Insights into adsorption rate constants and rate laws of preset and arbitrary orders

Adsorption kinetics is of great significance in a variety of scientific and engineering applications, for example, removal of toxic metal ions from aqueous solution by adsorption. The pseudo-first-order (PFO) and pseudosecond-order (PSO) models are most popularly used and the latter is claimed applicable to most experimental adsorption data. The aim of the present study is to explain the reason behind the claim, provide a quantitative understanding of these preset-order models and general ones of arbitrary orders by rigorously deriving their relationship. It is proven that the ratios of the adsorption time required to reach 0.95 of the fractional surface coverage (or adsorption progress factor) to the half-time are around 4.32, 19, 133, and 1143 for the 1st, 2nd, 3rd, and 4th orders, respectively. For the general rate laws of nth order (n >= 2), t(0.95)/t(0.5 )approximate to 1/(0.1(n-1) - 0.05(n-1)). Adsorption progress factor curves for different rate orders cross each other at the half-time point. When the time is shorter than the half-time, the adsorption progress factor increases with an increase in the rate order. However, this trend is opposite when the time passes over the half-time point. The higher the rate order, the longer the time is needed to reach equilibrium. We provide a guideline to determine the order of rate equation for describing the experimental data by deriving the relationship between the saturation time and the rate order. In addition, we propose a method and show how to employ it to validate the results obtained from the linearized PSO model and point out that the claimed wide applicability of the PSO might be caused by the ignorance or insufficient data points on the steep rising edge of the adsorption curve.

Automated summary

This study provides an explanation of the relationship between the pseudo-first-order and pseudo-second-order models in adsorption kinetics, as well as a quantitative understanding of arbitrary-order preset models. It was found that adsorption progress factor curves for different rate orders intersect at the half-time point, and the adsorption progress factor increases with an increase in rate order when the time is shorter than the half-time, but decreases with time passing over the half-time point. Furthermore, the higher the rate order, the longer the time needed to reach equilibrium, and a guideline is provided to determine the order of rate equation for describing experimental data.