PROFESSOR: This is section 2.3.
This is called The Ideal Gas Law.
We just saw how the collapsing can shrank really
fast when we put aluminum in ice water.
Well, the reason it happened so fast was heat transfer.
That was Section 2.2.
This is Section 2.3.
It's why it happened at all.
In this section, we're going to use the ideal gas
law to figure out how changing the temperature changes pressure and volume
and vice versa.
We're going to talk about how changing the pressure and volume
affects the temperature of stuff.
We also need to do this thing called moles.
It's a way to find out how many atoms of a gas do I have in a specific volume.
All right, we've got a lot to do in Section 2.3, Ideal Gas Laws.
Let's get started.
OK, this is section 2.3, The Ideal Gas Law.
First thing we need to start with is this thing
known as the kinetic molecular theory.
I typed it out for you so you wouldn't have
to listen quite so closely to everything I say.
All right, the kinetic molecular theory--
it's this idea that gases can be understood in terms of molecules having
elastic collisions with each other as well as the walls of the container
that they're in.
If you can, imagine for a second that you
can see a container with a whole bunch of molecules of gas in it.
OK, if it was at absolute zero, these molecules
wouldn't have any kinetic energy.
As we started to warm this up and give it a little heat,
they would begin to move around a little bit randomly.
They would bounce off the sides of the container.
They would smash into each other and then bounce off
the sides of the container.
If we warmed it up more, they'd go faster.
They would have more collisions with the side of their container.
Those collisions with the side of the container is what creates pressure.
OK, let's see.
Temperature, we said before, was a way to measure
that amount of kinetic energy.
Energy is the ability to do work.
And when it comes to gas laws, work done by a gas
is some change in pressure or volume.
All right, so we see from this kinetic molecular theory of gases
that pressure, volume, temperature, changes in energy,
and work are all related to each other.
That's our goal in this section.
We want to find out how does changing the pressure affect the volume?
How does changing the volume affect the pressure?
How does changing the temperature affect those things?
We are going to, by experiment, try to come up
with our own relationship for the ideal gas law.
OK, here we go.
Let's get started.
OK, first we're going to find out how lowering the pressure while keeping
the volume constant affects the temperature.
Our goal for this section is to figure out
how pressure and temperature and volume are all related.
It's called the ideal gas law.
I could just tell you what it is, but it's
a little more interesting to try and figure it out for ourselves.
What we have here is a commercial food sealer.
This pump removes air from this cylinder.
So the volume stays the same, but the pressure
is going to drop as fast as I can make the pressure drop.
We should then be able to figure out what happens to the temperature.
It appears as though right now, under all these lights,
that the temperature is somewhere between 80 and 82 degrees.
It's pretty warm.
All right, let's see what happens if I drop the pressure as fast as we can.
OK, let's see what that did.
Well, now the green part is 78 and the brown part is 80.
It dropped a couple degrees.
So what we saw as we removed air from the can,
the pressure dropped and the temperature went down with it.
Now we know when it comes to come up with our own formula for pressure
and temperature, they are directly related to each other.
Pressure goes down, temperature goes down.
All right, let's see what happens if we let
the pressure come back up to ambient pressure real fast.
Jumps up to 82 degrees there for a second.
All right, now we know how temperature and pressure are related.
All right, well now we know how pressure and temperature
are related to each other.
That food sealer showed us how lowering the pressure
also lowered the temperature.
When it comes time to make up our own equation,
we would say that this pressure and temperature
are directly proportional to each other.
And we saw that if you lower the pressure of something,
there was a corresponding decrease in the temperature.
And when we opened it up and let the pressure increase really fast,
the temperature went up also.
All right, so these are directly proportional to each other.
I'm going put one on this side of the equation
and one on this side of the equation.
Next, we need to figure out where volume fits in.
We're going to see what happens to the volume
if we lower the pressure but keeping the temperature relatively constant.
What this showed us was as we lowered the pressure,
the temperature dropped accordingly.
We have a relationship between pressure and temperature now.
We know that as the pressure decreases, the temperature drops also.
That's just one of the relationships we need
to figure out for our ideal gas law.
What this apparatus will show us is the relationship
between pressure and volume.
It's a great big heavy duty piece of glass on a thick rubber base
and a very powerful pump that will slowly
remove air and drop the pressure in here.
We'll be able to see what happens to the volume of the balloon
as the pressure gets lower.
Just to help us compare, I have another balloon for comparison.
All right, let's lower the pressure in this glass
and see how that affects the volume of the balloon.
It doesn't take very long at all to see what's going on here.
As the pressure lowered, the volume of the balloon got a whole lot bigger.
Started out it was this big, and it's a whole lot bigger now.
The volume increased as the pressure got lower.
Let's see what happens if we lower the pressure even more.
And that's as far as it can go.
The balloon got so big it covered up the hole where
the air is pulled out from this jar.
So we can definitely tell as the pressure got lower,
the volume got bigger.
Let's see what happens if we bring the pressure
back up to the atmospheric pressure as fast as we can.
As the pressure increased, the volume of the balloon got a whole lot smaller.
Now we have one more of the relationships
we need in order to figure out the ideal gas law.
