Essentially exact closures are presented for the second‐ and third‐moment rate equations for diffusion of a pollutant released at a particular time in homogeneous turbulence. The quasi‐Gaussian closure is found to require addition of a down‐gradient diffusion term. Explicit dependence upon the time after release is found to be retained by the highest moment gradient term. The second‐ and third‐moment approaches are therefore concluded to be inherently incapable of accurately solving the general problem of pollutants released at different places at different times.