Atmospheric Chemistry for Astrophysicists: A Self-consistent Formalism and Analytical Solutions for Arbitrary C/O

TL;DR

This paper develops a self-consistent formalism for exoplanet atmospheric chemistry, deriving analytical models that explain how chemical mixing ratios depend on the C/O ratio, temperature, and pressure, with implications for astrophysical observations.

Contribution

It introduces a unified physical and mathematical framework connecting thermodynamics, equilibrium constants, and chemical kinetics for exoplanet atmospheres, including analytical solutions for C-H-O systems.

Findings

Derived a general form of the Arrhenius equation.

Reproduced key trends in chemical abundances versus C/O ratio.

Showed that mixing ratio variations are mainly due to stoichiometry.

Abstract

We present a self-consistent formalism for computing and understanding the atmospheric chemistry of exoplanets from the viewpoint of an astrophysicist. Starting from the first law of thermodynamics, we demonstrate that the van't Hoff equation (which describes the equilibrium constant), Arrhenius equation (which describes the rate coefficients) and procedures associated with the Gibbs free energy (minimisation, rescaling) have a common physical and mathematical origin. We address an ambiguity associated with the equilibrium constant, which is used to relate the forward and reverse rate coefficients, and restate its two definitions. By necessity, one of the equilibrium constants must be dimensionless and equate to an exponential function involving the Gibbs free energy, while the other is a ratio of rate coefficients and must therefore possess physical units. We demonstrate that the Arrhenius equation takes on a functional form that is more general than previously stated without recourse to tagging on ad hoc functional forms. Finally, we derive analytical models of chemical systems, in equilibrium, with carbon, hydrogen and oxygen. We include acetylene and are able to reproduce several key trends, versus temperature and carbon-to-oxygen ratio, published in the literature. The rich variety of behavior that mixing ratios exhibit as a function of the carbon-to-oxygen ratio is merely the outcome of stoichiometric book-keeping and not the direct consequence of temperature or pressure variations.