Particle tracking algorithms are very useful methods to model conservative transport in surface and subsurface hydrological systems. Recently, a novel ad hoc particle-based method was proposed to account for multicomponent reactive transport by Benson and Meerschaert (2008). This one-dimensional particle method has been shown to match theoretical predictions, but, to date, there has been no rigorous demonstration that the particle method actually matches the governing equations for chemical transport. We generalize this particle method to two-dimensional and three-dimensional systems and rigorously demonstrate that this particle method converges to the diffusion-reaction equation at the limit of infinitely small time step. We also investigate the numerical error associated with the method.