Download Calculating the conductivity of natural waters

LIMNOLOGY
and
OCEANOGRAPHY: METHODS
Limnol. Oceanogr.: Methods 6, 2008, 489–501
© 2008, by the American Society of Limnology and Oceanography, Inc.
Calculating the conductivity of natural waters
Rich Pawlowicz*
Department of Earth and Ocean Sciences, University of British Columbia, 6339 Stores Road, Vancouver, British Columbia,
Canada, V6T 1Z4.
Abstract
An algorithm is developed to compute the conductivity of lake and dilute ocean water from measured chemical composition at arbitrary temperature and pressure. The complex mixed electrolyte is considered as a sum
of binary electrolytes rather than a sum of ions. Effects of ion association are included, and it is found that pairing effects are important in natural freshwaters. Bounds on the accuracy of the algorithm for specific classes of
binary electrolytes are assessed and it is estimated that the algorithm has an overall accuracy of better than 2%
for salinities less than about 4 g L–1. Comparison with seawater conductivities is much better than 1%, but predicted conductivities of some published analyses of river waters are about 3% too high. Some of this difference
may be due to a lack of data on ion pairing effects between bivalent metals and bicarbonate, but also may result
from uncertainties in the measured chemical composition and measured conductivity. An iterative procedure
incorporating this algorithm is used to compute reference conductivity at 25°C and salinity from in situ measurements of conductivity in waters where only relative amounts of ions are known. It is found that the conversion to reference conductivity is reasonably independent (to within about 1%) of the ionic composition for
most world river waters, but is somewhat different than that for KCl solutions. However, derived salinities are
quite sensitive to the composition, and the ratio of ionic salinity to reference conductivity varies between 0.6
and 0.9 mg L–1 (μS cm–1)–1.
practice is to convert measurements of conductivity κ (μS
cm–1) at a temperature T (°C) to a reference conductivity κ25 at
a fixed temperature of 25°C using the approximate temperature dependence of a typical electrolyte:
κ
κ 25 =
(1)
1 + 0.0191ΔT
where ΔT =T −25 is the temperature anomaly from 25°C
(Clesceri et al. 1998, hereafter SMEWW). Note that most freshwater has T < 5°C. Reference conductivity then is converted to
a measure of total dissolved solids (TDS, mg L–1) using a relationship such as
Introduction
The salinity (mass of dissolved salts per unit mass of solution) of seawater has for many years been estimated operationally by measurements of its conductivity, and empirical
equations exist to convert between the two (Lewis 1980). As a
practical matter, this is possible because the relative ratios of
the ionic constituents are similar enough in all oceans that
similar conductivities yield similar salinities (with an error
<<0.1%). The situation is somewhat different in freshwater
systems. Dissolved ions are present in amounts large enough
that a conductivity (in the range of 5 to 7,000 μS cm–1) can be
easily measured, but, as the relative ratios of these ions can
vary widely, a conversion to salinity is not straightforward.
However, once salinity is available, density and a number of
other thermodynamic properties can be computed using standard parameterizations which, for many purposes, are insensitive to exact composition (Chen and Millero 1986). A standard
TDS = 0.65κ25
(2)
(Snoeyink and Jenkins 1980), where the scale factor for different systems is known to vary between 0.55 and 0.9 mg L–1 (μS
cm–1)–1 in general usage (SMEWW), and as high as 1.4 in
meromictic saline lakes (Hall and Northcote 1986). The error
arising from Eq. 1 when applied to natural waters is not
known, and the error arising from Eq. 2 could be as much as
30%. Accurate determinations of salinity, which are required
in many deep and weakly stratified systems where salinity is
an important contributor to density, and also in those of more
exotic chemistry, require some knowledge of the ionic composition and/or are restricted to specific systems for which
empirical governing relationships can be determined.
*Corresponding author: E-mail: [email protected].
