An isothermal porous-electrode model of a discharging lead-acid battery is presented, which includes an extension of concentrated-solution theory that accounts for excluded-volume effects, local pressure variation, and a detailed microscopic water balance. The approach accounts for three typically neglected physical phenomena: convection, pressure diffusion, and variation of liquid volume with state of charge. Rescaling of the governing equations uncovers a set of fundamental dimensionless parameters that control the battery's response. Total volume change during discharge and nonuniform pressure prove to be higher-order effects in cells where variations occur in just one spatial dimension. A numerical solution is developed and exploited to predict transient cell voltages and internal concentration profiles in response to a range of C-rates. The dependence of discharge capacity on C-rate deviates substantially from Peukert's simple power law: charge capacity is concentration-limited at low C-rates, and voltage-limited at high C-rates. The model is fit to experimental data, showing good agreement.