## 3 Results and Discussions
### Kinetic Models for the Absorption of Heavy Metals
For Pb\({}^{2+}\), PA--DVB showed a better fit with the first-order (PFO) model, with an R\({}^{2}\) value of 98.3% and a \(K1\) constant of 0.0377 min\({}^{-1}\). In the case of cobalt (Co\({}^{2+}\)), PA--DVB showed a good fit to the PFO model, with an R\({}^{2}\) value of 97.8% and a \(K1\) constant of 0.127 min\({}^{-1}\). For Ni\({}^{2+}\), the PFO model was the best (R\({}^{2}\) = 97.88%), with a \(K1\) constant of 0.126 min\({}^{-1}\). In conclusion, the trends obtained using the PA--DVB resin seem to be better explained by the PFO model for some metals, such as Pb\({}^{2+}\), Co\({}^{2+}\), and Ni\({}^{2+}\), while for Cu\({}^{2+}\), it seemed to be better explained using the PSO model. With the use of the TFSA resin, Cu fit better to the PFO model, with an R\({}^{2}\) value of 98% and a rate constant \(K1\) of 0.07047 min\({}^{-1}\), suggesting faster adsorption. For Pb\({}^{2+}\), the PSO model fit well (R\({}^{2}\) = 99.5), with a rate constant \(K2\) of 0.0008 g/(mg*min), suggesting moderate adsorption kinetics. Finally, the OGL resin showed limited data for most metals, indicating poor affinity or limited adsorption capacity. The lack of data may suggest that OGL is unable to significantly adsorb metals or that its structure is not particularly suitable for these applications. This makes this particular OGL less versatile compared to the other resins tested, and potentially useful only for highly specific or selective applications. However, further physical-chemical modification of lignin may alter its properties to strengthen the adsorption of metal ions. For certain metal-resin pairs, it was not possible to calculate the statistical parameters (R\({}^{2}\)) and kinetic constants (\(K1\) and \(K2\)). This occurs when the experimental data do not follow the trends predicted by either the pseudo-first-order (PFO) or pseudo-second-order (PSO) kinetic models. When this happens, it can be inferred that no significant chemical-physical interactions develop between the resin and the specific metal, making the adsorption process inefficient. In other cases, such as the Ni\({}^{2+}\)-OGL pair, where the graph shows a clear reduction in Ni concentration, the experimental data did not fit well with either of the selected kinetic models. Careful observation shows that for both the OGL and TFSA resins, the adsorption and desorption phenomena occurred within the observed time range, which may complicate the application of kinetic models that better describe this type of process. Further investigations will be conducted to explore this behavior. However, some researchers argue that the pseudo-first-order (PFO) model is particularly suited for describing the initial phases of adsorption kinetics but fails to adequately represent the entire process, which is more comprehensively described by the pseudo-second-order model [33]. Nevertheless, this interpretation does not explain the superior fit of one model over another, and these empirical models cannot provide mechanistic insights into the adsorption phenomena. Adsorption kinetics are governed by multiple concurrent mechanisms, encompassing both diffusive and chemical processes, and the pseudo-first and pseudo-second-order models alone are insufficient to identify all the involved mechanisms [34; 35]. While empirical kinetic models are widely employed to describe resin adsorption processes, it is crucial to acknowledge their interpretative limitations. A comprehensive understanding of the adsorption process necessitates an integrated approach that considers both mass transport phenomena (external diffusion, intraparticle diffusion) and specific chemical interactions between the adsorbate and adsorbent. Only through this broader perspective can an accurate description of the governing mechanisms be achieved, thereby enabling optimization of the system's operational parameters.