____________________________________o_c_E_A_N_O_L_O_G_I_C_A_A_C_T_A_1_9_8_2_-_v_o_L_._5_-_N_o_2__
·
On the evaluation
of potentiornetric titrations
of seawater with hydrochloric acid
·~----~--
Alkalinity
. Non-linear curve-fitting
Potentiometric titration
pH scales
Seawater
Alcalinité
Ajustement de courbe non linéaire
Titrage potentiométrique
Échelles de pH
Eau de mer
O. Johansson, M. Wedborg
Department of Analytical and Marine Chemistry, Chalmers University of Technology
and University of Gôteborg, S-412 96 Gôteborg, Sweden.
Received 21/4/81, in revised form 16/11/81, accepted 26/11/81.
ABSTRACT
Direct minimisation and modified Marquardt algorithms are used for non-linear curvefitting procedures of evaluation of the potentiometric titration of seawater with
hydrochloric acid. The procedures are tested on theoretical titration data. Effects of
introducing various errors are studied. The procedure based on the modified Marquardt algorithm is also applied to experimental data from potentiometric titration of
natural seawater samples with hydrochloric acid. Other methods of evaluation and the
use of various pH scales are discussed.
Oceanol. Acta, 1982, 5, 2, 209-218.
De l'évaluation des dosages potentiométriques de l'eau de mer
par l'acide chlorhydrique
RÉSUMÉ
Une méthode d'ajustement directe et des algorithmes dérivés de celui de Marquardt ont
été utilisés comme méthodes de lissage des courbes non linéaires obtenues par titrage
potentiométrique de l'eau de mer par l'acide chlorhydrique. Ces méthodes ont été testées à l'aide d'une courbe de titrage théorique. L'influence de l'introduction de différentes erreurs a été étudiée. La méthode basée sur l'algorithme de Marquardt a été également appliquée à des données expérimentales obtenues par dosage potentiométrique
par l'acide chlorhydrique d'échantillons d'eau de mer naturelle. D'autres méthodes
d'évaluation ainsi que l'utilisation de différentes échelles de pH sont discutées.
Oceanol. Acta, 1982, 5, 2, 209-218.
Non-linear minimisation methods have been used at
our department for severa! years (e.g. Hansson, 1972;
Wedborg, 1979; Johansson, 1981). In spring 1980 we
worked with the application of the non-linear curvefitting algorithm Stepit (Chandler, 1965) to the mass
balance equations describing the alkalinity titration.
During a visit to our department, Dickson (1981) suggested that a modified Marquardt procedure according
to Nash (1979) might be faster and more reliable. As a
consequence, we decided to try both algorithms.
The purpose of this paper is to show how non-linear
curve-fitting ·works on theoretical, as weil as practical
titration data. Effects of introducing various errors are
also studied. In doing so, ·we arrive at a procedure and
conclusions that deviate from the result of Dickson
(1981) at sorne essential points.
INTRODUCTION
In 1965 Dyrssen suggested that the alkalinit,Y (A,) and
total carbonate (C,) of seawater could be determined by
the evaluation of the equivalence point according to
Gran (1952). This method was treated by Dyrssen and
Sillén (1967) who suggested that A, and C. should be
given on the p-T-independent concentration scale moles
per kg of seawater (Mw). Hansson and Jagner (1973)
showed that the Gran method needed a refinement
which was made feasible in practice by computer evaluation. Their procedure was used by Almgren et al.
(1977) during a cruise on the Pacifie Ocean in 1973. The
recent work of Bradshaw and Brewer (1980) and Bradshaw et al. (1981) is based on the same procedure.
0399-1784/1982/209/ $ 5.00/
t)
Gauthier-Villars
209
O. JOHANSSON, M. WEDBORG
THEORY
Definition of alkalinity
The. definition of illkalinity, which has hitherto been
somewhat diffuse, was brought up for discussion by A.
Dickson during his visit to the Department of Analytical and Marine Chemistry in Gôteborg in April-May
1980. After careful consideration (L. Anderson, A.
Dickson, D. Dyrssen, D. Jagner and the present
authors), we agreed on a stringent· definition which
does not necessitate a change of the practical evaluation
of the titration of alkalinity in seawater.
With this definition (Dickson, 1981), it is very easy to
see that, for instance, fluoride and sulphate should not
be included in the alkalinity.
based. As a consequence, the total concentrations to be
given for the constitu~nts depend on the choice.of components.
· When this concept is applied to the protolytic equilibria
. of natural waters, choosing the dominant forms of the
protolytes in an exactly neutralised solution · (i.e.
A. = 0) as components, together with either the hydrogen or the hydroxide ion, is in best accordance with the
practical procedure. Defining either H• or OH- as a
component is equivalent to adopting one of the concepts "acidity" or "alkalinity". The choice is not
important. For our equations, we have chosen H 2C0 3,
B(OHh, Si(OH) 4, H 2P04, SOl-, F- and H• as components. The system of mass balance equations (notation,
see Table 1; charges have been omitted).
c, =
[CO:J + [HCO:J + [H 2COJ
[B(OH)J + [B(OH)J
[Si(OH)J + [SiO(OH):J
P, = [POJ + [HPOJ + [H 2POJ + [H 3POJ
Su,·= [SOJ + [HSOJ
FI,= [F) + [HF]
H, = [H] - [HCO:J - 2[C0 3] - [B(OH)J [SiO(OH)J + [H 3POJ - [HPOJ - 2[POJ [OH] + [HSOJ + [HF].
