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21. Kinetic Theory of Ideal Gases

Speed Distribution of Ideal Gases


Speed Distribution & Special Speeds of Ideal Gases

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Hey guys, so in previous videos we've seen the equations for a special type of speed called the RMS speed, but that isn't the only type of speed that you need to know for an ideal gas. So in this video, I'm gonna show you a couple more special speeds that you need to know. But first we're gonna take a look at a sort of visual way to understand these speeds, which is called a speed distribution. So let's check this out and we're gonna work this example out together. So the whole idea here is that gas particles don't all travel at the same speed if you have a container of gas, they're all bouncing off of each other, they're speeding up and slowing down. And so what happens is if you took all of the gas particles and measured their speeds and then plotted on a graph the number of particles versus their speeds. You would get these kinds of curves here depending on the temperature. So these curves are called distributions. Now, specifically these types of distributions are called Maxwell bolts, hman distributions and they basically measure those sort of speeds that you could possibly measure in an ideal gas. So the way this works is that for certain temperatures like 300 kelvin or kelvin at certain speeds, you might be more likely to find a particle with that specific speed. That's how this works. So let's check out our example here. So we've got these speed distributions for certain temperatures, we've got 300 kelvin and 600 in the first part. And the first part of this problem, we want to calculate the most probable the average and the RMS speed. So these are three speeds that we have to calculate for a sample here at 300 kelvin. So let's get started. So for part a, what we want here is that we're gonna be using the blue curve because the temperature here is 300 kelvin, we're gonna ignore the red curve just for now. What we want to do is we want to calculate the most probable speed, which is one of the three special speeds that we need to know. We've already seen the equation for the RMS speed. So we really only have two new ones that we're gonna look for or that we're gonna we're gonna learn. So the most probable or most likely speed is the easiest one to identify on these graphs, because the most probable meaning the highest probability is going to correspond to the peak of the curve. So, when you see this curve here, the most probable speed is going to be the highest point on the curve. So this is gonna be uh want me to write this in blue, this is gonna be V M. P. So we want to do is we want to actually calculate what the speed is here on the X axis. All right. So it turns out that the equation for the most likely speed and most probable speed is gonna be very similar to the equation for the RMS speed instead basically of the three inside of this equation, we're going to use a two. So the equation here is gonna be two KB T divided by little M. But remember anytime we see a KB over M, we can rewrite this as big R over big M. So the way I like to remember this is that the most probable this mp here has two letters and two letters means that there's a to that goes inside of the square roots. Likewise, for the RMS, RMS has three letters and that's why it goes, it has a three inside of the square roots. Right? That's just one way I remember this. All right. So, we're just gonna use this equation one of these two. And it really just depends on whether we have the mass or the molar mass of this sample here. If we look at the problem, we have the molar mass which is 28 g per mole. So, we actually were given, what we're given here is big M. So big. M. Is equal to 0.28. Therefore, we're gonna be using this equation here. So, the most probable speed of 300 kelvin is gonna be the square roots of two. R. T, divided by big M. So this is gonna be a big M. Here. So, if you work this out, you're gonna square root of two times 8. Times the temperature, which is 300. And then you're gonna divide this by 0.028. If you go out and work the sandwich you're gonna get is 422 meters per second. All right. So basically what happens is that this number here is equal to 422. That's the most probable speed. Alright. Pretty simple. So let's move on now and calculate the average speed. That's where that's that's the next speed on our list here. So this is the other special speed that you need to know which is a true average. Now this isn't exactly the same thing as the RMS speed because the RMS speed is a type of average that's skewed towards higher numbers because you're sort of averaging the squares of the numbers. The average is actually just a true average. And the equation for this is gonna be eight KB. T divided by pi little M. So we can also rewrite this one as well. This is gonna be eight R. T. Divided by pi times big M. All right. So basically we're just gonna use the exact same numbers except now. We're just gonna change what's inside the square root here. So this is gonna be eight R. T divided by pi times big M. And you're gonna do square root of eight times 8.314 times 300 divided by pi times 0.28. Alright. Just make sure that you enclose your apprentices and stuff like that. So if you calculate this you're gonna get 476.3 m/s. All right. So that means that over here this line here actually corresponds to the average speed. So this is gonna be let's see 476. And this is gonna be your v. Average. Now the last thing we want to do is calculate v. RMS and I'm just gonna fly through this because we've already seen this equation before. This is gonna be three R. T. Divided by big M. And this is going to be the square root of three. 8.314. You're basically just going to do exactly what you did for the first one is just plugging in a three and then divided by 0.28. So when you plug this in you're gonna get 516 m per second. Alright, so that means that this V. R. M. S is going to be even higher than the V average. That makes sense. It's skewed towards higher numbers here and this is gonna be 516. That's gonna be your V. RMS. Alright. So in general what happens is that for any given temperature, the most probable speed is gonna be the peak and that's actually sort of the least speed. It's gonna be the least of the three values. So the V. M. P. Is going to be less than the average as we can see here and the average is going to be less than the R. M. S. So it's always gonna go mp. It's going to be the most probable to average. RMS. In terms of at least to greatest. Alright so let's move on now to the second part here. Part B. And part D. What we wanna do is we want to calculate the speed that we would most likely measure but now for the 600 kelvin gas sample. So which one of these three speeds is this? Well if the speed we would most likely measure is going to be the most probable speed. So we're really just going to use for t equals 600. So now we're gonna be looking at the red curve. What is V. M. P. Alright so we just use the same equation that we did here. So this is gonna be two R. T. Over a big M. Except now we're just R. T. Is gonna be different. So this is gonna be the square root of two times 8.314 And divided by 0.028. Uh and whoops I'm sorry I forgot the 600 up here and if you go ahead and work this out what you're gonna get is 596. m/s. So notice how this number here is actually greater than all of these three numbers. So what we got here is that the most probable Speed is equal to 596, which is greater than the other three. And so therefore what happens is that the average speed would be somewhere over here and the RMS speed would be somewhere here. We're not going to calculate them, but you can if you wanted to. So basically what happens here is that in general all the three speeds you're going to increase as the temperature increases and the curve gets a little bit flatter. So this kind of makes some sense, if the temperature is higher, there is sort of more possible speeds, there's more energies. And so what happens here here is that these uh these particles are speeding up and slowing down and the distribution gets a little bit more flat. So it's very sort of the probability kind of gets spread out a little bit and um all of these velocities sort of go towards the right on the time, on the on the X axis. Alright guys, so that's it for this one. Let me know if you have any questions

