## 2 Materials and Methods
### Calculation and Analysis of Data  
The quantity of dye that was eliminated from water at equilibrium (\(q_{e}\); mg/g), and time \(t\) (\(qt\); mg/g) was estimated using equations (Equations (1) and (2)) of the mass balance as follows:  
\[q_{e} =\frac{(C_{o}-C_{e})}{m}V \tag{1}\] \[q_{t} =\frac{(C_{o}-C_{t})}{m}V \tag{2}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**C.I. Name** & **Chemical Structure** & **Molecular Formula** & **MW (g/mol)** & **Density (g/cm\({}^{3}\))** & **Solubility** \\ \hline Crystal violet & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Chemicals characteristics of crystal violet dye. where \(C_{o}\) (mg/L), \(C_{l}\) (mg/L), and \(C_{e}\) (mg/L) are the CV dye concentration at beginning, at time \(t\), and at equilibrium; \(V\) (L) is the volume of solution, whereas \(m\) (g) represents the mass of DSD used. The kinetic data were described mathematically using three reaction kinetic models and the diffusion model. The general forms of the pseudo-first-order, pseudo-second-order [20], and Avrami fractional-order [21] models are given by Equations (3)-(5), respectively. The linear form of intraparticle diffusion model was expressed in Equation (6) [22]. Moreover, the adsorption isotherms were investigated to gain insights on the distribution of adsorbate at the adsorbent-solution interface. The experimental data of CV adsorption isotherm at different temperatures were fitted to the [PERSON] [20], [PERSON] [23], [PERSON] [24], and [PERSON] [25] isotherm models (Equations (7)-(10)). All equations and relevant parameters of the aforementioned models (for kinetic and isotherm) have been listed in Table 2.  
The best model representing the sorption mechanism can be selected by calculating the difference between the qe in mg/g (or \(q_{l}\)) values of the experiment (\(q_{e,\text{exp}}\) or \(q_{l,\text{exp}}\)) and \begin{table}
\begin{tabular}{l c c} \hline \hline
**Model** & **Equation** & **Description of Parameters** \\ \hline \hline \multicolumn{3}{c}{\(qe\) and \(qt\) (in mg/g) are the adsorbed quantity of CV dye at equilibrium and at time t, respectively. \(K_{1}\) (L/min) and \(K_{2}\) (g/mg.min): rates constants. \(k_{AV}\) (1/min): is the Avrami kinetic constant. \(n_{AV}\): is a fractional adsorption order corresponding to adsorption mechanism. \(Kip\) (mg/g/min\({}^{1/2}\)): intraparticle diffusion rate constant. \(C\) (mg/g): constant related to the thickness of the boundary layer. \(C_{e}\): concentration of adsorbate at equilibrium. \(Q^{Q}_{max}\) (mg/g): maximum monolayer adsorption capacity of Langmuir. \(K_{L}\) (L/mg): Langmuir constant. \(K_{F}\) [(mg/g)/(mg/L)\({}^{n}\)]: Freundlich constant. \(n_{F}\): dimensionless Freundlich intensity parameter. \(K_{RP}\) (L/g) and \(a_{RP}\) (mg/L) \({}^{n}\) are the Redlich–Peterson constants. \(g_{RP}\): dimensionless Redlich–Peterson exponent, g should be \(\leq 1\). \(Q^{Q}_{max}\) (mg/g): the Liu maximum adsorption capacity. \(K_{g}\) (L/g): is the Liu equilibrium constant. \(n_{\text{L}}\): is dimensionless exponent of the Liu equation. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Equations, relevant parameters, and references of the applied kinetic and isotherm models. values of the model (\(q_{e,cal}\) or \(q_{t,cal}\)), which are reflected through the coefficient of correlation (\(R^{2}\)), standard deviation (SD), and chi-squared test (\(\chi^{2}\)) [7; 26]:  
\[R^{2}=1-\frac{\sum\left(q_{e,\text{exp}}-q_{e,\text{cal}}\right)^{2}}{\sum\left( q_{e,\text{exp}}-q_{e,mean}\right)^{2}} \tag{11}\]  
\[SD=\sqrt{(\frac{1}{n-p})\times\left[\sum_{i=1}^{n}\left(q_{e,\text{exp}}-q_{e, \text{cal}}\right)^{2}\right]} \tag{12}\]  
\[\chi^{2}=\sum\frac{\left(q_{e,\text{exp}}-q_{e,\text{cal}}\right)^{2}}{q_{e, \text{cal}}} \tag{13}\]  
where \(q_{e,\text{ }mean}\) (mg/g) is the mean of \(q_{e}\) experimental values; \(n\) is the number of observations; \(p\) is the number of parameters in the selected equation.