Statistical Mechanics · Problem 3.12

The Grand Canonical Ensemble — R. K. Pathria

Physics — Statistical Mechanics · free-volume-classical-system

Problem

If the ``free volume''

\bar{V}

of a classical system is defined by the equation

\bar{V}^N = \int e^{[\bar{U} - U(\mathbf{q}_i)]/kT} \prod_{i=1}^{N} d^3q_i,

where

\bar{U}

is the average potential energy of the system and

U(\mathbf{q}_i)

the actual potential energy as a function of the molecular configuration, then show that

S = Nk\left[\ln\left\{\frac{\bar{V}}{N}\left(\frac{2\pi mkT}{h^2}\right)^{3/2}\right\} + \frac{5}{2}\right].

In what sense is it justified to refer to the quantity

\bar{V}

as the ``free volume'' of the system? Substantiate your answer by considering a particular case, e.g. the case of a hard sphere gas.