(adsbygoogle = window.adsbygoogle || []).push({}); 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 [itex]\sum[/itex] p_{i}ln(p_{i})

3. The attempt at a solution

I honestly have no idea where to start! I tried letting p_{i}= 1/Ω, to obtain,

S = k [itex]\sum[/itex] (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 [itex]\sum[/itex] p_{i}ln(p_{i}) ==>

S = k [itex]\sum[/itex] (1/Ω) ln(Ω) ==>

S = k (1/Ω) ln(Ω)[itex]\sum[/itex] 1, [itex]\sum[/itex] 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!!

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# Homework Help: Prove that Entropy is Extensive

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