Environmental Engineering Reference
In-Depth Information
a
i
denotes the activity of the
i
th species, which is the product of concentration and a
species-dependent activity coefficient
g
i
:
a
i
¼ g
i
c
i
(8.10)
For more details concerning activities see Sidebar 8.3 below. For low
concentrations, the activity coefficients are close to one and the activities in the
law of mass action can be replaced by concentrations. Taking the logarithm
2
of
(
8.9
) one obtains:
X
N
s
i¼
1
ðm
ðj
i
n
ðj
i
Þ
log
aðÞ¼
log
ðK
j
Þ j ¼
1
; :::N
r
(8.11)
It is convenient to write (
8.11
) in matrix form. In order to do that, the convention
is used that stoichiometric coefficients related to reactants obtain a positive expo-
nent and those related to reaction products obtain a negative exponent:
n
ji
¼ m
ðj
i
n
ðjÞ
(8.12)
i
¼
(
n
ji
) results:
With the help of matrix S
0
1
0
1
a
1
:::
a
N
s
K
1
:::
K
N
r
@
A
and k
@
A
S
log
ð
a
Þ¼
log
ð
k
Þ
with a
¼
¼
(8.13)
S has
N
r
rows and
N
s
columns. Each line in (
8.13
) represents one reaction. With
respect to the calculation of
N
s
unknown activities, (
8.13
) provides
N
r
conditions.
For low ionic strength, the activity coefficients are approximately one, and it is
allowed to replace activities by concentrations:
S
log
ð
c
Þ¼
log
ð
k
Þ
(8.14)
Usually it is required that there is no reaction that can be obtained as combina-
tion of the other reactions. The mathematical expression for that requirement is
that the rank
3
of matrix S is maximal. The maximal rank of the
N
r
N
s
reaction
matrices is
N
r
. With MATLAB
it is easy to check if the condition is fulfilled, by
®
using the
rank
command.
2
In this chapter, we deal with logarithms to the basis 10; in the text we do not introduce a different
notation to distinguish from the logarithm to the basis e, which is used in other chapters. However,
in MATLAB
the
log10
command has to be taken!
3
The rank of a matrix is the maximum number of linear independent column or row vectors in the
matrix; see textbooks on Linear Algebra.
®
