# Preparation and Physicochemical Properties of Tannin-Imobilized Membrane Adsorbent
## 3 Results and Discussion
In addition, the results of fitting using the Elovich model (q\({}_{\text{f}}\) = 1/8\({}^{\circ}\)ln(1 + \(\alpha\)gt), where q\({}_{\text{t}}\) (mmol/g) is the adsorption capacity of UO\({}_{2}\)\({}^{2+}\) at time \(t\), \(\alpha\) is the initial adsorption rate constant, and \(\beta\) is the chemical adsorption activation energy parameters) are shown in Figure 5f, and the correlation coefficient R\({}^{2}\) > 0.99 obtained by fitting indicates that the Elovich model is more appropriate to describe the adsorption of UO\({}_{2}\)\({}^{2+}\) on PA-BWT, which is a chemical adsorption process. The Weber-Morris model was also used for fitting. As shown in Figure S3, the adsorption process is divided into two stages, namely the surface diffusion process and the intraparticle diffusion process. The straight line through the origin indicates that surface diffusion and intra-particle diffusion jointly control the adsorption process of PA-BWT on UO\({}_{2}\)\({}^{2+}\). The thermodynamic behaviors of UO\({}_{2}\)\({}^{2+}\) ion adsorption onto PA-BWT were evaluated by employing the following equations [15]:  
Figure 1:0  
Figure 1:1  
Figure 1:2  
where R (8.314 J/K mol) is the ideal gas constant; T (K) is the temperature; Kd (L/g) is the distribution coefficient; C\({}_{0}\) (mg/L) is the initial concentration; \(\Delta\text{H}^{\circ}\), \(\Delta\text{S}^{\circ}\), and \(\Delta\text{G}^{\circ}\) (kJ/mol) are the enthalpy change, the entropy change, and the Gibbs free energy change in a given process, respectively; b is the Langmuir parameter, whereas b' is a constant; and x(g/mol) is the adsorbent molecular weight. Thermodynamic parameters were calculated according to Equations (6)-(8) and illustrated in Figure 1:3 \(\Delta\text{G}^{\circ}\) values are negative for UO\({}_{2}\)\({}^{2+}\) adsorption onto the PA-BWT surface, indicating that adsorption is a spontaneous process in standard conditions [34]. The positive value of \(\Delta\text{S}^{\circ}\) suggests the affinity of PA-Tan toward UO\({}_{2}\)\({}^{2+}\) ions. The positive value of \(\Delta\text{H}^{\circ}\) confirms the endothermic nature of the adsorption process. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Temperature (K)**} & \multicolumn{4}{c}{**Langmuir**} & \multicolumn{4}{c}{**Freundlich**} \\  & **q\({}_{\text{max}}\)** & **B (L/mmol)** & **q\({}_{\text{max,exp}}\)** & **R\({}^{2}\)** & **K** & **1/n** & **R\({}^{2}\)** \\  & **(mmol/g)** & & **(mmol/g)** & & & & \\ \hline
303 & 0.209 & 12.982 & 0.192 & 0.9825 & 6.72 & 0.5132 & 0.9665 \\
313 & 0.282 & 24.201 & 0.269 & 0.9992 & 14.53 & 0.4610 & 0.9635 \\
323 & 0.314 & 40.043 & 0.317 & 0.9975 & 23.21 & 0.4662 & 0.9256 \\
333 & 0.401 & 41.565 & 0.405 & 0.9933 & 28.34 & 0.4956 & 0.9434 \\ \hline \hline \end{tabular} \(q_{\text{max,exp}}\): the maximum adsorption capacity determined by experiments.  
\end{table} 
Table 1: Langmuir and Freundlich parameters of UO\({}_{2}\)\({}^{2+}\) adsorbed on PA-BWT. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Temperature**} & \multicolumn{4}{c}{**Pseudo-Second-Order**} & \multicolumn{4}{c}{**Pseudo-First-Order**} \\  & **R\({}^{2}\)** & **k\({}_{2}\)** & **q\({}_{\text{e,cal}}\) (mmol/g)** & **q\({}_{\text{e,exp}}\) (mmol/g)** & **R\({}^{2}\)** & **k\({}_{1}\)** \\ \hline
303 & 0.9975 & 0.101 & 0.206 & 0.201 & 0.7708 & 0.553 \\
313 & 0.9978 & 0.123 & 0.246 & 0.240 & 0.7899 & 0.566 \\
323 & 0.9998 & 0.142 & 0.269 & 0.262 & 0.8789 & 0.664 \\
333 & 0.9998 & 0.223 & 0.284 & 0.279 & 0.9701 & 0.605 \\ \hline \hline \end{tabular} \(q_{\text{e,exp}}\): the maximum adsorption capacity determined by experiments. \(q_{\text{e,cal}}\): the maximum adsorption capacity determined by calculation.  
\end{table} 
Table 2: Fitting parameters of the pseudo-first-order and pseudo-second-order models. The stability experiment of M-TANs was carried out by immersion oscillation experiment under different temperatures and pH values.