## 3 Methods
### Models of Radial and Flat Diffusion Based on Fick's Laws
#### 3.3.1 Model Based on Arrhenius Theory  
The more commonly used model in the case of mercury emissions from soils or debris is based on the Arrhenius theory. This approach states that the rate constant k of chemical reactions is related to temperature, as shown in the equation:  
\[\mathrm{k(T)}=\ A_{\mathrm{f}}\ \mathrm{exp}\bigg{(}-\frac{\mathrm{E_{a}}}{ \mathrm{R\ T}}\bigg{)} \tag{15}\]  
where,  
k: kinetic constant  
E\({}_{a}\): activation energy (J/mol)  
A\({}_{\mathrm{f}}\): frequency factor (of the collisions between molecules)  
R: ideal gas constant (8.3144 J/mol K = 8.3144 Pa m\({}^{3}\) /mol K)  
T: temperature (K)  
Several authors have used the Arrhenius equation to model the flow of mercury from a solid surface, such as [39; 40; 41; 42]. In this case, the relationship is:  
\[\mathrm{F=\frac{G}{A}=\ c_{f}\ \mathrm{exp}\bigg{(}-\frac{\mathrm{E_{a}}}{ \mathrm{R\ T}}\bigg{)}} \tag{16}\]  
where,  
F: gas flux per unit area perpendicular to the flow (g/sm\({}^{2}\))  
E\({}_{a}\): apparent activation energy (J/mol, kcal/mol)  
c\({}_{f}\): factor related to the Arrhenius A\({}_{\mathrm{f}}\) constant  
G: emission rate G (g/s). A: area (m\({}^{2}\))  
R: ideal gas constant (8.3144 J/mol K = 8.3144 Pa m\({}^{3}\) /mol K)  
Usually, the theoretical activation energy would be equal to the gas evaporation enthalpy. However, since in this case it is not a pure gas evaporation reaction, this activation energy takes a different value and may vary with the types of solids from which the gas is released. Finally, it should be noted that in no case has UVA solar radiation been evaluated separately. As pointed out by many authors [37; 43; 44], emissions increase with solar UVA radiation. On cloudy days, emissions would decrease. In this case, the most unfavorable conditions corresponding to the highest emissions were sought out and the measurements were carried out on clear days. Estimation of the Apparent Activation Energy E\({}_{a}\) and the Coefficient c\({}_{\mathrm{f}}\)  
The Arrhenius model can be defined for a specific case by determining two parameters, factor c\({}_{\mathrm{f}}\) and apparent activation energy E\({}_{a}\), from the actual measurements taken. By applying Neper logarithms to the Arrhenius equation, we have a linear relationship between mercury flux F and the variable 1/RT:  
\[\mathrm{ln(F)=ln\bigg{(}\frac{G}{A}\bigg{)}=\ ln(c_{f})-\frac{E_{a}}{\mathrm{R \ T}}} \tag{17}\]  
Hg concentration measurements C\({}_{9}\) made at point 9 have allowed for the determination of the value for mercury emission rate G (ng/s) and mercury flux F (ng/sm\({}^{2}\)), as summarized in Table 1. When fitting by least squares to a straight line (Figure 7), we obtain the following expression with a correlation coefficient of R\({}^{2}\) = 0.64:  
\[\mathrm{ln(F)=16.16-\frac{48,562}{\mathrm{R\ T}}} \tag{18}\]From this, it follows that the value of the parameter is c\({}_{\rm f}\) =1.04 \(\times\) 10\({}^{7}\) and that of the apparent activation energy is E\({}_{\rm a}\) = 48,562 J/mol = 48.56 kJ/mol. This value of the activation energy, although of the same order, is relatively low compared to the data found in the literature. For example, the study in [45] found values of 124.8 kJ/mol, the study in [46] provided values between 80 and 110 kJ/mol, the study in [37] reported values between 30 and 75 kJ/mol depending on the origin and concentration of Hg in the soil, and the study in [47] showed values of 131 kJ/mol. Validation of the Model Based on the Arrhenius Equation  
To check the model based on the Arrhenius theory, the model is first used to estimate the gaseous Hg flux and then the results are compared to the experimental data.