# Preparation of Freeze-Dried Porous Chitosan Microspheres for the Removal of Hexavalent Chromium
## 3 Results
### The Adsorption Properties of Porous Chitosan Microspheres
#### 3.3.3 Adsorption Kinetic Analysis
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Kinetic Model**} & \multirow{2}{*}{**Parameters**} & \multicolumn{3}{c}{**Freezing Time of the Chitosan Microspheres**} \\ \cline{3-5}  & & **24 h** & **48 h** & **72 h** \\ \hline \multirow{3}{*}{Pseudo first order} & \(q_{e}\), exp (mg g\({}^{-1}\)) & 590.6 & 723.6 & 945.2 \\  & \(q_{e}\), cal (mg g\({}^{-1}\)) & 581.7 & 708.12 & 937.8 \\  & \(k_{1}\) (min\({}^{-1}\)) & 0.0111 & 0.0122 & 0.0155 \\  & r\({}^{2}\) & 0.982 & 0.961 & 0.992 \\ \multirow{3}{*}{pseudo second order} & \(q_{e}\), cal (mg g\({}^{-1}\)) & 701.3 & 847.9 & 1070.5 \\  & \(k_{2}\) (g mg\({}^{-1}\)) & 0.0000176 & 0.0000165 & 0.0000183 \\  & min\({}^{-1}\)) & r\({}^{2}\) & 0.996 & 0.995 & 0.997 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cr(VI) adsorption kinetic parameters of the chitosan microspheres prepared over different freezing times for PFO and PSO kinetics at a solution pH of 4. Figure 7: The adsorption data of porous chitosan microspheres prepared for different freezing times for (**a**) PFO and (**b**) PSO kinetics. where \(q_{t}\) (mg g\({}^{-1}\)) is the adsorption amount at contact \(t\); \(k_{wm}\) (mg g\({}^{-1}\) min\({}^{-0.5}\)) is the intraparticle diffusion rate constant, and C represents the intercept of the plot \(q_{t}\) vs. \(t^{0.5}\). \(C\) represents the effect of the boundary layer. If the plot \(q_{t}\) vs. \(t^{0.5}\) was a single straight line across the origin (\(C=0\)), the intra-particle diffusion is only controlled the adsorption process. The multi-linear parts graph represented multiple steps participating in the adsorption process. Figure 8 demonstrates the plots \(q_{t}\) vs. \(t^{0.5}\) of Cr(VI) removal amount versus \(t^{0.5}\). Obviously, no straight line crossed the origin and the intra-particle diffusion plots of each porous chitosan microsphere were divided into three linear regions with three different slopes (\(k_{wm}\)) and intercept value \(C\). The first sharp part was for the boundary-layer (film) diffusion and the second linear one was due to pore diffusion, while the third portion was a steady section with interactions between the active site and Cr(VI) ions [34; 47]. On the other hand, no single line crossed the origin, indicating that many factors participated in controlling the adsorption rate, but not the intra-particle diffusion. In this case, the boundary-layer (film) diffusion rate, pore diffusion rate, interaction between the active site and Cr(VI) ions, and the intra-particle diffusion rate, together, determined the overall adsorption speed. In addition, the different deviations from the origin (different \(C\) values in the graph for one type of microsphere prepared for the same freezing time) might be attributed to the different Cr(VI) ion concentrations in the initial and final stages of adsorption [47]. Further, in the first linear region, the value of k\({}_{\text{wm}}\) increased with the increase in freezing time (as shown in Table 2). The faster the diffusion rate of the boundary-layer (film), the smaller the effect of boundary-layer. This may be related to the rough surface (as shown in Figure 4), which influenced the diffusion ability of the surface liquid film. In the slopes of the three parts of the plots, k\({}_{\text{wm}}\) gradually decreased, suggesting that the first portion of boundary-layer (film) diffusion was the main factor controlling the Cr(VI) adsorption reaction [47]. Figure 8: Plots of \(q_{t}\) vs. \(t^{0.5}\) of the Cr(VI) removal amount versus \(t^{0.5}\).