00:01
In this problem, we're going to go over problem 2 .11 in schroeder's thermal physics textbook.
00:09
In this problem, we have a system composed of two paramagnets.
00:14
So this red box indicates that the system is closed.
00:20
We have paramagnet a and paramagnet b, where both pair of magnets have 100 dipoles each, where i've represented these dipoles as these half arrows.
00:33
So i'm just going to specify the number of dipoles as in.
00:56
And these paramagnets are allowed to exchange energy between each other.
01:03
So within this system there's 80 units of energy, where whatever amounts of energy, paramagnet a doesn't have, paramagnet b will have, and vice versa.
01:23
What we want to find in this problem is, is the most likely macro state and the least likely macro state and the probabilities of both of these macro states.
01:35
A macro state in the system will specify how many units of energy are in paramagnet a and how many are in paramagnet b.
01:45
To find these least and most probable macrostates, we're going to use excel, where in the column on the right, omega total, each column we're going to to have represent a different macro state.
02:05
Now the most probable macro state is going to be the one with the highest multiplicity and the least probable macro state will be the one with the lowest multiplicity, where multiplicity for the entire system gives the number of different ways that the energy can rearrange itself within the both of the paramagnets.
02:32
So each macro state will have a different amount of energy in paramagnet a.
02:39
So these possible amounts of energy is anywhere from 0 to 80.
02:47
So in this first column, we're going to type in those macro states from 0 to 80, where we can just type in a few in the cells and then highlight these three cells by clicking and dragging.
03:05
And then we're going to go to the bottom right corner where this black cross appears, and we're going to click and drag it down, which is called filling down.
03:18
What this does is excel detects a pattern between the numbers that we've already typed in, and it will fill in the rest.
03:33
And what this corresponds to is here in column c, we're going to fill out the units of energy for pagmatic.
03:42
So when paramagnet a has zero units of energy, that means paramagnet b must have 80 since it has whatever paragmatic magnet a does not have.
03:57
So for one unit of energy in paramagnet a, there will be 79 units of energy in paragagnet b.
04:04
And with two units of energy in a, there will be 78 and b and so on.
04:09
So we're going to do the same thing we did before by highlighting the fields and clicking on the bottom right and dragging down so now we've specified all of our macro states now in order to find the total here the total multiplicity of the system for each individual macro state of the system we need to first find the multiplicity of paramagnet a for each of these states for each of these macrostates and then find the multiplicity of b for each of these macro states.
04:55
And then we will multiply them together to get the total.
05:00
So first, we want to find the multiplicity.
05:06
We know, let's go back to the blackboard where the multiplicity is equal to this standard abbreviation for the multiplicity, which is not exactly how we actually how we calculate it, but we're going to have n diples and out of those n dipoles we're going to choose some of them to be up which is this value right here and why we're going to do that is because n up is actually equivalent to the number of units of energy in that paramagnet because the parac magnet will be down when it does not have energy but it needs a unit of a unit of a energy in order to flip upwards, which is equivalent to saying in choose q.
06:12
We're going back to our excel sheet where q is in the column to the left of the multiplicity.
06:23
So we're going to write the function of multiplicity into excel, where we're going to write equals combine without an e, and then in parentheses.
06:38
Parentheses we're going to first put n, which in this case we have n is 100, there's 100 dipoles, and then comma q or in up, where we're going to reference the cell to the left of it.
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In this case is a2, and then hit enter.
07:07
So now if we want to find the multiplicities easily for the rest of the cells, we're just going to click on this cell, go to the bottom right, and drag down where the function will be modified, such that here, after the comma, we have cell a2 referenced, but in the next ones, it modifies the formula after the comma to be consistent with what the cell is to the left of it.
07:59
So now we're going to do the same with paramagnet b.
08:03
We're going to find the multiplicities of each of these macro states that specifies a different number of units of energy in the paramagnet.
08:14
So we have equals combine parentheses in the number of dipoles, which is 100, comma, that number of dipoles we're going to choose to be in that position, which is just equal to the units of energy.
08:33
Which is in the cell to the left of it.
08:37
So right here, this multiplicity we need to reference c2, cell c2.
08:46
And then we're going to modify it for the rest of the cells so that we get the multiplicities for each of the columns, for each of the rows.
09:04
Now we can just find the total multiplicity of each of these macrostates, the overall, this is the macrostates of the the system by multiplying the multiplicities.
09:20
So we have to reference.
09:24
So for this particular macro state, we're going to have cell b2 times cell d2.
09:38
And then we're going to hit enter.
09:41
So we can manually just type that in, redo the formula for each cell.
09:46
Like we can go ahead and say equals for this column.
09:53
I mean for this row, we can say cell b3 times cell d3 and hit enter.
10:03
But it's a lot easier if we just again click on this cell and then just have excel change the function such that it can be applied to every cell following it.
10:27
So now we have all of our multiplicities for every macro state.
10:31
So our most probable macro state is the one with the highest multiplicity, which looking through we see that it must be one of the powers to the 56...