## 3 Results
### Dimethoate Interaction with IG and GO Surfaces
#### 3.1.2 Adsorption Isotherms and Standard Gibbs Free Energy  
For the proper understanding and interpretation of the adsorption process on selected surfaces, the equilibrium of adsorption of DMT on 0.25 and 2 g dm\({}^{-3}\) graphene adsorbents was evaluated using Langmuir, Liu, and Dubinin-Raduskevich (DR) isotherm models [17; 25]. The following equations (Equations (6)-(8)) were applied to the experimental data presented as points in Figure 4:  
\[q_{e}=\frac{q_{max}K_{L}c_{eq}}{1+K_{L}c_{eq}} \tag{6}\]  
\[q_{e}=\frac{q_{max}\left(K_{g}c_{eq}\right)^{1/n_{g}}}{1+\left(K_{g}c_{eq} \right)^{1/n_{g}}} \tag{7}\]  
\[q_{e}=q_{DR}exp\left(-K_{DR}\left(RT\ln\left(1+\frac{1}{C}\right)\right)^{2}\right) \tag{8}\]  
Here, \(q_{e}\) represents adsorption capacity (\(c_{eq}/\gamma\)), \(C=c_{s}/c_{eq}\), \(c_{eq}\) is the equilibrium concentration of adsorbate, \(c_{s}\) is the saturation concentration of dimethoate in water (for diluted solutions \(c_{eq}/c_{s}<<1\)), \(R\) is the universal gas constant, \(T\) is temperature, and \(q_{max}\), \(q_{DR}\), and \(n_{g}\) are parameters obtained from fitting using equations in Equations (6)-(8). \(K_{L}\), \(K_{g}\), and \(K_{DR}\) represent Langmuir, Liu, and Dubinin-Radushkevich isotherm constants, respectively, related to adsorption's energy. The experimental results were fitted to the above-listed adsorption models and are presented in Figure 4 as lines. The high values \begin{table}
\begin{tabular}{c c c} \hline
**Adsorption Parameters** & **IG** & **GO** \\ \hline \(C_{max}/\text{mol dm}^{-3}\) g\({}^{-1}\) & (3.16 \(\pm\) 1.20) \(\times\) 10\({}^{-4}\) & (2.44 \(\pm\) 1.45) \(\times\) 10\({}^{-5}\) \\ \(K_{dis}/\text{mol dm}^{-3}\) & (6.78 \(\pm\) 2.5) \(\times\) 10\({}^{-4}\) & (5.09 \(\pm\) 3.01) \(\times\) 10\({}^{-5}\) \\ \(K_{ads}/\text{mol}^{-1}\) dm\({}^{3}\) & 1.47 \(\times\) 10\({}^{3}\) & 1.96 \(\times\) 10\({}^{4}\) \\ \(n_{H}\) & 1.94 \(\pm\) 0.36 & 1.82 \(\pm\) 0.25 \\ \(N\) & 5.47 \(\times\) 10\({}^{-2}\) & 0.57 \\ \(R^{2}\) & 0.9994 & 0.9916 \\ \hline \end{tabular}
\end{table} 
Table 1: Maximal DMT concentration per g of adsorbents (\(C_{max}\)), apparent dissociation and adsorption constants of DMT-adsorbent composite (\(K_{dis}\), \(K_{ads}\)), Hill coefficient (\(n_{H}\)) and number of occupied binding sites as mol BS per g of adsorbents (\(N\)) for DMT binding on IG and GO surfaces. Figure 4: Experimental data (points) for (**a**) 2 g dm\({}^{-3}\) and (**b**) 0.25 g dm\({}^{-3}\) IG and GO fitted with Langmuir (blue), Liu (red), and Dubinin–Raduschevich (green) adsorption isotherms. of the correlation coefficient \(R^{2}\) suggest that the values calculated for both adsorbents are reliable. The thermodynamic parameters of adsorption were evaluated using \(K_{L}\) and \(K_{S}\) constants, as reported previously [43]. Gibbs free energy of adsorption (\(\Delta G\)) was calculated according to the equation proposed by [PERSON] [44]:  
\[\Delta G=-RT\ln\left(K_{S}C^{0}\right) \tag{9}\]  
In water solutions, \(C^{0}\) represents the standard activity of water (55.5 mol dm\({}^{-3}\)). DR isotherm equation parameter \(K_{DR}\) was used to calculate the mean adsorption energy (absolute value) according to the relation [45; 46; 47]:  
\[\left|E_{ads,\ mean}\right|=\left(-2K_{DR}\right)^{-1/2} \tag{10}\]  
The mean values of isotherm parameters and energy calculated from these parameters are listed in Table 2. A satisfactory agreement was achieved by various models, suggesting the monolayer adsorption on the energetically homogeneous surface [25]. All three quantities--\(\Delta G\) (from Langmuir and Liu models), \(E_{ads}\) (from DFT calculations), and \(\left|E_{ads,\ mean}\right|\) (from DR model) in general describe the strength of surface/adsorbate interaction. Obtained negative \(\Delta G\) values of adsorption in all investigated cases confirm that adsorption at 298 K is exothermic. Moreover, \(\Delta G\) values from Langmuir and Liu isotherms are in good agreement with DFT calculated values of adsorption energy \(E_{ads}\) (Section 3.2.1, Table 3).