# The Virial Effect--Applications for SF\({}_{6}\) and CH\({}_{4}\) Thermal Plasmas
## 3 Virial Coefficients
### Results and Discussion  
#### 3.2.1 Choice of Parameters  
From the values of \(\sigma\) and \(\frac{\varepsilon}{k_{B}}\) obtained with the Stiel formulation, it is possible to calculate the plasma composition and the mass density at high pressure. For validation, experimental gas densities at high pressure are not often available. Nevertheless, experimental mass densities are more widely available. Thus, a validation of the calculations for theoretical composition were performed with this data. As the species densities from our composition tool were previously presented, the mass density of the gas could be calculated according to Equation (36):  
\[\rho=\sum_{i=1}^{N}n_{i}.m_{i} \tag{36}\]  
where \(N\) is the total number of species, \(n_{i}\) represents the densities of the species \(i\), and \(m_{i}\) is the mass of the species. Table 5 shows a comparison of the calculated mass densities (from our tool and from using the Stiel formulation for the parameters of the Lennard-Jones potential) at different pressures, using the experimental data for SF\({}_{6}\). The theoretical data were compared over a temperature range of 300-490 K. In order to study the influence of the virial coefficients, the calculation was performed without taking them into account. It can be observed in Table 5 that the theoretical results with and without the virial corrections become closer over higher temperatures. This indicates that the virial corrections were predominantly relevant at low temperatures for high pressures. Nevertheless, despite the good agreement between the Stiel formulation-derived values of \(\sigma\) and \(\frac{\varepsilon}{k_{B}}\) and the experimental values (observed in Table 4), these values led to quite large differences in the theoretical mass densities, compared to the experimental values given by [PERSON] [19] and [PERSON] [57] at around the critical point (\(Pc\approx 37.5\) bar and \(Tc\approx 318.8\) K) (Table 5). Table 6 shows the same comparison for CH\({}_{4}\). The mass densities for this gas were less sensitive for the values of \(\sigma\) and \(\frac{\varepsilon}{k_{B}}\). Indeed, the virial effect was not as preponderant as that observed in Table 6.  
Even with the consideration of the virial correction, for SF\({}_{6}\) (Table 5), large differences could be observed between the calculated and experimental values for some points. In order to better understand this, we tried to calculate the mass density through another method. Indeed, [PERSON] [19] gives the values of the second and third virial coefficients in his paper, deduced from measurements in a homogeneous area of SF\({}_{6}\). Thus, by using the state equation proposed by [PERSON], it was possible to re-calculate the mass densities on the points given in the article. These values of \(\rho_{cal}\) are reported in Table 7. We expected that the mass densities calculated from the virial coe \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Species** & \((\sigma_{cal})[\dot{\mathbf{A}}]\) & \((\frac{\varepsilon}{k_{B}})[\mathbf{K}]\) & **Reference** & \(\sigma_{it}\) [\(\dot{\mathbf{A}}\)] & \((\frac{\varepsilon}{k_{B}})_{it}\) [\(\mathbf{K}\)] & **Reference** \\ \hline Ar & 3.451 & 116.035 & [52] & 3.465 & 116 & [18] \\ N\({}_{2}\) & 3.694 & 94.837 & [53] & 3.681 & 91.5 & [18] \\ O\({}_{2}\) & 3.474 & 115.008 & [54] & 3.433 & 113 & [18] \\ SF\({}_{6}\) & 4.993 & 215.04 & [55] & 5.199 & 212 & [45] \\ CH\({}_{4}\) & 3.828 & 143.02 & [51] & 3.796 & 144 & [18] \\ CO\({}_{2}\) & 3.993 & 191.143 & [56] & 3.996 & 190 & [18] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: A comparison of the LJ (12-6) potential parameters, based on their use in experimental measurements, as calculated by several authors from the formulation given by [PERSON] et al. [51]. would lead to values that were close to the experimental ones given in the same paper. Unfortunately, this was not the case, and a perfect agreement was only reached for P = 20 bar.