Acknowledgments
Discussions with Roger Pieters, Bernard Laval, and the comments of
an anonymous reviewer greatly improved this work, which was supported by the Natural Sciences and Engineering Research Council of Canada
through grant 194270-04.
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Rich Pawlowicz
Conductivity of natural waters
ne = ν1z1 = ν2z2. νi are the moles of the ith constituent ion
formed from each mole of electrolyte, and zi the algebraic
valency (absolute value of charge) on that ion. At low concentrations, the equivalent conductivity of a dissociated electrolyte can be represented by a limiting Debye-HückelOnsager equation of the form
SMEWW present a semi-empirical method for relating salinity
and κ25 (i.e. determining Eq. 2) by summing up infinite dilution
conductivities for all constituents and then scaling by a specified function of ionic strength. This provides good results for
North American fresh waters. Sorensen and Glass (1987)
assumed constancy of the product of viscosity and a power of
conductivity (a modified Walden product) and developed a pHdependent formula for adjusting conductivities for lakes in
Minnesota, Wisconsin, and Michigan (i.e. to avoid Eq. 1).
McManus et al. (1992) avoided both problems by developing an
empirical equation directly relating measured conductivity to
salinity in Crater Lake. This was done by fitting a polynomial to
simultaneous measurements of conductivity and temperature
from two dilutions of lake water over a range of temperatures
(similar to the method used for seawater). Wüest et al. (1996),
hereafter W96, presented a general parameterization for predicting the conductivity of lake water from measurements of
ionic composition and used this to develop polynomial relationships between measured conductivities and salinity for Lake
Malawi. A similar approach that requires some speciation information a priori was described by Talbot et al. (1990).
In general, little information has been presented about the
general reliability and limitations of the different approaches
(although they proved useful in the particular cases considered), nor have they been formally developed. The physical
chemistry of limnological conductivity is still poorly understood (Hall and Northcote 1986). Below, we review existing
theory, and then formally develop and assess an algorithm for
predicting the conductivity of natural waters of known composition at arbitrary temperature and pressure, and for determining reference conductivity, using principles of physical
chemistry. The algorithm has two novel features: first, the complex electrolyte is considered as a sum of binary electrolytes
rather than a sum of ions, and second, the effects of ion association (as well as relaxation and electrophoresis) are included
in the parameterization of binary electrolytes. All numerical
parameters (other than pressure dependence) are determined
from basic chemical data. We also outline a procedure for estimating salinity from measured conductivities using our algorithm. Comparisons are made to previous formulations and
the accuracy of the algorithm is evaluated using predictions of
freshwater and dilute seawater conductivities.
(
Λ = Λ o − AΛ o + B
I.
(4)
where Λ0 is the infinite dilution equivalent conductivity, different for every electrolyte, and I (mol L–1) the stoichiometric
ionic strength:
N
N
1 c
c c
I = ∑ ν i zi2 = ∑ ci zi2
(5)
2 i
2 i
Here c i= νic are the ionic concentrations, and Nc = 2 the number of ionic constituents. The first form for I is useful when
dealing with simple electrolytes; the second will be useful
when dealing with natural waters (i.e. when Nc > 2). A,B are
constants that can be developed from theory in a dilute solution, depend on physical factors such as the viscosity and
dielectric constant, and respectively represent the leading
terms in an expansion of the relaxation effect as the solution
rearranges behind the moving ions, and an electrophoretic
effect from the viscous drag of neighboring moving ions. Formal expansions to higher orders of I have been developed
(e.g., Quint and Viallard 1978), but the analytic forms are
complex. In addition, the expansions are of terms such as
1/(1+ka’) where a’ is an “ionic diameter” which varies by less
than a factor 2 and k can be expressed as a function of other
fundamental parameters. Numerically, ka’ is of order
,which must formally be <<1. Validity is thus
marginal in lakewaters for ionic constituents with a valency
>1 for any order of expansion. More complex theoretical
approaches also have been applied to less dilute systems (e.g.,
Anderko and Lencka 1997). As a practical matter many different quasi-empirical forms parameterizing finite dilution
effects have been proposed (see, e.g., Robinson and Stokes
1970, Horvath 1985, or Barthel et al. 1998 for summaries) to
provide useful predictions at greater concentrations than
those modeled by Eq. 4. All of these approaches generally are
limited to simple binary electrolytes that dissociate into an
anion and a cation and require constants which are determined from data.
