# Cationic and Anionic Dye Adsorption on a Natural Clayey Composite
## 2 Materials and Methods
### Adsorption Isotherm Modeling  
The adsorption equilibria data is crucial to understand how much of the adsorbate can fit onto the solid adsorbent surface. The adsorption isotherm models express a mathematical relation between the dye adsorbed and the remained dye concentration in solution. There are many possible interactions between the adsorbent and the adsorbate. Various isotherm models consist of different model parameters [21]. Therefore, we studied the adsorption equilibria data by analyzing the Langmuir, Freundlich, Tempkin, and Dubinin-Radushkevich isotherm models' feasibility. The criterion for choosing the more feasible isotherm model was that it should be a good fit between the isotherm function and the experimental data. The best-fit isotherm model was determined by carrying out linear regressions of the isotherm linear equation. Data evaluation was accomplished by using R\({}^{2}\) value analysis. The Langmuir model (LM) was developed using the following assumptions [22; 23; 24]: (1) a fixed number of accessible sites are available on the adsorbent surface, and all active sites have the same energy, (2) adsorption is reversible, (3) once the adsorbate occupies a site, no further adsorption can occur on that site (the adsorbate forms a monolayer on the adsorbent surface), and (4) there is no interaction between adsorbate species. The nonlinear and the Lineweaver-Burk linearization of the LM are described in Equation (4). The LM parameters are found when \(1/q_{e}\) plotted vs. \(1/C_{e}\) yielding a straight line with \(1/q_{m}C_{e}\) the slope and \(1/q_{m}\) intercept. \[q_{e}=\frac{q_{m}K_{L}C_{e}}{1+K_{L}C_{e}}\text{ and }\frac{1}{q_{e}}=\left( \frac{1}{q_{m}K_{L}}\right)\frac{1}{C_{e}}+\frac{1}{q_{m}} \tag{4}\]  
where \(q_{e}(\text{mg g}^{-1})\) is the adsorbed amount uptake at equilibrium, \(q_{m}\big{(}\text{mg g}^{-1}\big{)}\) is the maximum possible adsorption (mg g\({}^{-1}\big{)}\) or maximum adsorption capacity of an adsorbent, \(K_{L}\Big{(}\text{L mg}^{-1}\Big{)}\) is a constant related to the affinity between an adsorbent and adsorbate. The LM's essential characteristic can be expressed in terms of a dimensional constant called the separation factor or equilibrium parameter \(R_{L}\), defined in Equation (5). The adsorption nature is: linear when \(R_{L}=1\), favorable when \(R_{L}<1\) (concave shape), and unfavorable when \(R_{L}>1\) (convex shape) [24]. \[R_{L}=\frac{1}{1+K_{L}C_{0}} \tag{5}\]  
The Freundlich model (FM) describes the equilibrium data and adsorption characteristics for a heterogeneous surface [25]. The nonlinear and the linear forms of the FM are described in Equation (6). The LM parameters can be found when \(\log q_{e}\) plotted vs. \(\log C_{e}\) yielding a straight line with \(n\) the slope and \(\log K_{F}\) intercept. \[q_{e}=K_{F}C_{e}^{n}\text{ and }\log q_{e}=n\log C_{e}+\log K_{F} \tag{6}\]  
In Equation (6) \(K_{F}\big{[}(\text{mg}/\text{g})/(\text{mg}/\text{L})^{n}\big{]}\) is the Freundlich constant and \(n\) is the Freundlich intensity parameter. The adsorption nature is linear when \(n=1\), favorable when \(n<1\) (concave shape), and unfavorable when \(n>1\) (convex shape). The Temkin model (TM) assumes that adsorption heat (\(\Delta H_{ads}\)) as a function of all molecules' temperature in the layer declines linearly rather than logarithmically due to the surface coverage increase [21; 26]. Equation (7) describes the nonlinear and linear forms of the TM. By plotting \(q_{e}\) vs. \(\ln C_{e}\), both \(A_{T}\) and \(b_{T}\) constants can be obtained. \[q_{e}=\frac{RT}{b_{T}}\ln(A_{T}C_{e})\text{ and }q_{e}=\frac{RT}{b_{T}}\ln A_{T}+ \left(\frac{RT}{b_{T}}\right)\ln C_{e} \tag{7}\]In Equation (7), \(R\) is the constant of the ideal gases (8.314 J mol\({}^{-1}\) K\({}^{-1}\)). \(T\) is the temperature at which the adsorption was performed (K). The \(b_{T}\) is a dimensionless constant related to the heat of adsorption. The \(A_{T}\) is the isothermal equilibrium binding constant (L g\({}^{-1}\)). The \(RT/b_{T}\) is the adsorption energy (J mol\({}^{-1}\)).