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## Homework Statement

If we assume entropy is a function of the multiplicity, [tex]\Omega[/tex], (S=k*f([tex]\Omega[/tex])) show that that function f([tex]\Omega[/tex]) is ln([tex]\Omega[/tex]).

## Homework Equations

## The Attempt at a Solution

[tex]\Omega[/tex] can be written as N!/ni!. By using stirling's approximation, this becomes [tex]\Omega[/tex]= ((N/e)^N)/((n1/e)^n1*(n2/e)^n2*...(ni/e)^ni). We know that the probability pi=N/ni so this reduces to W=1/(p1^n1*p2^n2*...*pi^ni). To make this user friendly take the log so ln([tex]\Omega[/tex])=-[tex]\Sigma[/tex]pi*ln(pi).

I just started down the road of trying to use definition of multiplicity and probabilities and I did get to ln([tex]\Omega[/tex]), but it doesn't seem like I'm really doing a solid proof and I'm not sure what's missing/ how to tie it together.