## 2 Research Method
### Method 1 of 4, Simplified Frost Functions: \(\Pi_{e}=1.5\vec{T}\)  
The first method used herein to estimate the temperature-dependent electron-neutral collision cross sections is based on a published model [37] for computing the electric conductivity of plasma that may be encountered in magnetohydrodynamic (MHD) electric power generation. Such plasma is formed as combustion product gases seeded with an alkali metal vapor (either cesium, Cs, or potassium, K). The temperature range of the presented computational model was from 2000 K to 4000 K. In the published model for plasma electric conductivity, a list of expressions for the quantity (\(v_{e-n}/N\)) as functions of the monoenergetic (directly dependent on the instantaneous electron speed) electron energy (\(u_{e}\)) is given. These expressions were derived such that they approximately matched the corresponding experimental data available in the earlier literature, with one or two references identified for the source of data used to arrive at each expression. One expression was given for each of the 13 gaseous neutral species. The variable (\(v_{e-n}\)) represents the electron-neutral momentum transfer collision frequency for a particular species, and (\(N\)) represents the number density of that species in the plasma gas mixture. In the published electric conductivity model, the quantity (\(v_{e-n}/N\)) is equal to (\(v_{e}Q\)), which is the product of the electron speed and the electron-neutral momentum transfer cross section for the species of concern. Thus,  
\[v_{e-n}(u_{e})=v_{e}QN \tag{8}\]  
Solving for the momentum transfer collision cross section gives  
\[Q(u_{e})=\frac{v_{e-n}(u_{e})}{v_{e}N} \tag{9}\]  
or  
\[Q(u_{e})=\frac{v_{e-n}}{N}(u_{e})\frac{1}{v_{e}} \tag{10}\]  
This can be rewritten as  
\[Q(u_{e})=\frac{\psi(u_{e})}{v_{e}} \tag{11}\]  
where \(\psi(u_{e})\) is the electron-energy-dependent quantity (\(v_{e-n}/N\)), for which an expression was made in the published electric conductivity model. In the present study, an approximation is made by utilizing Equation (11) with an average electron thermal speed of (\(\overline{v}_{e}\)) and an average electron kinetic energy (\(\overline{n}_{e}\)). The resulting electron-neutral momentum transfer collision cross section is designated as the simplified value (\(\overline{Q}\)), which is the approximated temperature-dependent alternative to the monoenergetic collision cross section. This simplified value of the collision cross section may be referred to as the \"average\", \"mean\", or \"constant\", with the meaning that it no longer depends on the distribution of the electron thermal speed and is not computed for a particular electron energy. Instead, it is interpreted as a constant approximate value of all the electron speeds and electron energies, which depends on the electron temperature as the sole independent variable. Thus,  
\[\overline{Q}(T_{e})=\frac{\psi(\overline{u}_{e})}{\overline{v}_{e}} \tag{12}\]  
In order to select proper values for the average electron thermal speed of (\(\overline{v}_{e}\)), and a proper average electron kinetic energy (\(\overline{u}_{e}\)), the Maxwell-Boltzmann probability distribution function for electron thermal speeds is employed. In this distribution for the thermal speed of electrons, the electron temperature (\(T_{e}\)) is the only adjustable parameter, as follows [38; 39]:  
\[f(v_{e};T_{e})=4\pi v_{e}^{2}\bigg{(}\frac{m_{e}}{2\pi k_{B}T_{e}}\bigg{)}^{1. 5}\exp\bigg{(}-\frac{m_{e}v_{e}^{2}}{2k_{B}T_{e}}\bigg{)} \tag{13}\]  
The integral of \(f(v_{e};T_{e})\) between two speed values (\(v_{e1}\)) and (\(v_{e2}\)), or \(\int_{v_{e1}}^{v_{e2}}f(v_{e};T_{e})\ dv_{e}\), is the fraction of electrons with thermal speeds between (\(v_{e1}\)) and (\(v_{e2}\)), when the electrons have an absolute temperature of (\(T_{e}\)).