LangmuirFreundlich

LangmuirFreundlich isotherm, commonly referred to as the Sips isotherm, is an empirical model used to describe adsorption on heterogeneous surfaces while allowing for a finite saturation capacity. It blends features of the Langmuir and Freundlich isotherms and is widely applied to adsorption from liquids and gases onto porous or heterogeneous solids.

The standard form of the LangmuirFreundlich isotherm is q_e = q_m (K C)^n / [1 + (K C)^n], where q_e is the amount adsorbed at equilibrium, q_m is the maximum adsorption capacity, K is an affinity constant, C is the equilibrium concentration in solution (or the gas-phase concentration/partial pressure), and n (0 < n ≤ 1) is the heterogeneity parameter. At high concentrations, q_e tends toward q_m, reflecting saturation. When n = 1, the equation reduces to the classical Langmuir isotherm, describing a homogeneous surface with a single adsorption energy.

Interpretation and use: The parameter n quantifies surface heterogeneity; smaller values indicate greater variability in adsorption site energies. The model is typically fit to experimental data by nonlinear regression; linearized forms exist but can bias parameter estimates. The LangmuirFreundlich isotherm is favored for systems where adsorption sites are unevenly distributed yet finite capacity must be captured, such as activated carbons, clays, and various porous materials in environmental engineering, catalysis, and materials science.

Alternative naming: the LangmuirFreundlich formulation is often equated with the Sips isotherm in the literature; some sources note extended variants that introduce additional exponents to better describe complex adsorption behavior.