# 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  
The adsorption kinetics of Cr(VI) adsorption were carried out to investigate the adsorption rate and mechanisms. The corresponding Cr(VI) removal data were obtained from adsorption experiments for different contact times. The pseudo-first order (PFO) mechanism from the Lagergren equation and the pseudo-second order (PSO) mechanism from the Ho equation were the most adequate adsorption models for metal ion removal [41]. The PFO model was based on the fact that the adsorbates from the solution to the adsorbent surface was controlled by the surface diffusion step, and the PFO is expressed as follows [42]:  
\[\ln(q_{e}-q_{t})=\ln(q_{e})-k_{1}t \tag{8}\]  
where \(k_{1}\) is the equilibrium adsorption rate constant (min\({}^{-1}\)); \(q_{t}\) and \(q_{e}\) (mg g\({}^{-1}\)) are the adsorption amount at contact time t and equilibrium, respectively. \(q_{e}\) and \(k_{1}\) can be calculated from the linear plots of \(\ln(q_{e}-q_{t})\) vs. \(t\). The PSO model was employed where the adsorption rate was determined by the square of the amount of unoccupied adsorptive vacancies on the adsorbent surface, and the process was controlled by a chemical adsorption mechanism. In other words, there was electron pair sharing or transfer between the adsorbent and the adsorbate [43]. The rate equation of the PSO can be described as follow [44]:  
\[t/q_{t}=1/(k_{2}q_{e}{}^{2})+t/q_{e} \tag{9}\]  
where \(k_{2}\) is the equilibrium adsorption rate constant (g mg\({}^{-1}\) min\({}^{-1}\)); \(q_{t}\) and \(q_{e}\) (mg g\({}^{-1}\)) are the adsorption amount at contact time \(t\) and equilibrium, respectively. \(q_{e}\) and \(k_{2}\) can be calculated from the linear plots of \(\ln t/q_{t}\) vs. \(t\). Figure 7 shows the adsorption data of chitosan porous microspheres prepared with different freezing times for PFO (Figure 7a) and PSO kinetics (Figure 7b). Correlation coefficient \(r^{2}\) was used to test the validity of two models. The results are shown in Table 1, and indicate that the PSO model was more suitable for Cr(VI) adsorption by the chitosan microspheres than PFO, which was consistent with previous reports [34; 41], where it was reported that chemical adsorption controlled the Cr(VI) removal process and significant sharing or exchange of electrons happened between the chitosan and Cr(VI) ions. In fact, the active functional groups (-NH\({}_{2}\)) of chitosan adsorbed heavy metal ions by complexation and electrostatic adsorption (which can be proven with the results of FTIR in Section 3.4) [45; 46]. Moreover, qe (cal) values obtained by the PFO were closer to qe (exp) measured by the experiment as opposed to the PSO, which was different from previous reports [34; 35]. However, both the PFO model and PSO model were inadequate to reflect the main controlling steps of the adsorbate moving from solutions to the solid adsorption surfaces, including the outer surface and pore inner surface, of adsorbents. The controlling steps generally include boundary-layer (film) or external diffusion, pore diffusion, interior surface diffusion and surface adsorption process [47]. For example, the intra-particle diffusion model was expressed where the amount adsorbed (\(q_{t}\)) was proportional to the square root of adsorption time t, which could indicate the main steps through fitting experiment data. The equation was given by [PERSON] and [PERSON] as follow [44]:  
\[q_{t}=k_{wm}t^{1/2}+C \tag{10}\]