The steady state and equilibrium approximations: A geometrical picture

Abstract

We present a geometrical picture of the steady state (SSA) and equilibrium (EA) approximations for simple chemical reactions. Geometrically, SSA and EA correspond to surfaces in the phase space of species concentrations. In this space, chemical reaction is represented by trajectory motion governed by first order differential equations. Initially, during transient decay, trajectories move quickly into the (narrow) region ℛ between the equilibrium (ℰ) and steady state (𝒮) surfaces, puncturing ℰ or 𝒮 orthogonal to a variable axis. This is rigorously what the differential equations say. Once in ℛ, trajectories approach another surface, the slow manifold ℳ, sandwiched between ℰ and 𝒮. SSA or EA is the restriction of the system motion to 𝒮 or ℰ as a ‘‘shadow’’ of its true, nearby motion along ℳ. We illustrate these ideas in relation to the Lindemann mechanism, showing how ℳ is related to 𝒮 and ℰ, and how it may be obtained by iteration, yielding higher order SSA and EA formulas. This procedure is asymptotic with an interesting error analysis. We briefly discuss the generalization of these ideas.