Bicarbonate
and carbonate
model of seawater at 25°C
ion-pairs
R. M. Pytkowicx
and J. E. Hawley
School of Oceanography,
Oregon
State University,
and a
Corvallis
97331
Abstract
The apparent (stoichiometric)
association constants for the formation
of ion-pairs between bicarbonate
and carbonate and the major cations of seawater at 25°C were dctcrmined experimentally.
The results wcrc used, in conjunction
with earlier data on sulfate
of the major species present in scawatcr.
ion-pairs, to calculate the concentrations
Here we measured the apparent (stoichiometric)
association constants for the
formation
of bicarbonate
and carbonate
ion-pairs with sodium, calcium, and magnesium ions at 25°C and 0.72 ionic strength.
The results were used, in conjunction with
the sulfate association constants determined
by Kester and Pytkowicz ( 1969)) to detcrmine the major ionic species present in seawater at 25°C and 34.8z0 salinity. We will
show later that the proportions of ion-paired
and free species are insensitive to the
salinity.
Activity coefficients are lower and solubility products higher in some mixed elcc.trolyte solutions such as seawater than in
chloride solutions ( Kester and Pytkowicz
1969). This is usually attributed to ionic
interactions, which can range from the formation of ion-pairs (outer sphere interactions due to electrostatic attraction) to that
of true complexes (inner sphere bonds with
loss of water of hydration).
Partial molal
volume data (Kestcr 1969) and results from
Raman spectra (Daly et al. 1972) suggest
that the major ions of seawater may form
although
the potentiometric
ion-pairs,
method we used does not distinguish between ion-pairs and complexes.
Ionic interactions are important because,
by affecting the charge distribution
of
solutes, they influence properties such as
electrical conductivity,
osmotic pressure,
solubility of minerals, and sound attenua‘This work was supported by National Science
Foundation
Grants GA-17011 and (X-36057X
and
by Office of Naval Research Contract N00014-07A-0369-0007.
LIMNOLOGY
AND
OCEANOGRAPHY
tion. The effect of interactions on activity
coefficients can be successfully accounted
by two approaches; the specific interaction
model of Br@nstcd ( 1922) and Guggenheim ( 1935)) in which interaction terms
of an unspecified nature arc measured in
bi.nary solutions and are applied to seawater ( Leyendclkers
1972; Robinson and
Wood 1972; Whitfield 1973) and the ionpair concept of Bjerrum, which was introduced to oceanography by Carrels and
Thompson (1962), and in which association constants are determined and applied
to seawater. The two methods differ in
their algebra but have much in common bccause they both require measurements in
simple solutions and their application to
seawater by the use of an extended ionic
strength principle, namely, that interaction
coefficients and association constants are
independent of the medium composition at
a given ionic strength. The validity of this
principle was tested successfully (Pytkowicz and Kcster 1969; Kcster and Pytkowicz 1969; ILeyendekkers 1972; Robinson
and Wood 1972; Whitfield
1973) because
species distributions
and activity coefficients calculated from measurements in
binary solutions agreed with results in seawater, and because apparent association
constants were found to be independent of
composition at a given ionic strength. Both
methods are compatible with Harned’s rule
(Pytkowicz and Kester 1969) and lead to
successful predictions
of activity
coefficients (Leyendekkers 1972; Whitfield 1973).
It should be emphasized that, although
apparent association constants can be mea-
223
MARCH
1974,
V.
19(2)
224
Pytkowicx
sured in simple binary solutions and applied to seawater, resulting total activity
coefficients and species distributions
are
functions of the medium composition. Theoretical grounds for the invariance of stoichiometric constants and the concepts of
free versus total activity coefficients were
discussed by Pytkowicz
(1969) and by
Kester ( 1969).
