## 4 Results and Discussion
Additionally, Table 5 presents parameter estimates of analytical models for carbon dioxide (CO\({}_{2}\)) adsorption with and without estimation of maximum adsorption capacity (q\({}_{\text{s}}\), N\({}_{0}\)). For CO\({}_{2}\) adsorption, the [PERSON], Adams-Bohart, and Yan models showed good fits in estimating the maximum adsorption capacity for the highest concentration (35 mg/L),  
Figure 3: (**a**) Breakthrough curves for concentration of 1.327 mg/L and (**b**) concentration of 2.577 mg/L with deterministic q\({}_{\text{s}}\) and N\({}_{0}\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Models**} & \multirow{2}{*}{**Parameters**} & \multicolumn{3}{c}{**(q\({}_{\text{s}}\), N\({}_{0}\)) Estimated**} & \multicolumn{3}{c}{**(q\({}_{\text{s}}\), N\({}_{0}\)) Deterministic**} \\  & & **(Present Work)** & & & [30] \\ \hline \multirow{7}{*}{Thomas} & C\({}_{0}\) (mg/L) & 18 & 29 & 35 & 18 & 29 & 35 \\  & q\({}_{\text{s}}\) (mg/g) & 8.3114 & 12.7682 & 13.2753 & 7.2617 & 10.43037 & 12.1468 \\  & k\({}_{\text{th}}\) & 0.0632 & 0.0801 & 0.0699 & 0.0205 & 0.0099 & 0.0213 \\  & (L-/min-mg) & 0.9811 & 0.9935 & 0.9955 & 0.6660 & 0.7059 & 0.8223 \\  & R\({}^{2}\) & 0.9799 & 0.9934 & 0.9951 & 0.6451 & 0.6863 & 0.8062 \\  & BIC & 550.4370 & 274.7252 & 153.2467 & 1.6583 \(\times\) 10\({}^{4}\) & 2.4228 \(\times\) 10\({}^{4}\) & 1.0195 \(\times\) 10\({}^{4}\) \\  & k\({}_{\text{jm}}\) (min) & - & - & - & 1.1780 & 2.4075 & 2.4467 \\  & \({}^{\text{\tau}}\) (min) & - & - & - & 21.2342 & 20.2541 & 17.4458 \\  & R\({}^{2}\) & - & - & - & 0.9809 & 0.9939 & 0.9955 \\  & R\({}^{2}\)Adjusted & - & - & - & 0.9797 & 0.9935 & 0.9951 \\  & BIC & - & - & - & 557.1840 & 275.0701 & 153.3209 \\  & k\({}_{\text{th}}\) & 6.4991 \(\times\) 10\({}^{-5}\) & 8.7282 \(\times\) 10\({}^{-5}\) & 7.0016 \(\times\) 10\({}^{-5}\) & 8.8822 \(\times\) 10\({}^{-6}\) & 5.7670 \(\times\) 10\({}^{-6}\) & 4.9664 \(\times\) 10\({}^{-6}\) \\  & N\({}_{0}\) (mg/L) & 18065 & 27771 & 28866 & 5954.6 & 85520-904 & 9960.3 \\  & R\({}^{2}\) & 0.9809 & 0.9940 & 0.9955 & 0.2020 & 0.3307 & 0.3665 \\  & R\({}^{2}\)Adjusted & 0.9797 & 0.9935 & 0.9951 & 0.1522 & 0.2861 & 0.3089 \\  & BIC & 557.3886 & 270.7467 & 153.3194 & 1.3121 \(\times\) 10\({}^{5}\) & 1.1801 \(\times\) 10\({}^{5}\) & 8.2006 \(\times\) 10\({}^{4}\) \\  & q\({}_{\text{s}}\) & 9.5851 & 11.3468 & 13.0378 & 7.2617 & 10.4304 & 12.1468 \\  & n\({}_{\text{j}}\) & 5.6501 & 5.0110 & 5.0643 & 7.4527 & 5.4479 & 5.4352 \\  & R\({}^{2}\) & 0.9750 & 0.9891 & 0.9937 & 0.9749 & 0.9891 & 0.9937 \\  & R\({}^{2}\)Adjusted & 0.9734 & 0.9884 & 0.9932 & 0.9733 & 0.9884 & 0.9932 \\  & BIC & 763.9940 & 559.1430 & 236.2257 & 766.4743 & 559.1664 & 236.2271 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameter estimation of analytical models for CO\({}_{2}\) gas. as indicated by the determination coefficients (R\({}^{2}\) e R\({}^{2}\)\({}_{\text{Adjusted}}\)) and the Bayesian Information Criterion (BIC). The Yoon-Nelson model, although it does not estimate q, also demonstrated a satisfactory fit for this concentration. By analyzing the selection metrics (R\({}^{2}\) e R\({}^{2}\)\({}_{\text{Adjusted}}\) e BIC), the Adams-Bohart models (estimating N\({}_{0}\)) and Yoon-Nelson (without estimating q\({}_{\text{s}}\)) showed the best fits for the 35 mg/L, with R\({}^{2}\) and R\({}^{2}\)\({}_{\text{Adjusted}}\) of 0.9951 and BIC varying between 153.3194 and 153.3209. In contrast, the Adams-Bohart model, without estimating N\({}_{0}\), exhibited the worst fit for the lowest concentration (18 mg/L), with an R\({}^{2}\) of 0.2020 and R\({}^{2}\)\({}_{\text{Adjusted}}\) of 0.1522. Generally, the maximum adsorption capacity for CO\({}_{2}\) concentrations increased with the rise of (C\({}_{0}\)). In the paper [37], the authors state that the concentration gradient between the adsorbate and the solute is higher at elevated concentrations, increasing mass transfer and consequently the adsorption capacity. The Thomas model provided q\({}_{\text{s}}\) estimates close to the deterministic q\({}_{\text{s}}\).