## 3 Theory  
The Sips isotherm model is effective in predicting adsorption on heterogeneous surfaces. At low adsorbate concentrations, it simplifies to the Freundlich model, while at high adsorbate concentrations, it resembles the Langmuir model [34]. Equation (2) expresses the Sips isotherm model, as follows:  
\[\mathrm{q}=\frac{q_{\mathrm{mS}}\cdot b_{\mathrm{S}}p^{\mathrm{ms}}}{1+b_{ \mathrm{S}}\cdot p^{\mathrm{ms}}}[\mathrm{mmol/g}] \tag{2}\]  
qms--the maximum adsorption capacity [mmol/g]  
b\({}_{\mathrm{S}}\)--the Sips constant [kPa\({}^{-1}\)]  
nS--the heterogeneity factor  
The Toth isotherm model is a different empirical equation designed to enhance the fitting of isotherms between experimental and predicted equilibrium data. This model  
Figure 5: Spark discharge reactor. 1— quartz case, 2—high-voltage electrode, 3—grounded electrode, 4—active carbon, 5—quartz wool, 6—silicone seal. is particularly effective for characterizing heterogeneous adsorption systems, accurately capturing both low- and high-end concentration boundaries [35]. Equation (3) outlines the Toth equation. \[\mathrm{q}=\frac{q_{mT}b_{T}\mathrm{p}}{\left(1+\left(b_{T}p\right)^{nT}\right)^ {\frac{1}{2T}}}[\mathrm{mmol/g}] \tag{3}\]  
\(\mathrm{q_{mT}}\)--the maximum adsorption capacity [mmol/g]  
\(\mathrm{b_{T}}\)--the Toth constant [kPa\({}^{-1}\)]  
\(\mathrm{n_{T}}\)--the heterogeneity factor  
The Radke-Prausnitz model [36] possesses several key attributes that render it the preferred option for most adsorption systems with low adsorbate concentrations. At low adsorbate concentrations, this model simplifies to a linear isotherm. Conversely, at higher adsorbate concentrations, it approaches the Freundlich isotherm, and when \(\mathrm{n_{RP}}=0\), it resembles a Langmuir isotherm. Another important characteristic of this isotherm is its ability to provide a good fit across a broad range of adsorbate concentrations. The Radke-Prausnitz equation is formulated as follows in Equation (4):  
\[\mathrm{q}=\frac{q_{mRP}\cdot b_{RP}\cdot\mathrm{p}}{1+b_{RP}\cdot p^{n_{RP}}}[ \mathrm{mmol/g}] \tag{4}\]  
\(\mathrm{q_{mRP}}\)--the maximum adsorption capacity [mmol/g]  
\(\mathrm{b_{RP}}\)--the Radke-Prausnitz constant [bar\({}^{-1}\)]  
\(\mathrm{n_{RP}}\)--Radke-Prausnitz model exponent  
The Langmuir isotherm was developed to explain gas-solid phase adsorption and remains a fundamental method for quantifying and comparing the maximum adsorption capacities of different sorbents. This theory assumes monolayer coverage of the adsorbate, with adsorption taking place at specific homogeneous sites, in which all sites are equal in their adsorption energies. Once an adsorbate molecule occupies a site, no additional adsorption can occur at that site. The sorbent has a finite capacity for the adsorbate [37]. The Langmuir isotherm equation is represented by Equation (5), as follows:  
\[q=\frac{q_{mL}\cdot b_{L}\cdot p}{1+b_{L}\cdot p}\left[\mathrm{mmol/g}\right] \tag{5}\]  
\(\mathrm{q_{mL}}\)--the maximum adsorption capacity [mmol/g]  
\(\mathrm{b_{L}}\)--the Langmuir constant [kPa\({}^{-1}\)]  
\(\mathrm{p}\)--pressure [bar]  
\(\mathrm{q}\)--the adsorbed quantity under \(\mathrm{p}\) pressure [mmol/g]  
The Freundlich isotherm is an empirical equation used to describe adsorption on heterogeneous surfaces with varying adsorption energies [38]. The Freundlich equation is represented by Equation (6), as follows:  
\[q=k_{F}\cdot p^{nF}\left[\mathrm{mmol/g}\right] \tag{6}\]  
\(\mathrm{k_{F}}\)--the Freundlich constant [mmol/g]  
\(\mathrm{n_{F}}\)--the heterogeneity factor  
To assess the optimal fittings of isotherm models to the experimental data, the hybrid error function (HYBRID) [39] has been employed. The sum of squared errors (SSE) is widely used to quantify the deviation of measured data from the true mean of the data. To improve the SSE at lower values of pressure, each of the sum of the squares of the error values was divided by the experimental value of CO\({}_{2}\) adsorption.