Measurements of solution-phase crystal growth provide mechanistic information that is helpful in designing and synthesizing nanostructures. Here, we examine the model system of individual Au nanocrystal formation within a defined liquid geometry during electron beam irradiation of gold chloride solution, where radiolytically formed hydrated electrons reduce Au ions to solid Au. By selecting conditions that favor the growth of well-faceted Au nanoprisms, we measure growth rates of individual crystals. The volume of each crystal increases linearly with irradiation time at a rate unaffected by its shape or proximity to neighboring crystals, implying a growth process that is controlled by the arrival of atoms from solution. Furthermore, growth requires a threshold dose rate, suggesting competition between reduction and oxidation processes in the solution. Above this threshold, the growth rate follows a power law with dose rate. To explain the observed dose rate dependence, we demonstrate that a reaction-diffusion model is required that explicitly accounts for the species H<sup>+</sup> and Cl<sup>-</sup>. The model highlights the necessity of considering all species present when interpreting kinetic data obtained from beam-induced processes, and suggest conditions under which growth rates can be controlled with higher precision.