## 4 Results and Discussion
### Isotherm, Kinetic, and Thermodynamic Studies
#### 4.4.3 Thermodynamic Parameters  
The thermodynamic studies were analyzed using the following temperature-dependent parameters: (\(\Delta\)S\({}^{\circ}\)), (\(\Delta\)H\({}^{\circ}\)), and (\(\Delta\)G\({}^{\circ}\)). Table 4 shows the results obtained for each parameter. The results shown for the enthalpy parameter were positive for all adsorbents and pollutants, indicating that the uptake process is endothermic. This result is consistent with the data obtained in the temperature effect study. The tabulated result of entropy values indicates that there was an increase in freedom during uptake due to the positive  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**adsorption**} & \multirow{2}{*}{**Parameters**} & \multicolumn{2}{c}{**CPW**} & \multicolumn{2}{c}{**APWH**} & \multicolumn{2}{c}{**APWN**} & \multicolumn{2}{c}{**APWP**} & \\ \cline{3-10}  & & **Cr(VI)** & **MB** & **Cr(VI)** & **MB** & **Cr(VI)** & **MB** & **Cr(VI)** & **MB** \\ \hline Langmuir & Qo (mg/g) & 8.621 & 22.69 & 23.27 & 26.14 & 29.01 & 29.50 & 25.73 & 25.62 \\  & B (L/mol) & 0.093 & 0.129 & 0.091 & 0.165 & 0.096 & 0.181 & 0.099 & 0.162 \\  & R\({}^{2}\) & 0.979 & 0.968 & 0.983 & 0.974 & 0.975 & 0.981 & 0.984 & 0.981 \\  & RMSE & 12.09 & 18.21 & 22.60 & 19.16 & 21.35 & 23.77 & 21.07 & 22.51 \\ Freundlich & K\({}_{\rm F}\) & 1.046 & 1.199 & 1.060 & 1.210 & 1.105 & 1.227 & 1.164 & 1.199 \\  & N & 1.102 & 1.124 & 1.109 & 1.130 & 1.088 & 1.142 & 1.097 & 1.128 \\  & R\({}^{2}\) & 0.990 & 0.995 & 0.991 & 0.993 & 0.997 & 0.995 & 0.995 & 0.996 \\  & RMSE & 8.321 & 7.207 & 7.829 & 6.115 & 7.937 & 5.812 & 9.520 & 8.103 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Isotherms for the removal of Cr(VI) and MB.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**adsorption**} & \multirow{2}{*}{**Parameters**} & \multicolumn{2}{c}{**CPW**} & \multicolumn{2}{c}{**APWH**} & \multicolumn{2}{c}{**APWN**} & \multicolumn{2}{c}{**APWP**} & \\ \cline{3-10}  & & & **Cr(VI)** & **MB** & **Cr(VI)** & **MB** & **Cr(VI)** & **MB** & **Cr(VI)** & **MB** \\ \hline PFOM & q\({}_{e}\) (mg/g) & 8.09 & 16.38 & 18.89 & 20.51 & 23.65 & 24.91 & 22.08 & 21.20 \\  & K\({}_{1}\) (min\({}^{-1}\)) & 0.004 & 0.008 & 0.009 & 0.010 & 0.011 & 0.012 & 0.010 & 0.011 \\  & R\({}^{2}\) & 0.948 & 0.950 & 0.941 & 0.957 & 0.944 & 0.957 & 0.949 & 0.953 \\ FSOM & q\({}_{e}\) (mg/g) & 15.18 & 25.90 & 29.10 & 30.16 & 33.52 & 34.11 & 31.15 & 31.35 \\  & K\({}_{2}\) (g.mg/min) & 0.812 & 0.920 & 1.077 & 1.059 & 1.071 & 1.082 & 1.066 & 1.069 \\  & R\({}^{2}\) & 0.994 & 0.996 & 0.992 & 0.996 & 0.993 & 0.998 & 0.994 & 0.997 \\ IPDM & C (mg/g) & 6.529 & 18.83 & 19.02 & 19.84 & 21.63 & 22.03 & 20.02 & 21.47 \\  & K\({}_{i}\) (g/g.min\({}^{1/2}\)) & 0.027 & 0.031 & 0.038 & 0.042 & 0.047 & 0.052 & 0.045 & 0.044 \\  & R\({}^{2}\) & 0.978 & 0.981 & 0.983 & 0.979 & 0.985 & 0.972 & 0.980 & 0.976 \\ \hline Experimental & q\({}_{e}\) (mg/g) & 15.10 & 25.93 & 28.83 & 30.01 & 33.46 & 34.00 & 31.06 & 31.30 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Kinetic studies for the removal of Cr(VI) and MB.  
values. It was also found that the Gibbs free energy values were all negative, indicating that adsorption between the adsorbents and the contaminants was favored [38].