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Correct option for n dependence of free energy f per unit

  1. Apr 4, 2017 #1
    1. The problem statement, all variables and given/known data
    The equation of state of an ideal gas is p = nkT, where p is the thermodynamic
    pressure and n = N / V is the thermodynamic variable for the number of particles per
    unit volume. The n dependence of the free energy f per unit volume of the ideal gas is
    obtained by the following expression , where c is temperature-dependent constant k is boltzmann constant.
    (a) nkT[In(n)+c]
    (b) 2nkT[n ln(n)+c.]
    (c) 3/2 nkT
    (d) 3nkT

    2. Relevant equations
    ∂f/∂n=μ
    pV=NkT
    p=NkT/V
    3. The attempt at a solution
    internal energy U=U(S,V,N)
    ∴μ=∂U/∂N
    and ∂μ/∂V=∂2U/∂N∂V= -∂p/∂N
    ∂μ/∂V=-kT/V
    ∴μ=-kTlnV+c
     
  2. jcsd
  3. Apr 5, 2017 #2
    If you know these are free particles (i.e. potential energy term in the Hamiltonian is 0) then the best start would be to calculate the N particle partition function, it's usually called Z or QN. Once you have the partition function, the Helmholtz free energy is given by: A(N,T,V) = -kTln(Z). You can then use the laws of logarithms as well as the Stirling Approximation (to estimate the term ln(N!)).
     
  4. Apr 18, 2017 #3
    thank you.
     
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