The Maxwell–Boltzmann distribution | AP Chemistry | Khan Academy

Summary

The video script discusses the Maxwell-Boltzmann distribution, a statistical model describing the speeds of particles in an ideal gas. It introduces James Clerk Maxwell and Ludwig Boltzmann, who independently developed this distribution. The script uses a thought experiment with nitrogen gas at different temperatures to illustrate how temperature relates to the average kinetic energy of gas particles. It explains that at 300 Kelvin, the most probable speed of nitrogen molecules is 422 meters per second, which is faster than the speed of sound, highlighting the vast range of molecular speeds and their impact on air pressure.

Takeaways

• 🔬 James Clerk Maxwell and Ludwig Boltzmann independently developed the Maxwell-Boltzmann distribution, which describes the distribution of speeds of particles in an ideal gas.
• 🌡️ Temperature, at a molecular level, is proportional to the average kinetic energy of the molecules within a system.
• 🌡️ At 300 Kelvin, the average kinetic energy of nitrogen molecules in a gas is higher compared to a system at 200 Kelvin.
• 🚀 Even though individual molecules may move faster in a cooler system, on average, the molecules in a system at 300 Kelvin will have greater kinetic energy and move faster.
• 📊 The Maxwell-Boltzmann distribution is a bell-shaped curve that shows the number of molecules at different speeds; the peak represents the most probable speed.
• 📉 As temperature decreases, the peak of the Maxwell-Boltzmann distribution shifts to lower speeds, indicating a decrease in average kinetic energy.
• 📈 Conversely, as temperature increases, the peak shifts to higher speeds, reflecting an increase in average kinetic energy.
• 🌀 The distribution curves for different temperatures have the same area under the curve, representing the same number of molecules in each system.
• 🏃‍♂️ At room temperature (300 Kelvin), the most probable speed of nitrogen molecules is approximately 422 meters per second, or about 944 miles per hour.
• ✈️ Despite the high speeds of some molecules, the mass of nitrogen molecules is so small that their collisions with objects or people do not cause harm and feel like ambient air pressure.

Q & A

Who are James Clerk Maxwell and Ludwig Boltzmann, and what is their contribution to physics?

James Clerk Maxwell was a titan of physics, famous for Maxwell's equations and foundational work on color photography. Ludwig Boltzmann is considered one of the founding fathers of statistical mechanics. Together, through the Maxwell-Boltzmann distribution, they independently described the distribution of speeds of gas particles.

What does the Maxwell-Boltzmann distribution represent?

The Maxwell-Boltzmann distribution represents the distribution of speeds of particles in an idealized gas at a given temperature.

How is temperature related to the average kinetic energy of molecules?

Temperature is proportional to the average kinetic energy of the molecules in a system. This means that as the temperature increases, so does the average kinetic energy of the molecules.

What does a thermometer reading of 300 Kelvin signify in terms of molecular motion?

A thermometer reading of 300 Kelvin indicates that the average kinetic energy of the molecules in the system is at a level that corresponds to that temperature. The molecules are moving with a certain average speed and energy.

How does the distribution of molecular speeds change with temperature?

As the temperature increases, the distribution of molecular speeds shifts towards higher speeds, with a broader and flatter curve. Conversely, at lower temperatures, the distribution shifts towards lower speeds, with a narrower and taller peak.

What is the most probable speed of nitrogen molecules at room temperature (300 Kelvin)?

The most probable speed of nitrogen molecules at room temperature (300 Kelvin) is approximately 422 meters per second, which is about 944 miles per hour.

Why do we not feel the impact of nitrogen molecules traveling at high speeds despite their high kinetic energy?

We do not feel the impact of nitrogen molecules traveling at high speeds because they are extremely small in mass. Even though they may be moving fast, the force exerted by each collision is minimal and feels like ambient air pressure.

How does the speed of sound compare to the most probable speed of nitrogen molecules at room temperature?

The most probable speed of nitrogen molecules at room temperature (422 meters per second) is faster than the speed of sound (approximately 340 meters per second).

What is the significance of the Maxwell-Boltzmann distribution in understanding gas behavior?

The Maxwell-Boltzmann distribution is significant because it provides a statistical description of the speeds of gas particles, which is crucial for understanding gas behavior, such as pressure and temperature effects on a gas.

Why does the distribution curve for a lower temperature system have a higher peak?

The distribution curve for a lower temperature system has a higher peak because, with the same number of molecules, if the distribution is narrower (fewer molecules at high speeds), it must be taller to ensure the total area under the curve remains the same, representing the total number of molecules.

How does the Maxwell-Boltzmann distribution help in understanding the microscopic view of temperature?

The Maxwell-Boltzmann distribution helps in understanding the microscopic view of temperature by illustrating how the speed of particles is distributed across different speeds at a given temperature, showing that temperature is a measure of the average kinetic energy of the particles.

Outlines

00:00

🔬 Introduction to Maxwell-Boltzmann Distribution

The script introduces the Maxwell-Boltzmann distribution, a statistical model that describes the distribution of speeds of particles in an ideal gas. It begins with a historical context, mentioning James Clerk Maxwell and Ludwig Boltzmann, who independently developed this distribution. The narrator uses a thought experiment involving a container of nitrogen gas at 300 Kelvin to explain the concept of temperature as it relates to the average kinetic energy of gas particles. The script visually represents this with a hypothetical container and thermometer, and then moves on to compare two containers at different temperatures, illustrating how temperature affects the average kinetic energy and speed of the particles.