We know that pressure and volume are inversely proportional to each other.
The volume got bigger because the pressure got smaller.
All right, let's go see what that looks like on paper.
OK, we're back.
That was pretty cool.
The whole point of that was to lower the pressure in this bell jar
and see what happens to the balloon.
Well, as we pumped air out, it lowered the pressure
on the inside and the volume of the balloon got bigger.
That's my horrible drawing of a bell jar.
But we know now where volume fits into the equation.
They are inversely proportional to each other.
When the pressure went down, the volume went up.
So we're going to put volume on the bottom of our equation right here.
They're inversely proportional.
We could stop right now.
But we haven't checked to see how temperature and volume are related.
We know how pressure and temperature and pressure and volume are related.
Let's see if we can figure out how temperature and volume tie in
to each other.
All we've got to do is remember the collapsing can.
There was a demonstration we did at the end of Section 2.2
where we took a really hot aluminum can and dipped it into some ice water.
What we just saw in that last section was
that different elements and different materials
have the ability to gain and lose heat at different rates.
Styrofoam is one of the best thermal insulators.
It holds onto its thermal energy for a really long time.
Aluminum is one of the best thermal conductors.
It can send heat really, really fast.
So what we have here is a beaker full of ice water and a few aluminum
cans with some boiling water inside.
I'm going to take this aluminum can, stick it in the ice water,
and it is going to get rid of all its thermal energy in under a second.
Let's see what happens.
All right, notice how fast that happened.
It takes less than a second.
And this aluminum can is already at room temperature.
So the reason it happened so fast was because it's aluminum.
Notice what happened to its volume.
We're going to learn about that in the next section,
about how temperature changes affect volume changes.
You want to see it again?
There we go, the collapsing can.
The collapsing can showed us how a rapid drop in temperature
caused an equally rapid drop in pressure that
rapidly decreased the volume of the can.
Now, we learned in that section why it happens so rapidly.
It's because aluminum has a very high thermal coefficient.
It has the ability to move heat through it very quickly.
So now we know for sure this is the relationship for our ideal gas law.
All right, let's see what we can do with these things now.
It looks like this.
Here's our gas law.
I'm going to have to rewrite it in its more common form.
I don't know why it's always written this way.
PV is proportional to temperature.
We're still missing two things.
One is the amount of gas that's present.
Now they use a little n for number of moles.
Moles is a chemistry thing.
It's just a way of measuring a quantity of something.
One is the amount of gas that's present, and that's measured in moles.
The second is a constant to make the units match up right.
It's called is R. R is our ideal gas constant.
So we're going to squeeze a little n and an R into our equation.
Now it's official.
That is the ideal gas law.
All right, R can be measured in different units, but for our purposes,
we are going to use 8.314, because that's
what it is if you measure things in joules and moles and Kelvin.
Remember, don't say degrees Kelvin.
It's just Kelvin.
The entire ideal gas law looks like this.
Got it.
Or there is another way to write it.
Rather than using number of moles, little n,
with a big R for gas constant, we could use the number of particles,
capital N, and Boltzmann's constant, kb.
They are equal to each other.
If you take the number of moles times the gas constant,
you get the same thing as taking the number particles
that there are times Boltzmann's constant.
Thus there are two different ways of writing the ideal gas law.
The first one with the little n, number of moles, and the second one,
big N, number of molecules.
OK, we could plug in a whole bunch numbers into our calculator now.
But more fun than that, let's see how these things are related to each other.
A lot of times on the AP test, they don't just give you numbers.
They ask you like, well, what if the volume stays constant
and the pressure increases?
How does that affect temperature?
Let's find out.
This volume is going to stay constant.
And we're going to change stuff over here.
Here's what it was before.
Here's what it is now.
We said the volume is constant.
If the temperature should increase, what's going to happen to the pressure?
One way to do it, you can look back and try and imagine
those little moving molecules bouncing all around.
And if you increase the temperature, they move faster and they collide more.
That should make the pressure go up.
Or you could just look at it mathematically and think, well,
whatever I do to the equation on this side has
to have an effect on the other side of the equation.
If this side of the equation gets bigger and the volume remains constant,
the pressure has to go up.
All right, let's try another one.
What would happen to the volume if the pressure doubles?
And, I don't know, let's just say the temperature triples.
If the pressure doubles and the temperature triples,
what happens to the volume?
Well, this is what everything was like the first time.
PV equals NRT.
And over here, we said the pressure doubles.
So I'm going to put a 2 right there.
And the temperature triples.
I'm going to put a 3 right there.
Well, if this side of the equation doubles and this side of the equation
triples, our volume is going to change by a factor of 3 over 2.
So it will be 1 and 1/2 times larger than the volume was originally.
All right, let's try another one down here.
What would happen to the temperature if the pressure quadruples
and the volume is suddenly 1/9 as much?
Well, the first time, we had our pressure, our volume,
and our temperature over here.
And we're wondering what happens to the temperature
if the pressure goes up by a factor of 4 and the volume is suddenly 1/9 as much?
Whatever happens on this side of the equation
has to happen on this side of the equation also.
So the temperature would now be 4/9 as much as it was before.
All right, there you go.
That's the ideal gas law.