# Homework Help: Stat phys - free energy from eq of state

1. Sep 20, 2012

### lol_nl

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
$\beta U(r) = 0 (r > \sigma)$
$∞ (r ≤ \sigma)$

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
$\frac{P_{liq}V}{Nk_{B}T}= \frac{1+ \eta + \eta^2 - \eta^3}{ (1 - \eta)^3 }$

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 ($\eta=0$, 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 $F = -k_{B}T Log(Z) \approx N k_{B}T (Log(\frac{N}{V n_Q}) - 1)$ where $n_{Q}$ is a (scaling?) constant.