## 3 Results and Discussion
### Adsorption Studies
#### 3.2.3 Effect of Temperature and Thermodynamic Studies  
Temperature typically affects adsorption capacity and efficiency. In this study, the effect of temperature on Cu\({}^{2+}\) adsorption was evaluated in the range of 293-333 K using solutions with an initial metal ion concentration of 200 mg L\({}^{-1}\) at pH 4 (Figure 7). The results showed that the adsorbed amount increased with increasing temperature for the biosorbents MO and M, indicating that the adsorption process is endothermic in nature. This behavior suggests that higher temperatures enhance the interaction between Cu\({}^{2+}\) ions and the sorbent surface. In contrast, a slight decrease in Qe was observed for LA and L as the temperature increased, which implies that the adsorption is exothermic, and elevated temperatures may reduce adsorption efficiency for these materials.  
For liquid-phase adsorption, the distribution coefficient (Kd) is expressed as the ratio of the equilibrium concentration of adsorbate on the adsorbent to its equilibrium concentration in solution, as given by  
\[\text{Kd}=\text{Qe/Ce}\]  
where Qe (mg L\({}^{-1}\)) is the amount of adsorbate adsorbed per unit volume of solution at equilibrium, and Ce (mg L\({}^{-1}\)) is the equilibrium concentration in solution.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} Biosor- \\ bent \\ \end{tabular} } & \multicolumn{3}{c}{**Pseudo-First-Order Model**} & \multicolumn{3}{c}{**Pseudo-Second-Order Model**} & \multicolumn{3}{c}{**Intraparticle Diffusion Model**} \\ \cline{2-9}  & **Qe (mg g\({}^{-1}\))** & **k\({}_{1}\) (min\({}^{-1}\))** & **r\({}^{2}\)** & **Qe (mg g\({}^{-1}\))** & **k\({}_{2}\) (g mg\({}^{-1}\))** & **r\({}^{2}\)** & **k\({}_{\text{id}}\) (mg g\({}^{-1}\))** & **C (mg g\({}^{-1}\))** & **r\({}^{2}\)** \\ \hline MO & 1.50 & 0.012 & 0.9353 & 13.55 & 1.720 & 0.9999 & 0.449 & 14.79 & 0.9531 \\  & & & & & & 0.009 & 16.73 & 0.8213 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Kinetic parameters for the adsorption of Cu(II) ions onto MO.  
Figure 6: Influence of contact time on the amount of adsorbed Cu(II) ions onto MO, M, LA, and L.  
The thermodynamic parameters for adsorption, such as Gibbs free energy (\(\Delta\)G\({}^{0}\)), enthalpy (\(\Delta\)H\({}^{0}\)), and entropy (\(\Delta\)S\({}^{0}\)), were determined using the van't Hoff equation:  
\[\ln(\text{Kd})=\Delta\)S\({}^{0}/\text{R}-\Delta\)H\({}^{0}/\text{RT}\]  
where R is the universal gas constant (J\(\cdot\)mol\({}^{-1}\cdot\)K\({}^{-1}\)) and T is the absolute temperature in Kelvin. A plot of \(\ln(\text{Kd})\) versus 1/T allows estimation of \(\Delta\)H\({}^{0}\) and \(\Delta\)S\({}^{0}\) from the slope and intercept, respectively. The standard Gibbs free energy change is calculated using  
\[\Delta\)G\({}^{0}=-\text{RT}\ln(\text{Kd})\]  
The calculated thermodynamic parameters are summarized in Table 3.  
The positive \(\Delta\)H\({}^{0}\) values for MO and M confirm that the adsorption process is endothermic, meaning it is enhanced at higher temperatures. Conversely, negative \(\Delta\)H\({}^{0}\) values for LA and L indicate exothermic adsorption, which slightly decreases with increasing temperature. Additionally, the relatively low magnitudes of \(\Delta\)H\({}^{0}\) (all <20 kJ/mol) suggest that the adsorption mechanism is likely physisorption. The negative values of \(\Delta\)G\({}^{0}\) across all biosorbents and temperatures confirm that Cu\({}^{2+}\) adsorption is spontaneous and thermodynamically favorable. These values range between \(-\)0.68 and \(-\)4.78 kJ/mol, which falls well within the expected range for physisorption (0 to \(-\)20 kJ/mol). The positive entropy changes for MO and M indicate an increase in randomness at the solid-liquid interface during adsorption, possibly due to the release of solvated water molecules. In contrast, negative \(\Delta\)S\({}^{0}\) values for LA and L suggest a slight ordering of the system, possibly due to specific interactions or structuring of adsorbed ions on the surface.