## 3 Results and Discussion
### Kinetics of Adsorption  
Knowledge of adsorption kinetics is a necessity to design kinetics models and to determine the adsorption mechanism. A number of kinetics models such as [PERSON]'s pseudo first order, [PERSON]'s pseudo second order, and [PERSON] and [PERSON]'s intra particle diffusion models have been tested by previous researchers [80]. Figure 9: Effect of other ions on adsorption of Cr(VI) and Ni(II). In the present study, the adsorption data were also fitted in the pseudo first order and second order kinetics models. These models are expressed by the following Equations (11) and (12) respectively. \[\log\left(q_{\text{e}}-q_{\text{t}}\right)=\log q_{\text{e}}-\frac{K_{1}t}{2.303} \tag{11}\]  
\[\frac{t}{q_{\text{t}}}=\frac{1}{K_{24}{}^{2}}+\frac{t}{q_{\text{e}}} \tag{12}\]  
where \(q_{\text{t}}\) (mg\(\cdot\)g\({}^{-1}\)) and \(q_{\text{e}}\) (mg\(\cdot\)g\({}^{-1}\)) are the amount of heavy metals adsorbed after time \"t\" and equilibrium time respectively, \(K_{1}\) and \(K_{2}\) are the rate constants of pseudo first order (min\({}^{-1}\)) and pseudo second order (g\(\cdot\)mg\({}^{-1}\) min\({}^{-1}\)) respectively. \"t\" is the time in minutes and \"C\" is constant (mg\(\cdot\)g\({}^{-1}\)). Figure 10a shows a pseudo first order graph obtained by plotting ln(qe - qt) vs. t by using adsorption data of Cr(VI) and Ni(II) on Fe\({}_{3}\)O\({}_{4}\), PANI, and composite materials respectively. From the slope of this graph, values of K1 from intercept qe values were determined as shown in Table 2. K1 is the time scaling factor and its values indicate how fast or slow the equilibrium is reached. In the present study, values of K1 for the adsorption of both Cr(VI) and Ni(VI) on Fe\({}_{3}\)O\({}_{4}\), PANI, and PANI/Fe\({}_{3}\)O\({}_{4}\) composites were of the following order (Fe\({}_{3}\)O\({}_{4}\) > PANI > PANI/Fe\({}_{3}\)O\({}_{4}\)). It indicates that time of equilibrium was reached first by PANI/Fe\({}_{3}\)O\({}_{4}\), then PANI, then Fe\({}_{3}\)O\({}_{4}\). These results are in close agreement with already reported work [81]. Although all plots of Cr(VI) and Ni(II) adsorbed on all the three adsorbents have good linearity, the values of R\({}^{2}\) indicate that pseudo first order equations do not fit closely to the adsorption data. Therefore, a pseudo-second order kinetics model was tested for the adsorption data. Figure 10b shows the pseudo-second order plot obtained by plotting t/qt vs t. Values of the rate constant (K2) and equilibrium adsorption (qe) were calculated from the intercept and slope respectively. The pseudo-second order model explained the behavior of adsorption over the whole range of time and closely agreed with the chemisorption mechanism. Values of R\({}^{2}\) indicated that the pseudo-second order kinetics model fit more closely to the adsorption data than the pseudo-first order model [82]. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Adsorbents**} & \multicolumn{4}{c}{**Pseudo 1 st Order**} & \multicolumn{4}{c}{**Pseudo 2 nd Order**} \\ \cline{3-8}  & & \(\mathbf{K_{1}}\) & \(\mathbf{q_{\text{e}}}\) & \(\mathbf{R^{2}}\) & \(\mathbf{K_{2}}\) & \(\mathbf{q_{\text{e}}}\) & \(\mathbf{R^{2}}\) \\  & & **min\({}^{-1}\)** & **mg\(\cdot\)g\({}^{-1}\)** & & **g\(\cdot\)mg\({}^{-1}\) min\({}^{-1}\)** & **mg\(\cdot\)g\({}^{-1}\)** & \\ \hline \multirow{2}{*}{**Fe\({}_{3}\)O\({}_{4}\)**} & Ni\({}^{2+}\) & 0.226 & 22.23 & 0.926 & 0.0893 & 52.771 & 0.989 \\  & Cr\({}^{6+}\) & 0.223 & 42.27 & 0.912 & 0.0551 & 62.973 & 0.994 \\ \multirow{2}{*}{**PANI**} & Ni\({}^{2+}\) & 0.202 & 52.43 & 0.930 & 0.0133 & 80.580 & 0.997 \\  & Cr\({}^{6+}\) & 0.127 & 64.38 & 0.645 & 0.0128 & 133.52 & 0.994 \\ \multirow{2}{*}{**PANI/ Fe\({}_{3}\)O\({}_{4}\)**} & Ni\({}^{2+}\) & 0.201 & 82.14 & 0.865 & 0.0181 & 90.744 & 0.990 \\  & Cr\({}^{6+}\) & 0.116 & 83.24 & 0.877 & 0.0078 & 145.98 & 0.980 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of kinetics study of Cr(VI) and Ni(II) adsorption.