## 3 Results and Discussion
### Equilibrium and Kinetics Studies
#### 3.3.1 Equilibrium Isotherms  
The adsorption isotherm expresses the specific relationship between the degrees of accumulation on the adsorbent surface and the concentration of sorbate at constant temperature [18]. Four isotherm models were used to investigate the adsorption process of TY on ADAV. These models included Langmuir, Freundlich, Temkin, and Dubinin-Radushkevich (DR) isotherm. Langmuir isotherm explains the single-layer homogeneous adsorption on a surface, as shown in Figure 9A and Table 6. The basic characteristic of Langmuir equation is shown in Equation (6). \[q_{e}=\frac{q_{m}\ K_{L}\ C_{e}}{1-K_{L}\ C_{e}} \tag{6}\]  
where \(q_{m}\) is the monolayer adsorption capacity and \(K_{L}\) is the Langmuir equilibrium coefficient. In addition, the Langmuir equation can be stated using the dimensionless equation:  
\[R_{L}=\frac{1}{1-K_{L}\ C_{0}} \tag{7}\]  
where \(R_{L}\) is separation factor, and \(C_{0}\) (mg/L) is the initial concentration. Based on previous reports [47; 48], the adsorption favorability is dependent on the value of \(R_{L}\). In general, if \(R_{L}\) is \(>\)1, the adsorption process is unfavorable, if \(R_{L}\) = 1, adsorption is linear, and with a value between 0-1, the adsorption is favorable (occurs spontaneously); if the value = 0, adsorption is irreversible. The calculated \(R_{L}\) values in the current investigation were \(<\)1 and \(>\)0, implying that the adsorption process was spontaneous with maximum adsorption (\(q_{max}\)) = 55.25 mg/g. Figure 8: SEM micrographs of ADAV (**A**), TTAV300 (**B**), as well as energy dispersive X-ray spectroscopy (EDX) analysis of ADAV (**C**) and TTAV300 (**D**). The Freundlich isotherm used to describe heterogeneous surface energies is given by the equation:  
\[q_{e}=K_{F}C_{e}^{\frac{1}{2}} \tag{8}\]  
\begin{table} \begin{tabular}{c c c c} \hline \multirow{2}{*}{**Isotherm**} & \multicolumn{2}{c}{**Equations**} & \multirow{2}{*}{**Parameters**} & \multirow{2}{*}{**Value**} \\  & & **(Generalized/Linearized Forms)** & & \\ \hline \multirow{2}{*}{**Langmuir**} & \(q_{e}=\frac{q_{e}}{1-K_{L}}\frac{C_{e}}{C_{e}}\) & \(q_{m}\) (mg/g) & 55.25 \\  & \(\frac{C_{e}}{q_{e}}=\frac{1}{q_{e}}\frac{C_{e}}{C_{e}}+\frac{C_{e}}{q_{e}}\) & \(K_{L}\) (L-mole\({}^{-1}\)) & 0.017 \\  & \(q_{e}=K_{F}C_{e}^{\frac{1}{2}}\) & \(\frac{1}{n}\) & 0.947 \\ \hline \multirow{2}{*}{**Freundlich**} & \(q_{e}=K_{F}C_{e}^{\frac{1}{2}}\) & \(\frac{1}{n}\) & 0.710 \\  & \(\log(q_{e})=\log(K_{F})+\big{(}\frac{1}{n}\big{)}\log(C_{e})\) & (mole/g)(L/mole)\({}^{1/n}\) & 1.486 \\  & \(\mathrm{R}^{2}\) & 0.984 \\ \hline \multirow{2}{*}{**Temkin**} & \(q_{e}=\frac{RT}{br}\ln(A_{T}\,C_{e})\) & \(b_{T}\)(J/mole) & 311.014 \\  & \(q_{e}=\frac{RT}{br}\ln(A_{T})+\frac{RT}{br}\ln(C_{e})\) & \(A_{T}\) (L/mole) & 0.473 \\ \hline \multirow{2}{*}{**DR**} & \(\ln(q_{e})=\ln(q_{m})-g\epsilon^{2}\) & \(\beta\) & \(1\times 10^{-8}\) \\  & \(\epsilon=RT\big{(}1+\frac{1}{C_{e}}\big{)}\) & \(E\) (kJ/mole) & 7.07 \\ \end{tabular} 
\begin{tabular}{c c c} \hline \multirow{2}{*}{**Parameters**} & \multirow{2}{*}{**Value**} \\  & & **(\(\mathrm{mole/g}\))** \\Here, the \(C_{e}\) is the equilibrium concentration of TY (mg:L\({}^{-1}\)), \(q_{e}\) is the amount of TY adsorbed/unit mass (mg:g\({}^{-1}\)), and \(K_{F}\) (mole:g\({}^{-1}\))(L-mole\({}^{-1}\))\({}^{1/n}\) andl/n are Freundlich coefficients (Figure 9B and Table 6). The Freundlich plot (Figure 9B) showed a good fit with a coefficient of determination, R\({}^{2}\) = 0.9843 with 1/n = 0.71 and n = 1.487. Therefore, the adsorption potential (A = nRT) = 4.12 kJ, thus any TY with a potential energy less than 4.12 kJ would be adsorbed on the surface of ADAV, and reactions would tend to be irreversible and favorable. These data confirm that the Freundlich isotherm was more applicable to study the adsorption process of TY on ADAV.