WEBVTT

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

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Maxwell-Boltzmann distribution.

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And this right over here, this is

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a picture of James Clerk Maxwell.

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

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wife Katherine Maxwell and I guess this is their dog.

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And James Maxwell, he is a titan of physics

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famous for Maxwell's equations.

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He also did some of the foundational work

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on color photography and he was involved in

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thinking about, "Well, what's the distribution

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of speeds of air particles

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of idealized gas particles?"

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And this gentleman over here, this is Ludwig Boltzmann.

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And he's considered the father or one

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of the founding fathers of statistical mechanics.

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And together, through the Maxwell-Boltzman distribution

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they didn't collaborate, but they

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independently came to the same distribution.

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

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distribution of the speeds of air particles?"

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

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do a little bit of a thought experiment.

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

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Let's say that I have a container here.

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And let's say it has air.

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And air is actually made up mostly of nitrogen.

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

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just to simplify things.

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So let me just draw some nitrogen molecules in there.

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And let's say that I have a thermometer.

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I put a thermometer in there.

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And the thermometer

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reads a temperature of 300 Kelvin.

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What does this temperature of 300 Kelvin mean?

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Well, in our everyday life, we have

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kind of a visceral sense of temperature.

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

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It's going to burn me.

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

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And that's how our brain

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processes this thing called temperature.

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

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Well, temperature, one way to think

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about temperature, this would be a very

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accurate way to think about temperature

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is that tempera-

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I'm spelling it wrong.

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Temperature is proportional to average kinetic energy

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of the molecules in that system.

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So let me write it this way.

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Temperature is proportional to average kinetic energy.

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Average

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kinetic

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energy

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in the system.

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I'll just write average kinetic energy.

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

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So let's say that I have two containers.

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So it's one container.

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Whoops.

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And two containers right over here.

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And let's say they have the same

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number of molecules of nitrogen gas

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And I'm just gonna draw 10 here.

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This obviously is not realistic

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you'd have many, many more molecules.

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

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

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And let's say we know that the

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temperature here is 300 Kelvin.

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So the temperature of this system is 300 Kelvin.

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And the temperature of this system is 200 Kelvin.

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So if I wanted to visualize what these molecules are doing

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they're all moving around, they're bumping

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they don't all move together in unison.

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The average kinetic energy of the molecules

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in this system is going to be higher.

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And so maybe you have

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this molecule is moving in that direction.

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So that's its velocity.

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This one has this velocity.

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This one's going there.

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This one might not be moving much at all.

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This one might be going really fast that way.

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This one might be going super fast that way.

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This is doing that.

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This is doing that.

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This is doing that.

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So if you were to now compare it to this system

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this system, you could still have a molecule

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that is going really fast.

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Maybe this molecule is going faster

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than any of the molecules over here.

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But on average, the molecules here

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have a lower kinetic energy.

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So this one maybe is doing this.

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

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

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That doesn't mean all of these molecules

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are necessarily slower than all of these molecules

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or have lower kinetic energy than all of these molecules.

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

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And we can actually draw a distribution.

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And this distribution, that is

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the Maxwell-Boltzmann distribution.

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So if we...

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Let me draw a little coordinate plane here.

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So let me draw a coordinate plane.

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

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If I were to put speed.

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

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Number of molecules.

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Right over here.

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For this system, the system that is at 300 Kelvin

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the distribution might look like this.

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So it might look

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the distribution...

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Let me do this in a new color.

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So, the distribution

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this is gonna be all of the molecules.

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The distribution might look like this.

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Might look like this.

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And this would actually be the Maxwell-Boltzmann

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distribution for this system

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For system, let's call this system A.

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System A, right over here.

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And this system, that has a lower temperature

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which means it also has a lower kinetic energy.

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The distribution of its particles...

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So the most likely, the most probable...

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You're going to have the highest number of molecules

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at a slower speed.

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

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right over here.

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So its distribution might look something like this.

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So it might look something like that.

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Now why is this one...

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It might make sense to you that

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okay, the most probable

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the speed at which I have the most molecules

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

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at which I have the most molecules in system A

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because I have, because on average

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these things have less kinetic energy.

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They're going to have less speed.

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But why is this peak higher?

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Well, you gotta remember we're talking about

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the same number of molecules.

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So if we have the same number of molecules that means

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

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

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And if I were gonna, if I were to somehow

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raise the temperature of this system even more.

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

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

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Well then my distribution would look

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something like this.

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So this is if I heated it up.

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Heated up.

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

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

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but really the idea of what it is.

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It's a pretty neat idea.

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And actually when you actually think about the actual

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

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

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

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That the most probable speed of

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if you picked a random nitrogen

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molecule around you right now.

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So the most probable speed.

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I'm gonna write this down

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'cause this is pretty mindblowing.

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Most probable speed at room temperature.

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Probable speed

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of N2 at room temperature.

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Room temperature.

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

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distribution for nitrogen at room temperature.

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

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we call room temperature 300 Kelvin.

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This most probable speed right over here

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the one where we have the most molecules

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the one where we're gonna have the most

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molecules at that speed.

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

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'cause it's actually mind boggling.

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Well, it turns out that it is approximately

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400, 400 and actually at 300 Kelvin

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it's gonna be 422 meters per second.

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422 meters per second.

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Imagine something traveling 422 meters in a second.

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

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this is approximately 944

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miles per hour.

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So right now, around you

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you have, actually

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the most probable, the highest number

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of the nitrogen molecules around you

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are traveling at roughly this speed

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and they're bumping into you.

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

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And not just that speed, there are actually ones

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that are travelling even faster than that.

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Even faster than 422 meters per second.

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Even faster.

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There's particles around you traveling faster

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than a thousand miles per hour

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and they are bumping into your body as we speak.

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

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

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of a nitrogen molecule is, that it can

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bump into you at a thousand miles per hour

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and you really don't feel it.

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It feels just like the ambient air pressure.

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

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wait, 422 meters per second?

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That's faster than the speed of sound.

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The speed of sound is around 340 meters per second.

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Well, how can this be?

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Well, just think about it.

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Sound is transmitted through the air

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through collisions of particles.

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So the particles themselves have to be moving

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or at least some of them, have to be moving

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faster than the speed of sound.

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So, not all of the things around you

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are moving this fast and they're

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moving in all different directions.

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Some of them might not be moving much at all.

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But some of them are moving quite incredibly fast.

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