Kinetic box models for the uptake of radionuclides and heavy metals by suspended particulate matter: equivalence between models and its implications

In recent years an increasing experimental effort has been paid to the study of the sorption process of radionuclides and heavy metals by particulate matter in aquatic environments. This has led to the development of different kinetic box models. Most of them are variations of two basic approaches: one containing several (up to three) parallel reactions while the other involves consecutive reactions. All the reactions are reversible (irreversibility is contained as a particular case) with concentration independent coefficients. The present work provides analytical solutions and demonstrates that both approaches are mathematically equivalent. That is, both models produce the same analytical solution for the uptake curve (time course of the concentrations in the dissolved phase), which is illustrated using literature data. This result unifies the description of the observed behaviour, but it brings up the question of the physical meaning of the involved coefficients. Finally, the mathematical relationship developed here serves to discuss some limitations found in recent attempts in literature devoted to distinguish the actual uptake mechanism.