Maxwell-Boltzmann Distribution - Video Tutorials & Practice Problems

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The Maxwell-Boltzmann Distribution is a probability distribution curve that describes the speed of a gas at a specific temperature.

Distribution Curve

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concept

Maxwell-Boltzmann Distribution

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So the Maxwell Boltzmann distribution is a probability distribution that describes the speed of ideal gases at a given temperature. So as temperature changes, the velocities of gas molecules will vary. Now a probability distribution itself is just the region of the curve that shows the relative number of gas molecules. So we choose a temperature and we can get the relative number of gas molecules traveling at that velocity. Now to do this, we use a distribution function to determine these varying velocities, but it's beyond the scope of this class. So this complex formula, don't worry about it too much. You're not gonna see it within this course. Instead, we're gonna use our memory tool that just remember 283 for your velocity. So we'll see different types of velocities when looking at a distribution curve, and we'll use 283 to help us remember which is which.

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concept

Maxwell-Boltzmann Distribution

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So like I said earlier, just remember 283 for your velocity. So here we have 3 different velocities or speed that is involved with a distribution curve. So here we have 2, then we have 8, and then we have 3. The first speed or velocity represents the speed at the top of the curve that represents the largest number of molecules with that speed. This is called your probable speed. So if you had to guess you would say a majority of gases within a container have this speed most likely. Next, we have the average speed of gaseous molecules. What's another word for average? Mean. So here mean speed. And it's equal to square root of 8 r t over m. Now next, we have the speed that is the square root of the average speed squared. So that's just our root mean square speed. So if you've watched this video earlier, you know it's pretty familiar to us. So here root mean square speed equals square root of 3 r t over m. Now when we're talking about m r t, m here represents the molar mass of the gas in kilograms per mole. So all of these are kilograms per mole. Because we're talking about speed or velocity, remember our r becomes 8.314 and it'll be joules over moles times k, And then temperature as always is in Kelvin when we're dealing with calculations. Now this probable speed, mean speed, and root mean square speed can be transfixed or put on to a distribution curve. So let's take a look at this distribution curve here. So we're gonna say here that in our distribution curve, on our y axis we have our probability distribution. This is can be seen as the likelihood of x number of gas molecules within within a container existing there. So And on our x axis we have our velocity from 0 all the way up to 13 50 meters per second. We can see at the very top we have our probable speed. So what this is telling me, it's telling me at 400 meters per second, most likely, if you're gonna look at a particular gas, it has the most likely chance of falling here around 400 meters per second, because that's where the curve is the highest. Next, right next to it is we have our average or mean speed. Notice here that our probable speed's around 400 meters per second and our average or mean speed is a little bit higher than that. And then even higher than both of them is our root mean square speed. We see that it kind of falls close to 600 meters per second. So here this distribution curve is just showing us the varying velocities for collection of gases. And what we need to understand in terms of velocity, of increasing velocity, we would say that root mean square speed we can see is the highest because it's closest to 600. Then we can see next would be our average or mean speed, and then we would see that our probable speed would be the lowest in terms of velocity when it comes to this chart. So that's the way we need to think about it in terms of the different types of velocities or speeds that exist at any given temperature. That's the whole idea behind the Maxwell distribution curve.

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example

Maxwell-Boltzmann Distribution Example 1

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1m

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So here it says to calculate the most probable speed of f 2 molecules at 335 kelvin. So remember, most probable speed is vp equals square root of 2 r t over the molar mass. So here we're gonna say that we have square root of 2 times 8.314. Now remember, 8.314 uses the units of joules over moles times k. But remember that 1 joule equals kilograms times meter squared over seconds squared. So I can substitute this in for joules. When we do that, that's gonna give me my units of kilograms times meter squared divided by seconds squared, and then bring down moles times Kelvin as well, times 335 Kelvin divided by the molar mass in kilograms per mole for f 2 molecules. We have 2 fluorines, and according to the periodic table their atomic masses is 19 grams respectively. So that gives me 38 grams per mole. Converting it converting the grams to kilograms means that we'll have point 038 kilograms per mole. Cancel out units, moles cancel out, kelvins cancel out, kilograms cancel out. So we'll have here meter squared over second squared and we take the square root. When we do that we get 383 meters per second as the most probable speed for our f 2 molecules.

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Problem

Problem

Calculate the molar mass of an unknown gas if its average speed is 920 m/s at 303 K.