## 3 Results
### Adsorption Kinetics  
Two major models are used in the study of adsorption kinetics: pseudo-first-order and pseudo-second-order kinetic models. Both models are expressed in Equations (7) and (8).  
\[\log\left(q_{e}-q_{t}\right)=\log q_{e}-\frac{K_{1}t}{2.303} \tag{7}\]  
\[\frac{t}{q_{t}}=\frac{1}{K_{2}q_{e}}^{2}+\frac{t}{q_{e}} \tag{8}\]  
where \(q_{e}\) and \(q_{t}\) are the amounts of adsorbate adsorbed at equilibrium and time \"\(t\)\" (mg/g), \(K_{1}\) and \(K_{2}\) are the pseudo-first-order and pseudo-second-order rate constants in 1/minute and g/mg min, respectively. A linear plot of \(\log(q_{e}-q_{t})\) against \(t\) (time) for this study, shows a low correlation  
\begin{table}
\begin{tabular}{c c c} \hline
**Adsorbent** & **Adsorption Capacity (mg/g)** & **References** \\ \hline Magnesium Titanate & 0.03 & [38] \\ Granula acid treated bentonite & 0.094 & [39] \\ Pumice & 0.31 & [40] \\ Zeolite & 0.47 & [41] \\ Regenerated spent bleach earth & 0.6 & [42] \\ Fe/Zr-Alginate microparticles & 0.981 & [43] \\ Gyspiferous limestone & 1.07 & [44] \\ Synthetic Siderite & 1.77 & [45] \\ Hydroous bismuth oxide & 0.06–1.93 & [46] \\ Iron impregnated granular ceramic & 1.70–2.16 & [47] \\ Fe-Al-Ce nanocomposite & 2.22 & [48] \\ CMPNS & 2.3 & [49] \\ Stilbite zeolite modified Fe3+ & 2.31 & [50] \\ HDTMA-VMT\({}_{0.2\text{M}}\) & 2.36 & Present study \\ \hline \end{tabular}
\end{table}
Table 2: Fluoride adsorption capacity of various adsorbents in literature.  
Figure 11: Freundlich plot of HDTMA-VMT\({}_{0.2\text{M}}\).  
coefficient of 0.2335, thus suggesting that the adsorption did not conform to pseudo-first-order model (Figure 12). On the other hand, a plot of \(t/q_{t}\) against \(t\) gave a linear plot with a correlation coefficient of 0.9964 (Figure 13). The results showed that the adsorption conformed to pseudo-second-order reaction, which essentially implies that the mechanism of adsorption is chemisorption [53]. The closeness of the calculated \(q_{c}\) (0.2896) to the experimental \(q_{c}\) (0.2950) values for HDTMA-VMT\({}_{0.2\text{M}}\) further confirms that it obeys pseudo-second-order reaction based on the study by [PERSON] et al. [17] (Table 3).  
The same mechanism was discovered in other anionic adsorption studies such as [PERSON] et al. [22], [PERSON] et al. [54], [PERSON] et al. [20], [PERSON] et al. [17], [PERSON] et al. [28], [PERSON] et al. [26], [PERSON] et al. [19], and [PERSON] et al. [29], amongst others. The formation of a positively charged bilayer on the clay surface was responsible for fluoride attraction [26; 55]. Further study by [PERSON] [56] also confirms this fact, as it states emphatically that sorption onto positively charged admicelle formation is a major mechanism for anionic component removal by an organomodified adsorbent.  
Figure 12: Pseudo-first-order plot of HDTMA-VMT\({}_{0.2\text{M}}\).  
Figure 13: Pseudo-second-order reaction of HDTMA-VMT\({}_{0.2\text{M}}\).  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \multirow{2}{*}{**adsorbent**} & \multirow{2}{*}{\(q_{c}\)**exp**} & \multicolumn{2}{c}{**Pseudo-First-Order Model**} & \multicolumn{2}{c}{**Pseudo-Second-Order Model**} \\ \cline{3-6}  & & \(K_{1}\) & \(q_{c}\)**cal** & \(K_{2}\) & \(q_{c}\)**calc** & \\  & & (\(\times\)**10\({}^{3}\) min\({}^{-1}\))** & **(mg/g)** & \(R^{2}\) & (\(\times\)**10 g\(\cdot\)mg\({}^{-1}\)**min\({}^{-1}\))** & **(mg/g)** & \(R^{2}\) \\ \hline HDTMA-VMT\({}_{0.2\text{M}}\) & 0.2950 & 0.0274 & 0.0143 & 0.0069 & 0.6757 & 0.2896 & 0.9964 \\ \hline \end{tabular}
\end{table}
Table 3: Parameters for pseudo-first-order and pseudo-second-order kinetic models.