Simple Metal Sorption Model for Heterogeneous Sorbents: Application to Humic Materials

A semiempirical equilibrium model to simulate proton and metal binding to heterogeneous sorbents is presented. In the simple metal sorption (SiMS) model, proton and metal binding reactions to a heterogeneous surface are conceptualized as reactions with a single, composite “site,” with empirical correction factors to the equilibrium constants that are represented as simple power functions of hydrogen ion concentration, metal-to-ligand ratio (Me_T/L_T), and ionic strength (I). That is, the observed metal-binding equilibrium constant, ${\mathit{K}}_{\mathit{Me}\text{,}\mathit{app}}$, is represented as ${\mathit{K}}_{\mathit{Me}\text{,}\mathit{app}}={\mathit{K}}_{\mathit{M}\mathit{e}}\left\{H^{+}{\}}^{\mathit{\alpha }}\right({\mathit{Me}}_{\mathit{T}}/{\mathit{L}}_{\mathit{T}}{)}^{\mathit{\beta }}{\mathit{I}}^{\mathit{\varsigma }}\text{.}$ The validity of this approach is tested by fitting the model to a hypothetical multiligand data set and three data sets from the literature involving proton and metal binding to humic materials (two data sets involving Cu²⁺ and H⁺ binding, and one data set for binding of Co²⁺ and H⁺). Independent data sets involving Cu²⁺ binding are used for model prediction. The fitted models are used to contrast the three humic materials in terms of acid/base characteristics and H⁺/Me exchange ratios. A theoretical limitation of the model is that it does not satisfy the Gibbs-Duhem equation for thermodynamic consistency. The major advantages of the SiMS model are simplicity (i.e., few fitting parameters), flexibility in describing proton and metal binding to heterogeneous sorbents, and ease of application (model results presented in this paper were done on a standard spreadsheet). The model is presented not as a new development in the conceptual understanding of metal-humate interactions, but rather a practical engineering tool that can easily be incorporated into general fate and transport models.