The Steady-State and Equilibrium Approximations ©
Background Reading
by
Flick Coleman
Department of Chemistry
Wellesley College
Wellesley, MA 02181
wcoleman@wellesley.edu
© Copyright Flick Coleman 1996. All rights reserved. You are welcome to use this document in your own
classes but commercial use is not allowed without the permission of the author.
When dealing with complex reaction mechanisms, you will frequently encounter situations where you
cannot easily solve the differential equations for the various changes in concentration with time, if indeed
you can solve them at all. There are two commonly used approximation methods which make the
mathematics of these systems somewhat more tractable. These are known as the steady-state and the
equilibrium approximations. The two methods take different approaches to approximating the
concentrations of various species in the reaction mechanism. Often, they lead to very similar
predictions, but that is not always the case.
In the steady-state approximation, the focus is on reaction intermediates, which are formed and
disappear during the course of the reaction. Consider the reaction A --> B --> C. The rate constants
for the two steps are given by k1 and k2, respectively. The equations for the change in the
concentrations of A, B and C can easily be written by referring back to the mechanism. The reactant,
A, disappears with rate k1*A, so that equation is:
d
A
dt
k1. A
The intermediate species B is formed with rate k1*A and disappears with rate k2*B. The
appropriate equation for the change in B with time is therefore:
d
B k1. A
dt
k2. B
In the steady-state approximation, B is assumed to reach a constant value, the so-called
steady-state value. Therefore, the change in B with time is set to zero. When that is done, the
result is:
k1. A k2. B
Creation Date: Fall 1995
Modified:
or
B
k1 .
A
k2
Flick22b.mcd
Author: Flick Coleman
Page 1
This result, when substituted into the expression for the rate of formation of C, gives:
k1 .
d
C k2. B k2.
A k1. A
k2
dt
One advantage of this treatment is that we now have an expression for the change of C
with time, which does not involve the intermediate B. Therefore, we can express the rate
of formation of the product in terms of A, whose concentration will most likely be easier to
measure than that of B. Consider some further implications of this approach. The
expression for B implies that B is directly proportional to A, and that at the start of the
reaction, B is some fraction (k1/k2) of A, when, in fact, we know that B is zero at the
start of the reaction. The expression for C, implies that C is formed at the same rate that
A disappears. This result is frequently interpreted by saying that the steady-state
approximation is most valid when applied to a species which is formed rapidly, disappears
slowly, and never reaches a very high concentration throughout the course of the reaction.
The latter of these criteria has recently been challenged (Gellene, G. L. , J. Chem. Educ.
1995, 72, 196).
Creation Date: Fall 1995
Modified:
Flick22b.mcd
Author: Flick Coleman
Page 2