00:01
For this problem on the topic of entropy, we have n -gas molecules contained in a box, and we want to consider the box divided into three equal parts, and write a formula for the multiplicity of any given configuration.
00:11
We then want to consider two configurations, a, with equal numbers of molecules in all three -thirds of the box, and b with equal numbers of molecules in each half of the box, if the box is now divided into two equal parts rather than three.
00:23
We then want to find the ratio of the multiplicity of configuration a to that of b, and then find this ratio again for n is equal to a hundred.
00:34
Now given n molecules, if the box is divided into m equal parts with n1 molecules in the first and n2 in the second, such that n1 plus n2 all the way up to addition of nm gives the total number of molecules capital n.
00:48
There are n factorial arrangements of the n molecules, but n1 factorial are simply rearrangements of the n1 molecules in the first part, and n2 factorials are rearrangements of the n2 molecules in the two factorials the second.
01:00
The rearrangements do not produce a new configuration and therefore the multiplicity factor w is equal to n factorial where n is a total number of molecules over n1 factorial times n 2 factorial times in 3 factorial all the way up until n m factorial and so if we are to suppose for part a that they are n molecules in the left that are the box, nc molecules in the center third and nr in the right third, using the argument above, we get the multiplicity w to be n factorial divided by n l factorial times n c factorial times n r factorial, where n l plus nc plus nr is equal to capital n...