319
The contact made by processes of dissolution
and precipitation with the solid phase of calcite is
described by the reaction:
CaCO 3,calcite ⇔ Ca 2+ + CO 3
2(9.9)
The equilibrium is described by the equation:
K calcite = [Ca 2+ ] · [CO 3
2- ] = 3.31E-9
(25 ° C, 1 atm)
9.10
Another essential condition for describing the
calcite-carbonate-equilibrium consists in the fulfillment of the neutral charge rule. This holds that
charges may be neither lost nor gained on reaching the equilibrium state.
The balance of the charges is done indirectly
on the basis of the ions that are the carriers of
these charges. Therefore, calculations must consider divalent ions twice:
2 (Ca 2+ ) + (H + ) = (OH - ) + (HCO 3
- ) + 2 (CO 3
2- )
(9.11)
It should be noted that the parentheses (...) in
this charge balance represent molar concentrations, since the chemical activities are not of importance here, instead the really existent amounts
of the substances and their charges. Equation 9.11
describes in a simplified form the conditions in
which the equilibrium state is reached starting
from pure water. Only in this case all final concentrations produce charges equal to zero in the
balance.
In an impure system, any initial concentration
(...) i or initial activity [...] i may be present. In this
context, the index 'i' stands for 'initial'. On the basis of these concentrations, those reached on attaining the final equilibrium state (...) f or [...] f will
then have to be sought for. The index 'f' in this
case stands for 'final'. The statement as to the
charge balance made in Equation 9.11 is not accomplished by observing the total concentration
of each reactant, but by the difference between
'initial' to 'final'. The generalized case described in
Equation 9.12 resumes the condition of Equation
9.11 in which all initial concentrations are set to
zero (pure water).
2 (Ca 2+ ) f – 2 (Ca 2+ ) i + (H + ) f – (H + ) i =
(OH - ) f – (OH - ) i + (HCO 3
- ) f – (HCO 3
- ) i +
2 (CO 3
2- ) f – 2 (CO 3
2- ) i
(9.12)
In a system following this description, the solution for all six variables (Ca
2+
), (H
+
), (OH
-
),
(H 2 CO 3
0
), (HCO 3
-
), (CO 3
2) is sought for the condition of a final equilibrium state determined by the
Equations 9.2, 9.4, 9.6, 9.8, 9.10, and 9.12. It should
be noted that in Equations 9.2, 9.4, 9.6, 9.8, and
9.10 the activities written in brackets [...] are to be
understood as final activities [...] f , which are products of an activity coefficient and the single concentrations in equilibrium (...) f (cf. Eq. 15.2 in Sect.
15.1.1). For a system at infinite dilution, however,
these activity coefficients are 1.0, and thus concentrations are (...) equal to activities [...].
The system of six non-linear equations with six
unknown variables can be solved mathematically
for (H
+
) f by inserting the equations into each
other until only constants, initial concentrations
(...) i , and pCO 2 describe the complete equation.
This way of solving the problem leads to a cubic
equation for which an analytical solution exists. In
practice, however, this strategy is irrelevant, since
such a set of equations cannot be analytically
solved for other boundary conditions (e.g. a system which is closed with respect to CO 2 , having
no contact with the atmosphere) or if more constituents are included (e.g. complex species like
CaHCO 3
+
aq
or CaCO 3
0
aq
). In such a case, it must be
solved by an iteration process, as will be described in the succeeding section. The application
of spreadsheet calculation programs make such an
iterative solution easy to find by pursuing the
procedures below:
First, the concentration of (H 2 CO 3
0
) f is calculated on the basis of the given pCO 2 using Equation 9.2
Then, the pH-value of the equilibrium is estimated. For a first approximation, the pH-value derived from the 'initial' concentration of (H
+
) i may be
used. But any other pH-value, for instance pH 7,
is permitted as well.
With this pH-value, the concentrations of (H
+
) f
and (OH
-
) f are calculated using Equation 9.8.
Successively, (HCO 3
-
) f and (CO 3
-
) f are calculated using the Equations 9.4 and 9.6 and the values so far determined. The concentration of (Ca
2+
) f
is obtained from Equation 9.12.
Now the saturation index (SI calcite ) can be calculated with Equation 9.10 (cf. Eq. 15.1 in Sect.
