  We study the equilibrium behaviour of a mixture of monodisperse hard sphere
colloids and polydisperse non-adsorbing polymers at their $\theta$-point, using
the Asakura-Oosawa model treated within the free-volume approximation. Our
focus is the experimentally relevant scenario where the distribution of polymer
chain lengths across the system is fixed. Phase diagrams are calculated using
the moment free energy method, and we show that the mean polymer size $\xi_{\rm
c}$ at which gas-liquid phase separation first occurs decreases with increasing
polymer polydispersity $\delta$. Correspondingly, at fixed mean polymer size,
polydispersity favours gas-liquid coexistence but delays the onset of
fluid-solid separation. On the other hand, we find that systems with different
$\delta$ but the same {\em mass-averaged} polymer chain length have nearly
polydispersity-independent phase diagrams. We conclude with a comparison to
previous calculations for a semi-grandcanonical scenario, where the polymer
chemical potentials are imposed, which predicted that fluid-solid coexistence
was over gas-liquid in some areas of the phase diagram. Our results show that
this somewhat counter-intuitive result arose because the actual polymer size
distribution in the system is shifted to smaller sizes relative to the polymer
reservoir distribution.
