  We examine the ability of stellar X-rays to desorb molecules from grains in
outer protoplanetary disks. In particular, we consider the possibility of spot
heating by X-rays and examine its effectiveness, compared to whole grain
heating by X-rays, in sustaining a gas phase abundance of CO. As found in
previous studies, whole grain heating is effective only for small grains (< 500
A). As a result, large grains are a permanent sink of CO and whole grain
heating cannot sustain an equilibrium gas phase abundance of CO. Spot heating,
in which the incident X-ray deposits energy in only a restricted region of the
grain volume and which may occur as a consequence of the aggregate nature of
grains, proves to be more promising. Assuming that grains are comprised of many
thermal subunits that are poorly connected thermally, we find that spot heating
is efficient at all grain sizes if large grains are effectively ``coated'' with
small grains < 100 A in size. We discuss the implications of X-ray spot heating
for the interpretation of millimeter emission line studies of outer
protoplanetary disks.