At infinite dilution, the equivalent conductivity of a binary
electrolyte Λ° is exactly the sum of infinite dilution ionic
equivalent conductivities for the cation (λ+°) and anion (λ°):
–
Theory
The problem of predicting the conductivity of dilute electrolytes is of general chemical interest and much theoretical
and experimental work has been carried out (e.g., Robinson
and Stokes 1970; Barthel et al. 1998). It is convenient to deal
with the equivalent conductivity Λ (μS cm–1 (mol L–1)–1), which
can be related to the measured conductivity κ of a binary electrolyte of arbitrary concentration by:
κ = Λcn e
)
Λ o = λ+o + λ−o
(6)
and so at small concentrations a modification of Eq. 3:
Nc
κ = ∑ ci zi λi ( I )
(7)
i
with Nc = 2 taken as a starting point for determining the conductivity of dilute systems in general, where the ionic equivalent conductivities λi are each dependent on ionic strength
and can be modeled by an equation which has a limiting form
(3)
where c is the concentration (mol L–1) of the electrolyte, and
490
Rich Pawlowicz
Conductivity of natural waters
virtually nonexistent. However, noting that most ions other
than H + and OH – have similar λ°i 25 (see, e.g. Table 1) we can estimate crudely that the conductivity of the right hand side of
this equilibrium is about one half that of the left hand side
solely because of the difference in the number of charges. Thus
pairing will still reduce the conductivity greatly.
To avoid the complexity and difficulties with lack of data in
the standard approach, which is required for investigations
into chemistry but, we believe, is overly complicated for the
more practical matter of predicting conductivity to reasonable
accuracy at the current state of knowledge, our approach will
therefore be to write
similar to Eq. 4. An obvious generalization proposed by W96
and Talbot et al. (1990) for natural waters is to use Eq. 7 with
Nc > 2 (i.e. to add together the effects of all ions present). A
problem with this approach is that the conductivity of some
systems (notably 2:2 electrolytes like MgSO4 but also a number
of 1:1 electrolytes involving NO 3– or ClO 4–) is somewhat less
than this sum at even very small concentrations. Ion pairing,
a mechanism not included in Eq. 4, provides a self-consistent
explanation for this effect.
Ion pairing or association occurs when oppositely charged
ions are closer than a certain critical distance. They then act as
a neutral molecule (in symmetrical electrolytes) and hence do
not contribute to conductivity. The standard way of dealing
with this situation is to presume a dissociated fraction α% ≤1,
so that
2
κ = α ( I )∑ ci zi λi ( I )
where α(I )≤1 now is defined as an ionic-strength-dependent
reduction factor for pairing and speciation effects in a binary
electrolyte. Its functional form (discussed below) will be
assumed constant for particular valency combinations, with
an amplitude that is correlated with the available association
constants when not available directly from data.
2
κ = α% ∑ ci zi λi (α% I )
(11)
i
(8)
i
Parameterization of α% involves both an association “constant”
KA for the particular ion pair involved through an equilibrium
of the form
Materials and procedures
with
KA =
1 − α%
α% 2cf 2
In order to take into account ion pairing, we generalize the
existing theory for binary electrolytes to one for complex electrolytes by considering the system as a weighted sum of binary
electrolytes formed by partitioning ions into all combinations
of available anions and cations, rather than a sum of ions as
in Eq. 7. The weighting of the partition is by ionic equivalent
fractions, which was found by Anderko and Lencka (1997) to
be effective in modeling ternary electrolytes. We have:
(9)
where f ≤ 1 is an activity coefficient for the ions parameterizing the deviations from ideal thermodynamic behavior of the
solution (Davies 1962). In fact both KA and f also are functions
of temperature and ionic strength. In very dilute solutions α%
decreases with ionic strength, but this decrease can slow or
(for ions with higher valencies) even pass through a minimum
at an ionic strength of around 0.1–1 mol L–1 and then increase.