Combining these equations, and taking addition of acid
from the burette into consideration, gives :
B,
=
Si, =
Chemical model
The seawater constituents included in our model are
shown in Table 1, where ali symbols used in this paper
are defined (note that ali equilibrium constants are
given as stability constants according to the concept of
Sillén and Marteli, 1964; 1971). It should be pointed
out.Ùtat the restrictions on the number of constituents
lncluded are of. a practical nàture. Th us, adding sulphide or ammonia to the equations for calculations on
anoxie waters, or any other bases present in a sample, is
an extremely simple procedure, which does not require
changes of the derivation of the equations, as will be
shown below.
A chemical system at equilibrium can be completely
described by defining a number of components and a
number of complexes formed by a combination of the
components .. If the total concentration of each component and the stability constant of each complex are
known, a system of equations can be formed, to which
there exists a unique, chemicaliy reasonable mathematical solution - the free concentrations of the components. This is the fundamental concept upon which the
computer program Haltafali (Ingri et al., 1967) is
VaHl + vHb
(v.
= [H.A)I[H]
K..
s
H.
=
-A.
C., Ktco Kzc
B,, Ks
P,, K,r, KzP, K:ll'
Si., Ks,
Su,, Ks.
F1., 1(..,
v.
(v.
v
H.
[H1, pit, EKI
[H)F, p~. EKF
[H]T, pHT, EKT
k
- Kj[H)
+
v)
(1)
where
C =
B =
Si =
p =
C, ([H]K 1c + 2)/(1 + K1c[H] + K1c Kzc [H]Z)
B,/(1 + KH[H])
Si,/(1 + Ks;[H])
P. (KlP[H] + 2 - K1P K2P K3P [H]3)/(1 + KlP[H]
+ KlP K2P [H]Z + KlP K2P K3P [HP)
Su = Su,/(1 +. 1/Ks.[H])
FI = Fl,/(1 + 1/KFI[H]).
The presence of any other base, Bas, can be taken into
account by adding the term Bas,/(1 + KH,..[H]) to the
right hand side of Equation (1), within the parentheses
(i.e. with a positive sign, analogously toC, B, etc.). The
alkalinity is, by definition, A. = - H,.
For our calculations, we have chosen to use the pH
scale based on the "free" concentration of hydrogen
ions in the ionie medium (pHF), as suggested by Bates
(1975). As is shown in the discussion below, Equation
(1) is, however, easily modified to any of the pH scales.
[H.-1A] consecutive stability constant of acid
H.A
ionie product of water
salinity ("'oo)
initial total concentration· of hydrogen
ion which equals the total alkaiinity with
reversed sign
:r~~:ate
= [H]
v)
_ _v_;_.__ (C +B +Si+ P-Su- FI),
Table 1
Definitions of symbols used. Concentration units, mo/.1 1 •
K...
+
l
initial total
phosphate
concentration
silicate
stability constants
sulphate
fluoride
initial volume in titration vesse! (ml)
volume acid added from burette (ml)
total concentration of hydrogen ion in
burette
"free" concentration of hydrogen ion,
pH and standard potential (mY) on
three different scales: infinite dilution
(1), ionie medium, "free" (F) and
ionie medium, "total" (T)
Nernstian slope (rn V.decade-1)
Choice of dependent variable
The experimental titration curve is obtained by reading
the emf after each addition of acid from the burette.
The relationship between the emf readings and the concentration of hydrogen ions is given by the Nernst
equation,
E = EK + k . log [H], where k
=
RTlnlO/F.
The pH scale used is defined by the procedure of standardisation, i.e. by the value chosen for EK. Mathema210
EVALUATION OF POTENTIOMETRIC SEAWATER TITRATIONS
tically, either of the experimental variables, v ml or
E rn V, can be selected as the dependent variable. Equation (1) is a linear equation with respect to v ml and of
higher degree with respect to [H] (i.e. IO<E- EK)/k).
Thus, the choice of dependent variable is important for
the effectiveness of the mathematical procedure. Since
it is simpler and less time-consuming to solve an explicit
linear equation, v ml is selected, and consequently
E mV is defined as the independent variable.
However, as stressed by Wolberg (1967), there could be
statistical objections. The simple form of the method of
least-squares used in the minimisation procedure (see
below) presupposes that the experimental uncertainty
in the independent variable be negligible as compared
with the uncertainty in the dependent variable. Since
there are experimental errors in both E and v, the
uncertainty introduced by the choice of dependent
variable is further investigated below, theoretically as
weil as experimentally.
Table 2
Concentrations and stability constants used for Haltafa/1 calculation
of a theoretical titration of a typical deep water. Al/ values ofstability
constants given on the pHF scale.
Concentration
S f/oo)
35
A. (/LM)
2 500
C {JtM)
2 400
B. (/LM)
412
P, (/LM)
3
Si, {JtM)
150
Su, (mM)
28.23
F, {JtM)
73
ionie product of water
Consecutive stability
constant
Reference
1.187.109, 9.6S0.10S
5.444.1()8
1.319.109, 1.187.1()6, 4.169.101
3.162.109
1.187.10
3.981.1()2
4.854.10'" 14
1
2'
3,4
s
6
7
1
Hansson (1973 a)
Hansson (1973 b)
Johansson and Wedborg (1979)
Dickson and Riley (1979)
S. Dyrssen (1975)
6. Khoo et al. (1977)
7. Smith and Martell (1976)
1.
2.
3.
4.
was generated with the Haltafall program (lngri et al.,
1967). Acid was added in 0.1 ml increments, and pHF
varied from 7.714 (v = 0 ml) to 3.438 (v = 3.0 ml),
i.e. 31 points were used for the least-squares minimisation. The whole titration curve, including points for
v = 3.1 - 4.0 ml, which can be used for Gran
methods, is given in Table 3. Theoretical emf's were
calculated with the Nernst equation using
EKF = 350 mV.
