## 3 Results and Discussion
### Mathematical Modeling  
Mathematical modeling and simulation is a very useful tool to predict the adsorption/desorption process in a fixed bed column [52]. \begin{table}
\begin{tabular}{c c c c c} \hline
**Relative** & **Adsorption** & **Total Adsorption** & **Time of** & **Time of** \\
**Humidity (\%)** & **Capacity (mg\(\cdot\)g\({}^{-1}\))** & **Percentage (\%)** & **Breakthrough (min)** & **Equilibration (min)** \\ \hline
40 & 410.9 & 48.8 & 55 & 435 \\
60 & 354 & 48.9 & 55 & 405 \\
80 & 203.3 & 33.8 & 5 & 335 \\ \hline \end{tabular}
\end{table} 
Table 2: The effects of different relative humidity on the operational parameters of toluene adsorption on ZIF-8. Figure 5: FT-IR spectra of ZIF-8/Air/300 \({}^{\circ}\)C after toluene adsorption at (**a**) 40% RH; (**b**) 60% RH; and (**c**) 80% RH. For successful design of a continuous adsorption system, the prediction of the breakthrough curve is very crucial. Maximum adsorption capacity of the adsorbent is another important parameter that is needed for a reactor design. Experimental data and mathematical modeling are used to obtain breakthrough curves of a desired adsorption system. In comparison with experimental method, mathematical modeling is simple, economic, and quick. Thus, they are used increasingly by researchers [53]. In this study, the experimental breakthrough curves were fitted with [PERSON] and Thomas models to predict adsorption parameters. Equations (3) and (4) describe the [PERSON] and [PERSON] models, respectively [52; 53]:  
\[\frac{C}{C_{0}}=1-\frac{1}{1+\left(\frac{C_{0}Q}{q_{y}m}t\right)^{A_{y}}} \tag{3}\]  
\[\frac{C}{C_{0}}=\frac{1}{1+exp\left(\frac{Tq_{T}m}{Q}-K_{T}C_{0}t\right)} \tag{4}\]  
where \(A_{y}\) is the [PERSON] models' constant and \(K_{T}\) is the [PERSON] models' constant. \(q_{y}\) and \(q_{T}\) denote adsorption capacity of the adsorbents, which can be estimated by [PERSON] and [PERSON] models (mg\(\cdot\)g\({}^{-1}\)). In the current work \(A_{y}\), \(K_{T}\), \(q_{y}\), and \(q_{T}\) parameters were estimated by fitting the collected experimental data into the Equations (3) and (4). Statistical parameters of fitting such as squared correlation coefficient (R\({}^{2}\)), adjusted squared correlation coefficient (R\({}^{2}\)adj), root mean square error (RMSE), and sum square error (SSE) were computed in order to evaluate the characteristics of fitting models. The Trust-Region algorithm was used to perform non-linear least square fitting of the adsorption parameters using MATLAB software (Mathwork Inc., Natick, MA, USA). The [PERSON] and Thomas mathematical models [52; 53] were used to fit the experimental breakthrough curves using non-linear curve fitting (Figures 6-8). Statistical parameters of the fitting are summarized in the Table 3.  
In case of adsorption by ZIF-8/Air/300 \({}^{\circ}\)C (Figure 6a), the values of R\({}^{2}\)adj and R\({}^{2}\) in the [PERSON] and Thomas models was 0.98, with low values of RMSE and SSE in both models (Table 3), which indicate good fitness of the experimental data with the mathematical models. In addition, the values of \(q_{y}\) and \(q_{T}\) that are estimated by the [PERSON] and [PERSON] models, respectively, are very close to the values obtained from the experimental data (Table 3). Figure 6: Experimental and predicted breakthrough curves of toluene adsorbed on (**a**) ZIF-8/Air/300 \({}^{\circ}\)C and (**b**) as-synthesized ZIF-8. Figure 8: Experimental and predicted breakthrough curves of toluene adsorption on ZIF-8/Air/300 \({}^{\circ}\)C at relative humidity (**a**) 40%; (**b**) 60%; and (**c**) 80%. Figure 7: Experimental and predicted breakthrough curves of toluene adsorbed on (**a**) ZIF-8/N\({}_{2}\)/300 \({}^{\circ}\)C and (**b**) ZIF-8/N\({}_{2}\)/400 \({}^{\circ}\)C.  
The experimental and simulated breakthrough curves of toluene adsorption on ZIF-8/N\({}_{2}\)/300 \({}^{\circ}\)C and ZIF-8/N\({}_{2}\)/400 \({}^{\circ}\)C samples are shown in Figure 7a,b.