# Adsorption of Hexavalent Chromium Ions Using Pine Sawdust Cellulose Fibres
## 3 Results and Discussion
### Kinetic Study for Adsorption of Chromium  
The pseudo-first-order and pseudo-second-order kinetic models (Figure 17a,b) were studied to understand the rate and type of adsorption. A linear relationship can be observed by plotting \(\log(\mathrm{q_{e}}-\mathrm{q_{t}})\) against time (t), which enables the determination of biosorption rate constant (k\({}_{1}\)), \(\mathrm{q_{e}}\) (cal), and the correlation coefficient (R\({}^{2}\)) [21]. The k\({}_{1}\) and \(\mathrm{q_{e}}\) (cal) values in mg/g of Cr (VI), calculated from the plot shown in Figure 17, are 8.2 \(\times\) 10\({}^{-3}\) and 1.5739 mg/g, respectively. Table 7 indicates that the correlation coefficient (R\({}^{2}\) = 0.9108) suggests the pseudo-first-order model did not provide a good fit. [PERSON] et al. [67] reported comparable findings when studying chromium removal from synthetic wastewater using the low-cost Odaracha adsorbent from Ethiopia. Additionally, the experimental adsorption result (\(\mathrm{q_{e}}\) (exp)) was exceptionally higher than that of \(\mathrm{q_{e}}\) (cal), which further revealed that the pseudo-first-order model was not suitable to explain the adsorption kinetics of chromium ions on cellulose fibres (CF)'s adsorbent. \begin{table}
\begin{tabular}{c c c c} \hline  & & **Linear** & **Nonlinear** \\ \hline
**adsorption Isotherm** & **Parameters** & **Cr (VI)** & **Cr (VI)** \\ \hline Langmuir isotherm & SSE & 1.84 \(\times\) 10\({}^{-2}\) & 0.1158 \\  & RSME & 1.457 \(\times\) 10\({}^{-2}\) & 0.1079 \\  & \(\chi^{2}\) & 8.40 \(\times\) 10\({}^{-2}\) & 3.14 \(\times\) 10\({}^{-7}\) \\  & R\({}^{2}\) & 0.9985 & 0.9992 \\  & SNE & 0.0278 & 6.976 \(\times\) 10\({}^{-4}\) \\ \hline Freundlich isotherm & SSE & 2.82 \(\times\) 10\({}^{-4}\) & 5.44 \(\times\) 10\({}^{-3}\) \\  & RSME & 7.132 \(\times\) 10\({}^{-3}\) & 3.46 \(\times\) 10\({}^{-2}\) \\  & \(\chi^{2}\) & 6.202 \(\times\) 10\({}^{-3}\) & 4.91 \(\times\) 10\({}^{-5}\) \\  & R\({}^{2}\) & 0.8854 & 0.9988 \\  & SNE & 7.52 \(\times\) 10\({}^{-3}\) & 5.44 \(\times\) 10\({}^{-3}\) \\ \hline \end{tabular}
\end{table} 
Table 6: Error analysis for isotherm models using linear and nonlinear regression for hexavalent chromium ions. Figure 17: Kinetics plots of (**a**) linear and (**b**) nonlinear PFO and PSO for the adsorption of hexavalent chromium ions on cellulose fibres (CF). Figure 17 shows the kinetics plots of (a) linear and (b) nonlinear PFO and PSO for the adsorption of hexavalent chromium ions on cellulose fibres (CF). Figure 17a reveals a linear trend in the plot of t/q\({}_{t}\) against time, indicating that the pseudo-second-order kinetic model produced the best fit, with a correlation coefficient value (R\({}^{2}\) = 0.9999) close to unity, as shown in Table 8. The experimental adsorption equilibrium value (q\({}_{e}\) exp.) of 9.7772 mg g\({}^{-1}\) agree with the calculated adsorption equilibrium value (q\({}_{e}\) cal.) of 9.7847 mg g\({}^{-1}\). Based on the results, it can be concluded that the pseudo-second-order model is a suitable fit for describing the adsorption kinetics of chromium using the cellulose fibres (CF)'s adsorbent. The obtained results suggest that the adsorption of chromium ions by the cellulose fibres (CF)'s adsorbent is controlled by chemisorption, which involves the exchange of metal ions with the functional groups present on the surface of the adsorbent [68]. The results in Figure 17b and Table 7 suggest that nonlinear kinetics models (Cr (VI)) are not suitable for describing adsorption kinetics. The experimental adsorption equilibrium value (q\({}_{e}\) exp.) of 0.706 mg g\({}^{-1}\) aligns with the calculated adsorption value, which is lower than the equilibrium value (q\({}_{e}\) cal.) of 9.7847 mg g\({}^{-1}\). Equation (10) defines the initial rate of sorption (h). \[\text{h}=\text{k}_{2}\;\text{q}_{e}{}^{2} \tag{10}\]  
where h (mg g\({}^{-1}\) min\({}^{-1}\)) can be regarded as the initial adsorption rate when the initial concentration of metal ions does not influence t \(\to\) 0, which is the initial adsorption rate (h).