## 3 Results and Discussion
### Adsorption Kinetics and Equilibrium Models
Figure 7: The fit of the pseudo-second-order adsorption kinetics to the experimental data for (**a**) LD, (**b**) RS, (**c**) CH, and (**d**) HH biosorbents. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{q_{eq,exp}}\) (mg/g) & **LD** & **RS** & **HH** & **CH** \\  & **88.5** & **75.0** & **86.5** & **83.4** \\ \hline Pseudo-first order & & & & \\ \(\mathbf{q_{eq,cal}}\) (mg/g) & 9.1 & 65.6 & 57.8 & 52.5 \\ \(\mathbf{k_{1,ad}}\) (1/min) & 0.167 & 0.292 & \(-\)0.093 & \(-\)0.090 \\ \(\mathbf{R}^{2}\) & 0.856 & 0.992 & 0.902 & 0.874 \\ \(\varepsilon\)*(\%) & \textgreater{}10 & \textgreater{}10 & \textgreater{}10 & \textgreater{}10 \\ \hline Pseudo-second order & & & & \\ \(\mathbf{q_{eq,cal}}\) (mg/g) & 88.5 & 75.8 & 88.5 & 85.5 \\ \(\mathbf{k_{2,ad}}\) (g/mg.min) & 0.075 & 0.009 & 0.005 & 0.006 \\ \(\mathbf{R}^{2}\) & 0.999 & 0.997 & 0.996 & 0.9964 \\ \(\varepsilon\)*(\%) & 0.1 & 1.1 & 2.2 & 2.5 \\ \hline Intra-particle diffusion & & & & \\ \(\mathbf{C}\) & 51.87 & 27.47 & 39.2 & 41.4 \\ \(\mathbf{k_{3,ad}}\) (mg/g.min\({}^{1/2}\)) & 7.178 & 6.87 & 6.84 & 5.95 \\ \(\mathbf{R}^{2}\) & 0.325 & 0.556 & 0.884 & 0.922 \\ \hline \hline \end{tabular}  
* \(\varepsilon\) is the mean absolute percentage error (MAPE).  
\end{table} 
Table 3: The experimental and estimated \(\mathbf{q_{eq}}\) values for the pseudo-first-order and pseudo-second-order adsorption kinetic models for an initial 100 ppm MB concentration. The pseudo-first-order model did not provide a satisfactory fit for the adsorption data across all biosorbents, as evidenced by the significant deviation of calculated \(q_{eq}\) values from experimental data and the anomalously high mean absolute percentage error (MAPE) values, \(>\)10%. Notably, the negative rate constants (\(-\)0.093 and \(-\)0.090) obtained for HH and CH biosorbents under this model are indicative of its inapplicability, which may arise from limitations of the model in capturing the complexity of the adsorption process. Conversely, Table 3 highlights that the mean absolute percentage error (\(\varepsilon\), MAPE) and the correlation coefficients (R\({}^{2}\)) for pseudo-second-order kinetics were much lower than those obtained for the pseudo-first-order and intra-particle diffusion kinetic models when fitted to the experimental data. These results demonstrated that the pseudo-second-order kinetic model has the best fit for the biosorbents and identifies the adsorption kinetics of MB onto these sorbents (LD, RS, CH, and HH). The pseudo-second-order kinetic model is often associated with chemisorption because it assumes that the rate-limiting step involves chemical bonding or sharing of electrons between adsorbent and adsorbate. However, since the enthalpy values are low (presented in Section 3.3: Adsorption Thermodynamics), this typically suggests that the adsorption process is more characteristic of physisorption, which involves weaker van der Waals forces rather than the stronger covalent bonds of chemisorption. This apparent discrepancy between kinetic modelling and thermodynamic data is not uncommon and can arise due to several reasons such as the complexity of  
Figure 8: Estimated adsorption capacities for a pseudo-second-order kinetic model with experimental results for (**a**) LD, (**b**) RS, (**c**) CH, and (**d**) HH biosorbents. (“th” is the estimated adsorption capacities from the pseudo-second-order model). adsorption mechanisms, limitations of kinetic models, and surface properties and adsorbate concentration. The adsorption of MB over these biosorbents could occur via a multi-step process involving both physisorption and chemisorption. The discrepancy could be attributed to an initial rapid physisorption, followed by slower chemisorption. Under these conditions, the pseudo-second-order model can provide a good fit with the overall kinetics because it is sensitive to the slower, rate-limiting step that could be chemisorption, even if the process also involves significant physisorption.