A number of studies have used 14C to estimate ages of unconfined ground water and to infer recharge rates. Some of these studies have been in arid or semi-arid regions, and the inferred recharge rates have been very low. The implicit assumption has been that the vertical movement of 14C within the unconfined ground water is entirely by convective processes. In this paper we extend the ground water 14C model of previous workers to include diffusion within the aquifer, and show that for recharge rates less than approximately 1 mm year-1, diffusion can be important in defining the 14C distribution. For this reason it is not possible to use 14C profiles to estimate recharge in these cases. We also show how neglecting diffusion can lead to serious underestimates of groundwater ages in unconfined aquifers where recharge rates are similarly low.