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Finding chemical potential with given thermodynamic relation

  1. Nov 2, 2016 #1
    1. The problem statement, all variables and given/known data
    Suppose you are given the following relation among the entropy S, volume V , internal energy U, and number of particles N of a thermodynamic system, where A is a constant.:
    $$ S = A(NVU)^{\frac 1 3} $$
    Find the chemical potential μ(T,P)
    2. Relevant equations
    $$ \frac μ T = -(\frac{∂S}{∂N})_{U,V} $$
    U = TS - PV + μN
    3. The attempt at a solution
    $$ \frac μ T = -(\frac{∂S}{∂N})_{U,V} = \frac 1 3 A N^{-\frac 2 3} (VU)^{\frac 1 3} $$
    This only solved μ(U,V,N)
    Since internal energy is a function of N,V,T : U(N,V,T)
    $$ ∴ μ(U,V,N) → μ(T,\frac V N) → μ(T,P) $$
    However, I don't know how to carry out the above transform in this case. I don't know U(N,V,T)

    Thanks!
     
  2. jcsd
  3. Nov 2, 2016 #2

    DrDu

    User Avatar
    Science Advisor

    Try to express p and T in terms of U, S and V, using dU=TdS-pdV.
    Then set up G(T,p,N)=U-TS+pV.
    Calculate mu from G.
     
  4. Nov 2, 2016 #3
    Oh! Thanks, I didn't think of that before
     
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