## 4 Results and Discussion  
In the development of the parameter estimation method, experimental data from [29] for H\({}_{2}\)S adsorption and from [30] for CO\({}_{2}\) adsorption were utilized. The Markov Chain Monte Carlo method with the Metropolis-Hastings algorithm was applied to obtain the parameters. The results obtained for the estimation of maximum adsorption capacity (q\({}_{\text{s}}\), N\({}_{0}\)) were compared to those obtained deterministically in the literature. Subsequently, based on selection metrics, it was determined which models showed better agreement with the experimental data. Thus, the number of states of the Markov chain (n) was set in a range from 2000 to 10,000, and the search step (w) used ranged from 0.003 to 0.03, ensuring the stabilization of the chains. For uncertainties, a variance of 1% of the maximum experimental measurement was applied, which is related to the ratio between concentrations (C/C\({}_{0}\)) in the breakthrough curves. The elements used for parameter estimation such as the number of states (n), search step (w), and heating states (aq) for each model, with different initial concentrations, are included in the Supplementary Material. Table 4 presents the parameter estimates of the analytical models for H\({}_{2}\)S gas adsorption with and without estimation of the maximum adsorption capacity. In the fitting analysis of the [PERSON], [PERSON], Adams-Bohart, and [PERSON] models for H\({}_{2}\)S gas adsorption, it was observed that all models, except Yoon-Nelson, showed good agreement in estimating the maximum adsorption capacity (q\({}_{\text{s}}\), N\({}_{0}\)) for the lowest initial concentration (C\({}_{0}\)) of H\({}_{2}\)S (1.327 mg/L). Although the Yoon-Nelson model does not include q\({}_{\text{s}}\) in its equation for this estimation, it also demonstrated a good fit for the respective concentration. Among the analyzed models, the Yan model showed the best fit using the estimation of q\({}_{\text{s}}\), with an R\({}^{2}\) of 0.9982, R\({}^{2}\)\({}_{\text{adjusted}}\) of 0.9981, and BIC of 11.2732. On the other hand, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Models**} & \multirow{2}{*}{**Parameters**} & \multicolumn{2}{c}{**(q\({}_{\text{s}}\), N\({}_{0}\)) Estimated**} & \multicolumn{2}{c}{**(q\({}_{\text{s}}\), N\({}_{0}\)) Deterministic**} \\  & & **(Present Work)** & & **[**29**]** \\ \hline \multirow{5}{*}{Thomas} & C\({}_{0}\) (mg/L) & 1.327 & 2.577 & 1.327 & 2.577 \\  & q\({}_{\text{s}}\) (mg/g) & 18.9861 & 17.8316 & 20.61 & 20.61 \\  & k\({}_{\text{th}}\) (L/min\(\cdot\)mg) & 0.0306 & 0.0204 & 0.0278 & 0.0168 \\  & R\({}^{2}\) & 0.9937 & 0.9700 & 0.9746 & 0.9488 \\  & R\({}^{2}\) Adjusted & 0.9932 & 0.9673 & 0.9728 & 0.9443 \\  & BIC & 249.1613 & 1.0900 \(\times\) 10\({}^{3}\) & 1.2473 \(\times\) 10\({}^{3}\) & 1.9488 \(\times\) 10\({}^{3}\) \\  & k\({}_{\text{yr}}\) (min) & - & - & 0.0412 & 0.0534 \\  & \(\tau\) (min) & - & - & 145.7883 & 69.9294 \\ [PERSON]– & R\({}^{2}\) & - & - & 0.9936 & 0.9698 \\ [PERSON] & R\({}^{2}\) Adjusted & - & - & 0.9732 & 0.9672 \\  & BIC & - & - & 251.3955 & 1.0958 \(\times\) 10\({}^{3}\) \\  & k\({}_{\text{ba}}\) (10\({}^{5}\)) & 3.0840 & 2.0333 & 2.4615 & 1.4473 \\  & N\({}_{0}\) (10\({}^{-4}\)) & 4.8053 & 4.5456 & 5.5729 & 5.5729 \\  & R\({}^{2}\) & 0.9937 & 0.9701 & 0.9331 & 0.9293 \\  & R\({}^{2}\) Adjusted & 0.9932 & 0.9675 & 0.9284 & 0.9232 \\  & BIC & 249.2162 & 1.0845 \(\times\) 10\({}^{3}\) & 3.6145 \(\times\) 10\({}^{3}\) & 2.7708 \(\times\) 10\({}^{3}\) \\  & q\({}_{\text{s}}\) & 18.5658 & 16.9291 & 20.61 & 20.61 \\  & a\({}_{\text{y}}\) & 5.7419 & 3.3525 & 5.8802 & 3.3550 \\ Yan & R\({}^{2}\) & 0.9982 & 0.9899 & 0.9671 & 0.9479 \\  & R\({}^{2}\) Adjusted & 0.9981 & 0.9890 & 0.9648 & 0.9434 \\  & BIC & 11.2732 & 316.4672 & \(1.6493\times\) 10\({}^{3}\) & \(1.9835\times\) 10\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameter estimation of analytical models for H\({}_{2}\)S gas.