## 3 Results
### Temperature Profiles of Average Collision Cross Sections
In the case of H\({}_{2}\)O, the nonlinearity involved in methods 1 and 4 is at its strongest level among all six species covered here. The magnitude of the negative slope of \(\overline{Q}_{\text{H2O}}(T_{e})\), or \(d\overline{Q}_{\text{H2O}}/dT_{e}\), becomes smaller as \((T_{e})\) increases, and the curvature of the respective profile curves can be visually noticed. The slope (the first derivative) of \(\overline{Q}_{\text{H2O}}(T_{e})\) with respect to \(T_{e}\) in method 1 (\(\overline{u}_{e}=1.5\overline{T}_{e}\)) is  
\[\left.\frac{d\overline{Q}_{\text{H2O}}}{dT}\right|_{\overline{u}_{e}=1.5 \overline{T}_{e}}\left[\frac{\text{\AA}^{2}}{\text{K}}\right]=-\frac{141,579.62 }{T_{e}\left[\text{K}\right]^{2}} \tag{39}\]  
For \(T_{e}\) = 2000 K, the above slope is \(-0.035395\)\(\text{\AA}^{2}/\text{K}\), which becomes \(-0.015731\)\(\text{\AA}^{2}/\text{K}\) at 3000 K. Thus, the magnitude of the slope drops at 3000 K to 44.44% of its value at 2000 K.  
The slope of \(\overline{Q}_{\text{H2O}}(T_{e})\) with respect to \(T_{e}\) in method 4 (quadratic) is  
\[\left.\frac{d\overline{Q}_{\text{H2O}}}{dT}\right|_{\text{Quadratic}}\left[ \frac{\text{\AA}^{2}}{\text{K}}\right]=-0.1842+5.066\times 10^{-5}T_{e}[\text{K}] \tag{40}\]  
For \(T_{e}\) = 2000 K, the above slope is \(-0.082880\)\(\text{\AA}^{2}/\text{K}\), which becomes \(-0.032220\)\(\text{\AA}^{2}/\text{K}\) at 3000 K. Thus, the magnitude of the slope drops at 3000 K to 38.88% of its value at 2000 K.  
Figure 11 is used to contrast the four predicted curves of the average electron-neutral collision cross sections for potassium vapor (K). Three of the four methods agree about the decrease in \(\overline{Q}_{\text{K}}\) as \(T_{e}\) increases (similar behavior to \(\overline{Q}_{\text{H2O}}\)). The outlier is method 2 (linear), which estimates that \(\overline{Q}_{\text{K}}\) is simply a constant value of 250 \(\text{\AA}^{2}\), independent of the  
Figure 10: Temperature dependence for the average electron–neutral collision cross sections of water vapor (H\({}_{2}\)O), as obtained using four different methods. electron temperature. For the three methods predicting a decrease in \(\overline{Q}_{\rm K}\) as \(T_{e}\) increases, there is a large gap between each pair of them, with method 1 (\(\overline{u}_{e}~{}=~{}1.5\overline{T}_{e}\)) giving the largest values of \(\overline{Q}_{\rm K}\), and method 3 (integration) gives the smallest values. At \(T_{e}\) = 2000 K, the computed \(\overline{Q}_{\rm K}\) using method 1 is 575.888 A\({}^{2}\), while method 3 gives 229 A\({}^{2}\); thus, the ratio is 2.5148:1. At \(T_{e}\) = 3000 K, the computed \(\overline{Q}_{\rm K}\) using method 1 is 470.210 A\({}^{2}\), while method 3 gives 161 A\({}^{2}\); thus, the ratio is 2.9206:1. Based on the arithmetic mean profile of the four individual profiles (including the zero-degree polynomial of method 2), the mean rate of change in \(\overline{Q}_{\rm K}\) is \(-0.0756318\) A\({}^{2}\)/K. The values of \(\overline{Q}_{\rm K}\) are the largest among the six species covered in the present study, being much larger than the values of \(\overline{Q}_{\rm H2O}\), which are themselves much larger than the values of \(\overline{Q}_{\rm CO}\), \(\overline{Q}_{\rm CO2}\), and \(\overline{Q}_{\rm H2}\). As in the cases of CO\({}_{2}\) and H\({}_{2}\), the nonlinearity of \(\overline{Q}_{\rm K}(T_{e})\) in methods 1 and 4 is small. Figure 12 is used to contrast the four predicted curves of the average electron-neutral collision cross sections for molecular oxygen (O\({}_{2}\)). Methods 1 (\(\overline{u}_{e}=1.5\overline{T}_{e}\)) and 2 (linear) predict a constant value of 5.0326598 A\({}^{2}\) and 3.9 A\({}^{2}\), respectively. Methods 3 (integration) and 4 (quadratic) predict that \(\overline{Q}_{\rm CO2}\) increases as \(T_{e}\) increases (the opposite behavior to \(\overline{Q}_{\rm K}\) for the three methods that showed a temperature-dependent \(\overline{Q}_{\rm K}\)).