Environmental Engineering Reference
In-Depth Information
@f =@c
is to be computed in order to apply Newton's
method. For function (
8.7
) the derivative is available, although a bit lengthy to write
down. In the following sub-chapter we present a general procedure, which can be
applied to general systems of multi-species interacting in several reactions.
Note that the derivative
8.2 The Law of Mass Action for Equilibrium Reactions
The most common unit for concentrations of components in the fluid phase is
[component mass / fluid volume]; in SI units [kg/m
3
]. Another way describing
concentrations is by using mass fractions [component mass / fluid mass], which in
physical units are dimensionless. In chemistry it is common to use molar or molal
concentrations, where the number of moles per fluid volume or fluid mass is
measured.
From the mathematical point of view a reaction is regarded as the transition from
one set of species to another. In the following, the total number of species is denoted
by
N
s
, and the number of reactions by
N
r
. In a reaction a subset of species has
the role of reactants and another set gathers the reaction products. Formally one
may write:
X
X
N
s
N
s
n
ðj
i
component
i
!
m
ðj
i
component
i
j ¼
; :::N
r
1
(8.8)
i¼
1
i¼
1
where the coefficients
n
ðjÞ
i
and
m
ðj
i
are stoichiometric integer numbers. Reversible
reactions are characterized by the observation that both the reaction and the back
reaction occur. If the reaction is fast in comparison to the other relevant processes
(as transport for example), the equilibrium between the reactions in both directions
is reached at every time instance. This can be assumed to hold for many geochemi-
cal processes, because transport and compaction in the pedosphere are usually
slow. According to the
Law of Mass Action
the equilibrium is characterized by
a constant
K
j
:
N
s
i
¼
1
a
i
m
ðjÞ
Q
i
K
j
:¼
(8.9)
Q
N
s
a
i
n
ðjÞ
i
i¼
1
Equation (
8.9
) is valid for fast reversible reactions. The
K
j
'
s are independent
from the geochemical surrounding but depend on temperature and pressure.
