# The Virial Effect--Applications for SF\({}_{6}\) and CH\({}_{4}\) Thermal Plasmas
## 3 Virial Coefficients
### The Lennard-Jones Potential
#### 3.1.2 Potential Parameters \(\sigma\) and \(\varepsilon\)  
The parameter \(\sigma\) corresponds to the distance at which it can be assumed that \(u(r)=0\); \(\varepsilon\) is an energy value, and it represents the depth of the potential. This last parameter is generally given by the expression \(\frac{\varepsilon}{k_{B}}\). These two parameters play a very important role in the determination of the virial coefficients. Currently, much data concerning these parameters can be found in the literature for several species [18; 32; 41; 43; 45; 46]. The determination of these parameters was obtained for almost all cases from experimental measurements, based on the transport coefficients (viscosity and diffusion), or the second virial coefficient [18; 43; 45; 46]. Nevertheless, for a given species, the values obtained by several authors [18; 47; 45; 46; 22] present very large differences, depending on the method used for their determination. An example of these disparities is presented for some species in Table 3. The LJ potential (12-6) has been considered in all relevant sources.  
In Table 3, the range for the values of \(\sigma\) is quite narrow, compared to the values of \(\frac{\varepsilon}{k_{B}}\). In the cases of SF\({}_{6}\) or CO\({}_{2}\), the \(\frac{\varepsilon}{k_{B}}\) values presented a large range, and it was difficult to make conclusions pertaining to the value to be used. In the literature, several approximation methods have been proposed for determining these parameters. Some of them are based on empirical rules, considering species interactions [18], and others are based on critical properties, such as the critical temperature, pressure, volume, or compressibility factor [45; 48; 49; 50; 51]. For example, [PERSON] [51] proposes the determination of the parameters based on Equations (34) and (35):  
\[\sigma=0.1866\;v_{c}^{\frac{1}{2}}\;Z_{c}^{\frac{-6}{5}} \tag{34}\]  
\[\frac{\varepsilon}{k_{B}}=65.3\;T_{c}\;Z_{c}^{\frac{18}{5}} \tag{35}\]  
where \(v_{c}\), \(Z_{c}\), and \(T_{c}\) are critical properties. In order to choose the better dataset, it is necessary to compare the results with the experimental data. [PERSON] [18] recommends the use of parameters deduced from experimental viscosity for calculations of transport properties, the experimental second virial coefficients for the state equation calculation, and the calculation of thermodynamic properties. Nevertheless, the data in this last case were scarce, and they were only available for some species.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Species** & \(\sigma\) **[Å]** & \(\frac{\varepsilon}{k_{B}}\) **[K]** & **References** \\ \hline Ar & 2.83–3.48 & 110.2–124 & [18; 45; 46] \\ N\({}_{2}\) & 3.681–4.21 & 26.9–95.9 & [18; 45; 46] \\ O\({}_{2}\) & 3.30–3.58 & 88–118 & [18; 45; 46] \\ SF\({}_{6}\) & 4.8–6.5 & 155–414.81 & [18; 22; 45; 47] \\ CH\({}_{4}\) & 3.22–3.882 & 137–176.8 & [18; 45; 46] \\ CO\({}_{2}\) & 2.84–4.486 & 152–594.4 & [18; 45; 46] \\ \hline \hline \end{tabular}
\end{table}
Table 3: The variation of potential LJ (12-6) parameters for some species.  
In Table 4, a comparison of the \(\sigma\) and \(\frac{\varepsilon}{k_{B}}\) parameters is given. The values from the literature used for the comparison with Stiel [51] are included in the range given in Table 3. These values were chosen, as they corresponded to the experimental values deduced from the viscosity. Consequently, it appears that the Stiel formulation [51] can be a good option for when data are missing.