## 3 Methods
### Models of Radial and Flat Diffusion Based on Fick's Laws
#### 3.3.2 Model Based on the Laws of Liquid Evaporation  
This model was proposed by [48] for the analysis of the formation of pollutant gas clouds (EPA) and is widely accepted; it follows the expression:  
\[\mathrm{G=K\;\frac{A\;M_{Hg}\;p_{v}}{R\;T}} \tag{22}\]  
G: emission rate of gases from the liquid to the atmosphere (g/s)  
Figure 8: Emission Rate vs. temperature values derived from measurements (dots) and calculated with model based on Arrhenius theory (line). Figure 9: Hg concentration C\({}_{10}\) vs. temperature; measured (dots) and calculated using the model based on Arrhenius theory (line). A: liquid surface area (m\({}^{2}\))  
p\({}_{\rm v}\): vapor partial pressure of gaseous mercury in the mixture (Pa)  
M\({}_{\rm Hg}\): molar mass of mercury (g/mol)  
T: temperature (K)  
R: ideal gas constant (J/mol K = Pa m\({}^{3}\) /mol K)  
K: mass transfer coefficient (m/s)  
The equation is derived from [PERSON]'s laws and parameter K is the same as previously explained. In effect, derived from the ideal gas equation, we have for gaseous Hg:  
\[C_{\rm A}=\frac{n}{V}=\frac{p_{\rm v}}{R\ T} \tag{23}\]  
where:  
CA: molar mercury concentration (mol/m3)  
n: number of moles of gaseous mercury in the mixture (mol)  
p\({}_{\rm v}\): partial vapor pressure of gaseous mercury in the mixture (Pa)  
V: volume of mixture considered (m\({}^{3}\))  
T: temperature (K)  
R: ideal gas constant R = 8.3144 (J/mol K = Pam\({}^{3}\)/mol K)  
Taking into account  
\[M_{\rm Hg}\ N_{\rm A}=F=M_{\rm Hg}\ K\ C_{\rm A} \tag{24}\]  
From Equations (23) and (24) and rearranging the terms, Equation (22) can be easily deduced. In this formula, K is the mass transfer coefficient related to the emission of gaseous Hg from the liquid surface due to the evaporation and it can be estimated as described in [48]. In the studied case, it is not a liquid surface, but the surface occupied by a pile of rubble on the ground; therefore, although the equation is the same, the mass transfer coefficient is different i.e., K\({}^{\prime}\). To develop the model, the vapor pressure of mercury gas over the debris, p\({}_{\rm v}\), and the transfer coefficient, K\({}^{\prime}\), governing the passage of mercury from the debris to the atmosphere, must be determined empirically. Parameter K\({}^{\prime}\) has been previously determined and p\({}_{\rm v}\) is estimated further on. Estimation of the Value of Mercury Vapor Pressure p\({}_{\rm v}\) over Debris  
From measurements of the mercury concentration at the center of focus C\({}_{10}\), an estimation can be made of the mercury vapor pressure p\({}_{\rm v}\) above the debris under working conditions. As the mercury evaporates and passes into the atmosphere, it diffuses into the gaseous mixture, which is air. Applying the equation of perfect gases to the gaseous mercury in the air above the debris, the vapor pressure will be:  
\[P_{\rm v10}=\frac{n\ R\ T}{V}=C_{\rm A10}\ R\ T=\ C_{\rm 10}\frac{R\ T}{M_{ \rm Hg}} \tag{25}\]  
where C\({}_{10}\) (g/m\({}^{3}\)) is the mercury concentration at point 10 over the debris and M is the molecular mass of mercury M\({}_{\rm Hg}\)= 200.59 g/mol. Table 3 summarizes the calculation of p\({}_{\rm v10}\) from the data. The value of p\({}_{\rm v10}\) is low, on the order of 10\({}^{-4}\) Pa in all cases. Since the temperature range is narrow (0 \({}^{\circ}\)C-30 \({}^{\circ}\)C or 273 K-303 K), the relationship between vapor pressure and saturation pressure remains almost constant. In effect, the saturation vapor pressure of gaseous mercury p\({}_{\rm s}\) is estimated as a function of temperature, assuming August's law (or a modified Antoine's law):  
\[\log_{10}(P_{\rm s})=a-\frac{b}{\Gamma} \tag{26}\]  
p\({}_{\rm s}\): saturation vapor pressure of mercury (Pa)T: absolute temperature in K  
a and b are two parameters that must be determined empirically. The values published by [48], a = 10.184 and b= 3210.29 are used here. Table 3 also shows the values of p\({}_{\rm s10}\), under these conditions, and the p\({}_{\rm v10}\)/p\({}_{\rm s10}\) ratio.