## 2 Methodology
### Adsorption Isotherm Models
#### 2.4.1 Langmuir Isotherm  
[PERSON] derived an isotherm adsorption equilibrium equation theoretically in 1916, and its formula is as follows:  
\[\mathrm{q_{e}}=\frac{\mathrm{q_{m}}\mathrm{K_{L}}\mathrm{C_{e}}}{1+\mathrm{K_{ L}}\mathrm{C_{e}}} \tag{2}\]  
where \(\mathrm{q_{m}}\) is the maximum saturated adsorption capacity (mg g\({}^{-1}\)) and \(\mathrm{K_{L}}\) is the Langmuir isotherm adsorption constant, which is related to the adsorption energy. By plotting \(1/\mathrm{q_{e}}\) and \(1/\mathrm{C_{e}}\), Equation (2) can be converted into a linear mode, and the constants \(\mathrm{q_{m}}\) and \(\mathrm{K_{L}}\) can be obtained from the intercept and slope of the line:  
\[\frac{1}{\mathrm{q_{e}}}=\frac{1}{\mathrm{q_{m}}}+\frac{1}{\mathrm{K_{L}} \mathrm{q_{m}}}\frac{1}{\mathrm{C_{e}}} \tag{3}\]  
[PERSON] and [PERSON] [37] used the Langmuir constant \(\mathrm{K_{L}}\) to define a dimensionless constant for this isotherm adsorption mode, called the separation factor or equilibrium parameter, expressed as \(\mathrm{R_{L}}\):  
\[\mathrm{R_{L}}=\frac{1}{1+\mathrm{K_{L}}\mathrm{C_{0}}} \tag{4}\]  
The type of adsorption isotherm curve can be observed by the value of \(\mathrm{R_{L}}\)[38]. Substituting Equation (1) into Equation (2) yields a second-order function, where C\({}_{\mathrm{e}}\) is a variable parameter:  
\[\mathrm{K_{L}}\times\mathrm{C_{e}^{2}}+\left(1+\frac{\mathrm{K_{L}}\times \mathrm{q_{m}}\times\mathrm{M}}{\mathrm{V}}-\mathrm{K_{L}}\times\mathrm{C_{0}} \right)\times\mathrm{C_{e}}-\mathrm{C_{0}}=0 \tag{5}\]  
If all variables except C\({}_{\mathrm{e}}\) are regarded as constants, Equation (5) can be regarded as a quadratic equation, so C\({}_{\mathrm{e}}\) can be solved [39]: \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Sample No.** & **Compounds** & **5 min** & **10 min** & **20 min** & **30 min** & **60 min** & **120 min** \\ \hline \multirow{2}{*}{Sample 1 (400 \(\mu\)L)} & TCE & 6.72 & 9.30 & 7.20 & 11.36 & 11.63 & 11.14 \\  & PCE & 2.12 & 0.94 & 3.15 & 3.54 & 0.68 & 2.74 \\ \hline \multirow{2}{*}{Sample 2 (450 \(\mu\)L)} & TCE & 1.50 & 4.03 & 1.78 & 1.36 & 4.10 & 2.61 \\  & PCE & 2.88 & 2.80 & 3.19 & 2.80 & 2.97 & 3.37 \\ \hline \multirow{2}{*}{Sample 3 (500 \(\mu\)L)} & TCE & 3.40 & 1.78 & 3.74 & 3.62 & 4.00 & 1.33 \\  & PCE & 1.90 & 1.66 & 1.69 & 3.43 & 8.28 & 1.62 \\ \hline \multirow{2}{*}{Sample 4 (550 \(\mu\)L)} & TCE & 2.64 & 2.78 & 3.34 & 2.88 & 5.22 & 2.80 \\  & PCE & 2.10 & 3.23 & 1.93 & 2.74 & 3.43 & 3.29 \\ \hline \multirow{2}{*}{Sample 5 (600 \(\mu\)L)} & TCE & 2.23 & 5.14 & 1.52 & 1.58 & 3.38 & 4.81 \\  & PCE & 2.06 & 0.88 & 1.53 & 3.54 & 4.66 & 4.45 \\ \hline \multirow{2}{*}{Sample 6 (650 \(\mu\)L)} & TCE & 2.53 & 1.87 & 2.10 & 1.14 & 2.32 & 1.20 \\  & PCE & 2.12 & 1.64 & 3.39 & 1.08 & 1.74 & 2.35 \\ \hline \multirow{2}{*}{Sample 7 (700 \(\mu\)L)} & TCE & 1.23 & 0.83 & 1.96 & 1.17 & 3.36 & 2.89 \\  & PCE & 2.27 & 2.32 & 0.76 & 2.60 & 1.19 & 1.63 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Relative standard deviation (R.S.D. ; %) of samples adsorbed by silica gel (n = 4). \[\mathrm{C_{e}}=-\left(1+\left(\mathrm{K_{L}}\times\mathrm{q_{m}}\times\frac{ \mathrm{M}}{\mathrm{V}}\right)-\mathrm{K_{L}}\times\mathrm{C_{0}}\right)+\frac{ \sqrt{\left(1+\left(\mathrm{K_{L}}\times\mathrm{q_{m}}\times\frac{\mathrm{M}}{ \mathrm{V}}\right)-\mathrm{K_{L}}\times\mathrm{C_{0}}\right)^{2}}+4\mathrm{K_{ L}}\times\mathrm{C_{0}}}{2\mathrm{K_{L}}} \tag{6}\]  
By obtaining a set of \(\mathrm{K_{L}}\) and \(\mathrm{q_{m}}\) from an adsorption experiment, the solute concentration (\(\mathrm{C_{e}}\)) at adsorption equilibrium can be estimated for a given initial solute concentration (\(\mathrm{C_{0}}\)) and fixed volume (\(\mathrm{V}\)) of the adsorbent mass (\(\mathrm{M}\)).