  We examine the composition of barium stars in the context of mass transfer
from an asymptotic giant branch (AGB) companion. We accrete between 0.01 and
0.5 M$_\odot$ of AGB ejecta on to low mass companions of [Fe/H] = -0.25 at the
ages expected for the end of the lives of AGB stars of 2.5, 3 and 4M$_\odot$.
In each case, we form a star of 2.5 M$_\odot$ which is thought to be a typical
barium star mass. We discuss the extent of dilution of accreted material as the
star evolves, and describe the impact on the surface abundances. For accretion
from a 2.5\ms\ primary, if the secondary's initial mass is 2.45 M$_\odot$ or
more, accretion takes place when the secondary is undergoing core helium
burning. Using data from the sample of De Castro et al., we attempt to fit the
observed properties of 74 barium giants using the models we have computed. We
find that all but six of these objects are best fit using ejecta from 2.5
M$_\odot$ (32 objects) or 3 M$_\odot$ (36 objects) AGB stars. Higher accretion
masses are typically required when accreting from a lower mass companion. We
find accretion masses that are broadly consistent with recent hydrodynamical
simulations of wind mass transfer, though the accretion efficiency is toward
the upper limit found in these simulations. For the 18 stars with reported
orbital periods, we find no strong correlations between period and accretion
mass.
