An exactly solvable model for depletion phenomena

The equilibrium properties of a one-dimensional mixture of hard rods of two different lengths are obtained. These are used to test the free-volume approximation, which has in the past been applied in the study of hard-sphere mixtures in three dimensions. Three different versions of the free-volume approximation are tested. In the most primitive of these the smaller spheres (or rods) are assumed not to interact with each other. In higher levels of the approximation the interactions between the smaller molecules are partly taken into account. The results show the ranges of compositions and densities over which the approximations are accurate and they identify one form of the approximation as the most reliable.