## 3 Methods
### Models of Radial and Flat Diffusion Based on Fick's Laws  
#### 3.2.1 Model of Radial Flow Diffusion and Determination of the Emission Rate G  
From the experimental data, it was found that the variation in the gas concentration outside the focus fits a radial flow model. Considering that the gas cloud is above the ground, the flow can be considered hemispheric. Surrounding the emitting focus of the hemispherical shape of radius r\({}_{0}\), the following relationship is fulfilled:  
\[\mathrm{y}_{\mathrm{A}}=\frac{\mathrm{G}_{\mathrm{A}}}{2\,\pi\,\mathrm{C}\, \mathrm{D}}\,\frac{1}{\mathrm{r}} \tag{4}\]  
Figure 5: Simplified sketch of the gaseous Hg emissions (flat and hemispherical flux). y\({}_{A}\): molar fraction of gas A in the mixture (dimensionless, mol/mol)  
G\({}_{A}\): emission rate (mol/s). C: molar concentration of the mixture (mol/m\({}^{3}\))  
D: diffusivity coefficient (m\({}^{2}\)/s)  
r: distance to the center of the focus (m) at the edge of the debris zone (considered the focus), with r\({}_{9}\) = 10 m, the emission ratio G\({}_{A}\) for different temperatures can be estimated from the concentrations. Thus,  
\[G_{A}=2\ \pi\ C\ D\ y_{AB}\ r_{9}=2\ \pi\ C_{A9}\ D\ r_{9} \tag{5}\]  
where C\({}_{A9}\) is the molar concentration of the gaseous Hg. By dividing both terms by the molecular mass of mercury (M\({}_{Hg}\) = 200.59 g/mol), the equation can be re-written:  
\[G=2\ \pi\ D\ C_{9}\ r_{9} \tag{6}\]  
where C\({}_{9}\) is in g/m\({}^{3}\) and G is in g/s (or alternatively ng/m\({}^{3}\) and ng/s). D is the diffusivity coefficient of mercury in the air, which is estimated by applying the formula proposed by [28]:  
\[D=D_{0}\ \left(\frac{P_{0}}{P}\right)\ \left(\frac{T}{T_{0}}\right)^{1.81} \tag{7}\]  
D\({}_{0}\) is the diffusion coefficient of mercury in the air, at standard conditions of pressure and temperature, (P\({}_{0}\) = 1 atm = 101.3 kPa and T\({}_{0}\) = 293 k), whose value, according to the same author, is D\({}_{0}\) = 0.122 cm\({}^{2}\) s\({}^{-1}\) = 1.22 \(\times\) 10\({}^{-5}\) m\({}^{2}\) s\({}^{-1}\). This same expression can be found in the bibliography used by other authors in recent studies related to atmospheric pollution, such as [29; 30]. At the limit of the focus, point 9, concentrations of up to 20,000 ng/m\({}^{3}\) were measured, similar to those found by [16; 31; 32] in other similar facilities studied. However, they are much larger than those measured by [33]: around 400 ng/m\({}^{3}\), possibly because they were not mineral processing facilities. On the other hand, if gaseous Hg is produced by the rubble over a floor area A (m\({}^{2}\)), the flux of gaseous Hg through this area F (ng/s m\({}^{2}\)) is:  
\[F=\frac{G}{A} \tag{8}\]  
Table 1 summarizes the calculation of G and F from experimental data T and C\({}_{9}\). It is assumed A \(\approx\pi\) 10\({}^{2}\) = 314 m\({}^{2}\) and P\({}_{0}\) = 101,325 Pa. The G emission rate varies from 0.9 ng/s on the coldest days to 15.5 ng/s on the hottest days. Since the emission surface is assumed to be a circle of a radius r\({}_{9}\) = 10 m with an area of 314 m\({}^{2}\), the emission flux varies approximately between 10 and 180 ng m\({}^{-2}\) h\({}^{-1}\). Therefore, although the measured concentrations are very high, the emission rate is rather low. The values of F obtained are in the typical range for the different situations, as seen in the review of [34; 35], or with those found by [10] at sites contaminated by mercury mining. Nevertheless, they are significantly less than the values reported by [10; 36]. On the other hand, there is a dependence of emission flux on temperature, as reported by authors such as [37; 38], based on laboratory tests.