Spectral width of the cloud droplet spectrum is important for radiative properties and drizzle/rain development in warm ice-free clouds. We use an adiabatic rising parcel model to study activation and diffusional growth of cloud droplets, focusing on the spectral width evolution, and contrasting clean and polluted environments. A comprehensive droplet growth equation is used that includes kinetic, solute, and surface tension effects. We show that those effects have an appreciable impact on the spectral width evolution above the cloud base. Without those effects, the droplet area standard deviation should not change once activation is completed. In contrast, simulation results show that the area standard deviation does increase with height, especially for weak updrafts and polluted environments. Implications of those results for cloud modeling, especially applying conventional bin microphysics, are discussed.