## 3 Results and Discussion
### Analysis of Sorption Kinetics
#### 3.5.2. Elovich Kinetic Model  
The model is expressed as:  
\[q_{t}=\frac{1}{\beta}*ln(\alpha*\beta)+\frac{1}{\beta}*\ln(t) \tag{7}\]  
where \(q_{t}\) is the sorption capacity at time t, in g/g; \(\alpha\) is the initial sorption rate, in g\(\cdot\)g\({}^{-1}\cdot\)min\({}^{-1}\); and \(\beta\) is the surface coverage parameter, in g/g (related to activation energy).  
For further analysis, the initial equation should be linearized (for plotting Figures 14 and 15):  
\[q_{t}=a+b*ln(t) \tag{8}\]  
Slope(\(b\)) = 1/\(\beta\);  
Intercept(\(a\)) = (1/\(\beta\)) * \(\ln(\alpha*\beta)\).  
For the calculation of \(\alpha\) and \(\beta\), the following equations were used:  
\[\beta=\frac{1}{b} \tag{9}\]  
\[\alpha=\frac{e^{a*\beta}}{\beta} \tag{10}\]  
Figure 14: [PERSON] kinetic model’s curve for MIP(DEGDMA).  
\[\beta(MIP(DEGDMA))=\frac{1}{1.3979}=0.7154\;\mathrm{g/g}\]  
\[\beta(MIP(DVB))=\frac{1}{1.3917}=0.7185\;\mathrm{g/g}\]  
\[\alpha(MIP(DEGDMA))=\frac{2.72(-4.0589*0.7154)}{0.7154}=0.0172\;\mathrm{min}^{-1}\]  
\[\alpha(MIP(DVB))=\frac{2.72(-4.2738*0.7185)}{0.7185}=0.0139\;\mathrm{min}^{-1}\]  
Higher \(\alpha\) indicates faster initial adsorption. MIP(DEGDMA) has a 24% higher initial rate than MIP(DVB) (0.0172 vs. 0.0139 min\({}^{-1}\)). MIP(DEGDMA) extracts silver more quickly at the early stages. These values correlate with the results calculated by the Radushkevich model.  
Possible reasons for MIP(DEGDMA) higher \(\alpha\) values:  
* Higher porosity/microporosity: Faster diffusion of adsorbate into pores.
* More active surface sites: Functional groups (e.g., -OH, -COOH) accelerate initial binding.
* Smaller particle size: Larger surface area-to-volume ratio.  
Thus, MIP(DEGDMA) saturates faster than MIP(DVB), which correlates with the results of the sorption experiments and the Radushkevich model.  
Both sorbents have nearly identical \(\beta\) values (\(\sim\)0.715-0.719 g/g), suggesting similar binding energies and surface heterogeneity. Nearly identical \(\beta\) values imply:  
* The same adsorption mechanism (e.g., chemisorption via ion exchange or complexation).
* Comparable surface heterogeneity (energy distribution of binding sites).  
The surface coverage parameter of MIP(DVB) is slightly more than for MIP(DEGDMA). A higher \(\beta\) indicates stronger adsorbate-adsorbent interactions (chemisorption). These results correlate with the results of the sorption experiments, where it was proposed that the binding of MIP(DVB) with silver is stronger.  
The results of both kinetic models (Radushkevich and Elovich) are similar and correlate with the results of the sorption experiments. The values of the sorption rate in both models are close to each other (Table 5); however, MIP(DEGDMA) has a higher sorption rate.  
Figure 15: Elovich kinetic model’s curve for MIP(DVB).  
Also, both kinetic models showed that MIP(DVB) sorbs a higher concentration of silver in comparison with MIP(DEGDMA).  
Thus, there is a similarity in the sorption process of both MIPs, so the sorption is provided mainly by complexation with monomer units, but slight differences are influenced by the differences in the structures of the crosslinkers.