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Stat phys - free energy from eq of state

  1. Sep 20, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider a mixture of hard spheres of diameter σ. The potential energy
    for a hard sphere system is given by
    [itex]\beta U(r) = 0 (r > \sigma)
    [/itex]
    [itex]
    ∞ (r ≤ \sigma)
    [/itex]

    The packing fraction (η) of the system is the amount of space occupied
    by the particles.

    (b) The equation of state for the hard sphere fluid is approximately
    [itex]
    \frac{P_{liq}V}{Nk_{B}T}= \frac{1+ \eta + \eta^2 - \eta^3}{ (1 - \eta)^3 }
    [/itex]

    What is the corresponding free energy?

    2. Relevant equations
    Hint: At very low packing
    fraction the hard sphere liquid acts like an ideal gas.

    3. The attempt at a solution
    Frankly, I have no idea how to calculate the free energy from an equation of state like the one given above. Even in the case of the ideal gas ([itex]\eta=0[/itex], I would suppose the free energy would have to calculated in a different manner. The way I learned the calculation for the ideal gas was quite complicated, beginning with a calculation of the partition function of a single molecule by looking at quantum densities. Once given the partition function, it was not difficult to show that the Helmholtz free energy for an ideal gas is given by [itex] F = -k_{B}T Log(Z) \approx N k_{B}T (Log(\frac{N}{V n_Q}) - 1) [/itex] where [itex] n_{Q} [/itex] is a (scaling?) constant.
     
  2. jcsd
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