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## AP®︎/College Chemistry

### Course: AP®︎/College Chemistry > Unit 9

Lesson 3: Arrhenius equation and reaction mechanisms# The Maxwell–Boltzmann distribution

AP.Chem:

SAP‑7 (EU)

, SAP‑7.B (LO)

, SAP‑7.B.1 (EK)

, SAP‑7.B.2 (EK)

, SAP‑7.B.3 (EK)

, SAP‑7.B.4 (EK)

The Maxwell–Boltzmann distribution describes the distribution of speeds among the particles in a sample of gas at a given temperature. The distribution is often represented graphically, with particle speed on the x-axis and relative number of particles on the y-axis. Created by Sal Khan.

## Want to join the conversation?

- If some air particles are moving faster than sound, wouldn't they perform a tiny sonic boom? Or is that just possible for objects with greater mass? This made me curious...(74 votes)
- Wow, mind blown! Cool question and I think the answer is "no sonicboom":

Particles in air will be able to travel at speeds greater than the speed of sound (300m/s) but I don't think this will cause a sonic boom because think about what sound is: sound is caused by the movement (compression and rarefractions) of molecules in the air. Sonic booms do not exist in space because there is no air and sound can therefore not travel. For the tiniest molecule in air, there is practically no such thing as air for sonic boom: the air molecule is floating in a vacuum with other air molecules but it does not have anything to "cause" a boom through any medium.(82 votes)

- If the particles around us are traveling so quickly, why don't we hear sonic booms? I understand that the particles are INCREDIBLY small, yes, but when you sum enough tiny things up, you get something of decent size...(14 votes)
- its a nice question.

the sonic boom is due to motion and pressure between air particles so the movement of the particles themselves would have nothing to 'boom' against.(26 votes)

- How often do air particles collide/interact? Wouldn't the kinetic energy of each particle approach the same velocity over time because faster particles would impart their momentum to slower particles?(13 votes)
- Its an interesting thought.

Now I am thinking you could be right. Intuitively it kind of makes sense.

On the other hand, I think that may only be the case if some energy is lost each time there is a collision. But, according to the theory, the total amount of ke remains constant. So your slow particle simply becomes the faster particle.

Can I suggest a thought experiment: How about thinking about just two partilces in a box. and considering the energy changes as they collide. What would be the mechanism by which they would tend towards equal energies over time??

I will chew on it further but let us know your thoughts

IM(13 votes)

- At06:05Sal says, " If we have the same no. of molecules, the areas under the curves need to be the same".

Why is that?(7 votes)- Because the area under the curve represents the number of molecules. This is why when you change the temperature, the graph changes shape but the area will always remain constant because the number of molecules does not change.(13 votes)

- At6:05Sal says that the area under the graphs has to be the same because the number of molecules is the same. But the green system has about 30 percent more energy. Should the green system have 30 percent more area?(5 votes)
- The area under the curve in this case refers to the mass (number of molecules) in the system, not the energy they have. More energy does not make molecules randomly appear in an ideal gas, so it would not affect the mass, or the area under the curve.

The number of molecules/mass here is the same, so the areas under the curves must be the same.

The "speed" on the x-axis just shows the speed which the different molecules fall into, but the area itself refers to the number of molecules, on the y-axis.(6 votes)

- At 6.02, I didn't really understand why the area of A must be equal to the area of B (In the coordinate plane).(4 votes)
- The two graphs are for the same container of gas at two different temperatures. The graphs is of the number of molecules at the various speeds. The total area of the graph is the number of molecules in the container so since it is the same container of gas the total number of molecules is the same so the areas have to be the same.(3 votes)

- Why does the area under the curve represent the number of molecules?(4 votes)
- because it is a distribution graph. so if you drew, straight down, a line from every point of the curve, the length of each line is the number of particles with a given speed. add up all the lines and you get the area under the curve which, then, is the same as the number of particles.(1 vote)

- I still don't understand why one curve is narrower while the other is wider and I've been struggling to understand it for a long time...Can somebody please help me understand? Thanks :)(2 votes)
- Since the number of particles in the gas no matter the temperature is constant, that means the area under the curve is constant. Therefore, if you shift the peak to the right, the height of the graph has to decrease in order to maintain the same total area under the curve. The same thing can be said when the peak moves to the left. Hope this helps!(2 votes)

- the area under the curve gives the total number of molecules but according to the graph drawn it seems like graph A has greater area than graph A but in reality both have same number of molecules as assumed i.e 10?(2 votes)
- Yep, both have same number of molecules.(2 votes)

- What is the equation used to calculate this most probable speed of air molecules 422 m/s that Sal mentions?(2 votes)
- He's using this equation here - https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution#Typical_speeds.

Note that R needs to be in units of J.K-1.mol-1 (ie, 8.314 J.K-1.mol-1) and molecular mass needs to be expressed in kg (ie, 0.028 kg/mol for nitrogen).

If you are wondering how speed, in m/s, falls out of this equation, it is because a Joule has the units of kg.m2.s-2. Prior to taking the square root, the units in the equation cancel out to give m2.s-2, which becomes m/s after taking the square root.(3 votes)

## Video transcript

- [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.