Critical analysis of adsorption/diffusion modelling as a function of time square root

- Detailed development of models for intraparticle mass diffusion.
- Definition of conditions that validates adsorption as function of time square root.
- Critical analysis of the quality of the time square root approach.
- Evaluation of external mass transfer resistance in time square root plots.
- Critical analysis of the multilinear segments in time square root plots.

A very common procedure to evaluate intraparticle diffusion in batch adsorption experiments consists in plotting the adsorbed concentration versus the square root of time. Furthermore, some works present in this plot a sequence of linear sections, attributing a specific phenomenon to each one, such as external mass transfer resistance and intraparticle diffusion into pores of different sizes. In spite of this frequent approach, the mathematical foundations of this relationship are rarely presented. In this work, it is presented a complete description of the physical-chemical and mathematical basis that leads to the relationship between adsorbed quantity and time square root, evincing the necessary hypotheses for this relationship to become meaningful. It was observed that this relationship could only be applied within very restrictive conditions: adsorption in a semi-infinite solid and constant liquid phase concentration. Moreover, this relationship is strictly valid in the absence of external mass transfer resistance, for linear adsorption isotherm and instantaneous equilibrium of fluid and solid concentrations inside the pores. It is clear that these restrictions are scarcely achieved under usual batch adsorption experiments. Besides, multilinear plots used to differentiate among external diffusion and macro, micro and mesoporous diffusion are evidently shown to be very subjective and without any mathematical support.