  Dust particles in protoplanetary disks experience various chemical reactions
under different physicochemical conditions through their accretion and
diffusion, which results in the radial chemical gradient of dust. We performed
three-dimensional Monte Carlo simulations to evaluate the dust trajectories and
the progress of fictitious irreversible reactions, of which kinetics is
expressed by the Johnson-Mehl-Avrami equation. The distribution of the highest
temperature that each particle experiences before the degree of reaction
exceeds a certain level shows the log-normal distribution, and its mode
temperature was used as the effective reaction temperature. Semi-analytical
prediction formulas of the effective reaction temperature and its dispersion
were derived by comparing a reaction timescale with a diffusive transport
timescale of dust as a function of the reaction parameters and the disk
parameters. The formulas reproduce the numerical results of the effective
reaction temperatures and their dispersions within 5.5 and 24 %, respectively,
in a wide temperature range (200-1400 K). We applied the formulas for the
crystallization of amorphous silicate dust and its oxygen isotope exchange with
the H2O vapor based on the experimentally determined kinetics. For sub-micron
sized amorphous forsterite dust, the predicted effective reaction temperature
for the oxygen isotope exchange was lower than that of crystallization without
overlap even considering their dispersions. This suggests that the amorphous
silicate dust in the protosolar disk could exchange their oxygen isotopes
efficiently with the 16O-poor H2O vapor, resulting in the distinct oxygen
isotope compositions from the Sun.