Note that the conductivity parameterization described by
SMEWW also uses the activity, but applies it somewhat inappropriately as a conductivity reduction factor for finite concentrations:
N+ N −
κ = ∑∑
j
Ceq
Λ jk
μS cm −1
(12)
where the equivalent concentration Ceq (mol L–1) is defined as
N
N+
⎛ N−
⎞
1 c
Ceq = ∑ c+j z j ⎜⎜= ∑ ck− z k , = ∑ ci zi ⎟⎟
(13)
2 i
j
⎝ k
⎠
Nc
κ 25 = f 2 ∑ ci zi λiο 25
k
c+j z j ck− z k
(10)
and can be computed from the sum of either cation or anion
ionic equivalent concentrations ( c+j z j or ck− z k respectively)
which are in principle identical in a neutral electrolyte. Here j
subscripts refer to cations with concentrations c+j (mol L–1), k
subscripts to anions with concentrations ck− , Λjk is the equivalent conductivity of a binary subsystem formed from the combination of one of the N+ cations and one of the N– anions,
and Nc = N+ + N–.
As we typically have N+, N– ~5 in lakes, using Eq. 12 implies
knowing the equivalent conductivity of ∼25 compounds over
a range of temperature, concentration, and pressure. Such data
is not available. Instead we write:
i
with f (I ) parameterized using the ion-independent monovalent Davies equation at 25°C (Davies 1962) and λ°i 25 infinite
dilution equivalent conductivities at 25°C. As we shall see, this
semiempirical procedure performs surprisingly well for lake
waters, but badly for seawater of the same conductivity.
If the electrolyte is a 2:1 system, say Mg(HCO3)2, then ion
pairs also may form, but the pair has a charge and therefore
can contribute to conductivity:
The theoretical basis for dealing with this situation in the standard approach is complex (Davies 1962; Barthel et al. 1998)
and as a practical matter, although association constants and
activities have been estimated for the constituents of natural
waters (Millero 2001), other necessary data, e.g., infinite dilution ionic equivalent conductivities for forms like MgHCO 3+, are
Λjk = (λj + λk)αjk Γ(p )
(14)
so that we must only parameterize the equivalent conductivities for ∼10 individual ions, adding an ion pairing reduction
factor αjk for those combinations that require it, and using a
491
Rich Pawlowicz
Conductivity of natural waters
Table 1. Ion-dependent numerical parameters. Limnologically important ions are listed at the top.
fi = F1ΔT + F2ΔT 2 + F3ΔT 3
zi
λi°25
ai
F1
F2
F3
-
mS cm–1 (mol L–1)–1
(mol L–1)–1/2
(°C) –1
(°C) –2
(°C) –3
Ca 2+
Mg 2+
Na +
NH 4+
K+
Cl –
HCO 3–
NO 3–
F–
CO 32–
SO 42–
2
2
1
1
1
1
1
1
1
2
2
59.47
53.00
50.08
73.50
73.48
76.31
44.50
71.42
55.40
69.30
80.00
0.85
0.90
0.80
0.62
0.62
0.62
0.91
0.74
0.80
0.74
0.74
2.213
1.901
2.189
1.931
1.941
2.012
1.909
1.874
2.089
1.772
2.119
H+
OH –
1
1
349.65
198.00
0.30
0.31
1.388 × 10–2
1.644 × 10–2
Ba2+
Sr 2+
Ag +
Li +
Br –
I–
ClO 4–
Zn 2+
Cu 2+
La3+
Fe(CN6)4+
2
2
1
1
1
1
1
2
2
3
4
63.60
59.40
61.90
38.66
78.10
76.80
67.30
52.80
53.60
69.70
110.4
0.83
0.84
0.62
0.96
0.60
0.61
0.66
0.80
0.80
0.81
0.62
2.084
2.096
1.969
2.294
1.960
1.950
1.724
2.200
2.329
2.191
Ion
general pressure dependence. Note that this formulation (Eq.