15.1.1). If the saturation index equals zero, then
the estimation of the pH-value in equilibrium, as
mentioned in step 2, has been correct, as well as
all other equilibrium concentrations (...) f calculated
in steps 3 and 4. If the saturation index (SI calcite ) is
9.3
The Calcite-Carbonate-Equilibrium in Marine Aquatic Systems
The contact made by processes of dissolution
and precipitation with the solid phase of calcite is
described by the reaction:
CaCO 3,calcite ⇔ Ca 2+ + CO 3
2(9.9)
The equilibrium is described by the equation:
K calcite = [Ca 2+ ] · [CO 3
2- ] = 3.31E-9
(25 ° C, 1 atm)
9.10
Another essential condition for describing the
calcite-carbonate-equilibrium consists in the fulfillment of the neutral charge rule. This holds that
charges may be neither lost nor gained on reaching the equilibrium state.
The balance of the charges is done indirectly
on the basis of the ions that are the carriers of
these charges. Therefore, calculations must consider divalent ions twice:
2 (Ca 2+ ) + (H + ) = (OH - ) + (HCO 3
- ) + 2 (CO 3
2- )
(9.11)
It should be noted that the parentheses (...) in
this charge balance represent molar concentrations, since the chemical activities are not of importance here, instead the really existent amounts
of the substances and their charges. Equation 9.11
describes in a simplified form the conditions in
which the equilibrium state is reached starting
from pure water. Only in this case all final concentrations produce charges equal to zero in the
balance.
In an impure system, any initial concentration
(...) i or initial activity [...] i may be present. In this
context, the index 'i' stands for 'initial'. On the basis of these concentrations, those reached on attaining the final equilibrium state (...) f or [...] f will
then have to be sought for. The index 'f' in this
case stands for 'final'. The statement as to the
charge balance made in Equation 9.11 is not accomplished by observing the total concentration
of each reactant, but by the difference between
'initial' to 'final'. The generalized case described in
Equation 9.12 resumes the condition of Equation
9.11 in which all initial concentrations are set to
zero (pure water).
2 (Ca 2+ ) f – 2 (Ca 2+ ) i + (H + ) f – (H + ) i =
(OH - ) f – (OH - ) i + (HCO 3
- ) f – (HCO 3
- ) i +
2 (CO 3
2- ) f – 2 (CO 3
2- ) i
(9.12)
In a system following this description, the solution for all six variables (Ca
2+
), (H
+
), (OH
-
),
(H 2 CO 3
0
), (HCO 3
-
), (CO 3
2) is sought for the condition of a final equilibrium state determined by the
Equations 9.2, 9.4, 9.6, 9.8, 9.10, and 9.12. It should
be noted that in Equations 9.2, 9.4, 9.6, 9.8, and
9.10 the activities written in brackets [...] are to be
understood as final activities [...] f , which are products of an activity coefficient and the single concentrations in equilibrium (...) f (cf. Eq. 15.2 in Sect.
15.1.1). For a system at infinite dilution, however,
these activity coefficients are 1.0, and thus concentrations are (...) equal to activities [...].
The system of six non-linear equations with six
unknown variables can be solved mathematically
for (H
+
) f by inserting the equations into each
other until only constants, initial concentrations
(...) i , and pCO 2 describe the complete equation.
This way of solving the problem leads to a cubic
equation for which an analytical solution exists. In
practice, however, this strategy is irrelevant, since
such a set of equations cannot be analytically
solved for other boundary conditions (e.g. a system which is closed with respect to CO 2 , having
no contact with the atmosphere) or if more constituents are included (e.g. complex species like
CaHCO 3
+
aq
or CaCO 3
0
aq
). In such a case, it must be
solved by an iteration process, as will be described in the succeeding section. The application
of spreadsheet calculation programs make such an
iterative solution easy to find by pursuing the
procedures below:
First, the concentration of (H 2 CO 3
0
) f is calculated on the basis of the given pCO 2 using Equation 9.2
Then, the pH-value of the equilibrium is estimated. For a first approximation, the pH-value derived from the 'initial' concentration of (H
+
) i may be
used. But any other pH-value, for instance pH 7,
is permitted as well.
With this pH-value, the concentrations of (H
+
) f
and (OH
-
) f are calculated using Equation 9.8.
Successively, (HCO 3
-
) f and (CO 3
-
) f are calculated using the Equations 9.4 and 9.6 and the values so far determined. The concentration of (Ca
2+
) f
is obtained from Equation 9.12.
Now the saturation index (SI calcite ) can be calculated with Equation 9.10 (cf. Eq. 15.1 in Sect.
15.1.1). If the saturation index equals zero, then
the estimation of the pH-value in equilibrium, as
mentioned in step 2, has been correct, as well as
all other equilibrium concentrations (...) f calculated
in steps 3 and 4. If the saturation index (SI calcite ) is
9.3
The Calcite-Carbonate-Equilibrium in Marine Aquatic Systems