Prove that Entropy is Extensive

[Tsar_183]

Homework Statement

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.

Homework Equations

S = -k $\sum$ p_i ln(p_i)

The Attempt at a Solution

I honestly have no idea where to start! I tried letting p_i = 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 $\sum$ p_i ln(p_i) ==>
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!

 

[TeethWhitener]

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.

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).