An analytical approach to transient homovalent cation exchange problems

SummaryCation exchange in groundwater is one of the dominant surface reactions that occurs in nature and it carries with it many important environmental implications. The mass transfer of cation exchanging pollutants in groundwater can be described by a series of coupled partial differential equations, involving both aqueous and adsorbed species. The resulting system is mathematically challenging due to the complex nonlinearities that arise, which in turn complicates analytical approaches. While some analytical solutions for simplified problems exist, these typically lack the mechanisms that allow the waters to change their global chemical signature (in terms of total cations present in aqueous form) over time. We propose a methodology to solve the problem of exchanging two homovalent cations by deriving the driving equation for one of the aqueous species. This equation incorporates explicitly a retardation factor and a decay term, both with parameters that can vary in space and time. While the full solution can only be obtained numerically, we provide a solution in terms of a perturbative approach, where the leading terms can be obtained explicitly. The resulting solution provides physical explanations for the possible existence of non-monotonic concentrations for a range of parameters governing cation exchange processes.