The escape velocity from the Earth is approximately 11.2 km/s. If the mass of helium atoms is 6.64 × 10-27 kg, at what temperature would the average speed of helium atoms be equal to the escape velocity?


Probability Distribution Graph

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everybody. So let's go ahead and check out this problem here. So, we have a graph that shows the probability distribution for oxygen gas here. Now, remember that this probability distribution applies to a specific temperature. So there's one specific temperature of this curve right now. So, for example, if the curve looks like this, it would have a different temperature and if it looked like this, it would have an even different temperature. What we want to do is we want to figure out what's the temperature of this sample of oxygen gas, given that the probability curve looks exactly like this. And the one thing that we're told here is that the farthest tick mark on the X axis is 1200 m meters per second. We're also told that the molar mass of oxygen is. So let's go ahead and get started here. Which equation are we going to use in our probability distributions? Remember there's three of them, there is the most probable, there's the average and then there's the R. M. S. So what happens here is if you remember the shape, the average isn't exactly always in the middle or like at the at the at the peak, it's kind of like somewhere beyond the peak like this. So this would be my V average and the RMS velocity is a kind of average that favors generally higher velocity. So the average to the V. RMS would probably be somewhere over here. The problem with this is that we're kind of just guessing at this point. There's no real way to figure that out. Would actually you have to use a really, really complicated equation. What's the one velocity that we actually can absolutely figure out where it is. It's the most probable velocity because remember, the most probable velocity occurs at the peaks of the curve. So this right here is the most probable velocity, it's the very very tip of the curve. So then how do we find what this velocity is? We can actually just use our scale here for the velocity. Alright, so working backwards, if this is 1200 there is equally spaced, this is going to be 11, 10, 98765 and m per second. Right? So this is 302 101 100. So we're kind of having to work backwards to figure out what the tick marks are. But then if you look, if you go ahead you'll see that this line corresponds perfectly to the dot that I just drew. So we can tell here from this graph, is that the most probable velocity here is 400 m/s. So now that we know what this most probable velocity is, we know we're gonna be looking at this equation over here because what's the target variable again? Remember we're looking for the temperature. So we're looking for capital T and if you look at this equation, you're the most probable velocity that's going to be in our equation. Alright, so let's get started. So we're gonna we're gonna use the V. M. P. Equation. Now remember which version are we gonna use? We're gonna use the one that has mass or molar mass. Well we're told what the molar mass of oxygen is. So we're just going to use this format over here. The one with capital M. So we have that VM P. Is equal to the square root of two. R. T. Divided by big M. So all we have to do now is just go ahead and solve and isolate this tea. Now I actually know what the V. M. P. Is. Remember it's just the it's just the speed that we found which is 400 m per second. However, I'm gonna have to square this to get rid of the square roots. This is gonna be 400. We're gonna square this and then the square root comes off of this. So this is gonna be two R. T. Divided by big M. Now all we have to do is just move this guy to the other side and then divide by two are so putting this all together here we have 400 squared times and then the moller mass. Now we're just be very careful here because we were given as grams per mole. So this M. Here if we convert this is going to be 0. kg per mole. So that's the number that we have to plug into our equation here. So this is gonna be times 0.32. And then we just divide by two are so we're gonna divide by two times 8.314. This is going to give you your temperature, and if you work this out, what you're gonna get is 307.9 degrees kelvin. And that's the answer. Alright folks, so that's it for this one.