The mathematical model for hydrogen production by the photosynthetic bacterium Rhodobacter capsulatus in a batch photobioreactor is discussed. An analytical method - the homotopy perturbation method - is presented to solve the nonlinear differential equations that describe the biomass formation, substrate utilization, and hydrogen production with respect to time. Approximate analytical expressions for the concentrations of biomass, substrate and hydrogen are derived for various values of the relevant parameters. The analytical results are compared with experimental data. In addition, the sensitivity of the kinetic parameters is also analyzed. The time required to achieve the maximum growth of biomass and complete degradation of the substrate is reported.