Curve-fitting procedure
The different members of the model equation are divided into variables, parameters and constants. The
members that are not changed during the experiment
and the values of which are known are defined as constants. Variables are the members thàt are changed in
the experiment, and the parameters are the ones to be
determined by the curve-fitting procedure. The criterion used as a measure of the degree to which the model
fits to the experimental curve is the sum of squares of
the differences between the calculated and experimental
values of the dependent variable E (d, 41 , - d ..p) 2 , for ali
experimental points.
·
Various strategies can be used to find the minimum in
the error square sum function (e.g. Nash, 1979). The
strategies triéd in this work are the modified Marquardt
algorithm (Nash, 1979; algorithm 23) and the direct
search procedure Stepit (Chandler, 1965). The major
difference in strategy is that direct search involves no
derivatives of the function to be minimised, while the
Marquardt algorithm uses the partial derivatives with
respect to the parameters in the search for the minimum.
Stepit refines only one parameter at a time. lt saves
information about the direction in which to find the
minimum, aliows the step size to increase ("giant
steps") and uses parabolic interpolation (i.e. the error
square sum function is approximated to a parabola),
close to the minimum, ali of which are important for
the effectiveness and reliability.
The modified Marquardt algorithm is described in
detail by Nash (1979) and by Dickson (1981).
Table 3
Theoretical titration data for the alkalinity ti/ration, obtainedfrom a
Haltafa/1 ca/culation with the total concentrations and stability constants given in Table 2.
v ml
p~
v ml
0
7.714003
7.444797
7.177020
6.%3528
6.798572
6.665382
6.552727
6.453830
6.364446
6.281753
6.203751
6.128935
6.056088
5.984164
5.912185
5.839169
5.764045
2.1
2.2
2.3
2.4
0.1
0.2·
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
5.685556
5.602114
5.511512
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
5.294948
5.155638
4.977153
4.728435
4.379065
4.039272
3.808492
3.649890
3.531619
3.437921
3.360536
3.294704
3.237464
3.186852
3.141506
3.100443
3.062929
3.028404
2.996432
2.966663
5.410811
Choice of parameters and calculation procedure
Important restrictions when selecting the parameters to
be refined are that they ·should appreciably influence
the process studied and that the number of parameters
should be kept rather smali. lt is also necessary to make
sure that the parameters to be determined are not severely interdependent, in which case gross errors may
arise. For the alkalinity titration, the choice should be
made among A, (H,), C,, EK, K1c and K 2c. When the
analytical information is the aim of the titration, A,
TESTING OF THE PROCEDURE
ON THEORETICAL DATA
Theoretical titration curve
A theoretical titration curve for a typical deep water
(concentrations and stability constants, see Table 2)
211
O.JOHANSSON,M.WEQBORG
and C, must al ways be parameters. Allowing EK to be a
parameter implies that separate standardisation of the
electrode couple in a buffer or in the region of excess
acid is superfluous. Since the titration curve to a larger .
extent is influenced by K 2c (i.e. the main reaction H• +
HC03 - C0 2 + H 20) than by K 1c, K 2c is a better
choice as the fourth parameter. For the theoretical
titration curve, Ktc can be refined as weil.
Fitting the perfect theoretical titration curve (Table 3)
to exactly the same chemical mode! as that used for the
Haltafall calculation gives back the values of ali five
parameters mentioned above. The sum of squares of
the errors increases from around w- 12 (ml) 2 to w-s (inl) 2
when the number of parameters is decreased from five
to three (A, c, EK). Also, the residuals (v;, cale _·V;, exp)
become less randomly distributed, and the errors in A,
and C, increase somewhat. These errors are, however,
still negligible (relative errors less than 20 ppm).
As pointed out by Dickson (1981), it is advisable to normalise the parameters to be of the same order of magnitude to achieve a well-conditioned minimisation procedure, i.e. to lessen the risk that small errors in input
data give rise to gross errors in the result. The parameters in our equations were ail normalised to be in the
interval 0.5-5.
The calculation procedure selected for the tests was the
modified Marquardt procedure combined with the
mass balance equations solved for v ml. However, comparison is made with the other two procedures in a few
tests. The reasons for this choice are further discussed
below.
Table 4
Effect of introducing errors into the perfect theoretica/ titration
curve. The corresponding errors in A, and C., M, and AC, are
expressed either in pM or as percent relative errors. These results are
further discussed in the text.
··
Superimposed random
error, max 0.2 rn V
Superimposed systematic
error of 1 rn V
Error in concentration of
acid, AH.
Error in initial volume, Av.
Error in standard potential,
ÂEK = 1 mV
Error in log K1c, Alog K1c
= 0.01
· Error in log Kzc, Alog K2c
= 0.01
Error in Nernstian slope,
Ak = 1 mV.dec- 1
Neglecting 3 pM phosphate
in minimisation
Neglecting 412 pM borate
in minimisation
Neglecting 150 pM silicate
in minimisation
Introduction of errors
In order to assess the sensitivity of the calculation procedure to experimental errors and uncertainties, we
superimposed random and systematic errors on the perfect titration curve and introduced erroneous values of
various constants into the equations. The following
tests were performed (a summary is given in Table 4) :
Random errors
The random number generator of a Commodore PET
3016 computer was used to generate random numbers
within the limits ± 0.1 and ± 0.2, respectively. These
numbers were added to the theoretical emf's in the
same order as they were generated by the computer.
Minimisation calculations using five as weil as three
parameters were performed for each one of the two
titration curves obtained. The corresponding relative
errors in A, and C, were of the order of magnitude of
0.01% for a random noise of± 0.2 mV.
