If the power lost by impurity radiation from the low-temperature boundary region of a confined plasma exceeds the power transported from the central region then temperature equilibrium is impossible and the temperature profile will collapse. This situation is studied on the assumption of coronal equilibrium and the results are used to predict the permissible impurity concentration at the edge of a tokamak reactor. The effect of neoclassical impurity transport is studied analytically and a 1D tokamak diffusion code is used to illustrate the detailed collapse of the temperature profile when the plasma density in a radiation-cooled tokamak is increased. Finally, the results are used to predict the limiting density of a radiation-cooled tokamak; this predicted density limit is compared with the density limits observed in current tokamak experiments.