Explicit and asymptotic solutions of simultaneous 1st-order and Riccati equations for a gas reaction system

Abstract

Systems of simultaneous or parallel chemical reactions of the type A → B → C → Other products are often treated as first order or pseudo-first order. For a system of simultaneous first and second order reactions — dB/dt = kABA − kBCB2 and dC/dt = kBCB2 - kCC, where A, B, and C are concentrations, t is time, and the reaction rate parameters kAB and kC in yr−1 are 1st-order and kBC in cm3 molecule−1 yr−1 is 2nd-order — no explicit solution is available, as far as we are aware. This paper presents explicit and asymptotic solutions of simultaneous 1st- and 2nd order Riccati equations and applies them to a simplified sequence of gas reactions in the atmosphere of Titan, the largest satellite of Saturn: CH4 methane (1st order, k12) → CH3 methyl (2nd order, k23) → C2H6 ethane (1st order, k3) → Other products. The Titan’s atmosphere contains methane (CH4) at the present-day partial pressure of 0.1 bar, out of a total atmospheric pressure made up by nitrogen (N2) of 1.5 bar, comparable to Earth’s. Methyl CH3 and ethane C2H6 are minor components. On Titan, methyl (CH3) is an intermediate product from methane to ethane, the latter raining out as liquid on Titan’s surface. The main points of this paper are: (1) the asymptotic solutions that approximate near-steady state of Titan’s atmosphere about 4.5 billion years after its accretion; (2) the computed present-day concentrations of the three gases in Titan’s scale atmosphere (i.e., scale atmosphere is a model of an isothermal well mixed reservoir); and (3) the agreement between Titan’s reported and computed atmospheric concentrations of CH4, CH3, and C2H6. The reaction rate parameters of the species are constants representative of their mean values during the satellite’s cooling history. The present-day concentrations of methyl (CH3) and ethane (C2H6) are several orders of magnitude lower than the concentration of methane (CH4). Since Titan’s accretion about 4.5 billion years B.P., steady-state concentrations of methane and ethane would have been attained in less than 500,000 years. This conclusion and the agreement of the model results with the reported concentrations underscores validity of the asymptotic solutions derived from the explicit solutions, and the long-term stability of the carbon cycle in Titan’s atmosphere.