Calculating Equilibrium Concentrations by Iteration Recycle Your Approximations E. Weltin
University of Vermont, Burlington, VT 05405
The dissociation of a weak acid HA is a classical application of an equilibrium calculation that is discussed in every chemistry textbook. With given dissociation constant Ka, total acid, [HA]
+
[A’]
=
c0
For
a
oA + bB +___= sS + fT
=
[AT
=
x
=-
ll)
or =
VXtt(c0
-
x)
(2)
If the degree of dissociation is small the solution of eq 2 is approximately
*-^T
approximation was justified (1-3). Of course eq 1 is equivalent to a quadratic equation that easily can be solved exactly for the one positive root. If we hate solving equations but do not mind the simpler problem of evaluating a function at a given argument with a scientific calculator, we can recycle the answer of eq 3 into the right-hand side of eq 2. If we are still not satisfied, we can recycle the next approximation, that is, iterate =
'(Ka(c0-xi)
(4)
The solution of eq 2 is a fixed point of the iteration; if the solution is substituted into the right side of eq 4, it is simply reproduced. This trick is well-known to experienced chemists but mentioned in only few textbooks (4). What is less well-known is that it can be easily generalized to any process described by a single chemical equation with a small equilibrium constant. Based on general convergence properties, it has the added great advantage of quality control; at each stage it furnishes an estimate of the maximum error. Systems with large equilibrium constants can be treated by exchanging reactants with products and replacing the equilibrium constant by its inverse. The method thus provides a convenient way to calculate equilibrium concentrations for a wide range of chemical problems. We have shown previously that the equilibrium concentrations for a given system are uniquely determined by the initial concentrations of all species (5). The condition that concentrations must be nonnegative limits the extent of reaction to a finite range xiow to Xhigh- Within this range all expressions associated with the equilibrium calculations show a simple systematic behavior, and any complications outside this range are of no chemical concern. 'The starting concentrations are obtained by assuming that the action goes to completion from right to left.
Journal of Chemical Education
Keq
...
-u0/u, ...)
Co-
x
uU4-
and
the equilibrium equation is A
m
=
Xjow
LH+]
+
with initial concentrations a0, b0, etc., the limits of the extent of reaction are given by the following two equations (5, 6).
and
36
general process
re-
If m is not already zero, it is convenient to make a change of variable from x to y where y
=
x
m
—
such thaty varies from 0 toymax M m. Define the starting concentrations1 at y 0 as a' a0 am, etc., for reactants and s' s0 + sm, etc., for products. Clearly there is at least one, possibly more species S, T,... with starting con=
-
=
-
-
=
centrations s', f,... equal to zero. There may or may not be additional products U... with starting concentrations ur... greater than zero. The following equilibrium table is obtained. species
A
B
S
T
U
starting
a'
tf
0
0
if
change
-ay
-by
+sy
+fy
+uy
final
(s'-ay)
(V
sy
ty
( ye. Forye
