Superelastic Heating in Treanor-Gordiets Plasmas: A Unified Analytic Closure

In thermally non-equilibrium plasmas, conventional harmonic models can significantly mispredict superelastic electron heating rates. When the vibrational temperature exceeds the gas temperature ($T_{\rm v}>T_{\rm g}$), these models underestimate energy transfer by several times; conversely, they overestimate heating when $T_{\rm g}>T_{\rm v}$. We show that this discrepancy arises from neglecting the exponential heating from overpopulated, high-lying states in anharmonic Treanor-Gordiets distributions, and their thermodynamic depopulation at high gas temperatures. To resolve this, we derive a closed-form, thermodynamically consistent macroscopic closure based on detailed balance and a second-order Dunham expansion. This unified framework introduces an analytic anharmonic correction factor that captures the kinetic competition between vibrational-vibrational (V-V) up-pumping and vibrational-translational (V-T) relaxation. By predicting the Treanor minimum, this formulation recovers the fidelity of full state-to-state kinetic benchmarks. Ultimately, this model provides a governing equation for heat exchange between electrons and excited states in non-equilibrium environments -- including plasma-assisted combustion and hypersonic flows -- enabling the development of accurate, rate-limited reduced-order models for macroscopic fluid solvers.