AA,
AC,
0.01 OJo
0.01 OJo
0.080Jo
0.50Jo
AH.OJo
- Av.OJo
AH.OJo
- Av.OJo
0.30Jo
0.1 OJo
- 0.0080Jo
0.030Jo
0.150Jo
0.150Jo
0.6(0.2)1 0Jo
1.6(1.0)1 OJo
0.01( -O.OOJ)IpM
3(3)1 pM
3(0.003)1 pM
9( -0.03)1 pM
-0.2 (negl.)1 pM
0.6( -0.03)1 pM
l. Values given for 3(5) parameters
shifted in the opposite direction to EK and K 2c. From
Table 4 it is evident that C, is much more sensitive to a
systematic error in E than -A,. Also, it was found that
~A, was about twice as large for three parameters as for
five, while ~C, was practically unaffected (the values in
Table 4 are val_id for five parameters).
The shifts in K 1c and K 2c were approximately 0.05 and
0.004logarithmic units per rn V of systematic error.
A constant displacement of the titration curve (i.e. the
same number added to each theoretical emf) is, of
course, compensated for by a corresponding change of
EK and does not cause errors in A, and C,.
A systematic error in v ml, caused by a constant leaking
of acid, gives rise to positive errors in A, and C, which
are equal to the leakage divided by the increment in v.
Thus, for a leakage of 1 ~tl per titration point and an
increment of 0.1 ml, the relative errors in A, and C, are
1%.
Error in concentration or acid or initial volume
From Table 4 it is obvious that the accuracy of the
determination of A, and C, will al ways be limited by the
accuracy with which Hb and v. are known.
Error in EK
Table 4 shows that A, is influenced to a larger extent
than C, by an error in EK. An uncertainty in EK of 0.3
rn V is enough to cause an error of 0.1 o/o in A, wh en EK
is not refined. The results in Table 4 are obtained for
three (A, C, EK) and four (A, C, EK, K2c) parameters.
If only A, and C, are allowed to float, the errors
become twice as large.
Systematic errors
A systematic shift in the emf readings such as the one
caused by a change of liquid junction potential was
simulated by displacement of the emf of the second
titration point by a certain number, the third emf by
twice that number, etc. The numbers used were 0.005,
0.01, 0.02 and 0.04 mV per point. Calculations were
·made for three and five parameters. A,, C, and K 1c were
Errors in K 1c and K 2c
As could be expected, A, and C, are more severely
affected by an error in K 2c than in K 1c. The errors, M,
and ~C, are approximately linear functions of the
212
. ·.~
EVALUATION OF POTENTIOMETRIC SEAWATER TITRATIONS
Table 5
Comparison of various procedures of calcula!ion. Results for perfect tilration curve. Errors given as the difference between refined and correct
values.
Modified Marquardt
M, (JtM)
AC. (JtM)
AEK (mV)
Alog K 1c
Alog K2c
error square sum
solved for v ml
solved forE mV
solved for v ml
6 .1o- 6
-2.1Q-4
-4.1Q- 6
- Io-s
- 1Q-4
2. 1Q- 12 (mlf
-3.1o-s
- 1Q-'
-7.1Q- 6
- 1o-s
- 1Q-4
8.1Q- 9 (mV)2
5.1o-4
-2.1Q-4
- 1Q-4
- 1o-s
- 1Q-4
2.1Q- 12 (ml)2
dent variable, were combined with the minimisation
su brou tine Stepit (Chandler, 1965).
Another change of procedure was to use the modified
Marquardt program and solve the mass balance equations for E mV as the dependent variable. A slightly
modified version of algorithm 18 in Nash (1979) was
used for this purpose.
When solved for v ml, the calculations were performed
in double precision on an IBM 3033N computer (ca. 15
significant digits), and when solved for E mV, on a
Commodore PET 3016 computer (ca. 9 significant
digits). Both modified Marquardt versions were coded
in Basic and the Stepit version in Fortran.
The results for five-parameter calculations with ali
three procedures are shawn in Table 5. The agreement
is quite good. Table 6 shows the effect of changing the
number of parameters as weil as the procedure of
calculation.
Table 6
Comparison between the two modified Marquardt procedures.
Results for theoretical curve with a superimposed systematic error
(0.04 m Vper titration point; 1.16 m Vover the whole titration curve).
Errors given as the difference between the rejined and correct values.
solved for v ml
AA (JtM)
AC. (JtM)
AEK (mV)
Alog K 1c
Alog K 2c
Stepit
solved forE mV
5 parameters
3 parameters
3 parameters
__: 2.3
-13.9
0.98
-0.059
- 5.1
-12.2
0.57
-2.8
-8.0
. 0.61
0.0046
logarithmic errors in the constants for Alog K .,;;; 0.1. If
K 2c is not refined, it should be known with an accuracy
better than 0.01 logarithmic units for the systematic
err or in A. and C, to be Jess th an 0.1 OJo.
Error in the Nernstian slope
C, is more sensitive than A,. The values given in Table 4
are obtained when A., C., EK (A., C., EK, K 2c, K1c) are
refined. If only A, and C, are parameters, ÂA, and ÂC,
become about 5-10 and 2 times larger, respectively.
TESTING
OF
THE
PROCEDURE
EXPERIMENTAL DATA.
MA TE RIALS AND METHODS
Effect of neg!ecting bases in the equations
As an assessment of this effect, the errors in A. and C,
were estimated from minimisation calculations where
the terms for phosphate, borate and silicate were excluded from the equations, one at a time. As is shawn
in Table 4, the errors in A. and C, depend on the
number of parameters. For silicate and borate the errors are negligible if the carbonate constants are allowed to float, in which case the program compensates by
accepting B(OH)4 and SiO(OH)j as COl- and producing erroneous values for especially K 1c (Âiog K 1c ""
' the er- O.llogarithmic units). With three parameters,
rors caused by silicate are still small, while for borate
they are substantial. Phosphate is a special case. Since
K 2p is very close to K 2c, the program compensates most
easily by increasing C, by a value equal to the total concentration of phosphate.. At the same time, the result is
hardly affeéted by the number of parameters, and ÂA,
is negligible.
