We investigate the connection between two classical models of phase transition phenomena, the (discrete size, Markov chain or infinite set of ODE) Becker-Döring equations and the (continuous size, PDE) Lifshitz-Slyozov equation. Contrary to previous studies, we use a weak topology that includes the boundary of the state space, allowing us to rigorously derive a boundary value for the Lifshitz-Slyozov model. This boundary condition depends on a particular scaling and is the result of a separation of time scales.