  We measure the concentration of carbon adatoms on the Ru(0001) surface that
are in equilibrium with C atoms in the crystal's bulk by monitoring the
electron reflectivity of the surface while imaging. During cooling from high
temperature, C atoms segregate to the Ru surface, causing graphene islands to
nucleate. Using low-energy electron microscopy (LEEM), we measure the growth
rate of individual graphene islands and, simultaneously, the local
concentration of C adatoms on the surface. We find that graphene growth is fed
by the supersaturated, two-dimensional gas of C adatoms rather than by direct
exchange between the bulk C and the graphene. At long times, the rate at which
C diffuses from the bulk to the surface controls the graphene growth rate. The
competition among C in three states - dissolved in Ru, as an adatom, and in
graphene - is quantified and discussed. The adatom segregation enthalpy
determined by applying the simple Langmuir-McLean model to the
temperature-dependent equilibrium concentration seriously disagrees with the
value calculated from first-principles. This discrepancy suggests that the
assumption in the model of non-interacting C is not valid.
