The eigenfunction solution to the problem of dispersion of neutrally buoyant material presented in Nokes et al. (1984) is extended to encompass the dispersion of particles with a rise or fall velocity in two dimensional open channel flow. Steady state conditions are modelled as this allows the turbulent diffusion equation to be separated and hence reduces it to a Sturm-Liouville eigenvalue problem. A general solution in the form of an eigenfunction expansion can be found for arbitrary velocity and diffusivity distributions but attention in this paper is centred on a comparison between a uniform velocity and diffusivity and the more realistic power law velocity and parabolic diffusivity. The power series technique of solution to ordinary differential equations is used to generate the eigenfunctions and eigenvalues for these two cases.