# The Virial Effect--Applications for SF\({}_{6}\) and CH\({}_{4}\) Thermal Plasmas
## 3 Virial Coefficients
### Results and Discussion
#### 3.2.1 Choice of Parameters
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{1}{c}{} & \multicolumn{6}{c}{**CH\({}_{4}\)**} \\ \hline \multicolumn{1}{c}{} & \multicolumn{2}{c}{**[PERSON] et al. [58]**} & \multicolumn{6}{c}{**This Work with the Stiel Formulation**} \\ \hline \multicolumn{1}{c}{} & \multicolumn{2}{c}{} & \multicolumn{6}{c}{\(\rho_{cal}\) [kg/m\({}^{3}\)]**} \\ \hline \multicolumn{1}{c}{} & \multicolumn{2}{c}{} & \multicolumn{6}{c}{**with virial**} & \multicolumn{2}{c}{**\% Error**} & \multicolumn{2}{c}{**without Virial**} & \multicolumn{2}{c}{**\% Error**} \\ \hline
305.236 & 50.01 & 34.173 & 34.011 & 0.5\% & 31.636 & 7.4\% \\
305.231 & 99.93 & 72.932 & 72.321 & 0.8\% & 63.216 & 13.3\% \\
338.049 & 50 & 29.983 & 29.848 & 0.5\% & 28.542 & 4.8\% \\
338.037 & 69.05 & 42.093 & 41.821 & 0.6\% & 39.416 & 6.4\% \\
338.103 & 99.69 & 62.054 & 62.391 & 0.5\% & 56.907 & 8.3\% \\
400.015 & 100.02 & 49.746 & 49.807 & 0.1\% & 48.246 & 3.0\% \\
450.115 & 344.92 & 136.647 & 136.563 & 0.061\% & 147.891 & 7.6\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: CH\({}_{4}\) mass densities for several pressure values. These large differences close to critical point for SF\({}_{6}\) caused us to search for better values for \(\sigma\) and \(\frac{\varepsilon}{k_{B}}\), by lowering the error between the theoretical and experimental values of mass density. Homemade optimization software was developed for obtaining such values. The software proposes values of \(\sigma\) and \(\frac{\varepsilon}{k_{B}}\) that minimize the difference between the theoretical and experimental mass densities, using all of the experimental data available. This minimization was based on a real genetic algorithm proposed by [PERSON] [59]. From this model, we obtained two values: \(\sigma\) = 5.21279 A and \(\frac{\varepsilon}{k_{B}}\) = 222.27 K. Different sources of SF\({}_{6}\) mass densities versus temperature are indicated in Figures 4-6, respectively, for pressure P = 20, 40, and 100 bar. These values were compared to the mass densities that were calculated from the parameters deduced from the genetic algorithm. Five curves are plotted in these figures:  
- the experimental mass densities measured by [PERSON] [19];  
- the mass densities calculated from the second and third virial coefficients given by [PERSON] [19];  
- the mass densities calculated from the values of \(\sigma\) and \(\frac{\varepsilon}{k_{B}}\) by [PERSON] et al. [18];  
- the mass densities calculated from the values of \(\sigma\) and \(\frac{\varepsilon}{k_{B}}\), deduced by the Stiel formulation [51];  
- the mass densities calculated from the values of \(\sigma\) and \(\frac{\varepsilon}{k_{B}}\), deduced from the optimization process and named \"fit\". The mass densities obtained from the optimization process are the closest to the experimental ones. The mass densities obtained from the other sources from the literature are sometimes in good agreement below the critical point, and sometimes agree at values over the critical point, but not over the full range of temperature and pressure. By using the optimized data (fit), the relative errors of the experimental values did not exceed 9%. Thus, when experimental data for the mass densities are available, obtaining \(\sigma\) and \(\frac{\varepsilon}{k_{B}}\) with the optimized software is preferable. This dataset \(\sigma\) = 5.21279 A and \(\frac{\varepsilon}{k_{B}}\) = 222.27 K gives a better approximation of the calculated mass densities for a pressure range of between 1 and 300 bar. The use of the Stiel [51] formulation is a good option for pressures and temperatures that are not too close to the critical point Pc \(\approx\) 37.5 bar and Tc \(\approx\) 318.8 K (cf. Figures 4 and 6), when no other data are available. The curves from Figures 4-6 show discrepancies in the variation of the LJ potential parameters on the mass densities deduced from the composition.