## 3 Results and Discussion
### Synthesis of MIP on Terbium Nitrate
#### 3.2.5 Radushkevich Kinetic Model  
The non-linear type of Radushkevich equation can be presented as follows:  
\[q_{t}=q_{e}\times\left(1-e^{-k_{\rm R}\times t}\right)\]  
where q\({}_{\rm e}\) is an equilibrium sorption capacity (from the experimental data), mg/g; q\({}_{\rm t}\) is an amount adsorbed at time t, mg/g; k\({}_{2}\)--is a Radushkevich rate constant, min\({}^{-1}\); and t is a time, min. After linearization, the equation is:  
\[\rm ln\left(1-\frac{q_{t}}{q_{e}}\right)=-k_{R}\times t\]  
Using the sorption data, the curve of \(\rm ln(1-q_{t}/q_{e})\) = f(t) is plotted (Figures 15 and 16), from which the following parameter could be found:  
\[\rm k_{R}=-Slope\ (S)\]  
\[\rm k_{R}(MIP)=-(-0.0267)=0.0267\ \rm min^{-1}\]  
\[\rm k_{R}(NIP)=-(-0.0258)=0.0258\ \rm min^{-1}\]A higher \(\mathrm{K_{r}}\) means faster sorption kinetics--in our case, the MIP has a slightly higher \(\mathrm{K_{r}}\) (0.0267 vs. 0.0258 min\({}^{-1}\)), which means that it reaches equilibrium marginally faster than the NIP. As mentioned above, the PSO model suggested that the NIP has a higher \(\mathrm{k_{2}}\) but much lower \(\mathrm{q_{e}}\). The Radushkevich model shows similar \(\mathrm{K_{r}}\) values, meaning the kinetics are comparable, but the MIP wins due to its higher capacity \(\mathrm{q_{e}}\). Kr reflects the diffusion rates in porous adsorbents--a higher \(\mathrm{K_{r}}\) means faster diffusion into pores, while a lower \(\mathrm{K_{r}}\) means slower diffusion (e.g., due to smaller pores or stronger adsorbate-adsorbent interactions). But the Kr alone does not determine the \"best\" sorbent--the equilibrium capacity (\(\mathrm{q_{e}}\)) is a critical parameter. Possible reasons why the MIP has a slightly higher \(\mathrm{K_{r}}\):  
1. Similar pore structures: If both sorbents have comparable porosity, their diffusion rates (\(\mathrm{K_{r}}\)) may differ only slightly. Figure 16: Radushkevich kinetic model’s curve for the NIP. Figure 15: Radushkevich kinetic model’s curve for the MIP. 2. Dominance of q\({}_{\rm e}\): The MIP may have more adsorption sites (higher q\({}_{\rm e}\)), even if the kinetics are similar. Also, for the evaluation of a material as a sorbent, desorption must be carried out along with adsorption. The desorption process shows the ability of the MIP to release the target molecules that were previously bound inside the polymer matrix. On the other hand, desorption means a process that allows the MIP to desorb the adsorbed molecules from its structure back into the reactive medium. At the same time, the specificity of the MIPs means that it is possible to extract ions for their further use and reuse them. To release bound molecules from the MIP, a solvent change desorption method was used. The work was performed with the same scheme as adsorption, but 0.05 M nitric acid was used as a solvent instead of water. The change of the Tb concentration in the solution and in the MIP and NIP is shown in Figure 17. In the case with the MIP, the concentration of Tb increases in the solution (along with a decrease in the polymer) for 150 min (2.5 h), and after that time, desorption stops. At this time, the concentration of Tb in solution is 0.32 g/100 mL. The desorption of Tb from the NIP matrix occurs very slightly, constant values of the Tb concentration are observed after 60 min (1 h), and the concentration of Tb in the solution is 0.026 mg/100 mL. These values remain constant up to the control measurement after 24 h.  
Figure 18 represents the desorption degree of Tb from the MIP and NIP. The desorption degree of Tb (MIP) increases sharply, reaching its final values 90.15% after 150 min, while this parameter for the NIP increases intensively for 1 h, after which the further increase is very slight--the final value of the parameter is 52.67%. In Table 8, the values of the desorption degree after 24 h are presented. Based on the results shown in Table 8, it can be concluded that the NIPs show low desorption properties compared to the MIPs.