Main features of the model estimated by
Garrels and Thompson (1962) have been
confirmed (Berner 1965; Thompson 1966;
Thompson and Ross 1966; Pytkowicz and
Gates 1968; Pytkowicz and Kester 1969;
Kester and Pytkowicz
1968, 1969) but
there are significant differences for several
species concentrations,
especially anionic
ones. This occurs because Garrels and
Thompson used estimated singIe ion activity
coefficients
which
are uncertain,
especially in the case of sulfate ions, as
these are associated in all solutions that
could be used for the mean salt method,
and because they assumed, as a result of
the scarce data available at the time, that
the activity coefficients of charged ionpairs were equal to those of bicarbonate
while those of ion-pairs with zero net
charge were the same as those of nonelectrolytes, even though ion-pairs are dipolar
entities.
We avoided questionable
assumptions
about activity coefficients in our choice of
experimental methods as far as possible.
In the earlier sulfate work we used ratios
of activity coefficients of cations, thus canceling errors in part, and at no time did
we have to resort to estimating activity
coefficients of anions and of ion-pairs. The
difficulty
of estimating such coefficients
accurately makes it difficult to apply the
thermodynamic association constants determined by Hostetler
(1963), Nakayama
( 1971), and Reardon and Langmuir
(in
prep.) to seawater.
In the sulfate work chloride solutions
were used as reference solutions (in which
association does not occur) and cation activities were compared in chloride and seawater to determine the extent of ion-pairing.
In this work on bicarbonate and carbon-
and Ha&e y
ate ions, changes in the dissociation equilibria of carbonic acid with composition at
constant ionic strength provide the ion.pairing information.
We have used reference electrodes with
salt bridges and assumed that liquid junc,tion potentials, to a first approximation,
are independent of composition at a given
ionic strength rather than avoiding salt
bridges and having to make assumptions
about the activity coefficient of chloride
ions. Either approach has to be viewed
with caution and the results considered
tentative until confirmed by other independent methods. Our assumption about
liquid junction potentials appears justified,
within the precision of the method, because
the association constants that we determined for sulfate ion-pairs (Pytkowicz and
Kester 1969) and for bicarbonate ion-pairs
( this work) are independent of composition and lead to a successful prediction of
the activity coefficients of cations (Kester
and Pytkowicz 1969). In the case of ca:cbonates we are forced to invoke triple ions
to explain the measurements although the
experimental results may reflect changing
liquid junction potentials.
Still, the relative strength of the several triple ions will
be justified from an energetics viewpoint
and the extent of carbonate association
shown to be compatible with data on carbonic acid dissociation and on calcium
carbonate solubility.
Earlier workers did
not find triple ions because they did not
measure the concentration dependence of
the association.
The ion-pair concept is used in preference
to the specific interaction model because
data on sound attenuation (Fisher 1967)
and from Raman spectroscopy (Daly et
al. 1972) indicate that ion-pairs may indeed
exist. Still, the specific interaction model
is a valuable tool for the study of multielectrolyte solutions.
Details of experimental and calculation
procedures not presented here are available in Hawley ( 1973).
We thank C. Culberson and S. Ingle for
assistance in this work and. S. Williams for
Bicarbonate
typing and proofing
the manuscript.
related to yIIco,, the free activity coefficient
which applies only to the nonassociated
ions, by (Pytkowicz et al. 1966; Pytkowicz
1969)
the many versions of
Theoretical
The methods used to determine the apparent ( stoichiometric)
association constants are based on the variation of K’1 and
K’2, the apparent dissociation constants of
carbonic acid ( Lyman 1956)) with the
major ion composition of the test solutions
at constant ionic strength.
The equations presented below were
used to derive the working relation for the
determination of the association constants.
The first apparent dissociation constant is
related to the first thermodynamic
dissociation constant, K1, by
K’ I = kf~o,a&G/f-tI~o,;
1.
