# Preparation and Physicochemical Properties of Tannin-Imobilized Membrane Adsorbent
## 3 Results and Discussion
To clearly understand the adsorption process, adsorption isotherm data were further analyzed by the Langmuir models (Figure 5b) [28; 29]:  
\[\text{Langmuir equation}:\ \frac{C_{e}}{q_{e}}=\frac{1}{b\cdot q_{\text{max}}}+ \frac{C_{e}}{q_{\text{max}}} \tag{2}\]  
\[\text{Freundlich equation}:\ln q_{\text{e}}=\ln K+\frac{1}{n}\ln C_{e} \tag{3}\]  
where \(q_{\text{e}}\) (mmol/g) is the equilibrium adsorption capacity; \(q_{\text{max}}\) (mmol/g) is the maximum adsorption capacity; \(C_{\text{e}}\) (mmol/L) is the equilibrium concentration; and \(b\) (L/mg) is the coefficient related to the affinity of the binding sites (\(b=k_{\text{a}}/k_{\text{d}}\), \(k_{\text{a}}\) is the rate constant of adsorption, \(k_{\text{d}}\) is the rate constant of desorption). \(K\) and \(1/n\) are the constants that are related to the adsorption capacity and the adsorption intensity, respectively. As summarized in Table 1, the Langmuir model provides a much better description of the isotherm data with a correlation constant higher than 0.99, and the calculated theoretical adsorption capacity is close to those obtained from the experiments. Consequently, the Langmuir model is more suitable for the description of adsorption isotherms of UO\({}_{2}\)\({}^{2+}\) on PA-BWT. Considering that the adsorption capacity was increased with the increase of temperature, it is reasonable to conclude that UO\({}_{2}\)\({}^{2+}\) is chemically adsorbed with monolayer coverage on the surface of PA-BWT. As shown in Figure 5c, it can be observed that the adsorption rate of PA-BWT is quite rapid, and adsorption equilibrium is attained at about 200 min. At higher temperatures, the time to attain adsorption equilibrium is also reduced. In order to understand the adsorption process of PA-BWT to UO\({}_{2}\)\({}^{2+}\), adsorption kinetic data were fitted by the Lagergren models (Figure 5d) [30; 31; 32]:  
Lagergren pseudo-first-order kinetic model:  
\[\ln(q_{e}-q_{t})=\ln q_{e}-k_{1}t \tag{4}\]  
Figure 4: (**a**)Water contact angle of membrane and BWT-immobilized membrane and (**b**) pure water flux of membrane and BWT-immobilized membrane. [PERSON] pseudo-second-order kinetic model:  
\[\frac{\text{t}}{q_{t}}=\frac{1}{k_{2}q_{e}^{2}}+\frac{t}{q_{e}} \tag{5}\]  
where \(q_{e}\) and \(q_{t}\) (mmol/g) are the adsorption capacity of UO\({}_{2}\)\({}^{2+}\) at equilibrium and at time \(t\), respectively; \(k_{1}\) (min\({}^{-1}\)) is the first-order rate constant, and \(k_{2}\) (g/(mmol\cdot min)) is the second-order rate constant. As listed in Table 2, the correlation coefficients obtained by the pseudo-second-order kinetic model fitting are very close to 1.0 compared to those of the pseudo-first-order model, suggesting that the chelating interaction between UO\({}_{2}\)\({}^{2+}\) and PA-BWT is the rate-control step of the adsorption process. Figure 5: Adsorption isotherms of PA-BWT to UO\({}_{2}\)\({}^{2+}\): UO\({}_{2}\)\({}^{2+}\) ion concentration 0.1–2 mmol/L, pH 5.0: (**a**) adsorption isotherms and (**b**) regression by the Langmuir equation. Adsorption kinetics of UO\({}_{2}\)\({}^{2+}\) on PA-BWT: UO\({}_{2}\)\({}^{2+}\) concentration 1 mmol/L, pH 5.0: (**c**) adsorption kinetics and (**d**) regression by the pseudo-second-order equation. (**e**) Regression by the pseudo-second-order model (PSO) expressed in a different linearized form. (**f**) Regression by the Elovich model. In order to further explore the adsorption kinetics of UO\({}_{2}\)\({}^{2+}\) on PA-BWT, we also used the pseudo-second-order model (PSO) expressed in a different linearized form (q\({}_{\text{f}}\)/t = k\({}_{2}\)q\({}_{\text{e}}\)\({}^{2}\) - k\({}_{2}\)q\({}_{\text{e}}\)q\({}_{\text{f}}\).) for fitting analysis [33]. As shown in Figure 5e, the data are not distributed along a straight line, so it follows that the PSO is not suitable for describing the adsorption kinetics.