12–14) can be rewritten as:
Nc
κ = ∑ ci zi λi ( I )α i Γ( p)
where αi ≤ 1 is an equivalent-fraction-weighted average of the
αjk associated with all the other cations for an anion and all
the other anions for a cation. This shows the relationship
between the procedure developed here with expansions of the
form Eq. 7.
There are many possible parameterizations for the ionic
equivalent conductivities. After testing several, we find that a
simple representation of the form
λiο
1 + ai zi I
(16)
is sufficiently accurate. Temperature and pressure dependence
in situations other than T = 25°C and p = 0 is parameterized
by multiplicative terms:
λiο = λiο 25 (1 + fi (ΔT )) μScm −1 (mol L−1 )−1
×
×
×
×
×
×
×
×
×
×
-
10–2
10–2
10–2
10–2
10–2
10–2
10–2
10–2
10–2
10–2
10–2
10–2
10–2
10–2
10–2
10–2
10–2
10–2
10–2
10–2
10–2
×
×
×
×
×
×
×
×
–6.001 ×
6.746 ×
9.854
2.096
9.159
1.113
5.786
6.217
1.573
6.764
10–5
10–5
10–5
10–4
10–5
10–5
10–5
10–5
10–5
10–5
–3.152 × 10–5
1.403 × 10–5
8.885
7.322
4.308
1.313
6.678
6.370
–2.359
1.287
1.676
8.119
×
×
×
×
×
×
×
×
×
×
-
10–5
10–5
10–5
10–4
10–5
10–5
10–5
10–4
10–4
10–5
–1.027 ×
–1.127 ×
2.546 ×
–1.593 ×
–4.921 ×
–1.320 ×
8.948 ×
-
10–6
10–6
10–6
10–7
10–8
10–6
10–7
–6.372 × 10–7
4.500 × 10–6
4.451 × 10–7
4.371 × 10–7
7.755 × 10–7
-
ai = ai25 (1 + βΔT ) (mol L−1 )−1/ 2
(18)
Γ( p) = 1 + γ p ( p)(1 + γ T (ΔT )) (−)
(19)
At constant temperature and pressure, there is only one
parameter (ai), usually slightly less than 1 in these units,
that must be determined empirically for each ion, but having this term in the denominator gives it a slightly wider
range of validity than a form like Eq. 4. This form is simpler
than the 4-term expansions generally recommended for
analysis of binary electrolytes (Barthel et al. 1998) and thus
cannot be valid at salinities or conductivities that are too
high, but as we must estimate the unknown coefficients for
some important ions (like CO 3–) without sufficient data it is
more robust. fi are polynomials that express approximate
doubling in λ°i over temperatures between 0° and 35°C, and
β a change of about 10% in ai over the same range. Pressurerelated changes are relatively small (of order 0.1% over 100
m), but are included for completeness. One advantage of
this parameterization is that the numerical parameters other
than λ°i 25 are generally very similar. This facilitates the
approximation of effects arising from more exotic ions not
included here, if desired.
(15)
i
λi =
×
×
×
×
×
×
×
×
×
×
×
(17)
492
Rich Pawlowicz
Conductivity of natural waters
Table 2. Numerical parameters taken as similar for all ions. See
various constituents. Alternatively, it may be useful to use
measurements of conductivity to estimate the salinity in a particular system. Rather than develop a second set of equations
specifically for this purpose, we can perform this calculation
by iteration. Note that salinity determination is not possible
without making additional assumptions about the chemical
composition of the system. In the simplest case, we assume
that the system is composed of dilutions or concentrations of
a known chemical composition, i.e. we have concentrations ĉi
(mol L–1) for each constituent (both ionic and nonionic) in a
standard composition, so that at any actual location the concentrations are ci=dĉi where d is an unknown scale factor. The
true salinity S' (in units of g L–1 of solution) is:
text for details.