ON
Rea gents
The solutions of 0.5 M hydrochloric acid were
prepared from Merck Titrisai ampoules.
Apparatus
Electrodes
Radiometer G 202 C glass electrode and Radiometer
K 4040 calomel reference electrode.
Voltmeter
Solartron 7055, resolution 0.01 mV.
Burette
Radiometer ABU 80, equipped with a stepper mot or,
resolution 1 ~tl.
·
Thermostat
A microcomputer.:controlled system (Intel SDK-85)
with semiconductor circuits as heater/cooler (Granéli et
al., 1981). The temperature was maintained at 25
± 0.01°C.
Changes of calculation procedure
In arder to further assess the reliability and effectiveness of the method of calculation suggested, the
mass balance equations, solved for v ml as the depen-.
213
O. JOHANSSON, M. WEDBORG
pK
Ti/ration vesse/
w
= 3 418.5/T + 2.0735- 0.012756.S
+ 7.8097.10- 5.5 2 (S = 20-400/oo)
log Ksu = 747.39/T-7.1833 + 0.020234.T
. -9.4466.10-3.S + 3.9813.10- 5.5 2
The titration vesse!, which was made of aluminium
covered with Teflon, holds ca. 150 ml of solution with
the magnetic stirring bar included. During a titration,
the vesse! was at !east 99% filled and closed in such a
way that no excess pressure arose. For this purpose, an
inflated rubber balloon was used instead of a piston.·
The magnetic stirrer motor gave good and constant stirring, which is essential in order to obtain fast and stable
readings from the glass electrode.
(S = 20- 45°/oo)
log K1P = 1 093.2/T + 6.0651- 0.033577.S
+ 3.5776.I0-4 .S 2 (S = 20-400/oo)
log K 2P = 620.53/T + 4.5038-0.030075.S
+ 3.3984.10-4 .S 2 (S = 5-400/oo)
log K3p = 1.62, maintained constant
log Ks, = 9.5, maintained constant
log KF, = 2.6, maintained constant.
Ail of the equilibrium constants for which the T -s-·
dependence is given above, except Ksu, were determined
on the pH, scale of Hansson (1973 c),
([H]r = [H)F + [HSOJ).
Thus the values calculated from the relations had to be
corrected by adding log (1 + Ksu·Su,), where Su, is the
total concentration of sulphate, so that the appropriate
values on. the pHF scale were obtained ("free" concentration of hydrogenions with the ionie medium as standard state).
Ti/ration system
A Commodore PET 3016 computer, programmed in
Basic, controlled voltmeter readings and additions
from the burette.
Procedure
Approximately 150 ml of sample was measured with a
temperature-calibrated plexiglass vesse! which was constructed as a syringe. By weighing ten separate fillings,
the volume of the syringe could be determined with an
accuracy·of ± 0.04 ml. The temperature of the sample
was measured and the volume corrected to 25°C with
the density equations of Millero (1976). Hydrochloric
acid was added in 0.03-ml aliquots up to a total volume
added of 0.75 ml (corresponding to a pH of ca. 4).
Emf readings were started after an initial delay of
40 seconds. A value was taken when the difference between two successive means of five readings was less
than 0.02 mV.
CALCULATI ONS
Calculations were performed for the titrations described above and for an on-board titration curve obtained
from 1 .L. Barron and E.P. Jones, Bedford Institute of
Oceanography, Dartmouth, Canada.
Our computer program calculates A, and C, on the
mol.l- 1 scale. A subrol]tine for calculating the density
was written so that the results could be converted to the
mol.kg- 1 (Mw) scale. The equations of Millero (1976)
were used for this purpose.
ADJUSTMENT OF EQUILIBRIUM CONSTANTS
The appropriate values for the constants were
calculated with the following equations, fitted to the
primary data of the sources in Table 2 (T is the
temperature in Kelvin and S is the salinity in permille):
EXPERIMENTAL RESUL TS
Laboratory titrations
1 376.4/T + 4.8256-0.018232.5
+ 1.1839.10-4 .5 2 (S = 20-400/oo)
log K2c
841.2/T + 3.2762- 0.010382.S
+ 1.0287.10-4.$ 2 (S = 20-400/oo)
log Kn
1 030.5/T + 5.5076- 0.015469.S
log K 1c
The results of two sets of titrations are shown in Table
7. Calculati ons were performed for 26 titration points
in the interval v = 0 - O. 75 ml, where O. 75 ml corresponds to pH 3.5 - 4.2. The concentrations of
phosphate and silicate were not known and thus set
equal to zero. The uncertainty in the salinity of the
+ 1.5339.10-4 .5 2 (S = 20-400/oo)
Table 7
Experimental results of two sets of titrations: standard seawater (IAPSO Standard Seawater Service, batch P77-770414, S = 35.006 °/oo) and
unfiltered seawater of 29.5 ± 0.5 %o salinity, taken outside G6teborg on the West Coast of Sweden. Results for four and three parameters (the
latter within parentheses). EKF is the standard potential on the pHF scale according to Bates (1975). pHF (start) is calculatedfrom the emffor v = 0
and EKF• using the Nernst equation.
A (pM.)
c. (pM.)