Equations
’
K1
*
aH20f
referred
= KI/1
= ~IICO,(HCOs-) = fmo,( HCOa-) T; (2)
aII00,
( HC03-) and ( HC03-) T are the molal concentrations of the fret and the total (free
plus ion-paired) bicarbonate ions. (I-ICOS-)T
is given by
( HCOa-)T = ( IIC03-) + ( NaHCOSO)
+ (MgI-ICOs+) + ( CaHC03+). (3)
The apparent association constant for a
generic ion-pair MHCOSC-l, where c is the
charge of the cation MC, is
(MHCO$-l)
K”
(1)
k is the correction in aII, the hydrogen ion
activity, which results from pH measurements in concentrated solutions ( Hawley
and Pytkowicz 1973)) fee, is the activity
coefficient of molecular COa, arI,o is the
activity of water, and frrcc, is the total activity coefficient of bicarbonate.
fIIeO, is
Table
225
and carbonate ion-pairs
MIICO,
=
(M”)(HCOa-)
= yMyI-Ico3 K YIIco,.
(4)
YMIICO,
Equation 5 in Table 1, the relation used
by us to determine K*~~rree3, was obtained
from equations 1 through 4. ZV1 = kK1/
levee, was determined experimentally
as
to in the text.
1 t K”
NaHC03
(Nat)
+ K*Mg,Ico
(W2’)
( Ca2+)
+ K*CaHCO
CO2
(5)
3
3
I
K;K’2
= K’;KN
*
aH20E
1 tK*
2
(Nat)
NaCo
+ K*Mgco
3
CO2
= KP;K/Z
(Ms2+)
t K*caco
(Nat)
1 t KxcNaCO
3
t K* MgGO
3
*
K CaCO
(Ca2f)
3
K*
Ca CO (Ca2t)2
2
3
Ws2?
(6)
lCa2’)
3
I
+
3
+ K*Mg
(Mg2+)’
co
2
t
3
’ K*MgCaCO
(Mg2’)(
3
(8)
Ca2+)
I
226
Pytkowicx
will be shown below. Equation 6 in Table
1, the working relation for the determination of K*dlCO, was obtained from a set of
equations similar to 1 through 4 but with
the use of K’lK’z instead of just K’, . K”l K”z
in equation 6 is kSKIKz/yco,.
The values of K*;lIITTCOy,K”,, K*cc,, and
K”lK”2 were determined by measuring K’1
and K’,K’2 in a series of solutions of varying composition but the same ionic strength.
This led to a system of equations such as
5 and 6 which could be solved for the desired quantities, although the solution of
the equations for K*c,,-,, required
extra
terms. K’l and K’lK’2 were obtained by the
methods used by Kester and Pytkowicz
(1967) for the apparent dissociation constants of phosphoric acid. In essence, K’1
was determined by titrating the solutions
with I-ICI and using the equation
and Hawley
LY!5r&d+
0
Magnesium
0.20
Concenlrotion , mold
0.:!4
Fig. 1. K’JuH,o~co~as a function of magnesolutions at
sium concentration in NaCl-MgCb
25°C and 0.72 ionic strength.
this assumption. For ASW, fco, was calculated from the solubility data of Murray
and Riley ( 1971). aiI,o was obtained from
osmotic coefficient
data ( Robinson and
TC02 K’l
Stokes 1968 ) and the additivity relation of
~
- Kfl
Robinson and Bower ( 1965).
ar1 = -CA
It can be shown that, in the absence of
where CA = ( HC0,7-) T + 2( COcj2-) Il is the triple ions, K’~/aI120f~0, should vary linearly
carbonate alkalinity.