Parameter
β
γp (p )
γT (ΔT )
1:1 I 0jk,
1:2 I 0jk,
2:1 I 0jk,
2:2 I 0jk,
1:4 I 0jk,
Value
1/300 (°C)–1
9.0718 × 10–6p –2.8013 × 10–10p 2 (–)
–2.5751 × 10–2ΔT + 8.46082 × 10–4ΔT 2 (–)
0.6 (mol L–1), 0.24 (–)
0.1 (mol L–1), 0.32 (–)
0.1 (mol L–1), 0.32 (–)
0.2 (mol L–1), 0.28 (–)
0.1 (mol L–1), 0.54 (–)
wjk
wjk
wjk
wjk
wjk
We parameterize the pairing reduction using a Gaussian curve:
α jk = 1 − Ajk e
− w jk ( log I −log
I 0jk )2
S′ =
(20)
∑
ci Mi = d
i
where the center I 0jk and width wjk are taken to be constant for
particular combinations of valencies, so that only an amplitude Ajk must be specified for each ion pair. The center I 0jk is
larger than typical ionic strengths of interest (i.e. we generally
need only a monotonically increasing segment of this functional form). Temperature dependence is again of the form
Ajk = A25
1 + βΔT )
jk (
Nc + N n
Nc + N n
∑ cˆ M
i
(22)
i
i
with Mi the molar mass of a particular constituent. The sum is
over both the Nc ionic and Nn nonionic constituents. We note
that in situ measured conductivity:
Nc
Nc
i
i
κ = ∑ λi ( I )αi ( I )ci zi = d ∑ λi ( I (d ))α i ( I (d ))ĉci zi
(23)
depends sensitively on d through a multiplicative term on the
RHS, and weakly through a dependence of λi and αi on ionic
strength I, which in turn depends on d. To determine d, we
take an initial guess d 0, compute I, and thence the summation
on the RHS of (23), dividing the measured conductivity κ by
this sum to generate a new estimate d 1, and then repeat until
values d 0, d 1, d 2, ..., d N, ... converge, which tends to happen
within a few iterations. The reference conductivity κ25 is not
actually needed to determine salinity, but if required it can then
be computed by setting T = 25°C in Eq. 23 once d is known.
Conversion of the salinity S' to salinity S (g kg–1 solution)
requires a determination of the density ρ which itself has a
salinity dependence (Chen and Millero 1986),
(21)
Numerical values for all ionic parameters are listed in Table 1,
and those that are taken as similar for all ions in Table 2. Interaction amplitudes are given in Table 3. In addition, lim, a toolbox containing programs implementing these procedures (as
well as all required ancillary calculations) in MATLAB, is available from the author at http://www.eos.ubc.ca/~rich.
Note that we have estimated these parameters for 24 ions
and not just the 11 limnologically important ones in order to
better assess the algorithm, and to facilitate application in
waters with exotic chemistry.
Equations 12–21 can be used to determine the conductivity
of a sample with known chemical composition at specified temperature and pressure. Salinity is found merely by summing the
S = S'/ρ (S,T ,p )
(24)
Table 3. Numerical parameters for ion pairing interactions. Numbers with two decimal places have been estimated in some way: trailing ‘+’ are estimates from data at 25°C, trailing ‘–’ are from lower quality data mostly at 18°C, and remaining values were estimated by
correlation and/or similarity. Single digit numbers are values set to zero in the absence of any quantifiable information, even though
some, e.g. combinations of F – and bivalent metals, are known to pair strongly.
