# Kinetics Study of Gas Pollutant Adsorption and Thermal Desorption on Silica Gel
## 3 Results and Discussion
### Toluene Thermal Desorption  
In order to evaluate the desorption performance of silica gel, series of experiments were conducted using a concentration of 609 ppm toluene at different temperatures of 313, 323, and 333 K. Figure 6 shows curves of the fraction of relative loading with time. From Figure 6, we can observe that the thermal desorption speed was obviously affected by the operating temperature. Based on the reaction in Equation (1), the thermal desorption kinetic model can also be calculated with Equation (5). For the silica gel regeneration processes, the initial condition at time \(t\) becomes:  
\[\left.\theta\right|_{t=0}=1 \tag{9}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Gas Pollutant** & \(C\) **(ppm)** & \(k_{a}\) **(s\({}^{-1}\))** & \(k_{a}C\) **(s\({}^{-1}\))** & \(R^{2}\) \\ \hline Carbon dioxide & 1000 & 0.240 & 0.0024 & 0.9999 \\ Methane & 1000 & 0.360 & 0.0036 & 0.9999 \\ Toluene & 609 & 0.053 & 0.00032 & 0.9903 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Adsorption parameters and model errors. Figure 6: Toluene thermal desorption curves of silica gel. By solving the difference equation of Equation (5), the thermal desorption kinetic model can be transformed into Equation (10):  
\[\theta(t)=\frac{k_{a}C}{k_{a}C+k_{d}}+\bigg{(}1-\frac{k_{a}C}{k_{a}C+k_{d}} \bigg{)}e^{-(k_{a}C+k_{d})t} \tag{10}\]  
Thus, the equilibrium relative loading, \(\theta_{e}\), becomes:  
\[\frac{1}{\theta_{e}}=1+\frac{K}{C} \tag{11}\]  
where \(K\) is the equilibrium constant, defined as \(K=k_{d}/k_{a}\). Table 3 lists the calculated reaction constants, equilibrium constant, and errors under thermal desorption conditions described in Section 2.2. The reaction constant values for the adsorption terms decrease with the increase of temperature. However, the reaction constant values for the desorption terms increase with the increase of temperature. The silica gel regeneration temperature promotes the reaction constant values to change reversely. The reaction constants are significantly influenced by temperatures, as shown in Table 3. Based on the Arrhenius equation, the reaction constant of the desorption terms reaction can be expressed in Equation (12):  
\[k_{d}=Ae^{-\frac{E_{a}}{RT}} \tag{12}\]  
where \(E_{a}\) is the activation energy for the desorption-terms reaction, J/mol; \(A\) is the pre-exponential factor for the desorption-terms reaction; and \(R\) is the universal gas constant, 8.314 J/(mol\(\cdot\)K). The Arrhenius plot for reaction constants is shown in Figure 7, which will be used to calculate the apparent activation energy. With Figure 7 and Equation (12), the apparent activation energy is calculated to be 45.83 kJ/mol for desorption terms. This value can reflect the difficult degree to deso \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Temperature (K)** & \(k_{a}C\) (s\({}^{-1}\)) & \(k_{a}\) (s\({}^{-1}\)) & \(k_{d}\) (s\({}^{-1}\)) & \(K\) & \(R^{2}\) \\ \hline
313 & 0.0486 & 7.9803 & 0.0141 & 0.00177 & 0.9989 \\
323 & 0.0216 & 3.5468 & 0.0231 & 0.00651 & 0.9952 \\
333 & 0.0108 & 1.7734 & 0.0406 & 0.02289 & 0.9955 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Thermal desorption parameters and model errors. Figure 7: The Arrhenius plot for reaction constants of silica gel thermal desorption. the adsorption site of the silica gel surface. The interaction between silica gel adsorbent and toluene is very probably physical adsorption, since its apparent activation energy is lower than 83 kJ/mol, which can characterize an absorption process controlled by chemical reactions [28]. The Van't Hoff equation was used to evaluate the enthalpy of thermal desorption. Thus, the equilibrium constant is:  
\[\ln K=-\frac{\Delta H}{RT}+\frac{\Delta S}{R} \tag{13}\]  
where \(\Delta H\) is the enthalpy of reaction, J/mol; \(\Delta S\) is the entropy of reaction, J/(mol\(\cdot\)K); and \(R\) is the universal gas constant, 8.314 J/(mol\(\cdot\)K).