00:01
Here we're going to be looking at the maxwell velocity distribution and learning some things about it.
00:08
So first of all, what it does is kind of takes a total number of, say, gas particles.
00:17
And so these are non -interacting point particles.
00:21
But it takes that number and kind of makes a histogram of where they are in terms of their velocity.
00:30
Now, the distribution in that number as a function of speed is not asymmetric bell -shaped curve, although it does come from a gaussian.
00:45
But there's a very long tail to the distribution at high speeds.
00:51
The average of that distribution then is not going to fall at the peak of the distribution because it is asymmetrical.
01:01
So the average actually falls off the average.
01:13
So what we're going to be doing is taking this distribution and getting a sense of where the most probable speed is, most probable.
01:26
And of course, that's where the distribution itself has a maximum.
01:32
So let's take a look at the distribution.
01:34
It's kind of ugly looking.
01:37
There's a constant out in front that involves m mass of the particles, mass of single particle, boltzman's constant, t is temperature, etc.
01:58
So we'll just kind of make a table here, boltzman's constant, and t is temperature, pi, of course, is pie, that's always true.
02:13
But it's just a constant which you can imagine comes from making sure that the area under that distribution curve adds up to all of the particles n.
02:28
You don't want to lose anybody.
02:31
But how do we find the most probable? what we're going to have to do is kind of the calculus thing, is to take the function, and of v, take its derivative with respect to v, and set that equal to zero.
02:49
And the part of it, i'm going to take the derivative of.
02:52
We're not going to worry about all the constants out in front.
02:56
The part we're going to take the derivative with respect to is the non -constant part, the part that depends on velocity.
03:05
And that's going to be hard enough.
03:09
But let's go ahead and do that and see if we can find that most part.
03:14
Probability solve to get most probable and we'll find some other things along the way but that derivative let's see so we'll have to kind of do that in steps so the first part comes from the derivative of v squared and then we have the e to the minus i'm just going to show it with the brackets and you'll see why later.
03:54
The second part, we're going to do the summation, the product rule...