Solving the Paramagnet Entropy Equation

[Dassinia]

Homework Statement

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) ) ]

Homework Equations

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)
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 .?

Thanks

 

[TSny]

Note that for any number x,

N*lnN = (1/2)(N+x)lnN + (1/2)(N-x)lnN

Try to choose an appropriate value for x that will lead to the desired result.