. 2020 Jul 2;10:10836. doi: 10.1038/s41598-020-67720-0

Table 7.

Kinetic models of heavy metal absorption used in this study.

Models	Equations	Variable descriptor
Pseudo first order	$l n (q_{e} - q_{r}) = l n (q_{e} - k_{1} t)$ $t_{1 / 2} = \frac{l n 2}{k_{1}}$	Dependence is established between $l n (q_{e} - q_{t})$ versus t where $q_{e}$ is the amount in grams of solute adsorbed per gram of adsorbent at equilibrium (mg/g); $q_{t}$ represents the grams of solute absorbed per gram of absorbent over time (mg/g); t is the time in min, $k_{1}$ is the pseudo first order absorption rate constant ( $m i n^{- 1}$ ) and $t_{1 / 2}$ is the mean lifetime in min
Pseudo second order	$\frac{1}{q_{t}} = \frac{1}{k_{2} q_{e}^{2}}$ $t_{1 / 2} = \frac{1}{q_{e} k_{2}}$ $h_{2} = k_{2} q_{e}^{2}$	Dependence is established between $1 / q_{t}$ versus t, where $q_{e}$ is the amount in grams of solute adsorbed per gram of adsorbent at equilibrium (mg/g); $q_{t}$ represents the grams of solute absorbed per gram of absorbent over time (mg/g); $h_{2}$ is the initial absorption rate (mg/gmin); $t_{1 / 2}$ represents the mean lifetime in min, and $k_{2}$ is the initial rate constant
Intraparticle diffusion	$q_{t} = k t^{1 / 2} + C$	Dependence is established between $q_{t}$ versus t, where $q_{t}$ is the amount of solute adsorbed per gram of absorbent over time (mg/g); $t^{1 / 2}$ represents the time variable in min; k is the intraparticle diffusion rate constant and C is a constant
Elovich model	$q_{t} = α + β l n t$	Dependence is established between $q_{t}$ versus lnt, where $q_{t}$ represents the milligrams of solute adsorbed per gram of adsorbent over time (mg/g); $α$ is the initial sorption rate (mg/min); $β$ is the sorption constant (mg/min) and t is the time in min