A Stochastic Model for Adsorption Kinetics

Abstract

A novel stochastic model is proposed to characterize the adsorption kinetics of pollutants including dyes (direct red 80 and direct blue 1), fluoride ions, and cadmium ions removed by calcium pectinate (Pec-Ca), aluminum xanthanate (Xant-Al), and reed leaves, respectively. The model is based on a transformation over time following the Ornstein–Uhlenbeck stochastic process, which explicitly includes the uncertainty involved in the adsorption process. The model includes stochastic versions of the pseudo-first-order (PFO), pseudo-second-order (PSO), and pseudo- $n$ -order (PNO) models. It also allows the estimation of the adsorption parameters, including the maximum removal capacity ( $q_e$ ), the adsorption rate constant ( $k_n$ ), the reaction pseudoorder ( $n$ ), and the variability $\left({\sigma }^2\right)$ . The model fitted produced $R^2$ values similar to those of the nonstochastic versions of the PFO, PSO, and PNO models; however, the obtained values for each parameter indicate that the stochastic model better reproduces the experimental data. The $q_e$ values of the Pec-Ca-dye, Xant-Al-fluoride, and reed leaf-Cd+2 systems ranged from 2.0 to 9.7, 0.41 to 1.9, and 0.04 and 0.29 mg/g, respectively, whereas the values of $k_n$ ranged from 0.051 to 0.286, 0.743 to 75.73, and 0.756 to 8.861 (mg/g)1-n/min, respectively. These results suggest a variability in the parameters $q_e$ and $k_n$ inherent to the natures of the adsorbate and adsorbent. The obtained $n$ values ranged from 1.13 to 2.02 for the Pec-Ca-dye system, 1.0–3.5 for the Xant-Al-fluoride system, and 1.8–3.8 for the reed leaf-Cd+2 system. These ranges indicate the flexibility of the stochastic model to obtain fractional $n$ values, resulting in high $R^2$ values. The variability in each system was evaluated based on ${\sigma }^2$ . The developed model is the first to describe pollutant removal kinetics based on a stochastic differential equation.