## 3 Results and Discussion
### Synthesis of MIP on Terbium Nitrate
#### 3.2.4 Pseudo-Second-Order Kinetic Model  
The non-linear PSO equation can be presented as follows:  
\[q_{t}=\frac{k_{2}\times q_{e}^{2}\times t}{1+k_{2}\times q_{e}\times t}\]  
where q\({}_{e}\) is an equilibrium sorption capacity, mg/g; q\({}_{t}\) is an amount adsorbed at time t, mg/g; and k\({}_{2}\) is a PSO rate constant, g/(mg \(\times\) min).  
For the calculation of k\({}_{2}\) and q\({}_{e}\), the above-mentioned equation should be linearized:  
\[\frac{t}{q_{t}}=\frac{1}{k_{2}\times q_{e}^{2}}+\frac{1}{q_{e}}\times t\]  
Using the sorption data, the curve of t/q\({}_{t}\) = f(t) is plotted (Figures 13 and 14), from which the following parameters could be found:  
\[\text{Slope(S)}=\frac{1}{q_{e}}\]  
\[q_{e}=\frac{1}{S}\]  
\[\text{Intercept(I)}=\frac{1}{k_{2}\times q_{e}^{2}}\]  
\[k_{2}=\frac{1}{I\times q_{e}^{2}}\]  
\[\text{q}_{e}(\text{MIP})=\frac{1}{0.2071}=4.83\;\text{mg/g}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Sorption Parameter** & **MIP** & **NIP** \\ \hline Sorption degree, \% & 70.80 \(\pm\) 0.48 & 9.95 \(\pm\) 0.07 \\ \hline Sorption capacity, g/g & 3.54 + 0.02 & 0.50 \(\pm\) 0.01 \\ \hline \end{tabular}
\end{table}
Table 7: Sorption parameters of the MIP and NIP in relation to Tb after 24 h.  
\[\mathrm{k}_{2}(\mathrm{MIP})=\frac{1}{11.126\times 4.83^{2}}=0.0039\ \frac{ \mathrm{g}}{\mathrm{mg}\times\mathrm{min}}\] \[\mathrm{q}_{\mathrm{e}}(\mathrm{NIP})=\frac{1}{1.5364}=0.65\ \mathrm{mg}/ \mathrm{g}\] \[\mathrm{k}_{2}(\mathrm{NIP})=\frac{1}{64.466\times 0.65^{2}}=0.0366\ \frac{ \mathrm{g}}{\mathrm{mg}\times\mathrm{min}}\]  
The obtained PSO kinetic rate constants for the MIP and NIP are realistic but seem counterintuitive at first glance. The higher values of \(\mathrm{k}_{2}\) for the NIP means faster sorption (but this is not always better). For a proper comparison of the MIP and NIP sorption properties, it is necessary to calculate the initial sorption rate:  
\[h=k_{2}\times q_{e}^{2}\]  
Figure 14: PSO kinetic model’s curve for the NIP.  
Figure 13: PSO kinetic model’s curve for the MIP.  
\[h_{MIP}=0.0038\times 4.83^{2}=0.0887\ \frac{\rm mg}{\rm g\times min}\]  
\[h_{MIP}=0.0366\times 0.65^{2}=0.0155\ \frac{\rm mg}{\rm g\times min}\]  
Also, the time taken to reach 50% saturation (t\({}_{1/2}\)) should be calculated:  
\[t_{1/2}=\frac{1}{k_{2}\times q_{e}}\]  
\[t_{1/2(MIP)}=\frac{1}{0.0038\times 4.83}=54.4\ \rm min\]  
\[t_{1/2(NIP)}=\frac{1}{0.0366\times 0.65}=42.1\ \rm min\]  
The MIP has a higher effective rate than the NIP despite the MIP having a lower k\({}_{2}\). The NIP reaches 50% saturation slightly faster (42.1 vs. 54.4 min), but its total capacity is 7.4 times lower (0.65 vs. 4.83 mg/g). The MIP sorbs slower per mass unit but removes far more target metal overall.  
The NIP has a lower q\({}_{\rm e}\), which indicates that the adsorbent saturates quickly (low capacity); it may show a high k\({}_{2}\) because equilibrium is reached faster, but the total uptake is poor. In other words, the NIP might bind adsorbates weakly but rapidly. It can be explained by binding to limited sites or weak surface interactions (high k\({}_{2}\) + low q\({}_{\rm e}\)), while in the case with MIP, slower but more extensive adsorption is observed (low k\({}_{2}\) + high q\({}_{\rm e}\)). Consequently, the NIP can be considered a sprinter (fast but tires quickly) while the MIP is a marathon runner (slower start but much higher total uptake).  
For the evaluation of the sorbent from the PSO kinetic model point of view, the following should be taken in account:  
1. Joint comparison of h and q\({}_{\rm e}\) for the studied sorbents;
2. High k\({}_{2}\
eq\) \"better\"--it is only better if the q\({}_{\rm e}\) is also high.