# The Virial Effect--Applications for SF\({}_{6}\) and CH\({}_{4}\) Thermal Plasmas
## 1 Introduction  
The calculation of plasma compositions under the assumption of local thermal equilibrium (LTE) is necessary to determine the thermodynamic properties and the transport coefficients that are needed for magneto-hydrodynamic models of thermal plasma processes and systems as circuit breakers, plasma torches, or thermal engines. These compositions are also necessary as initial equilibrium states for chemical kinetic studies and to determine departures from chemical equilibrium. In the literature, three methods are mainly used to calculate the evolution of species densities with temperature and pressure in thermal plasmas: the Gibbs free enthalpy minimization [1], the mass action law method [2; 3], and the collisional radiative model. Based on these methods, numerous papers have reported chemical plasma compositions for pure gases and mixtures, with or without the LTE assumption [2; 4; 5; 6; 16]. Nevertheless, in most of these papers, even at high pressure (>30 bar), only the first-order Debye-Huckel correction is taken into account [2; 17]. Although this correction is necessary at high temperatures to consider the Coulombic interactions between the charged particles and others, it is not sufficient for low temperatures (around 300 K), where molecule interactions are important. Under these conditions, it is necessary to correct the perfect gas behavior by considering virial pressure correction. However, in the thermal plasma community, only a few authors have considered this correction at high pressures [4; 6; 9; 14]. The virial pressure correction can be developed over several orders, named virial coefficients. They can be obtained experimentally ([18], for instance) by deducing them from measurements of the mass density, viscosity, or, in some cases, sound velocity [19; 20] or by theoretical calculation using intermolecular interaction potentials [18; 21; 22]. In the literature, we can find that, for some gases, the second and sometimes the third coefficients are given as a function of the temperature [19; 22; 23; 24; 25; 26]. These data are very dependent on the pressure and temperature of the gas, and they are not valid for a large range of temperatures and pressures. The densities of the species, and the mass density of the gases depend very strongly on the applied dataset. This mass density has a direct effect on convection modeling for its applications. In our research fields, at high pressures, we are particularly focused on three topics:  
* the plasma behavior in high-voltage circuit breakers (HVCBs) [27; 28; 29], where pressures in heating volumes can reach bar values of several tens;
* hydro-electro formation and hydraulic fracturing using an electric arc in water [30];
* the combustion of methane in thermal gas engines [31]. For each of these topics, the pressure considered and the gases used are different, and they include, within a high-pressure range, SF\({}_{6}\) for the first application, H\({}_{2}\)O for the second, and mixtures of air and CH\({}_{4}\) for the third. In the case of HVCB, some compositions and thermal properties are available in the literature; nevertheless, a complete databank for describing the current cutoff from the initial pressure to the one obtained during the high-current phase is not given, and authors seem to use the same parameters to calculate the Lennard-Jones potential for the entire pressure range [14; 32; 33]. This problem is the same for the case of hydro-electro formation and hydraulic fracturing: little data is available for H\({}_{2}\)O, and only few details on the parameters that are used to calculate the Lennard-Jones potential are given. For engine applications, we did not find data related to air-CH\({}_{4}\) mixtures in the literature. Therefore, papers from the literature generally consider pure air, and pressure corrections are not taken into account [34; 35; 36]. In the first part of this paper, the method that was used to calculate the plasma composition is presented.