason why there is
relatively little helium and

657
00:47:41 --> 00:47:45
hydrogen in our atmosphere.
That is because,

658
00:47:45 --> 00:47:49
at our temperatures,
there are enough molecules with

659
00:47:49 --> 00:47:55
high enough velocities to escape
the earth's gravitational pull,

660
00:47:55 --> 00:48:00
because they are here at the
tail end.

661
00:48:00 --> 00:48:05
And so, the helium and the
hydrogen leave our atmosphere.

662
00:48:05 --> 00:48:09
Unlike Jupiter,
which is 300 times more massive

663
00:48:09 --> 00:48:14
than the earth,
where the gravitational pull is

664
00:48:14 --> 00:48:16
greater.
And, in that case,

665
00:48:16 --> 00:48:22
then those helium atoms don't
have enough velocity to escape

666
00:48:22 --> 00:48:26
the earth's gravitational pull.
So, that is mass.

667
00:48:26 --> 00:48:32
What about temperature?
Now we are going to keep the

668
00:48:32 --> 00:48:35
mass constant and we are going
to look at temperature.

669
00:48:35 --> 00:48:39
Here is the distribution
function at 100 degrees Kelvin,

670
00:48:39 --> 00:48:42
pretty narrow.
Here is the distribution

671
00:48:42 --> 00:48:45
function for argon at
degrees Kelvin,

672
00:48:45 --> 00:48:47
broader.
Here is the distribution

673
00:48:47 --> 00:48:50
function at 1000 degrees kelvin,
broader again.

674
00:48:50 --> 00:48:55
You see that as we increase the
temperature, the average speed

675
00:48:55 --> 00:48:58
increases.
As we increase the temperature,

676
00:48:58 --> 00:49:03
the width of that distribution
function increases.

677
00:49:03 --> 00:49:07
Now, one thing to note here is
this is a probability.

678
00:49:07 --> 00:49:12
The areas under these curves
have to all add up to one.

679
00:49:12 --> 00:49:17
If this distribution function
is going to move to higher

680
00:49:17 --> 00:49:21
velocities as we increase the
temperature, well,

681
00:49:21 --> 00:49:26
of course this maximum
probability is going to have to

682
00:49:26 --> 00:49:31
go down because we cannot lose
any molecules.

683
00:49:31 --> 00:49:35
All the area under this curve,
at 100 degrees Kelvin,

684
00:49:35 --> 00:49:40
has to equal the area under
this curve, at 1000 degrees

685
00:49:40 --> 00:49:45
Kelvin, because this is a
probability that we are plotting

686
00:49:45 --> 00:49:48
here.
That is our description of the

687
00:49:48.582 --> 49:51
Maxwell-Boltzmann distribution.
See you on Friday.