Paramagnet entropy

1. The problem statement, all variables and given/known data

As a model of a paramagnet, consider a system of N fixed particles with spin 1/2 in a magnetic fiels H along z axis. Each particle has an energy e=μH (spin up) or e=-μH

Using S=kln(Ω), show that

S=k [ (N-E/e)/2 ln( 2N/(N-E/e) ) + (N+E/e)/2 ln( 2N/(N+E/e) ) ]

2. Relevant equations

3. The attempt at a solution

Ω= N!/(N+!N-!)
I used the Stirling approximation
ln(Ω)= NlnN – ( n+ ln(n+) + n ln(n-) )
Then replaced
n+=1/2 (N+E/e)

S= N*lnN + 1/2(N+E/e) ln (2/ (N+E/e)) + 1/2(N-E/e) ln (2/(N-E/e)

Then I don’t know what to do with the N*lnN to get the (2N) in the numerator inside the ln .?


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