# Kinetics Study of Gas Pollutant Adsorption and Thermal Desorption on Silica Gel
## 3 Results and Discussion  
### Adsorption Kinetic Model  
The equilibrium adsorption performances of silica gel for carbon dioxide, methane, and toluene were obtained from the TGA experiments. Figures 3-5 show the adsorption curves for carbon dioxide, methane, and toluene as a function of time. It takes 20-30 min to reach equilibrium for carbon dioxide and methane at a concentration of 1000 ppm. In contrast, it takes about 165 min to reach equilibrium for toluene at a concentration of 609 ppm. Some conclusions can be observed from Figures 3 and 4. The results display that a large amount of toluene was adsorbed at any given time and, at the simulated equilibrium condition, compared with methane, which may be caused by the different surface interactions. The silica gel adsorbent-toluene interaction is dominated by the hydrogen bond between the hydrogen atom bonds to the Si-OH and the aromatic delocalized pi electrons [25], which hold a much higher adsorption force than the Van der Waals force. Although the silanol groups can also interact with methane, the silica gel adsorption-methane interaction is so weak that the bands for adsorbed methane completely disappear when the temperature is over 213 K. This phenomenon was confirmed by [PERSON] et al. [26]. Thus, the silica gel adsorbent showed a stronger affinity for toluene than methane. Figure 4: Experimental and modeling adsorption curves for methane. Figure 3: Experimental and modeling adsorption curves for carbon dioxide. An appropriate kinetic model is required to understand the adsorption process and to evaluate its thermodynamics. The Langmuir isotherm is frequently assumed to comprehend the adsorptive processes [27], which considers that the adsorption and desorption of a single reaction is completely reversible. The equilibrium will be reached when the adsorption rate equals the desorption rate. The reaction is expressed as:  
\[\mathrm{P}+\mathrm{V}\rightleftharpoons\mathrm{PV} \tag{1}\]  
where P is the gas pollutant molecule in the gas phase, V is the vacant adsorption site on adsorbent solid surface, and PV implies that P is adsorbed. The fraction of relative loading, \(\theta\), is typically calculated as:  
\[\theta(t)=\frac{q(t)}{q^{*}} \tag{2}\]  
where \(q\) is the amount of adsorbate adsorbed, kg/kg; \(q^{*}\) is the equilibrium amount of adsorbate adsorbed, kg/kg, and \(t\) is the time, \(s\). The rate of adsorption is proportional to the percentage of unoccupied active sites and the concentration of gas pollutant in the bulk flow, and the rate of desorption is proportional to the percentage of the covered surface. Thus, the rate of adsorption terms, \(r_{a}\), and desorption terms, \(r_{d}\), may be written as [27]:  
\[r_{a}=k_{a}C(1-\theta) \tag{3}\]  
\[r_{d}=k_{d}\theta \tag{4}\]  
where \(k_{a}\) is the reaction constant of adsorption terms, s\({}^{-1}\); \(C\) is the concentration of gas pollutant in bulk flow and is given in molar fraction; and \(k_{d}\) is the reaction constant of desorption terms, s\({}^{-1}\). Thus, the overall rate of process is:  
\[r=\frac{d\theta}{dt}=k_{a}C(1-\theta)-k_{d}\theta \tag{5}\]  
For the silica gel adsorption processes described in Section 2.2, the initial condition at time \(t\) is:  
\[\theta|_{t=0}=0 \tag{6}\]  
By solving Equation (5), the adsorption kinetic model can be transformed as a function of time:  
\[q(t)=q^{*}\cdot\frac{k_{a}C}{k_{a}C+k_{d}}\Big{(}1-e^{-(k_{a}C+k_{d})t}\Big{)} \tag{7}\]  
Figure 5: Experimental and modeling adsorption curves for toluene. The fitting validity can be ensured only by ensuring adequate agreement between the kinetic model results and the experimental data. Here, the coefficient of determination, \(R^{2}\), is calculated with Equation (8):  
Figure 3:0  
where \(q_{\exp,i}\) and \(q_{sim,i}\) are the amounts of adsorption under experimental and predicted conditions, respectively, \(\overline{q}_{\exp}\) is the mean of experimental data, and \(N\) is the total number of fitted points.