TC02, the total dis- with the concentration
of cations. This
solved inorganic carbon, was held con- can be proven by introducing
the ionic
stant by filling the cell completely with
strength relationship
into equation 5 for
solution. K’1 remains constant as long as binary solutions and eliminating
all ion
the major ion composition of the solutions
concentrations but one. A linear relationremains essentially constant ( Weyl 1961; ship was indeed found in the case of bicarKester and Pytkowicz
1967; Pytkowicz
bonate association, as illustrated in Fig. 1
1969). K’lK’2 was obtained by adjusting
for a NaCl-MgC12 solution, This did not
the solution to a ~1-1 that was invariant to happen, however, for carbonate association,
the addition of NaHC0,7. The relation
as the corresponding plots were curved.
pH = 0.5 ( PK’~ + PK’~) is valid when this We were led therefore to add triple-ion
pH is obtained.
terms to equation 6, as is shown in equation
8 (Table 1). Triple ions involving (COs2-)2
f co, was calculated from the relation
fco = So/S, in which So and S are the solu- were not required because the concentrabihties of CO2 in distilled water and in the tion of carbonate ions was much lower than
test solution, and the solubility
data of those of the cations, rendering the probMarkham and Kobe ( 1941). We assumed ability of such triple ions negligible. Also,
that the activity coefficient of CO2 in the the data fit equation 8 better without the
solutions was the same as that in a NaCl
term K*C112c;o,(
Ca2+) 2, as was expected beof calcium ions
solution of the same ionic strength and cause the concentration
tested this assumption in NaCl-MgCl2 solu- was kept low relative to those of the other
of calcium
tions. WC found ftJo, = 1.20 2 0.04 in 0.72 cations to prevent precipitation
carbonate.
m NaCl, 1.20 * 0.04 in 0.36 m NaCl-0.12
The fitting of the measured values of K’1
m MgClz, and 1.14 c 0.04 m in 0.24 m
and
K’lK’3 to the systems of equations such
MgC12. Also, as will be shown later, the
as 5 and 8 was done by solving for K”‘1,
results for bicarbonate are compatible with
Bicarbonate
Table
Composition
2.
Test
solution
numbe r
of the test solutions in molal units.
(NaHC03)TQ
(CaC12JT
----
--.,-
0.7200
0.4800
0.2400
---0.0800
0.1600
0.2400
------0.1900
0.1400
0.0400
-m-w
0.2200
0.0597
0.3702
Added
227
(Na2S04)T
2
3
4
5
6
7
8
9
10
11
12
ASW
hk
and carbonate ion-pairs
----
---”
----
----
----
0. 6300
0.4800
----
----
--se
-m-w
0.6000
0.5700
----------
-m-m
0.0315
as Na2C03
due to the reaction
varying
pairs,
0.4659
but expressed
with
amounts
C02.
as NaHC03
because
When the solutions
of NaHCO
3
were
w---
------------0.0300
0.0800
0.0500
0.1000
---0.0500
0.0200
0.01195
of the c:onversion
were
used for
of C032-
the study
added to find the pH which
x 103
5. 00
5. 00
5.00
5. 00
5. 00
5. 00
5.00
5. 00
5. 00
5.00
5.00
5.00
5. 00
to HC03 -
of carbonate
was invariant
ion-
to further
additions.
K”lK”2, and the association constants, with
the condition that these quantities had to
be positive.
The calculation procedures used to determine the distribution of species were similar to those of Kester and Pytkowicz ( 1969).
A system of equations of the type
(Na+)T = (Na+) + (NaS04-)
+ ( NaHCOso) + ( NaC03-)
K*
(9)
(NaSOd-)
Nas04=
(Na-‘)(S(-&2-)
(10)
extended to all the species present was
solved for the concentrations of free ions,
ion-pairs, and triple ions.
Experimental
procedures
The compositions of the test solutions
are shown in Table 2. The solutions were
prepared by weight, corrected to weight
in vacuum.
Reagent grade NaCl and
NagSO4 were dried and weighed as solid
salts while MgC12, CaCl2, and Na2C03 were
added from stock solutions standardized
by the Mohr titration (Blaedcl and Meloche
1957). The standard Na&03 solution was
prepared by weight from Mallinckrodt primary standard Na2COs (99.96% pure)
which had been dried for 2 hr at 280°C
an d cooled in a vacuum desiccator. The
I-ICI titrant was standardized by potentiometric
titration
of primary
standard
Na2C0s.
