Tsar_183
Oct8-11, 06:08 PM
1. The problem statement, all variables and given/known data
Show explicitly that Entropy as defined by the Gibbs Entropy Formula is extensive. That is, for two independent (noninteracting) systems A and B,
S(A,B) = S(A) + S(B)
where S(A,B) is the entropy of A and B considered as part of a larger system.
2. Relevant equations
S = -k \sum pi ln(pi)
3. The attempt at a solution
I honestly have no idea where to start! I tried letting pi = 1/Ω, to obtain,
S = k \sum (1/Ω)ln(Ω), and then tried summing S(A) and S(B) together to obtain S(A,B), but it didn't work out. I also tried just summing up S(A) and S(B) without writing in terms of Ω...didn't work either. I then tried,
S = -k \sum pi ln(pi) ==>
S = k \sum (1/Ω) ln(Ω) ==>
S = k (1/Ω) ln(Ω)\sum 1, \sum 1 = Ω
S = k (1/Ω) ln(Ω)Ω
S = k ln(Ω)
and then I summed up S(A) and S(B) which WORKED,
S(A,B) = k ln(Ω(A))+k ln(Ω(B)) = k ln(Ω(A)Ω(B)) = k ln (Ω(A,B)), but I don't think this argument works. Plus the prof derived the Gibbs Entropy Formula from k ln Ω... so I don't think i'm even on the right track! Any ideas or suggestions? Thanks!!
Show explicitly that Entropy as defined by the Gibbs Entropy Formula is extensive. That is, for two independent (noninteracting) systems A and B,
S(A,B) = S(A) + S(B)
where S(A,B) is the entropy of A and B considered as part of a larger system.
2. Relevant equations
S = -k \sum pi ln(pi)
3. The attempt at a solution
I honestly have no idea where to start! I tried letting pi = 1/Ω, to obtain,
S = k \sum (1/Ω)ln(Ω), and then tried summing S(A) and S(B) together to obtain S(A,B), but it didn't work out. I also tried just summing up S(A) and S(B) without writing in terms of Ω...didn't work either. I then tried,
S = -k \sum pi ln(pi) ==>
S = k \sum (1/Ω) ln(Ω) ==>
S = k (1/Ω) ln(Ω)\sum 1, \sum 1 = Ω
S = k (1/Ω) ln(Ω)Ω
S = k ln(Ω)
and then I summed up S(A) and S(B) which WORKED,
S(A,B) = k ln(Ω(A))+k ln(Ω(B)) = k ln(Ω(A)Ω(B)) = k ln (Ω(A,B)), but I don't think this argument works. Plus the prof derived the Gibbs Entropy Formula from k ln Ω... so I don't think i'm even on the right track! Any ideas or suggestions? Thanks!!