## 3 Results and Discussion
### Adsorption Thermodynamics Studies  
The thermodynamic parameters, such as the Gibbs free energy change (\(\Delta G^{\circ}\)), enthalpy (\(\Delta H^{\circ}\)) and entropy (\(\Delta S^{\circ}\)), were calculated using the following equations [42]:  
\[K=\frac{Q_{e}}{C_{e}} \tag{11}\]  
The value of \(\Delta G^{\circ}\) can be determined from the following equation:  
\[\Delta G^{\circ}=-RT\ ln(K) \tag{12}\]  
where K is the thermodynamic equilibrium constant. The effect of temperature on thermodynamic constant is determined by the following equation:  
\[\frac{\text{d}(\ln(\text{K}))}{\text{dt}}=\frac{\Delta H^{\circ}}{RT^{2}} \tag{13}\]  
Integrating and rearranging Equation (14), we obtain  
\[ln\ (K)=\frac{\Delta S^{\circ}}{R}-\frac{\Delta H^{\circ}}{RT} \tag{14}\]  
Additionally, Gibbs free energy is also given by  
\[\Delta G^{\circ}=\Delta H^{\circ}-T\Delta S^{\circ} \tag{15}\]  
The \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) values were calculated from the slope and intercept of the linear plot of \(ln(K)\) versus 1/T. Such parameters reflect the feasibility and spontaneous nature of the process [43]. Experiments were carried out using 50 mg L\({}^{-1}\) dye solutions with 50 mg of BC-HVL or BS-HVL for 90 min at different temperatures. The apparent equilibrium constant K of the adsorption is calculated by Equation (10). The free energy of Gibbs of the adsorption \(\Delta G^{\circ}\) is calculated from Equation (11), the relationship between K and the \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Adsorbate** & **Adsorbent** & **Dose (mg)** & **C\({}_{\theta}\) (mg L\({}^{-1}\))** & **pH** & **Kinetic** & **Isotherm** & **q\({}_{\text{m}}\) (mg g\({}^{-1}\))** & **Ref.** \\ \hline Biochar BC CLS & AO-52 & 50 & 300 & 7 & Pseudo-second-order & Langmuir & 333.33 & [22] \\ Biochar (BCCLSh) & & & & & & & & \\ \hline Silica-Chitosan Composite & RR-23 & 40 & 60 & 7 & Pseudo-second-order & Langmuir & 128.2 & [38] \\  & RR-23 & & & & & & & \\ \hline Chitosan Composite MCs/MS & RR-23 & & & & & & & \\ \hline Activated Carbon derived from & MO & 50 & 500 & ** & Pseudo-second-order & Langmuir & 333.33 & [22] \\ Phragmites Australia & MV & 50 & 400 & & & & & \\ \hline Carbon nanotubes (CNTs) & MO & 200 & & & & & & \\  & mg L\({}^{-1}\) & 10 & ** & & & & \\ \hline Carbon of Quercus Brantiti (Oak) & ACT & 1 g L\({}^{-1}\) & 100 & 7 & Pseudo-second-order & Freundlich & 45.45 & [41] \\ \hline BS–HVL & X–PR & 50 & 50 & 7 & Pseudo-second-order & Langmuir-Freundlich & 71.642 & \\ \hline BS–HVL & MGC & 50 & 50 & 7 & Pseudo-second-order & Freundlich & 44.843 & This work \\ \hline BS–HVL & X–PR & 50 & 50 & 7 & Pseudo-second-order & Freundlich & 44.843 & This work \\ \hline BS–HVL & MGC & 50 & 50 & 7 & Pseudo-second-order & Freundlich & 121.95 & \\ \hline \hline \end{tabular}  
* undetermined.  
\end{table} 
Table 4: \(Cont.\)temperature is given by the Van 't Hoff equation [44] and the enthalpy and entropy can be gained from the slope and the interception of the Van 't Hoff plot of \(ln(K)\) with respect to 1/T, which are shown in Figure 18.  
The Temkin isotherm equation is  
\[q_{e}=B_{1}\ ln\ KT+B_{1}\ ln(C_{e}) \tag{16}\]  
The Dubinin-Radushkevich isotherm is expressed as follows:  
\[ln(q)=ln(q_{0})-KE^{2} \tag{17}\]  
\[\varepsilon\ =\ \text{RT}\ \ln\left(1+\frac{1}{C_{e}}\right) \tag{18}\]  
where \(\varepsilon\) is the Polanyi potential, \(K\) is the Dubinin-Radushkevich constant, \(R\) is the gas constant (8.31 Jmol\({}^{-1}\) k\({}^{-1}\)), T is the absolute temperature and E is the mean adsorption energy. In order to determine the thermodynamic parameters, the sorption studies were carried out at various temperatures (294, 303, 323 and 343 K). The values of \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) were calculated from the slope and intercept of plot between \(\Delta G^{\circ}\) vs. \(T\).