Electrode potentials were measured using a potentiometric
circuit similar to the
one described by Kester and Pytkowicz
( 1967). The sensitivity of the system was
within ~O.OI mV.
Titrations for the determination
of K’1
were performed in the cell shown in Fig. 2,
constructed from a 200-ml Berzelius beaker
fitted with a water jacket to allow temperature control of +O.O3”C. The barrel of a
5-ml ground glass syringe was cut off and
mounted horizontally in the cell with epoxy
cement. The piston of the syringe was displaced as titrant was added, eliminating the
need for an air space to take up the titrant
volume; it displaced about 4 ml of test solu-
228
Pytkowicx
and Hawley
Thermomeier
TO
E/et fronfcs
GiUSS
E/ecrook
‘1 ,.=; , I )
0
I
2
3
5
I/CA
Ground G/ass
syringe
Fig. 2. Water-jacketed
titration
cell. pH electrodc, Sargent S-30050-15C;
reference
electrode,
Sargent
S-30080-15C;
syringe
buret,
Gilmont
s 1200.
tion and was lubricated by a thin film of
solution. A pH electrode, a reference electrode having a porous ceramic plug liquid
junction, and a thermometer were fitted
tightly into a rubber stopper which had
been machined to fit into the beaker. A
small hole in the stopper held the syringe
needle of the titration buret and allowed
for overflow of test solution upon filling
of the cell. Electrode potentials in a standard buffer solution were the same in the
titration cell as in an ordinary thermostated
beaker.
The titration procedure was as follows.
First, the pH electrode pair was standardized in buffer solution (National Bureau
of Standards buffer 185d having an assigned pH of 4.008 at 25°C). Next, the titration cell was completely filled with test
solution and the electrodes were allowed
to equilibrate until the potential changed
by less than 0.1 mV per hour. Then, the
solution was titrated with standard HCl
Fig. 3. UH versus l/CA
from the titration
of
a 0.7200 m NaCl and 0.005 m NaHCO,
solution
with HCl.
from a calibrated syringe buret. Finally,
the electrodes were returned to the buffe:r
solution and the amount of test solution
titrated was determined by weight. The
electrodes were thoroughly rinsed with the
new solution when they were transferred.
The linearity of a rI versus I./CA, expected
from equation 7, is shown in Fig. 3.
For the determination of K’,KIZ the cell
was completely filled with ,test solution to
avoid exchange of CO:! and the pH was
adjusted to about 0.5( pK1 -t p&) by addition of a few hundredths of a milliliter of
0.02 N N&OS.
This brought the alkalinity
to about 0.005 meq liter-‘. Then lo-15 mg
of reagent grade NaHCOs ‘were placed in
a dry, gastight, 2.5-ml IIamilton
syringe
and about 0.3 ml of COZ-free distilled water
was pulled into the syringe to dissolve the
salt. The NaIICOs was injected slowly into
the test solution. The solution was stirred
for 5 min and the electrode potential was
then recorded for 5 min in the absence of
stirring. At least five additions of NaHC03
were made to establish that a constant pH
had been obtained. The final total alkaIinity was about 3.5 meq liter-‘. The constancy of the pH, within the experimental
precision of kO.003 pH units, was aster-
Bicarbonate
Table 3. Measured values of K’1 and K’s, calculated
Kpa calculated from equations 5 and 8 and the speciation
Test
solution
number
K’l
1
x106
Measured
/ /
KlK2xlO
1.161
1.161
1.072
1.084
1.085
1.140
1.160
1.199
1.243
1.244
1. 250
1. 278
1.145
1.170
1.290
1.307
4
5
6
7
8
9
10
11
12
ASW
229
and carbonate ion-pairs
values of aIt,0 and fCOZ, and values of K’I and
model.
