Maxwell Boltzmann distribution

- [Voiceover] So let's think a little bit about the

Maxwell-Boltzmann distribution.

And this right over here, this is

a picture of James Clerk Maxwell.

And I really like this picture, it's with his

wife Katherine Maxwell and I guess this is their dog.

And James Maxwell, he is a titan of physics

famous for Maxwell's equations.

He also did some of the foundational work

on color photography and he was involved in

thinking about, "Well, what's the distribution

of speeds of air particles

of idealized gas particles?"

And this gentleman over here, this is Ludwig Boltzmann.

And he's considered the father or one

of the founding fathers of statistical mechanics.

And together, through the Maxwell-Boltzman distribution

they didn't collaborate, but they

independently came to the same distribution.

They were able to describe, "Well, what's the

distribution of the speeds of air particles?"

So let's back up a little bit or let's just

do a little bit of a thought experiment.

So let's say that I have a container here.

Let's say that I have a container here.

And let's say it has air.

And air is actually made up mostly of nitrogen.

Let's just say it just has only nitrogen in it

just to simplify things.

So let me just draw some nitrogen molecules in there.

And let's say that I have a thermometer.

I put a thermometer in there.

And the thermometer

reads a temperature of 300 Kelvin.

What does this temperature of 300 Kelvin mean?

Well, in our everyday life, we have

kind of a visceral sense of temperature.

Hey, I don't wanna touch something that's hot.

It's going to burn me.

Or that cold thing, it's gonna make me shiver.

And that's how our brain

processes this thing called temperature.

But what's actually going on at a molecular scale?

Well, temperature, one way to think

about temperature, this would be a very

accurate way to think about temperature

is that tempera-

I'm spelling it wrong.

Temperature is proportional to average kinetic energy

of the molecules in that system.

So let me write it this way.

Temperature is proportional to average kinetic energy.

Average

kinetic

energy

in the system.

I'll just write average kinetic energy.

So let's make that a little bit more concrete.

So let's say that I have two containers.

So it's one container.

Whoops.

And two containers right over here.

And let's say they have the same

number of molecules of nitrogen gas

And I'm just gonna draw 10 here.

This obviously is not realistic

you'd have many, many more molecules.

One, two, three, four, five, six, seven, eight, nine, ten.

One, two, three, four, five, six, seven, eight, nine, ten.

And let's say we know that the

temperature here is 300 Kelvin.

So the temperature of this system is 300 Kelvin.

And the temperature of this system is 200 Kelvin.

So if I wanted to visualize what these molecules are doing

they're all moving around, they're bumping

they don't all move together in unison.

The average kinetic energy of the molecules

in this system is going to be higher.

And so maybe you have

this molecule is moving in that direction.

So that's its velocity.

This one has this velocity.

This one's going there.

This one might not be moving much at all.

This one might be going really fast that way.

This one might be going super fast that way.

This is doing that.

This is doing that.

This is doing that.

So if you were to now compare it to this system

this system, you could still have a molecule

that is going really fast.

Maybe this molecule is going faster

than any of the molecules over here.

But on average, the molecules here

have a lower kinetic energy.

So this one maybe is doing this.

I'm going to see if I can draw...

On average, they're going to have a lower kinetic energy.

That doesn't mean all of these molecules

are necessarily slower than all of these molecules

or have lower kinetic energy than all of these molecules.

But on average they're going to have less kinetic energy.

And we can actually draw a distribution.

And this distribution, that is

the Maxwell-Boltzmann distribution.

So if we...

Let me draw a little coordinate plane here.

So let me draw a coordinate plane.

So, if on this axis, I were to put speed.

If I were to put speed.

And on this axis, I would put number of molecules.

Number of molecules.

Right over here.

For this system, the system that is at 300 Kelvin

the distribution might look like this.

So it might look

the distribution...

Let me do this in a new color.

So, the distribution

this is gonna be all of the molecules.

The distribution might look like this.

Might look like this.

And this would actually be the Maxwell-Boltzmann

distribution for this system

For system, let's call this system A.

System A, right over here.

And this system, that has a lower temperature

which means it also has a lower kinetic energy.

The distribution of its particles...

So the most likely, the most probable...

You're going to have the highest number of molecules

at a slower speed.

Let's say you're gonna have it at this speed

right over here.

So its distribution might look something like this.

So it might look something like that.

Now why is this one...

It might make sense to you that

okay, the most probable

the speed at which I have the most molecules

I get that that's going to be lower than the speed

at which I have the most molecules in system A

because I have, because on average

these things have less kinetic energy.

They're going to have less speed.

But why is this peak higher?

Well, you gotta remember we're talking about

the same number of molecules.

So if we have the same number of molecules that means

that the areas under these curves need to be the same.

So if this one is narrower, it's going to be taller.

And if I were gonna, if I were to somehow

raise the temperature of this system even more.

Let's say I create a third system or I get this

or let's say I were to heat it up to 400 Kelvin.

Well then my distribution would look

something like this.

So this is if I heated it up.

Heated up.

And so this is all the Maxwell-Boltzmann distribution is.

I'm not giving you the more involved, hairy equation for it

but really the idea of what it is.

It's a pretty neat idea.

And actually when you actually think about the actual

speeds of some of these particles, even the air around you

I'm gonna say, "Oh, it looks pretty stationary to me."

But it turns out in the air around you is mostly nitrogen.

That the most probable speed of

if you picked a random nitrogen

molecule around you right now.

So the most probable speed.

I'm gonna write this down

'cause this is pretty mindblowing.

Most probable speed at room temperature.

Probable speed

of N2 at room temperature.

Room temperature.

So let's say this that this was the Maxwell-Boltzmann

distribution for nitrogen at room temperature.

Let's say that that's, let's say we make

we call room temperature 300 Kelvin.

This most probable speed right over here

the one where we have the most molecules

the one where we're gonna have the most

molecules at that speed.

In fact, guess what that is going to be before I tell you

'cause it's actually mind boggling.

Well, it turns out that it is approximately

400, 400 and actually at 300 Kelvin

it's gonna be 422 meters per second.

422 meters per second.

Imagine something traveling 422 meters in a second.

And if you're used to thinking in terms of miles per hour

this is approximately 944

miles per hour.

So right now, around you

you have, actually

the most probable, the highest number

of the nitrogen molecules around you

are traveling at roughly this speed

and they're bumping into you.

That's actually what's giving you air pressure.

And not just that speed, there are actually ones

that are travelling even faster than that.

Even faster than 422 meters per second.

Even faster.

There's particles around you traveling faster

than a thousand miles per hour

and they are bumping into your body as we speak.

And you might say, "Well, why doesn't that hurt?"

Well, that gives you a sense of how small the mass

of a nitrogen molecule is, that it can

bump into you at a thousand miles per hour

and you really don't feel it.

It feels just like the ambient air pressure.

Now, when you first look at this, you're like

wait, 422 meters per second?

That's faster than the speed of sound.

The speed of sound is around 340 meters per second.

Well, how can this be?

Well, just think about it.

Sound is transmitted through the air

through collisions of particles.

So the particles themselves have to be moving

or at least some of them, have to be moving

faster than the speed of sound.

So, not all of the things around you

are moving this fast and they're

moving in all different directions.

Some of them might not be moving much at all.

But some of them are moving quite incredibly fast.

So, I don't know, I find that a little bit mindblowing.