Atmospheric Chemistry for Astrophysicists: A Self-consistent Formalism and Analytical Solutions for Arbitrary C/O
Heng et al
We present a self-consistent formalism for computing and understanding the atmospheric chemistry of exoplanets. 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 correct an ambiguity associated with the equilibrium constant, which is used to relate the forward and reverse rate coefficients, and rigorously derive 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. To avoid confusion, we simply term them the dimensionless and dimensional equilibrium constants. We demonstrate that the Arrhenius equation takes on a functional form that is more general than previously thought without recourse to tagging on ad hoc functional forms. Our formulation of the evolution equations for chemical kinetics correctly enforces the book-keeping of elemental abundances, reproduces chemical equilibrium in the steady-state limit and is able to explain why photochemistry is an intrinsically disequilibrium effect. Finally, we derive analytical models of chemical systems with only hydrogen and with carbon, hydrogen and oxygen. For the latter, 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.