Calculated
16
2.070
2. 041
2.245
2. 239
8.120
7.860
16.79
17.01
16.65
28.16
28.54
6.390
34.52
K’l
fC02
x 106
0.983
1. 292
1.160
2. 06
0.972
1.178
1.089
2.26
0.977
1.178
1.152
8.12
0.981
1,178
1.211
16. 8
0.986
1.178
1.274
28. 3
0.974
0.977
0.986
0.986
1.178
1.178
1.178
1.178
I. 122
1.176
1.290
1.302
6.71
34. 0
5.030
9.840
29.61
8. 02
1.094
1.140
1.136
1.143
This is the value calculated
of the measured
values.
from
16
K’lK;xlO
aH20
4. 84
9.70
29.7
7. 87”
the model
which
agrees
within
3 percent
with
the average
Table 4. Values of the apparent (stoichiometric)
association constants determined in this work 1(l),
those based on the work of Garrels and Thompson (1962) (2), and those measured by Butler and IIuston
(1970) (3) and by Dyrssen and Hansson (1973) (4).
Ionic
Strength
(1)
(2)
0072
0. 66
(3)
(4)
0. 5
1.0
0. 39
0.21
NaH C03’
0.280
o. 26
MgHC03+
1.62
5.22
1.04
CaHC03+
1.96
5.10
1.04
NaC03-
4. 25
4.16
MgC03’
W2C03
112
2-k
CaC03’
MgCaC03
1.86
32.5
387
162
2t
160
1.38
1,040
78
32. 5
230
Pytkowicx
and Hawley
Table 5. Chemical model for the major species in seawater at 25°C calculated
from the sulfate
association constants @ester and Pytkowicx
1969) and the bicarbonate
and carbonate association c(?nstants (this work: Table 4). The constants for magnesium and calcium fluoride association were taken
from Elgquist (1970). The number of significant
figures is used for mass balance purposes and does
not reflect the accuracy which is unknown because of the assumptions made.
=
N2
Total
molality
0.4822
Mg
2f
0. 05489
Ca2 I0. 01063
97.70
89.11
88.35
% MS04
2.25
10.35
10.87
7’0 MHC03
0.05
0.24
0. 29
Yc MC03
0.01
0.17
0.41
“/c Free
metal
‘-30Mg2C03
0.03
% MgCaC03
0.01
0.07
% MF
0.07
0.02
so42-
cog2-
HC03-
d
-
0.01062
100.00
F
-
Total
molality
“/c Free
anion
0.02906
0. 00213
39.19
81.33
0.000171
7.99
% NaX
37.29
10.73
15.99
% MgX
19.55
6.44
43.86
“/c CaX
3.97
1.50
20.96
% Mg2C03
7.39
% MgCaC03
3.82
0. 000080
51.04
46.94
2.02
-
tained after correction for dilution.
The
pH = 0.5( pK’1 + pE2) ranged from 7.231
to 7.845 in going from MgQ-CaC12
to
NaCI solutions.
The electrodes ( Fig. 1) were standardized with NBS buffers 185d (pH = 4.008 at
25°C) and 186-l-b and 186-11-b (pH =7.413
at 25°C).
Results
Values of K’l, K’JX’Z, aE120,and fee, are
shown in Table 3. The values of the apparent association constants are presented
in Table 4. The percentages of the ions
present as free and as associated species are
shown in Table 5. The calculations were
made at pH 8.0 and 8.1 with no significant
difference in the results. In Table 6 we
present the model if an association constant
for KS04- is added to the calculations. This
constant should, however, be measured.
Discussion
The validity of the model was tested by
measuring K’, K’ 2 in a solution (ASW in
Table 3) with the cation composition of
seawater and comparing it to K’,K’2 calculated from equation 8, used in conjunction
with the association constants determined
in the simpler test solutions. The measured
value was 8.02 X lo-l6 while the calculated
value was 7.87 x 10-l”; the two agree within
3%. In addition, the value of KllK’2 r’eccntly measured in seawater by Mehrbach
et al. (1973) at 25°C and 35s0 salinity was
Ca2CO:12+
7.68 x 10-‘6. A model including
yielded 8.13 X lo- Is; thus, it did not agree
as well with measurements and, as the cal-
Bicarbonate
231
and carbonate ion-pairs
Table 6. Chemical model of Table 5 modified when an association
the results of Garrels and Thompson (1962), is added to the calculations.
theirs.
