  In this work we study the $D^*$ and $D$ multiplicities and how they change
during the hadron gas phase of heavy ion collisions. With the help of an
effective Lagrangian formalism, we calculate the production and absorption
cross sections of the $D^*$ and $D$ mesons in a hadronic medium. We compute the
time evolution of the abundances and the ratio $D^* /D$. They are approximately
constant in time. Also, assuming a Bjorken type cooling and using an empirical
relation between the freeze-out temperature and the central multiplicity
density, we estimate $D^* /D$ as a function of $ dN /d \eta (\eta =0)$, which
represents the system size. We find that, while the number of $D^*$'s and $D$'s
grows significantly with the system size, their ratio remains approximately
constant. This prediction can be compared with future experimental data. Our
results suggest that the charm meson interactions in the hadron gas do not
change their multiplicities and consequently these mesons are close to chemical
equilibrium.