EKF (mV)
Kzclo- 5 (M- 1)
pH,. (start)
standard seawater (six titrations)
arithmetic mean
standard deviation
coefficient of variation, CJ!o
2 376 (2 378)
2.3 (1.9)
0.093 (0.082)
2 420 (2 420)
11.2 (11.2)
0.46 (0.46)
371.3 (372.2)
0.3 (0.08)
0.07 (0.02)
9.37
0.10
1.0
7.33 (7.35)
0.03 (0.03)
Seawater (six titrations)
arithmetic mean
standard deviation
coefficient of variation, CJ!o
2 129 (2 131)
1.8 (1.6)
0.086 (0.075)
2 052 (2 053)
10.9 (10.5)
0.53 (0.51)
363.6 (364.1)
0.4 (0.5)
0.12 (0.14)
9.76
0.13
1.4
7.73 (7.74)
0.03 (0.03)
214
·J
EVALUATION OF POTENTIOMETRIC SEAWATER TITRATIONS
Table 8
Resu/ts of calcu/ations on an on-board titration of Baffin Bay Deep water (34.321 %o salinity, 1.64 p.M phosphate and 18.36 p.M silicate). The
uncertainties of the /east-squares parameters were obtained from the variance-covariance matrix. Results for four and three parameters (the latter
within parentheses).
A~M.)
c. ~M.)
EKF
pH range 7.7-3.9,
41 titration points
2 261 ± 4.6
(2 263 ± 3.2)
2 147 ± 3.5
(2 147 ± 3.5)
pH range 7.7-3.2,
54 titration points
2 260 ± 2.7
(2 260 ± 5.3)
2 148 ± 3.2
(2 168 ± 5.6)
seawater off the Swedish West Coast of ca. ± 0.5°/oo
does not seem to be important for the goodness of fit.
The su rn of the squares of the errors is typically 5.10- 6
(ml) 2 for both sets. As can be seen from Table 7, the
precisions of the two sets are also quite similar. The
uncertainties of the !east squares parameters A, C,
EKFo and K 2c, as obtained from the variance-covariance
matrix, are approximately 1(~tM), 1(~tM), 0.1 (rn V) and
0.1.10-s (M- 1), respectively.
KZ< .• I0-5 (M-t)
pHF (start)
411.7 ± 1.0
(412.3 ± 0.2)
9.53 ± 0.4
7.69
(7.70)
410.4 ± 0.1
(411.0 ± 0.3)
9.09 ± 0.05
7.67
(7.68)
(mV)
of Nash. Since Stepit is coded in Fortran, a translation
into Basic would be necessary if a small microcomputer-based titration system without dises is used.
The algorithm of Nash (1979) is, however, more compact, a fact which is also important when microcomputers are used.
Thus we suggest that the modified Marquardt method
be used for the evaluation, although other minimisation algorithms may be useful for checking the procedure.
On-board titration •
Dependent variable
Results of the on-board titration of Baffin Bay deep
water are given in Table 8. The sum of the squares of
the errors is substantially larger than that of the
laboratory titrations, 5.10- 3 (ml) 2 (41 points), as compared with 5.10-6 (ml) 2 (26 points).
The results in Table 6 indicate that solving for E gives a
calculation procedure that is slightly less sensitive to a
systematic error in E. Table 9 shows that, for experimental titrations, change of dependent variable
shifts the resulting A, and C, 1 - 3 ~tMw. If, therefore,
reliable procedures for standardisation are developed,
care should be taken to check the dependence of the ac- . · ..
curacy on the choice of dependent variable.
When solving for E, ca. 1 1/2 hours is required, as
compared to 15 minutes when solving for v (PET 3016).
For practical reasons, we have therefore chosen to solve
for v.
DISCUSSION.
CHOICE OF PROCEDURE OF EVALUATION
Evaluation of the alkalinity titration with methods
employing non-linear curve-fitting should be tested
more extensively in practical work before any definite
conclusions as to their value and to the detailed choice
of procedure can be drawn. It is possible, however, to
use the combination of theoretical and practical results
presented in this work to make preliminary suggestions
of procedure.
Number of parameters
The results in Table 4 suggest that the five parameters
A, C, EK, K 2c and K 1c be used. However, calculations
on experimental titrations diverge rather easily with
five parameters, a fact which is not unexpected, since
the titration curve depends only·slightly on K 1c.
From the results of the laboratory titrations (Table 7),
it is difficult to draw any conclusions as to the suitable
number of parameters being three or four. The results
of the on-board titration, however, indicate th at threeparameter calculations might be less reliable (Table 8;
C, deviates markedly for 3 parameters, 54 points).
The residuals v = v <ulc- vexp• are less randomly
distributed for three parameters if the whole titration
Minimisation algorithm
In general, direct search methods have the advantage of
being comparatively easy to code and to be applicable
when functions cannot be written as analytical expressions. However, they tend to need larger computer
storage and to be less reliable and effective than
methods making use of the gradient (Nash, 1979).
There is very little difference between thé CPU-time
needed by Stepit and the time needed by the algorithm
Table 9
Comparison of ca/cu/ations performed with v ml andE rn V as the dependent variable. Resultsfor E rn V as the dependent variable are given within
parentheses.
theoretical titration
theoretical titration with superimposed systematic
error of 1.16 mY
laboratory titration (standard seawater)
on-board titration
A~M.)
C.~)
2 443 (2 443)
2 345 (2 345)
350.0 (350.0)
9.903 (9.903)
2 440 (2 441)
2 372 (2 374)
2 257 (2 254)
2 336 (2 338)
2 426 (2 425)
2 141 (2 138)
350.9 (351.0)
371.2 (371.9)
409.8 (409.3)
9.818 (9.821)
9.36 (9.56)
8.87 (8.74)
215
O. JOHANSSON, M. WEDBORG
1..
curve is used (pH range 7.7- 3.2, Table 8). When only
titration points above pH 4 are used (pH range 7. 7 3.9, Table 8), the residuals for three and four
parameters are almost identical..
