## 3 Results
### Adsorption Isotherm  
The Langmuir isotherm describes quantitatively the formation of a monolayer of the adsorbate (fluoride) on smooth surface of the adsorbent (HDTMA-VMT\({}_{0.2\text{M}}\)) [35]. After the surface of the adsorbent is completely occupied, no further adsorption of fluoride from solution takes place. The linear form of the Langmuir equation is represented by Equation (3). \[\frac{C_{e}}{q_{e}}=\frac{C_{e}}{q_{o}}+\frac{1}{q_{o}K_{L}} \tag{3}\]  
where \(C_{e}\) and \(q_{e}\) have been defined in Equations (1) and (2), \(q_{o}\) is the maximum monolayer coverage (mg/g), and \(K_{L}\) is Langmuir isotherm constant (L/mg). Figure 8: Optimisation of pH (2 g/50 mL, 70 min, 25 \({}^{\circ}\)C, 200 rpm). Figure 9: Optimisation of temperature (2 g/50 mL, 70 min, 200 rpm). A plot of \(C_{e}/q_{e}\) against \(C_{e}\) for this study gave a correlation coefficient of 0.8902 but did not give a linear plot (Figure 10). The adsorption capacity/maximum monolayer coverage was calculated from the slope to be 2.3669 mg/g (Table 1) and it can be inferred that only a degree of monolayer adsorption occurred on the surface of the adsorbent. The conformity of the adsorption to Langmuir isotherm was confirmed using Equation (4). \[R_{L}=\frac{1}{(1+bC_{o})} \tag{4}\]  
where \(R_{L}\) is the separation factor, \(C_{o}\) has been defined in Equation (1), and \(b\) is the Langmuir constant (L/mg). \(R_{L}>1\) represents an unfavourable monolayer adsorption, \(R_{L}=1\) represents a linear adsorption, \(0<R_{L}<1\) represents a favourable adsorption process, and \(R_{L}=0\) represents an irreversible adsorption process [29]. The \(R_{L}\) value of this study was calculated to be 0.9294 (Table 1), thus monolayer adsorption took place which conforms to Langmuir isotherm. The Freundlich isotherm models multilayer adsorption on heterogeneous surfaces. The Freundlich equation is formulated in Equation (5) as:  
\[q_{e}=K_{F}{C_{e}}^{1/n} \tag{5}\]  
where \(K_{F}\) is the constant of the system and is known as the Freundlich adsorption coefficient, \(n\) is the Freundlich constant, while the value of \(n\) should be between 1 and 10 for favourable adsorption [36]. By taking logarithm of Equation (5):  
\[\log q_{e}=\log K_{F}+\frac{1}{n}\log C_{e} \tag{6}\]  
A plot of \(\log q_{e}\) against \(\log C_{e}\) for this study gave a linear plot with a correlation coefficient of 0.9985 (Figure 11). This shows that the adsorption conformed more to Freundlich isotherm than Langmuir. The value of \(1/n\) was also calculated to be 0.8357 (\(n=1.1966\)), which implies an affinity between the surface of the adsorbent and fluoride [37]. The adsorption capacity of HDTMA-VMT\({}_{0.2\text{M}}\), though on the higher side, is approximately similar to the capacities of many other fluoride adsorbents reported in \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Adsorbent**} & \multicolumn{4}{c}{**Langmuir Isotherm**} & \multicolumn{4}{c}{**Freundlich Isotherm**} \\ \cline{2-9}  & \(q_{o}\) (mg/g) & \(K_{L}\) (L/mg) & \(R^{2}\) & \(R_{L}\) & \(K_{L}\) (L/mg) & \(1/n\) & \(R^{2}\) \\ \hline HDTMA-VMT\({}_{0.2\text{M}}\) & 2.3669 & 0.0095 & 0.8902 & 0.9294 & 0.0293 & 0.8357 & 0.9985 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Adsorption isotherm constants. Figure 10: Langmuir isotherm plot of HDTMA-VMT\({}_{0.2\text{M}}\). literature as shown in Table 2. This can simply be attributed to a higher amount of adsorbate (fluoride) adsorbed per gram of HDTMA-VMT\({}_{0.2\text{M}}\) than the other adsorbents used for comparison in this study. Also, in comparison with other adsorption methods (such as the use of alumina), vermiculite is a naturally occurring clay mineral whereas alumina has to be purchased commercially, making it more expensive than vermiculite.