Lithium iron phosphate (LiFePO4 or LFP) is a common active material in lithium-ion batteries. It has been observed that this material undergoes phase transitions during the normal charge and discharge operation of the battery. Electrochemical models of lithium-ion batteries can be modified to account for this phenomenon at the expense of some added complexity. We explore this problem for the single particle model (SPM) where the underlying dynamic model for diffusion of lithium ions in phase transition materials is a partial differential equation (PDE) with a moving boundary. We derive a novel boundary observer to estimate the concentration of lithium ions together with a moving boundary radius from the SPM via the backstepping method for PDEs, and simulations are provided to illustrate the performance of the observer. Our comments are stated on the gap between the proposed observer and a complete state-of-charge (SoC) estimation algorithm for lithium-ion batteries with phase transition materials.