If the uncertain points around the first buffer
minimum, v = 0,0.03 and 0.06 ml are excluded, A, is
hardly affected, but C, increases by ca. 10 p.M,.. Thus it
seems essential for the determination of A, and C, that
readings around the two buffer minima are used.
Considering the above results, we suggest that the four
parameters A, C, EK and K 2c be used for the calculations and that the titration curve should include the two
buffer minima.
a(A,)
negligible
1
· 3".:5 ..
theoretical titration
laboratory titration
on-board , titration
a(C,)
negligible
1.
.
-·3- 4
Experimental precision
The coefficient of variation for A, is slightly less than
0.1 OJo and for C, around 0.5% (Table 7). Values .of approximately 0.1%, have been previously reported for
bothA. and C, (Almgren et al., 1977; Takahashi et al.,
1980). Our experimental procedure has not yet been
thoroughly examined and no particular precautions
were taken to prevent carbon dioxide exchange between
the samples and air. We would therefore suggest that
the slightly poor precision in C. is more likely caused by
sample handling, or possibly experimental procedure,
than by the method of calculation.
Neglecting the presence of a base
The presence of a base, Bas, which is titrated partly or
completely within the pH region 8 - 4.5 and which is
neglected in the minimisation equations, will cause a
systematic error in C, which is equal to the total concentration of Bas if the stability constant of the corresponding acid, Kn"'' is close to K 2c. This error becomes
smaller as Kn"' increases.
The alkalinity is always much less affected since, of
course, the base is included in the alkalinity by definition. The small effect observed should mainly be due to
a poorer fit caused by neglecting a term in the model
equation.
Omission of phosphate will cause a measurable
systematic error only for deep water samples. In surface water, P, is too low (often around 0.1 p.M) to be of
any importance. Also, it appears to be a quite acceptable procedure to omit phosphate in the evaluation
and instead correct the value of C, by subtracting P,.
This implies that P, need not be known when the titration is evaluated.
Omission of silicate will not cause measurable errors
unless Si, approaches 1 mM.
From our results it is also possible to estimate the effects of neglecting sulphide and ammonia in the titration of anoxie seawater. The stability constants of H 2S
and NH;, K25 and KN, are approximately 6.72 and 9.39
(logarithmic values, pHF scale; Almgren et al., 1976;
Johansson, Wedborg, 1980). Sulphide is, therefore, expected to cause an error which is somewhat smaller
than the total concentration, S, (ca. O. 7 x S, according
to our results). Ammonia should give an effect similar
to that of silicate, the difference between log Ks, and
log KN being only 0.1. ,
ERROR IN DETERMINATION
<iF pH
Dyrssen and Hansson (1974) studied the relation between the error in the determination of pH in seawater
and the error in A, - C, dpH/ d(A, - C,). They obtained an uncertainty in pH of 0.16 - 0.30 logarithmic
units per 100 p.Mw uncertaintly in A, - C,. If the standard deviations for A, and C, in Table 7 are added, d(A,
- C,) = 13p.Mw is obtained. If the Dyrssen-Hansson
relation is used, dpH is calculated to be 0.02 - 0.04
logarithmic units. This result fits very weil with our experimental standard deviation of 0.03 logarithmic units
in pHF (start) (Table 7).
COMPARISON
DICKSON (1981)
WITH
THE
METHOD
OF
Dickson (1981) suggests that carbonic acid be treated as
a mixture of two monoprotic acids in order to simplify
Equation (1). We do not feel, however, that this implies
a useful simplification. Accordingly, we have used the
exact equation for our calculations.
A minor difference that should be of no practical importance is that instead of using EK as a parameter,
Dickson defined a parameter, f. lt is defined as the proportion with which the hydrogen ion concentration is in
error as a result of a systematic error in EK.
ln this work we have employed two minimisation
algorithms and solved for v as weil as E as the dependent variable. The methods employing the Marquardt
procedure have been tested on theoretical, laboratory
and field data. Dickson (1981) used the Marquardt
algorithm in combination with solving for v, and tested
his procedure on theoretical data.
Goodness of fit
The mean values of the residuals, ~v. for a theoretical,
laboratory and on-board titration are 2 . w-1, 3 . w- 4
and 6 . w- 3 ml, respectively. The residuals of the experimental titrations are somewhat less randomly
distributed than for the theoretical titration, but no obvious difference was observed between the laboratory
and on-board titrations. When the goodness of fit, expressed as the uncertainty in the least-squares
parameters from the variance-covariance matrix is
comp~red, a difference is detected :
pH scales
Any of the pH scales can be used with the non-linear
curve-fitting method. The modifications that must be
made in the equations when pH scales are changed and
216
EVALUATION OF POTENTIOMETRIC SEAWATER TITRATIONS
The results of the second test are ÂA, = 0.3 ~tM and
ÂC, = 0.1~tM. 1t was also found that the minimisation
procedure becomes Jess effective and diverges more
easily when, as Dickson (1981) suggests, the titration
points around the second equivalence point are excluded.
when EK is refined are clearly seen from the Nernst
equation :
Infinite dilution:
E = EKI + k.log [H]~o
where [H]1 = 'YH·[H]F
Ionie medium, "free":
E = EKf. + k.log [H]F
Ionie medium, "total":
E = EKr + k.log [H]Tt
where
[H]r = [H]F + [HSOJ + [HF].
