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Prove that Entropy is Extensive

  1. Oct 8, 2011 #1
    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] 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 [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] pi ln(pi) ==>
    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!!
    Last edited: Oct 8, 2011
  2. jcsd
  3. Jan 28, 2017 #2


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    Science Advisor
    Gold Member

    This looks fine to me. The number of microstates for two noninteracting systems Ω(A,B) is the product of the number of microstates for each system individually Ω(A)Ω(B).
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