And in that balloon I have a bunch of

particles bouncing around.

They're gas particles, so they're floating freely.

And they each have some velocity, some kinetic energy.

And what I care about, let me just draw a few more, what I

care about is the pressure that is exerted on the surface

of the balloon.

So I care about the pressure.

And what's pressure?

It's force per area.

So the area here, you can think of it as the inside

surface of the balloon.

And what's going to apply force to that?

Well any given moment-- I only drew six particles here, but

in a real balloon you would have gazillions of particles,

and we could talk about how large, but more particles than

you can really probably imagine-- but at any given

moment, some of those particles are bouncing off the

wall of the container.

That particle is bouncing there, this particle is

bouncing there, this guy's bouncing like that.

And when they bounce, they apply force to the container.

An outward force, that's what keeps the balloon blown up.

So think about what the pressure is going to be

dependent on.

So first of all, the faster these particles move, the

higher the pressure.

Slower particles, you're going to be bouncing into the

container less, and when you do bounce into the container,

it's going to be less of a ricochet, or less of a change

in momentum.

So slower particles, you're going to

have pressure go down.

Now, it's practically impossible to measure the

kinetic energy, or the velocity, or the direction of

each individual particle.

Especially when you have gazillions

of them in a balloon.

So we do is we think of the average

energy of the particles.

And the average energy of the particles, you might say oh,

Sal is about to introduce us to a new concept.

It's a new way of looking at probably a very familiar

concept to you.

And that's temperature.

Temperature can and should be viewed as the average energy

of the particles in the system.

So I'll put a little squiggly line, because there's a lot of

ways to think about it.

Average energy.

And mostly kinetic energy, because these particles are

moving and bouncing.

The higher the temperature, the faster that these

particles move.

And the more that they're going to bounce into the side

of the container.

But temperature is average energy.

It tells us energy per particle.

So obviously, if we only had one particle in there with

super high temperature, that's going to have less pressure

than if we have a million particles in there.

Let me draw that.

If I have, let's take two cases right here.

One is, I have a bunch of particles with a certain

temperature, moving in their different directions.

And the other example, I have one particle.

And maybe they have the same temperature.

That on average, they have the same kinetic energy.

The kinetic energy per particle is the same.

Clearly, this one is going to be applying more pressure to

its container, because at any given moment more of these

particles are going to be bouncing off the side than in

this example.

This guy's going to bounce, bam, then going to go and

move, bounce, bam.

So he's going to be applying less pressure, even though his

temperature might be the same.

Because temperature is kinetic energy, or you can view it as

kinetic energy per particles.

Or it's a way of looking at kinetic energy per particle.

So if we wanted to look at the total energy in the system, we

would want to multiply the temperature times the number

of particles.

And just since we're dealing on the molecular scale, the

number of particles can often be represented as moles.

Remember, moles is just a number of particles.

So we're saying that that pressure-- well, I'll say it's

proportional, so it's equal to some constant,

let's call that R.

Because we've got to make all the units work out in the end.

I mean temperature is in Kelvin but we eventually want

to get back to joules.

So let's just say it's equal to some constant, or it's

proportional to temperature times the number of particles.

And we can do that a bunch of ways.

But let's think of that in moles.

If I say there are 5 mole particles there, you know

that's 5 times 6 times 10 to the 23 particles.

So, this is the number of particles.

This is the temperature.

And this is just some constant.

Now, what else is the pressure dependent on?

We gave these two examples.

Obviously, it is dependent on the temperature; the faster

each of these particles move, the higher pressure we'll

have. It's also dependent on the number of particles, the

more particles we have, the more pressure we'll have. What

about the size of the container?

The volume of the container.

If we took this example, but we shrunk the container

somehow, maybe by pressing on the outside.

So if this container looked like this, but we still had

the same four particles in it, with the same average kinetic

energy, or the same temperature.

So the number of particles stays the same, the

temperature is the same, but the volume has gone down.

Now, these guys are going to bump into the sides of the

container more frequently and there's less area.

So at any given moment, you have more force and less area.

So when you have more force and less area, your pressure

is going to go up.

So when the volume went down, your pressure went up.

So we could say that pressure is inversely

proportional to volume.

So let's think about that.

Let's put that into our equation.

We said that pressure is proportional-- and I'm just

saying some proportionality constant, let's call that R,

to the number of particles times the temperature, this

gives us the total energy.

And it's inversely proportional to the volume.

And if we multiply both sides of this times the volume, we

get the pressure times the volume is proportional to the

number of particles times the temperature.

So PV is equal to RnT.

And just to switch this around a little bit, so it's in a

form that you're more likely to see in your chemistry book,

if we just switch the n and the R term.

You get pressure times volume is equal to n, the number of

particles you have, times some constant times temperature.

And this right here is the ideal gas equation.

Hopefully, it makes some sense to you.

When they say ideal gas, it's based on this little mental

exercise I did to come up with this.

I made some implicit assumptions when I did this.

One is I assumed that we're dealing with an ideal gas.

And so you say what, Sal, is an ideal gas?

An ideal gas is one where the molecules are not too

concerned with each other.

They're just concerned with their own kinetic energy and

bouncing off the wall.

So they don't attract or repel each other.

Let's say they attracted each other, then as you increased

the number of particles maybe they'd want to

not go to the side.

Maybe they'd all gravitate towards the center a little

bit more if they did attract each other.

And if they did that, they would bounce into the walls

less and the pressure would be a little bit lower.

So we're assuming that they don't attract

or repel each other.

And we're also assuming that the actual volume of the

individual particles are inconsequential.

Which is a pretty good assumption, because they're

pretty small.

Although, if you start putting a ton of particles into a

certain volume, then at some point, especially if they're

big molecules, it'll start to matter in terms of their size.

But we're assuming for the purposes of our little mental

exercise that the molecules have inconsequential volumes

and they don't attract or repel each other.

And in that situation, we can apply the ideal gas equation

right here.

Now, we've established the ideal gas equation.

But you're like, well what's R, how do I deal with it, and

how do I do math problems, and solve chemistry

problems with it?

And how do the units all work out?

We'll do all of that in the next video where we'll solve a

ton of equations, or a ton of exercises with

the ideal gas equation.

The important takeaway from this video is just to have the

intuition as to why this actually does make sense.

And frankly, once you have this intution, you should

never forget it.

You should be able to maybe even derive it on your own.