00:01
In this problem on the topic of entropy, we are told that n molecules exist in a box, and there are two configurations, a, which has an equal division of the molecules between two halves of the box, and b with 60 % of the molecules in the left half and 40 % of the right half of the box.
00:16
For a value of n is equal to 50, we want to find the multiplicity wa of configuration a, the multiplicity wb for configuration b, the ratio f of b slash a of the time the system spends in configuration b to the time it spends in configuration a.
00:34
And then for n is equal to 100, we want to again find wa, wb, fb slash a, and then repeat this for n is equal to 200.
00:45
And lastly, with increasing n, we want to know if f increases, decreases, or remains the same.
00:53
Now, for configuration a, we know that the multiplicity w is equal to n, factorial over the number in the left half n over 2 factorial times the number in the right half n over 2 factorial, which in this case is 50 factorial divided by 25 factorial times 25 factorial, which is a multiplicity of 1 .26 times 10 to the power 14.
01:31
Now we need to do the same for configuration b.
01:38
And in configuration b, we have wb to be n factorial, in this case is divided by 0 .6n factorial, multiplied by 0 .4n factorial.
01:58
And so in this case, this is 50 factorial, divided by 0 .6 times 50 all factorial.
02:08
Times 0 .4 times 50 or 40 % or factorial, which gives us a multiplicity of 4 .71 times 10 to the power 13...