# Adsorption of Hexavalent Chromium Ions Using Pine Sawdust Cellulose Fibres
## 2 Materials and Methods
### Batch Adsorption Studies  
Stock solutions (100 ppm) of chromium (VI) ions were prepared by dissolving 0.1 g K\({}_{2}\)Cr\({}_{2}\)O\({}_{7}\) in 1 L of ultrapure water. The effect of agitation time, with intervals of 30, 45, 60,90, and 120 min, was used with a standard solution of 100 ppm. Then, 1 g of adsorbent was weighed and transferred into 100 mL capped sample bottles. Metal ion (Cr (VI)) solution of 100 mL was added to the sample bottles and equilibrated by agitation at 200 rpm. Thereafter samples were centrifuged at 6000 rpm for 5 min and then filtered. The filtrate was later used for Cr (VI) ions analysis using ICP-OES. The same procedure was also followed for the effect of mass (0.05, 0.5, 1, 1.5, 2, and 2.5 g), concentration (20, 40, 60, 80, 100, and 110 mg/L), and temperature (299, 303, 308, 313, and 318 K) by agitation at 200 rpm for 2 h. The effect of pH was determined by measuring 100 mL of metal ion solution and pouring it into five beakers. The initial pH of each solution was measured, and then the solution pH was adjusted to 2, 4, 6, 7, 8, and 12 using HCl (0.1 M) or NaOH (0.1 M). The adsorption capacity (q\({}_{\mathrm{e}}\)), expressed in mg/g, was used to determine the adsorbed chromium ions' quantity. Removal percentage (% R) was used in Equations (1) and (2), where (C\({}_{\mathrm{o}}\)) and (C\({}_{\mathrm{e}}\)) represent the initial and final concentrations of the contaminants in the solution (mg/L), respectively. The volume of the solution (v) was measured in mL, and the mass of the adsorbent (m) was measured in grams (g) [21]. \[\mathrm{q_{e}}=\frac{(\mathrm{C_{0}-C_{e}})\mathrm{V}}{\mathrm{m}} \tag{1}\]  
\[\mathrm{\%R}=\frac{(\mathrm{C_{0}-C_{e}})\times 100}{\mathrm{C_{0}}} \tag{2}\]  
Pseudo-first-order and pseudo-second-order models were used for chromium (VI) adsorption kinetics. The equations in Table 1 provide a comprehensive and accurate characterisation of reaction rates and mechanisms. The amounts of metal ions adsorbed in (mg/g) are (q\({}_{\mathrm{e}}\)) and (q\({}_{\mathrm{t}}\)) at equilibrium and time t (minutes). The rate constant is k\({}_{1}\) of the PEO in (min\({}^{-1}\)), and k\({}_{2}\) is the rate constant of the PSO (g mg\({}^{-1}\) min\({}^{-1}\)). Similarly, linear and nonlinear equations are used in Table 2 to calculate the adsorption isotherm for both Langmuir and Freundlich models, enabling a more comprehensive and accurate representation of the adsorption behaviour. where q\({}_{\mathrm{e}}\) is the adsorption capacity (mg. g\({}^{-1}\)), C\({}_{\mathrm{e}}\) is the equilibrium concentration of the adsorbate (mg. L\({}^{-1}\)), q\({}_{\mathrm{m}}\) is the maximum adsorption capacity of adsorbents (mg. g\({}^{-1}\)), and k\({}_{\mathrm{L}}\) is the Langmuir adsorption constant (L mg\({}^{-1}\)). Freundlich constants are k\({}_{\mathrm{F}}\) and n, which are adsorption capacity and strength, respectively [15; 16]. Thermodynamic parameters such as equilibrium constant (k\({}_{\mathrm{C}}\)) values were calculated using Equations (3)-(5):  
\[\mathrm{k_{c}}=\frac{\mathrm{q_{e}}}{\mathrm{C_{e}}} \tag{3}\] \begin{table}
\begin{tabular}{c c c} \hline
**Kinetic Models** & **Linear Equation** & **Nonlinear Equation** \\ \hline Pseudo-first-order (PFO) & \(\mathrm{ln}\left(\mathrm{q_{e}-q_{t}}\right)=\mathrm{lnq_{e}-k_{1}t}\) & \(\mathrm{q_{t}=q_{e}(1-e^{-k_{1}t})}\) \\ Pseudo-second-order (PSO) & \(\mathrm{t}/\mathrm{q_{t}=1/k_{2}q_{e}^{2}+1/q_{e}t}\) & \(\mathrm{q_{t}=q_{e}^{2}K_{2}t}\) \\ \hline \end{tabular}
\end{table} 
Table 1: The linear and nonlinear pseudo-first-order and pseudo-second-order models. \[\mathrm{lnk_{c}}=\frac{\Delta H^{\circ}}{RT}-\frac{\Delta S^{\circ}}{R} \tag{4}\]  
\[\Delta G^{\circ}=-RT\mathrm{lnk_{c}} \tag{5}\]  
Equilibrium adsorption data of the following temperatures, 298, 303, 308, 313, and 318 K, and other thermodynamic functions, namely, enthalpy (\(\Delta H^{\circ}\)), entropy (\(\Delta S^{\circ}\)), and Gibbs free energy (\(\Delta G^{\circ}\)), are incorporated into Equations (8) and (9).