The goal here is to study the voltage and expansion response of lithium-ion batteries at different charging rates. Specifically, the goal is to capture the observation of the smoothing of the peaks in dV/dQ and retention of the peaks in d^2 \delta/dQ^2 at higher C-rates. The retention of the peaks at higher charging rates enables better estimation of the cell capacity. To achieve this goal a reduced order electrochemical and mechanical model with multiple particles with a size distribution is developed. This allows us to capture the smoothing and preservation of the phase transitions in the voltage and expansion measurements at high C-rates, respectively. The model is written in Matlab software.