Any account of the chemistry within dense clouds must consider the propensity of molecules to stick to grain surfaces. Consider a volume V of cloud gas with a number density nd of spherical grains, each with radius ad. In the reference frame of a molecule, the grains are all moving with Vtherm, the molecule's thermal speed:
Here A is the molecule's mass relative to hydrogen. In a time interval At, each grain sweeps out the cylindrical volume na2d Vtherm At, and all grains sweep out a volume larger by ndV. Hence, the probability per unit time that the molecule is struck by some grain is the ratio of this total volume to V, or nd na?d Vtherm. Inverting this probability gives the average time for a collision to occur:
Here we have used equation (2.42) for the total geometric cross section of the grains per hydrogen atom.
The quantity tcoll measures the time to deplete significantly a given molecule, provided there is a high probability of sticking upon collision. Such is the case for all molecules except H2, which does not readily adhere to grain mantles. Consider CS, for which Vtherm = 5.3 x 103 cms-1 at T = 10 K. From equation (5.4), we find that tcon is only 6 x 105 yr at the center of a dense core with nH = 104 cm-3. Once again, we face the dilemma that the disappearance time is brief compared to the expected cloud age. To put the matter another way, chemical models without grain depletion of molecules give a reasonable match to the observed CS abundance.
There evidently must exist some mechanism for reinjecting molecules from grain surfaces back into the gas phase. Ultraviolet photons would serve the purpose, but too few of them penetrate dark cloud interiors. In sufficiently small grains, the heat from surface chemical reactions could raise the grain temperature enough to sublimate many species. However, for standard grains within dark clouds, the problem of rapid depletion of the molecules remains unsolved.
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