A j2k5(–)
F–
Cl –
Br –
I–
NO 3–
ClO 4–
HCO 3–
SO 42 –
CO 32 –
Fe(CN6 )4 –
Li +
Na+
K+
Ag +
NH 4+
Mg 2+
Ca2+
Sr 2+
Ba 2+
Zn 2+
Cu 2+
0
0.00–
0.00–
0
0.00–
0.00+
0
0.00+
0.00–
0.00+
0.00+
0.00+
0
0
0
0
0
0.00+
0
0
0
0.00+
0
0
0
0.00+
0
0
0
0.00+
0
0
0
0.00+
0
0
0
0.04–
0
0
0
0.04
0
0
0.00–
0.00+
0
0.015–
0.11
0
0.00–
0.00+
0.00+
0.01+
0.11+
0.04–
0.00+
0.02+
0.00+
0.02–
0.12+
0.10+
.010+
0
0
0.06–
0.11
0
0.01–
0
0
0.04–
0.11
0
0.00–
0
0.10
0.27–
0.35
0
0.01–
0
0.10
0.32–
0.45
0
0.03
0
0.10
0.13
0.20
0
0.05–
0
0.10
0.30
0.42
0
0
0
0.20
0.33+
0.66
0
0
0
0.30
0.36+
1.00
0
493
Rich Pawlowicz
Conductivity of natural waters
Ion pairing for bivalent metal sulfates was parameterized
using data published by Tomšič et al. (2002) and Bešter-Rogač
et al. (2005) over a range of temperatures, as well as Lide
(2007) at 25°C. Data for some other electrolytes for which
pairing effects needed to be estimated were available only at a
temperature of 18°C in Dean (1999); the general quality (accuracy, number of values) of this data appeared lower than that
found in Lide (2007). When no conductivity data was available, we used correlations between fitted Aj2k5 and association
constants in electrolytes of like valency, tabulated in Smith
and Martell (1976), Millero (2001), and Millero and Hawke
(1992), to estimate other Aj2k5.
The pressure dependence was determined from measurements of the conductivity of seawater with salinities in
the range of 2 to 35, temperatures from 0 to 30°C and pressures from 0 to 4000 dbar (Bradshaw and Schleicher 1980). As
salinity dependence is weak over the limnological range and
including it would complicate the method, we ignore it by
extrapolating data to a salinity of zero.
An independent data set for binary compounds at a temperature of 18°C was taken from Dean (1999) for verification
of predictions. Practical accuracy was determined by comparison with measured conductivities and chemical compositions
of lake and river waters listed in Wetzel (2001) and Hamilton
(1978). For seawater comparisons, we computed a salinity
using equations determined by Perkin and Lewis (1980) with
the low-salinity correction of Hill et al. (1986) for a given conductivity and temperature. We then used a salinity-dependent
chemical composition described by Millero (2001) to predict
conductivity using our algorithm at the same temperatures.
This produced substantially better results than direct comparisons with an equation for conductivity as a function of salinity (Poisson 1980) due to a lack of low-salinity data in derivation of the latter.
Again this can be done iteratively, substituting numerical values of S' for S on the RHS, computing a new S and then resubstituting on the RHS until convergence occurs.
In particular situations other assumptions and/or knowledge of the system may apply. For example, the system may be
a mixture of two different standard compositions. The above
procedure could be modified to estimate the mixing faction.
In addition, computation of density (or other derived parameters such as static stability) to a useful accuracy may sometimes depend on details of the chemical composition rather
than on salinity alone (e.g., W96).