05:01

📊 Exploring the Maxwell-Boltzmann Distribution

This section delves deeper into the Maxwell-Boltzmann distribution by discussing how it graphically represents the relationship between particle speed and the number of molecules at that speed. The narrator explains that at lower temperatures, most molecules are at slower speeds, but the distribution curve is taller and narrower, reflecting the same number of molecules as at higher temperatures. The script then explores the effect of increasing temperature on the distribution, showing how the peak shifts to higher speeds. The narrator also highlights the surprising fact that at room temperature, the most probable speed of nitrogen molecules is approximately 422 meters per second, which is faster than the speed of sound, yet we don't perceive it due to the small mass of the molecules. This part of the script aims to give a clearer understanding of the distribution's practical implications and the high speeds at which some particles move, contributing to air pressure and our perception of the environment.

Keywords

💡Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann Distribution is a statistical law that describes the distribution of speeds of particles in a gas at a certain temperature. In the video, it is central to understanding the behavior of nitrogen molecules in a container at various temperatures. The distribution curve shows how many molecules are moving at different speeds, with the peak of the curve indicating the most probable speed at which the highest number of molecules are moving.

💡James Clerk Maxwell

James Clerk Maxwell is a pivotal figure in physics, known for formulating Maxwell's equations, which describe the behavior of electric and magnetic fields. In the context of the video, Maxwell is credited with foundational work on the distribution of speeds of gas particles, which later became part of the Maxwell-Boltzmann distribution.

💡Ludwig Boltzmann

Ludwig Boltzmann is considered one of the founding fathers of statistical mechanics. He independently developed the distribution that now bears his name alongside Maxwell's, which describes the statistical distribution of speeds of particles in a gas. The video highlights his contribution to the understanding of molecular motion and energy.

💡Idealized Gas Particles

In the video, 'idealized gas particles' refers to the theoretical model of gas molecules that behave according to the laws of classical mechanics. The script uses nitrogen as an example of such particles to simplify the discussion of the Maxwell-Boltzmann distribution. These particles are assumed to have no volume and to collide elastically.

💡Temperature

Temperature, as discussed in the video, is a measure of the average kinetic energy of the molecules in a substance. It is proportional to the average kinetic energy, which is a key concept in understanding the Maxwell-Boltzmann distribution. The video uses the example of a thermometer reading 300 Kelvin to illustrate how temperature relates to the motion of gas particles.

💡Kinetic Energy

Kinetic energy, in the context of the video, refers to the energy possessed by an object due to its motion. For gas particles, the average kinetic energy is directly related to the temperature of the gas. The video explains that temperature is a measure of this average kinetic energy, which is crucial for understanding the distribution of particle speeds.

💡Nitrogen Molecules

Nitrogen molecules are used in the video as a specific example of gas particles to illustrate the concepts of the Maxwell-Boltzmann distribution. The video simplifies the composition of air to only nitrogen to focus on how these molecules distribute their speeds at different temperatures.

💡Molecular Scale

The term 'molecular scale' refers to the level at which molecules interact and move. In the video, the molecular scale is used to explain the concept of temperature and how it relates to the motion of gas particles. Understanding what happens at the molecular scale helps to visualize and comprehend the Maxwell-Boltzmann distribution.

💡Speed Distribution

Speed distribution, as discussed in the video, is the statistical representation of how the speeds of gas particles are spread out at a given temperature. The Maxwell-Boltzmann distribution curve shows this spread, with the peak indicating the most probable speed of the particles.

💡Room Temperature

Room temperature, in the video, is used as a reference point for discussing the most probable speed of nitrogen molecules. The script mentions that at room temperature (approximately 300 Kelvin), the most probable speed of nitrogen molecules is about 422 meters per second, which is faster than the speed of sound.

💡Air Pressure

Air pressure is mentioned in the video as a result of the collisions of gas particles, such as nitrogen molecules, with objects. The video explains that even though individual molecules may be moving very fast, the collective effect is what we perceive as air pressure, which is the force exerted by the gas particles on the surfaces they collide with.

Highlights

James Clerk Maxwell is a titan of physics, famous for Maxwell's equations and foundational work on color photography.

Maxwell was involved in determining the distribution of speeds of idealized gas particles.

Ludwig Boltzmann is considered one of the founding fathers of statistical mechanics.

Maxwell and Boltzmann independently came to the same distribution, known as the Maxwell-Boltzmann distribution.

The Maxwell-Boltzmann distribution describes the distribution of speeds of air particles.

Temperature is proportional to the average kinetic energy of the molecules in a system.

At 300 Kelvin, the average kinetic energy of nitrogen molecules is higher compared to 200 Kelvin.

The distribution of molecular speeds at 300 Kelvin is broader and has a higher peak than at 200 Kelvin.

The Maxwell-Boltzmann distribution can be visualized using a coordinate plane with speed on one axis and number of molecules on the other.

At higher temperatures, the peak of the Maxwell-Boltzmann distribution shifts to higher speeds.

The most probable speed of nitrogen molecules at room temperature (300 Kelvin) is approximately 422 meters per second.

The most probable speed of nitrogen molecules at room temperature is faster than the speed of sound.

Despite the high speeds of some molecules, the small mass of nitrogen molecules prevents us from feeling their impact.

The air pressure we feel is due to the collisions of nitrogen molecules moving at various speeds.

The Maxwell-Boltzmann distribution provides insight into the behavior of particles in a gas at different temperatures.

Transcripts

- [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.