The Free Volume for Rigid Sphere Molecules

The free volume is calculated for non-interacting rigid sphere molecules taking into account the exact geometry imposed by the face-centered cubic packing. The size and shape are quite different from that of the inscribed spheres which correspond to the Lennard-Jones and Devonshire approximation. At high densities the equation of state obtained from the exact treatment agrees well with the Eyring or Lennard-Jones and Devonshire equation. When the specific volume is greater than twice the cube of the collision diameters, the molecules are no longer confined to cages formed by neighboring molecules. At these low densities the free volume concept is ambiguous. The equation of state depends upon the shape and orientation of the cells with respect to the lattice positions of the molecules. A particular choice is considered which leads to an equation of state that at low densities is accurate through the second virial coefficient. There are other shapes and orientations of the cells which would lead to other equations of state having this same property.