We developed a new cell voltage model which depends on the current density response of the proton exchange membrane fuel cell (PEMFC). The proposed model focuses on the conductivity of a PEMFC fuel cell. The model for conductivity of a polymer electrolyte membrane, Nafion, uses the thermodynamic function to represent water activity. We observed membrane conductivity by monitoring the chemical potential of the swelling of the hydrogel. The swelling equilibrium of the ionized hydrogel is governed by the free energy of mixing, the free energy of elasticity, and the concentration of counterions with fixed charges on the gel network. A cell voltage equation with reasonable boundary conditions, including the swelling effect, which plays a major role in determining the cell voltage of a PEMFC, was newly developed.