## 3 Results and Discussion
### Theoretical Models
#### 3.3.4 Intraparticle Diffusion Model
 } \\ \cline{1-1}  & & & \(\mathbf{K_{1}}\) & 0.031 & \(\mathbf{K_{2}}\) & 37.8 & \(\mathbf{a}\) & \(2.33\times 10^{3}\) & \(\mathbf{C_{i}}\) & 0.0012 \\ \cline{1-1}  & & & \(\mathbf{q_{e}}\) & \(2.75\times 10^{-3}\) & \(\mathbf{\beta}\) & \(5.13\times 10^{-4}\) & \(\mathbf{K_{i}}\) & \(1.22\times 10^{-4}\) \\ \cline{1-1} \cline{2-7}  & & \(\mathbf{R^{2}}\) & 0.907 & \(\mathbf{R^{2}}\) & 0.98026 & \(\mathbf{R^{2}}\) & 0. In this sense, the Elovich model (Figure 5c) also represents a chemisorption phenomenon. Nonetheless, this model is applied to systems with heterogeneous surfaces and different activation energies [9,34]. Consequently, the correlation coefficients obtained by the Elovich model (\(0.88164<R^{2}<0.93829\)) exhibit less agreement with the experimental values obtained. The intraparticle diffusion model (Figure 5d) presents \(R^{2}\) values in the range of \(0.911<R^{2}<0.9523\) (Table 2). This result can be attributed to the homogeneous porous structure of the zeolite. Therefore, this model also describes the transport and diffusion mechanism of the solute through the internal structure of the pores of the adsorbent (clinoptilolite). Figure 6 presents the correlation values (\(R^{2}\)) obtained for the theoretical models presented. This graph shows that the highest \(R^{2}\) values are associated with the pseudo-second-order model. Typically, this model describes a chemisorption process, which involves valence forces generated between the zeolite surface and the active sites of the AgNPs. In this regard, it can be assumed that  
Figure 5: Graphs of the theoretical models corresponding to (**a**) the [PERSON] pseudo-first-order model; (**b**) the second-order model; (**c**) the Elovich; and (**d**) the intraparticle diffusion models. the ionization of Ag nanoparticles promotes the superficial adhesion of the AgNPs on the zeolite outer surface. In order to study the distribution of silver nanoparticles on the surface of the zeolite, a study by TEM was performed. Figure 7 shows a (a) bright-field TEM and (b) STEM images, in which the distribution of the AgNPs on the zeolite outer surface can be appreciated. This result verifies that the chemisorption phenomenon describes the adsorption process carried out by the mineral. This is due to the AgNPs' interaction with the active sites of the zeolite, and the formation of a uni-molecular layer uniformly distributed on the zeolite outer surface is observed. In this sense, it can be deduced that the process of impregnation of the particles was carried out efficiently since the adsorbate was distributed throughout the external surface of the zeolite. For additional evidence of the chemisorption process, Figure 8 schematizes the adhesion mechanism of the AgNPs on the clinoptiolite surface, which was carried out by the interaction of the  
Figure 6: Analysis of the \(R^{2}\) correlation values obtained from the theoretical models employed to describe the AgNPs adsorption process on zeolite. Figure 7: **(a)** Bright-field TEM and **(b)** STEM images corresponding to the AgNP distribution onto the zeolite surface. valence forces as describe in the pseudo-second-order model. Also, this proposed model is in accordance with the results observed by the TEM images. The model was performed using the software QUANTA (Accelrys, San Diego, CA, USA).