The mechanism for isotopic exchange by proton hopping and subsequent reorientation of H2O molecules in vapor-deposited, 0.1 mol % 2-naphthol containing amorphous solid water [M. Fischer and J. P. Devlin, J. Phys. Chem. 99, 11584 (1995)] has been reconsidered and an alternative mechanism in terms of diffusion controlled reactions is proposed. In this mechanism, H2O molecules diffuse within the H-bonded clusters, with or without a net increase in the number of H bonds, and isotopic exchange occurs by two processes: (i) Random diffusion of molecules in two clusters, one containing D2O and the other H3O+, leads to formation of an intercluster H bond, which in turn provides a path for proton hopping and converts one D2O (plus one H2O) to two coupled HODs by proton translocation and subsequent orientation. (ii) One H bond between two HOD neighbors in a cluster breaks and reforms with another H2O in the same cluster or in a different cluster, and hence a coupled HOD is converted to an uncoupled HOD. The decrease in D2O and the increase in HOD concentrations with time follow a stretched exponential kinetics, with exponent of 0.65 for the former and 0.54 for the latter process at 122 K. This is equivalent to the time-dependent rate constant in Plonka's formalism [J. Chem. Phys. 96, 1128 (1992)] and is seen as characteristic of dispersive kinetics. Because fluctuation of the environment is slower than the time scale of overall barrier crossing, the mass-controlled kinetics equations do not apply to a consecutive reaction scheme. The known variation of the isotopic species concentration with time seems to be consistent with this reaction kinetics.