The peculiarities of the Rayleigh problem (RP) governing equations of a rarefied gaseous plasma (RGP) are analyzed and proven to conform to the entropic performance using the moment method, separation of variables, associated with traveling-wave techniques. Maxwell's and Boltzmann equation (BE) were solved. The BE considerable advantage is that it allows us to analyze the depth performance of the equilibrium electrons' velocity distribution function (EVDF) and the perturbed EVDF and their implementation to determine how far the system is from the equilibrium state (ES). As a result, the contrast between the equilibrium EVDF and the perturbed EVDF was conceptually elucidated at various periods. For this purpose, the derived EVDF should be employed in the entropy, its production, and others thermodynamic important variables. After analyzing the results, we discovered that H-theorem, thermodynamic principles, and Le Chatelier's law were all consistent with our model. The Gibbs rule was used to express how the various influences of the forces acting on the system's internal energy modification (IEM) are expressed. The findings showed that the proposed model could accurately capture the performance. The suggested type could accurately predict RGP helium and argon gases performance in the upper atmosphere's ionized belts. 3D-Graphics representing the physical parameters were generated using analysis of variance calculations, and the results are thoroughly presented. The importance of this research stemmed from its broad array of utilization in micro-electro-mechanical systems (MEMS), physics, electrical engineering, and nano-electro-mechanical systems (NEMS) technologies in a variety of commercial and industrial utilization.