How Does Entropy Relate to the Arrangement Factor in Statistical Mechanics?

[bobey]

based on this question :

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i tried to answer the question as follow :

My answer for the first part of the question :

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My answer for the second part of the question :

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The assumption is bad. This is because the most probable distribution will effectively count, in this case the {4,1,0,1,0} state is the most probable state, and we will have S=k ln WD*., where WD* is 30.

my problem is really i didn't understand the last part of the question...this is my other attempt for the second part:
Since all the states are equiprobable to occur, the probability of the 1st level is 1/6 x 5 = 5/6 and the probability of the 5th level is 1/6 x 1 = 1/6 while the arrangement factor, W for the 1st level is (6 x 5 x 4 x 3 x 2)/5! = 6 and the arrangement factor, W for the 5th level is = 6!/1! = 720... is that means the probability for 1st level is 6/726 and the probability for the 5th level is 720/726? i get confuse with the earlier probability and the arrangement factor? what it has to be related with entropy? anyone can clarify it?

and the arrangement factor = probabalility iff all the molecules are at T = 0 which means S = 0... based on the example. probability = 1/6 x 6 = 1 and the arrangement factor, W = 6! / 6! = 1...

is my understanding towards the question is correct or I just misunderstanding it?
ANYONE CAN CLARIFY IT? PLZ3X

 

[Amok]

The arrangement factor just gives you the number of possible arrangements (microstates) given a set of assumptions (in this case, the ones used to derive Boltzmann statistics). I'm not sure I get the last question either. Given how the arrangement factor is described here, this is will lead to a Boltzmann distribution in a system where there's no degeneracy. The probability of a particle being in state i (energy level i) is given by:

$$P_i = (n_i/N) = \frac{g_i e^{-E_i/K_b T}}{Z}$$
where:
$$Z = \sum_{i=0}^{4} g_i e^{-E_i/K_b T}$$

T is the temperature, Kb is the Boltzmann constant, Ei is the energy of the ith state and gi is the degeneracy of the ith state (always = 1 in your case).

EDIT: W8, I was wronng. I'll get back to this ASAP. I do hope this helps in the meanwhile. I have to be honest and say I don't get what you're doing to compute those probabilities. Could you explain it?