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The pseudo-second-order (PSO) kinetic model has become among the most popular ways to fit rate data for adsorption of metal ions, dyes, and other compounds from aqueous solution onto cellulose-based materials. This review first considers published evidence regarding the validity of the mechanistic assumptions underlying application of the PSO model to adsorption kinetics. A literal interpretation of the model requires an assumption that different adsorption sites on a solid substrate randomly collide with each other during a rate-limiting mechanistic step. Because of problems revealed by the literature regarding the usual assumptions associated with the PSO model, this review also considers how else to account for good fits of adsorption data to the PSO model. Studies have shown that adsorption behavior that fits the PSO model well often can be explained by diffusion-based mechanisms. Hypothetical data generated using the assumption of pseudo-first-order rate behavior has been shown to fit the PSO model very well. In light of published evidence, adsorption kinetics of cellulosic materials is expected to mainly depend on diffusionlimited processes, as affected by heterogeneous distributions of pore sizes and continual partitioning of solute species between a dissolved state and a fixed state of adsorption.

The kinetics of adsorption processes at the solid/solution interface has been studied by using a fractal-like concept. For the first time, one possible physical meaning for fractal-like adsorption kinetics which indicates that sorption rate coefficient at solid/solution interface is a function of time is presented. So, new kinetic models have been presented by a combination of the fractal-like approach and integrated kinetic Langmuir equation (IKL), mixed 1,2-order equation (MOE), and statistical rate theory (SRT). The fractal-like SRT rate equation has been solved at two limiting conditions, one at initial times of adsorption and the other close to equilibrium. The new parameter (α) which has appeared in these models affects the rate of sorption. Fractal-like IKL (or fractal-like MOE) and also fractal-like SRT models have been used to analyze different experimental data. The obtained results indicate that the variation of initial concentration of solute and stirring rate of solution affect both "α" and the rate coefficient.

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