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 going to show you a couple more special speeds that you need to know, but first, we're going to 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 going to work this example out together. 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 it on a graph, the number of particles versus their speeds, you would get these kinds of curves here depending on the temperature. These curves are called distributions. Now specifically, these types of distributions are called Maxwell Boltzmann distributions, and they basically measure those sorts 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 600 Kelvin, at certain speeds, you might be more likely to find a particle with that specific speed. That's how these work. So let's check out our example here. We've got these speed distributions for certain temperatures. We've got 300 Kelvin and 600. In the first part of this problem, we want to calculate the most probable, the average, and the RMS speed. So these are 3 speeds that we have to calculate for a sample here at 300 Kelvin. So let's get started.

For part a, what we want here is that we're going to be using the blue curve because the temperature here is 300 Kelvin. We're going to ignore the red curve just for now. What we want to do is calculate the most probable speed, which is one of the 3 special speeds that we need to know. We've already seen the equation for the RMS speed, so we really only have 2 new ones that we're going to learn. 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 going to be \( V_{MP} \). What we want to do is actually calculate what the speed is here on the x-axis.

It turns out that the equation for the most likely speed or most probable speed is going to be very similar to the equation for the RMS speed. Instead, basically of a 3 inside of this equation, we're going to use a 2. So the equation here is going to be the square root of \( \frac{2k_BT}{m} \). But remember, anytime we see a \( \frac{k_B}{m} \), we can rewrite this as \( \frac{R}{\bar{M}} \). So the way I like to remember this is that the most probable, this MP here has two letters, and two letters mean that there's a 2 that goes inside of the square roots. Likewise, for the RMS, RMS has three letters and that's why it has a 3 inside of the square root. Right? That's just one way I remember this.

We're just going to use this equation, one of these 2, 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 grams per mole. So we actually were given what we're given here is \( \bar{M} \). So \( \bar{M} \) is equal to 0.028. Therefore, we're going to be using this equation here. So the most probable speed of 300 Kelvin is going to be the square root of \( \frac{2RT}{\bar{M}} \). So this is going to be a big \( M \) here. So if you work this out, you're going to get the square root of \( 2 \times 8.314 \times 300 \) and then you're going to divide this by 0.028. If you go ahead and work this out, what you're going to get is 422 meters per second.

Alright. 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 the next speed on our list here. 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 going to be the square root of \( \frac{8k_BT}{\pi m} \). So we can also rewrite this one as well. This is going to be the square root of \( \frac{8RT}{\pi \bar{M}} \).