Nat
Mg
2-b
constant for KHSO?, based on
The values in parentheses are
Ca2+
97.71
(99)
89.15
(87)
88.39
(91)
70 MS04
2.24
(1.2)
10.31
(11)
10.82
(8)
7’0 MHC03
0.05
(0.01)
% MC03
0.01
(
$J Free
metal
)
0.24
(1)
0.29
(1)
0.17
(0.3)
0.41
(0.2)
% Mg2C03
0.03
% MgCaC03
0.01
0.07
% MF
0.07
0.02
HC o3 -
so4z% Free
anion
39.01
(54)
81.33
(69)
% NaX
37.13
(21)
10.73
% MgX
19.47
(21.5)
6.45
1.50
$3 CaX
3.96
(3)
% Kx
% Mg2C03
% MgCaC03
0.42
(0.5)
cium concentration was kept low to prevent
precipitation of CaCOs, was rejected.
Another test of the validity of the model
is that it can be used to explain the solubility of calcium carbonate in seawater.
The apparent solubility product of calcium
carbonate, K’sIJ is related to the thcrmodynamic solubility
product, KxIJ, by the
equation K’ w = &dfc,fco,. ye the free
ion activity coefficient
obtained by the
mean salt method, is 0.255 as calculated
by C. Culberson in our laboratory.
The
use of an equation such as 2 for carbonate
ions yields fen= 0.225 as 88.35% of the calcium ions are free. f,&k2 = (KlKJK’lK’g)
= 0.031 according to the ion assoan,0f
co,
ciation model. KNgp= 3.98 X 10-O (Langmuir 1968) and, thus, we calculate K’gp
= 5.67 x 10-7. The experimental values of
Kgp are 5.4 X 1O-i ( MacIntyre 1965) and
4.93 X 10d7 (Ingle et al. in press). The
last value is expressed here as moles
co3z-
K+
98.85
1.15
(99)
(1)
F-
7.98
(9)
(8)
15.98
(17)
(19)
43.86
(67)
(4)
20.96
(7)
51.03
46.95
2.02
7.39
3.82
kg-I120-2; Ingle et al. present it in units
of moles kg-SW-“.
The agreement between the calculated
and measured results is fair considering
the uncertainties in the model and in the
measurement of solubility products in seawater. Still, the differences suggest, when
the data of Ingle et al. are used, that the
percent free carbonate may be closer to
9.1% than to 8. This is a slight difference
which could be resolved, for instance, if
the extent of MgCOao association were off
by only 2% of its value.
The model of Garrels and Thompson
( 1962), although it yields a percent free
carbonate of 9, does not lead to a satisfactory prediction of KfBP. They reported
an d used yea = 0.28 and ycoI = 0.20 which,
in conjunction with their model, yields for
the total activity
coefficients
fCn= 0.91
x 0.28 = 0.255 and fCO,
= 0.09 x 0.20 = 0.018
and results in Kap = 11.1 X 1O-7 instead of
232
Pytkowicx
a value near 5 X 10-7. The error appears
to arise because they used a rough estimate
of YCO~, the free activity coefficient, by
Walker et al. (1927). ‘Our value of ycos
and Hawley
evaluated because they did not examine
all the possible pairs. It may be, however,
that their results are low because ours explain
the apparent dissociation constants
is fco,( CO&/(
C03) = 0.388.
of carbonic acid and the apparent soluOur association constant for NaEIC030
bility of calcium carbonate. We found less
is in good agreement with those of Garrels
MgCOsO ion-pairing and considerably more
and Thompson (1962) and of Butler and CaCOsO ion-pairing than was the case in
Huston
(1970) but the constants for the model of Garrels and Thompson (TaMgHCOs+ and CaHCOs+ differ from those ble 6).
