Using a straightforward extension of the Onsager-theory for hard rods, we consider the ther-
modynamic stability of the isotropic (I) and nematic (N) phase of binary mixtures of thick and
thin hard rods of the same length. We show that such mixtures not only exhibit the expected
I{N ordering transition and the previously predicted depletion driven I{I demixing transition,
but also a N{N demixing transition driven by the orientation entropy of the thinner rods. For
various values of the diameter ratio of the two species we present the phase diagrams, which
exhibit I{N, I{I and N{N coexistence, I{N{N and I{I{N triple points, and I{I and N{N critical
points. We also present the results of computer simulations of the I{I and I{N coexistence for
diameter ratio 1 : 10, which are in good agreement with the theory.
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