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 bolts man distribution is a probability distribution that describes the speed of ideal gasses 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 to a three 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 to a three for your velocity. So here we have three different velocities, or speed that is involved with a distribution curve. So here we have to. Then we have eight, and then we have three. 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 gas is 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 eight rt over em 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 three rt over em. Now 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 our becomes 8.314 and will be jewels over moles times K. And then temperature is always is in Kelvin. When we're dealing with calculations Now, this probable speed means speed and root mean square speed can be transfixed or put onto a distribution curve. So let's take a look at this distribution curve here. So we're gonna say here that our distribution curve on our Y axis we have our probability distribution. This is can be seen as the likelihood off X number of gas molecules with within a container existing there. So and on our X axis, we have our velocity from zero all the way up to 13 50 m 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 m per second, most likely if you're gonna look at a particular gas, it has the most likely chance off following here around 400 m 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 are probable speeds around 400 m 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 routing square speed. We see that it kind of falls close to 600 m per second. So here this distribution curve, this is showing us the varying velocities for collection of gasses 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 are probable. Speed would be the lowest in terms of velocity when it comes to this chart. Okay, so that's the way we need to think about it in terms off the different types of velocities or speeds that exists 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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So here it says, to calculate the most probable speed of F two molecules at 335 Kelvin. So remember most probable speed is V p equals square root of two R t over the Mueller Mass. So here we're gonna say that we have square root of two times 8314 Now remember 8.314 uses the units of jewels over moles times K. But remember that one Jewell equals kilograms times meter squared over second squared, so I can substitute this in for Jules. When we do that, that's gonna give me my units off 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 two molecules, we have two florins and according to the periodic table, there are atomic masses are 19 g respectively, so that gives me 38 g per mole. Converting eight. Converting the grams to kilograms means that we'll have 80. kg per mole. Cancel out units, moles Cancel out Calvin's 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 3 83 m per second as the most probable speed for F two molecules.

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Problem

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

A

0.0105 kg/mol

B

0.0136 kg/mol

C

0.0238 kg/mol

D

0.0262 kg/mol

E

0.0281 kg/mol

Additional resources for Maxwell-Boltzmann Distribution

PRACTICE PROBLEMS AND ACTIVITIES (3)

- Consider the following graph. (c) For each curve, which speed is highest: the most probable speed, the root-me...
- A mixture of chlorine, hydrogen, and oxygen gas is in a container at STP. Which curve represents oxygen gas? ...
- (c) Calculate the most probable speed of an argon atom in the stratosphere, where the temperature is 0 °C.