# Prove that Entropy is Extensive

1. Oct 8, 2011

### Tsar_183

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!!

Last edited: Oct 8, 2011
2. Jan 28, 2017

### TeethWhitener

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).