of Garrels .and Thompson ( 1962) and DyrsOur model is not complete. Ca2C0,7:!+
sen and Hansson ( 1973). Our results sug- triple ions were not detected because the
gest that calcium associates more strongly
concentration of calcium was kept low to
than magnesium with bicarbonate, in con- prevent precipitation
of CaC03. In additrast to those based on the estimates by tion, triple ions involving sodium may ocGarrels and Thompson (1962), and that
cur but were not necessary to explain our
the extent of association for both cations is results, in part because of the composition
slighter than they predicted. Greenwald’s
of the test solutions and in part because the
( 1941) calculations also indicate that cal- association of sodium is weaker than those
cium .association
with bicarbonate
is of magnesium and calcium.
stronger than magnesium association, and
The association reactions possible when
TKester and Pytkowicz (1969) found that
triple ions form are of the following charge
pK*cnso, was slightly larger than K*MgSO,. types :
The relative strengths of magnesium and
2NA+ + COs2- = Na2C0,0;
(11)
calcium association probably reflect the
radii of the hydrated cations, as was sug(12)
Na+ + Mg2+ + COs2- = NaMgC03+;
gestcd by Nancollas (1966).
Robinson
2Mg”’ + COs2- = Mg2C03”+.
(13)
and Stokes (1968) presented hydration
numbers and ion size parameters for MgC12 The important factor in determining the
b and CaC12, derived from the hydration
relative stability of triple ions is their tota.
theory activity coefficients.
These paraminteraction energy. Consider Na2C030 with
eters suggest that the hydrated radius of a linear Na+-CO 32--Na-b configuration
and
calcium ions is smaller than that of mag- with the centers of the ions separated by
nesium ions and, therefore, that calcium
the distance r. Assuming point charges for
ions should interact more strongly than rough estimates, the interaction energy is
magnesium with anions, as we found to
2ZmZcop2
&&&”
be the case. Again, we found K*C,C03 to
+
Ah,,co, = (14)
be larger than KQAIgOO,in contrast to the
Dr
2Dr ’
constants based on the results of Garrels
where ZNn and Zco, are the valence charges,
and Thompson (1962). The concentration
e
is the electronic charge, and D is the diof MgCO,O is larger than that of CaCOs”,
electric
constant. Thus, AUN~,CO, = -7e2,/
but this is because there is more magneDr. The interaction energies for all the possium than calcium in seawater.
sible triple ions, taking the values of r in
Our value for K*NaCO:,agrees with that
centimeters, based on the effective hybased on Garrels and Thompson (1962)
drated ionic diameters presented by Stumm
but is larger than that of Butler and Huston
( 1970). Our values for K*MgCO,and K*c~co, and Morgan ( 1970)) are listed in Table 7.
The free energy AF is AH-TAS which
indicate significantly
more ion association
If it is assumed
than those of Dyrssen and Hansson ( 1973). is approximately AU-TAS.
The results of Butler and Huston and of that the entropy charge AS is roughly the
same for all triple ions then, as AF = -RT
Dyrssen and Hansson cannot be definitively
Bicarbonate
and carbonate ion-pairs
Table 7. Interaction
energies for triple
ions
calculated as explained in text. D is the dielectric
constant and e is the electronic charge.
Triple
ion
Na2C03
Interaction
energy
x lo8
D/e2
-0.83
NaMgC03
-1.29
NaCaCO3
-1.44
MgCaC03
-1.70
Mg 2C03
-1.54
Ca2C03
-1.87
233
constants of ion-pairs that have yet to be
explained.
An alternative theory to ion-pair formation, the specific interacTion theory of Br@nsted and Guggenheim, provides results comparable to those of the ion-pair theory. Both
approaches are equally valuable numersound attenuation
and
ically; although
Raman spectra data indicate the actual
existence of ion-pairs.
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witer
Submitted: 13 July 1972
Accepted: 30 November 1973