A Superior but Equally Convenient Alternative to the Steady-State Approximation and Secular Equilibrium

Abstract

The steady-state approximation (hereafter abbreviated as SSA) consists in setting $dy/dt=0$, where $y$ denotes the concentration of a short-lived intermediate subject to first-order decay with a rate constant $k$. The sole reason for enforcing SSA is to convert the rate equation for $y$ into an algebraic equation. The conditions under which SSA becomes trustworthy are now well understood, but a firm grasp of the physical content of the approximation requires more maturity than few teachers, let alone their students, may be expected to possess. Furthermore, there is no simple way to gauge the accuracy of the results derived by imposing SSA. The purpose of this note is to demonstrate that a better, but equally simple, approximation results if, instead of setting $dy/dt$ to zero, one substitutes $y(t+\tau)$ for $y+\tau dy/dt$, where $\tau=1/k$; SSA is a cruder approximation because it neglects the second term. For systems modelled as damped harmonic oscillators, the "reverse Taylor approximation" can be extended by retaining one more term in the Taylor expansion. The utility of the approximation (or its extension) is demonstrated by examining the following systems: radioactive equilibria, Brownian motion, dynamic response of linear first- and second-order systems.