# Chromium(III) Removal from Wastewater by Chitosan Flakes
## 3 Results  
### Adsorption Dynamics  
The uptake of metal ions from the solution involves several steps, necessary for the transfer of the solute from the liquid phase to the specific sites within the chitosan particles (e.g., external diffusion and intraparticle diffusion).  
In the case of chitosan, its chains have a large number of the -NH\({}_{2}\) and -OH groups distributed throughout the structure, making the kinetic or mass transfer representation likely to be global. The -NH\({}_{2}\) groups are the most important binding sites for metal ions [24], yet the hydroxyl groups can also contribute as coordinator groups, especially those in C-3 position [15; 22]. To examine the adsorption mechanism of the metal ion of interest, two kinetic models were tested:  
1. the pseudo-first-order equation described by [PERSON] [25], which can be rearranged to obtain a linear form as shown by Equation (1): \[Log(q_{e}-q_{t})=Log(q_{e})-(k_{1}/2.303)t\] (1)
2. a pseudo-second-order equation based on the equilibrium adsorption capacity, which can be expressed as in Equation (2): \[t/q_{t}=\left(1/k_{2}q_{e}^{2}\right)+(1/q_{e})t\] (2)  
In the above equations, \(q_{e}\) (mg g\({}^{-1}\)) represents the quantity of Cr(III) adsorbed when the system is at equilibrium, \(q_{t}\) (mg g\({}^{-1}\)) is the quantity of Cr(III) adsorbed at time \(t\), and \(k_{1}\) (min\({}^{-1}\)) and \(k_{2}\) (g mg\({}^{-1}\) min\({}^{-1}\)) are the rate constants of the pseudo-first and pseudo-second order kinetic models, respectively.  
Given that Equations (1) and (2) are not able to provide information on the adsorption mechanism, the simplified intraparticle diffusion model [26] was also tested, being \(k_{i}\) (g mg\({}^{-1}\) min\({}^{-1}\)) the rate constant of the model:  
\[q_{t}=k_{i}\,t^{1/2} \tag{3}\]  
The validity of these models was assessed by analyzing the slopes and intercepts of \(Log(q_{e}-q_{t})\) vs. \(t\), \(t/q_{t}\) vs. \(t\), and \(q_{t}\) vs. \(t^{1/2}\) for each of the linearized equations.  
The results obtained with different concentrations of chromium(III) are shown in Table 1 in terms of correlation coefficients (R\({}^{2}\)) as well as calculated and experimental adsorption capacity values.  
The correlation coefficient R\({}^{2}\) for the pseudo-second-order adsorption model was the highest and, in fact, its estimate of the equilibrium adsorption capacity \(q_{e,\ cal}\) was quite close to the experimental \(q_{t}\) values (23-28.5 mg g\({}^{-1}\) as shown in Figure 2 for the two grain-sizes). These results suggest that a second-order mechanism is predominant and that the overall Cr(III) adsorption rate is controlled by a chemisorption process.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**1 st Order Kinetic Model**} & \multicolumn{2}{c}{**2 nd Order Kinetic Model**} & \multicolumn{2}{c}{**Intraparticle Diffusion Model**} \\ \hline \(k_{1}\) & \(q_{e,\ cal}\) & R\({}^{2}\) & \(k_{2}\) & \(q_{e,\ cal}\) & R\({}^{2}\) & \(k_{i}\) & R\({}^{2}\) \\ (min\({}^{-1}\)) & (mg g\({}^{-1}\)) & & (g mg\({}^{-1}\) min\({}^{-1}\)) & (mg g\({}^{-1}\)) & (g mg\({}^{-1}\) min\({}^{-1}\)) & \\ \(1.07\times 10^{-2}\) & 9.32 & 0.7838 & \(5.07\times 10^{-3}\) & 23.89 & 0.9987 & 1.711 & 0.9199 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Values of the adsorption kinetic constants at T = 20 \({}^{\circ}\)C, pH = 3.8, C\({}_{0}\) = 0.5 g L\({}^{-1}\).