Our conclusions from these tests are that the systematic
errors that might arise from the choice between the pHr
and pHF scales, when the region pH = 5 - 3.4 is included in the calculations, are small compared to, e.g.,
errors introduced by uncertainties in v. and Hb.
It is our opinion that the two ionie medium pH scales
are to be preferred to the infinite dilution scale, mainly
because uncertainties about a change of Iiquid junction
potentials are practically eliminated and because 'YH
need not be involved in the calculations. For the
modified Gran method of evaluation, the pHr scale has
the advantage of allowing EK to be determined from
severa! titration points in the region of excess acid,
whereas the pHF scale implies a buffer calibration.
When a non-linear curve-fitting method is used,
however, EK can be chosen as a parameter, disregarding
the choice of pH scale. Since choosing the pHF scale
should then make the calculations Jess sensitive to
systematie errors when the whole acid region of the
titration curve, down to pH 3, is included, we believe
that a slight preference for this scale is justified.
The pHr scale requires that the terms Su and FI on the
right hand side of Equation (1) be excluded. At the
same time, ali stability constants that are not refined
must be given on the pHr scale, i.e. the logarithmic
values should be ca. 0.13 lower than on the pHF scale
(at 35 %o salinity). When the infinite dilution scale is
applied, the terms Su and FI should, of course, be
included. The activity coefficient, ')'H, must be known
for the calculation of [H]/. The stability constants used
should be apparent constants, where the standard state
of H is .infinite dilution and of ali other species, the
ionie medium. Bates (1975) gives log 'Y ± (HCI)
==" -0.136 at the ionie strentgh of seawater, and
thus the apparent stability constants should be on the
order of 0.13 - 0.14 logarithmie units larger than the
constants defined on the pHF scale (35 °/oo salinity).
Change of pH scale
In order to check the validity of the equations in our
program, we calculated on one of the titrations of standard seawater on the pHr scale for v = 0 - 0.75 ml.
The differences between the calculations on the pH,
and the pHr scales are:
Choice of pH scale
When forming the mass balance equations, we chose
the "free" ionie medium scale of Bates (1975). This
choice implies that the formation of HF and HS0 4 is
accounted for in the mass balance equation for H, and
th us that titration points in the acid region, between pH
3 and 5, can be used without any risk of introducing
systematic errors. Dickson (1981) suggests that the
"total" ionie medium scale should be used. As a consequence the equation for H, is not strictly valid below
pH 5.
ÂA,
ÂC,
ÂEK
Âiog K 2c
-0.02~tM ..
=
0.07 ~tM ..
7.93 mV
0.1338.
Theoretically, ÂEK and Âiog K 2c should be 7.98 and
0.1349. The agreement is th us good.
Dickson points out that the major advantage of his
choiee of scale is that the values calculated for A, and
C, become independent of uncertainties in Ks" and KFf.
We have tried to estimate the systematic errors introduced into the determination of A, and C. as a result
of the choice of pH scale by making two simple calculations (five parameters):
1. The perfect titration curve is fitted {o the perfect
mode! equation, on the pHF scale, where an erroneous
value of Ksu is inserted.
2. The mode! et)uation is transformed to the pHr scale
and used for fitting the whole titration curve (i.e. the
region pH = 5 - 3.4 is included).
COMPARISON WITH OTHER METHODS OF
EVALUATION
Barron et al. (1982) compared our non-linear curvefitting program (CF) with three other computer programs, one of which was based on the original Gran
method (ALCT) and the other two on the modified
Gran method (Geosecs and Alkfini). Alkfini is the
refined version of the modified Gran method (Bradshaw, Brewer, 1980; Bradshaw et al., 1981). Not surprisingly, the results of Barron et al. (1981) showed the
best agreement between the CF and Alkfini methods.
The maximum differences obtained between these two
methods for natural seawater samples were 4 ~tM .. for
A, and 8 ~tM .. for C, and the minimum differences were
1 ~tM .. for both A, and C.. The maximum differences
are obviously larger than could be tolerated if an ac-
The first test shows that even if Ksu is wrong by as much
as a factor of 5, the errors introduced are less than 1~tM
(ÂA, = -0.3 ~tM, ÂC, = -0.6 ~tM for Ksu = 59.35
and ÂA, = 0.05 ~tM, ÂC, = 0.1 ~tM for Ksu = · 2.37).
If Ksu is wrong by a factor of 2, the systematie errors
are reduced to around ± 0.1 ~tM.
217
O. JOHANSSON, M. WEDBORG
curacy and precision of 0.1% is accepted as a goal.
These results clearly show the need for an alkalinity
standard to be found.
Acknowledgements
We wish to thank A. Dickson who generously shàred
his knowledge of the mathematical procedures used for
non-linear curve-fitting. He also provided us with his
Basic version of the modified Marquardt algorithm.
The Fortran version of Stepit was sùpplied by L.
Lyhamn, Department of lnorganic Chemistry, University of Lund. M. Aronsson collaborated in coding and
modifying the Basic version of algorithm 18 in Nash
(1979) as a project within the undergraduate chemistry
program at the University of Gôteborg. Also; we would
like to thank D. Dyrssen for valuable discussions and
helpful criticism and J.L. Barran and E.P. Jones for
making their field data available to us. The work on the
seawater carbonate system at our department is supported by the Swedish Natural Science Research Council.
CONCLUSIONS
Our conclusion from the results obtained in this work,
by Dickson (1981) and by Barron et al. (1981) is that
non-linear curve-fitting is weil suited for evaluating the
alkalinity titration. Since a reliable alkalinity standard
is not available, it is not possible to decide if the curvefitting method makes possible an improved accuracy,
compared to the modified Gran treatment. We
therefore believe it is of value to use both these methods
until such a standard is available.
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218