## 3 Results and Discussion
### Adsorption Kinetics  
Figure 7a displays the effect of contact time on P adsorption onto Al-Ti oxide. The adsorbed amount of P increased with an increase in contact time. The adsorption equilibrium for P was achieved after about 12 h. The rapid adsorption observed during the initial stage was probably due to the abundant availability of adsorption sites on the Al-Ti oxide surface; then, the adsorption became less efficient with gradual occupancy of the adsorption sites.  
To predict the adsorption kinetics of P onto Al-Ti oxide, two adsorption kinetic models, including pseudo-first-order and pseudo-second-order models were applied to examine the adsorption process [41; 42; 43]. The pseudo-first-order and pseudo-second-order models can be represented by Equations (4) and (6), and their linear forms as Equations (5) and (7) when integrating the two equations for the boundary conditions of \(q_{\rm t}=0\) at \(t=0\) and \(q_{\rm t}=q_{\rm t}\) at \(t=t\), respectively:  
\[\frac{\mathrm{d}q_{\rm t}}{\mathrm{d}t}=k_{1}(q_{\rm e}-q_{\rm t}) \tag{4}\]  
\[\frac{\mathrm{d}q_{\rm t}}{\mathrm{d}t}=k_{2}(q_{\rm e}-q_{\rm t})^{2} \tag{5}\]  
\[q_{t}=q_{e}(1-e^{-k_{1}t}) \tag{6}\]  
\[\frac{t}{q_{t}}=\frac{1}{k_{2}q_{e}}^{2}+\frac{1}{q_{e}}t \tag{7}\]  
where \(q_{e}\) (mg/g) and \(q_{t}\) (mg/g) are the amount of P adsorbed onto Al-Ti oxide at equilibrium and at time \(t\) (min), respectively. \(k_{1}\) (1/min) and \(k_{2}\) (g/mg min) are the rate constants of the pseudo-first-order and pseudo-second-order models, respectively.  
The fitting of the two kinetic models for P adsorption with various initial concentrations are shown in Figure 7b-c. Kinetic parameters fit by the pseudo-first-order and pseudo-second-order models were calculated and are listed in Table 2. The coefficients of determination (r\({}^{2}\)) of the pseudo-second-order model were higher than those of pseudo-second-order mode, and the calculated \(q_{\rm e}\) values were in good agreement with experimental results, verifying that the adsorption process could be better described by the pseudo-second-order model.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**BET Surface Area (m\({}^{2}\)/g)**} & \multicolumn{1}{c}{**Micropore Area (m\({}^{2}\)/g)**} & \multicolumn{1}{c}{**Average Pore Diameter (nm)**} \\ \hline Al-Ti oxide & 3.94 & 1.79 & 12.1 \\ Al oxide & 5.52 & 3.43 & 20.2 \\ Ti oxide & 2.36 & 1.26 & 10.6 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Surface area measurements of Al-Ti oxide, Al oxide, and Ti oxide.  
Figure 6: (a) N\({}_{2}\) adsorption desorption isotherm and (b) pore size distribution curve of Al-Ti oxide.  
Figure 7: (**a**) Effect of contact time and initial concentration on P adsorption on Al-Ti oxide adsorbent; (**b**) adsorption kinetics fit by pseudo-first-order (dashed lines) and (**c**) pseudo-second-order (solid lines) models with P initial concentration of 2.5, 5.0.7,5.10.0, and15.0 mg/L, adsorbent dosage = 0.2 g/L, T = 298 K.  
Figure 8: Adsorption isotherm of phosphorus on Al-Ti oxide composite sorbent: (**a**) adsorptive experiments with various equilibrium concentrations of P, (**b**) Langmuir, (**c**) Freundlich, (**d**) D-R, (**e**) Temkin, (**f**) Frumkin, and (**g**) Harkin–Jura model. The average value in (**a**) was used for calculations in (**b–g**), adsorbent dosage = 0.2 g/L, equilibrium time = 8 h, pH = 6.8.