  The first goal of this article is to study the validity of the Zwanzig model
for liquid crystals to predict transitions to inhomogeneous phases (like
smectic and columnar) and the way polydispersity affects these transitions. The
second goal is to analyze the extension of the Zwanzig model to a binary
mixture of rods and plates. The mixture is symmetric in that all particles have
equal volume and length-to-breadth ratio, $\kappa$. The phase diagram
containing the homogeneous phases as well as the spinodals of the transitions
to inhomogeneous phases is determined for the cases $\kappa=5$ and 15 in order
to compare with previous results obtained in the Onsager approximation. We then
study the effect of polydispersity on these phase diagrams, emphasizing the
enhancement of the stability of the biaxial nematic phase it induces.
