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Paramagnet entropy

  1. Mar 18, 2014 #1
    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)
    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
     
  2. jcsd
  3. Mar 18, 2014 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    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.
     
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