This calculator finds the concentration versus radial position for two cases, assuming a constant, position independent diffusivity (in other words, this is a single medium problem).
Case 1: INITIAL condition is constant concentration external to a sphere of radius R, inside the sphere the concentration is zero. The number of impurities is "fixed", so when diffusion begins there is an initial depletion in concentration just outside the surface of the sphere as impurities begin to diffuse into the sphere. At long times the impurities from far away ultimately replace those flowing into the sphere, leading to a final constant concentration equal to the initial external concentration.
Case 2: BOUNDARY condition is constant concentration at the surface of the sphere. Obviously now there is no external depletion. The mathematical solution ONLY APPLIES FOR r < R; outside this the solution produces a profile that would be necessary to produce the flux required to produce the constant surface concentration boundary condition.