Stat phys - free energy from eq of state

[lol_nl]

Homework Statement

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?

Homework Equations

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

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.

 

[NateD]

I'm guessing there is a simpler way to calculate the free energy of a hard sphere fluid (given the equation of state) , but I can't seem to understand it. Any help would be greatly appreciated.