Abstract. This paper presents the mathematical proof of the relationship between kinetic energy and temperature at microscopic scales, offering new insights into the fundamental principles governing atomic and molecular behavior. Central to this study is the introduction of a new constant, derived from fundamental natural constants, which serves as a crucial link between energy and temperature. A notable contribution of this work is the derivation of a relationship connecting the kinetic energy of an atom to that of an electron bound to its nucleus. Using this relationship and established equations describing electron behavior, new equations are derived that govern atomic state-dependent properties previously undefined or poorly understood. These equations not only deepen the understanding of atomic behavior but also provide a foundation for deriving state-dependent properties of molecules, broadening the scope beyond individual atoms. The goal of this research was to develop a universal equation that enhances the predictability of the energy-temperature relationship and the properties of materials in chemistry and materials science. For example, by knowing the bond-dissociation energies of various compounds, one can predict the temperature at which covalent bonds will dissociate and vice versa. Additionally, the equations can be used to determine ionization temperatures for various atoms at different stages of ionization, offering more precise insights into ionization processes. This advancement in theoretical modeling improves predictive capabilities in areas such as reaction kinetics, material science, and plasma physics. The work also paves the way for the design of materials with specific thermal properties, such as those with tunable ionization thresholds. Ultimately, the derived framework provides a unified mathematical model linking atomic and molecular behavior with thermal properties, with wide-ranging implications for both fundamental research and practical applications in materials science and chemistry.

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