If the "free volume" \(\bar{V}\) of a classical system is defined by the equation

\[
\bar{V}^{N}=\int e^{\left\{\bar{U}-U\left(\boldsymbol{q}_{i}ight)ight\} / k T} \prod_{i=1}^{N} d^{3} q_{i}
\]

where \(\bar{U}\) is the average potential energy of the system and \(U\left(\boldsymbol{q}_{i}ight)\) the actual potential energy as a function of the molecular configuration, then show that

\[
S=N k\left[\ln \left\{\frac{\bar{V}}{N}\left(\frac{2 \pi m k T}{h^{2}}ight)^{3 / 2}ight\}+\frac{5}{2}ight] .
\]

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 - for example, the case of a hard sphere gas.