  Protoplanetary disks are thought to evolve viscously, where the disk mass -
the reservoir available for planet formation - decreases over time as material
is accreted onto the central star. Observations show a correlation between dust
mass and the stellar accretion rate, as expected from viscous theory. However,
the gas mass inferred from 13CO and C18O line fluxes, which should be a more
direct measure, shows no such correlation. Using thermochemical DALI models, we
investigate how 13CO and C18O J=3-2 line fluxes change over time in a viscously
evolving disk. We also investigate if the chemical conversion of CO through
grain-surface chemistry combined with viscous evolution can explain the
observations of disks in Lupus. The 13CO and C18O 3-2 line fluxes increase over
time due to their optically thick emitting regions growing in size as the disk
expands viscously. The C18O 3-2 emission is optically thin throughout the disk
for only a subset of our models (Mdisk (t = 1 Myr) < 1e-3 Msun). For these
disks the integrated C18O flux decreases with time, similar to the disk mass.
The C18O 3-2 fluxes for the bulk of the disks in Lupus (with Mdust < 5e-5 Msun)
can be reproduced to within a factor of ~2 with viscously evolving disks in
which CO is converted into other species through grain-surface chemistry driven
by a cosmic-ray ionization rate zeta_cr ~ 5e-17 - 1e-16 s^-1. However,
explaining the stacked C18O upper limits requires a lower average abundance
than our models can produce and they cannot explain the observed 13CO fluxes,
which, for most disks, are more than an order of magnitude fainter than what
our models predict. Reconciling the 13CO fluxes of viscously evolving disks
with the observations requires either a combination of efficient vertical
mixing and a high zeta_cr or low mass disks (Mdust < 3e-5 Msun) being much
thinner and/or smaller than their more massive counterparts.