Data sources
The parameterization (12–21) is carefully written to allow
us to determine the unknown numerical parameters in a
straightforward way. First, we parameterize the relationships
at T = 25°C and p = 0 dbar using data available from numerous sources. Values for λ°i 25 are readily available for a large variety of ions and generally can be taken from a single modern
source; we use Lide (2007). To determine ai25, data is required
for λi25 at different concentrations, sources were Lide (2007),
except for carbonate data from Dean (1999). As such data is
provided only for binary electrolytes and not individual ions
(i.e. for Λj2k5 = λj25 + λk25) an arbitrary initial choice must be
made. We take ai25 to be identical for Cl – and K+. Thereafter the
ionic equivalent conductivity for all other ions was obtained
by difference, e.g., λ(Na+) = Λ(NaCl ) – λ(Cl –), where we use
(12–21) for λ(Cl –). Although not obvious in the original tables,
during data reduction, a small number of points appeared to
show aberrant behavior compared to trends through neighboring points. These were deleted from the analysis. An inconsistency arose with tabulations of K2CO3 and Na2CO3 which
were listed as being measured at 18°C in both Dean (1999),
Dobos (1975), and Milazzo (1963), but whose trend using
Eq. 4 was more consistent with Λ0 at 25°C from those and
other sources.
Coefficients for the temperature dependence fi are obtained
by fitting polynomials to data available in older sources for a far
smaller number of ions in the range 0–35°C. Tabled values of λj
for a given ion sometimes are numerically different in different
sources, likely by scale factors due to changes in the definitions
of standard physical constants and/or experimental issues, so it
is more satisfying to use them only to determine relative
changes. For most ions we take values at seven temperatures
from Robinson and Stokes (1970). Values at three temperatures
for CO32–, HCO3– and several metals were found in Dean (1999),
and a few more exotic ions were listed in Horvath (1985).
A small joint temperature/concentration dependence
occurs through variations in ai. Examination of this joint
dependency is more speculative as tabulated data is scarce.
However, on theoretical grounds, we expect the temperaturedependency in ka’ terms to be somewhat independent of the
particular ions so we take a best fit for tabulated KCl data (Lide
2007) as being generally applicable.
Assessment
We first assess the degree to which the available data shows
the decomposition into ionic conductivities (Eq. 7) to be valid
when pairing does not occur, irrespective of the parameterizations, by finding sets of four binary electrolytes whose equivalent conductivities should sum to zero at concentrations
where this is true. For example, the sum Λ(KCl )−Λ(KI )−
Λ(NaCl )+Λ(NaI ) computed from tabulated values is less than
0.2% of Λ(KCl ) at concentrations up to 0.1 mol L–1, but for
some other quads, the residual was less than 0.4% only for
concentrations less than 0.01 mol L–1. This would then represent a maximum achievable accuracy. We use such comparisons to identify binary electrolytes with weakest pairing
effects, and then add/subtract them to estimate equivalent
conductivities for particular ions at tabulated concentrations.
Ionic equivalent conductivities at finite concentrations
Having estimated values for λi25, we rearrange (Eq. 16) to
form an expression for ai25zi as a function of ionic strength:
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Rich Pawlowicz
Conductivity of natural waters
ai25 zi =
1 ⎛ λiο 25 ⎞
⎜ 25 − 1⎟
I ⎝ λi
⎠
(25)
0.3%, although this likely represents a slight overfitting (i.e.
the polynomial order is higher than required), which may
mask inaccuracies in the data. Differences in the order of the
predicting polynomial arise because some ions are tabulated
less extensively than others so third order fits are not always
possible.
The effect of joint temperature and concentration dependence is small but systematic. Figure 2 shows the relative error
in predicted conductivity at a variety of temperatures for a
0.01 mol kg–1 KCl solution, with and without a β = 1/300 (°C)–1
dependence. In order to correctly account for this effect, it is
necessary to convert concentrations into units of mol L–1
using density computed according to Chen and Millero (1986)
as the systematic error from neglecting this is larger than the
joint dependence. The remaining underestimate of about
0.2% arises from the ai25 parameterization. We find that an
identical parameterization adequately reduces the temperature dependence of pairing effects in metal sulfates and thus
assume it to be general.
Results are shown in Fig. 1 for all ions considered. Note the
general clustering of lines for ions of similar valencies, with
clusters at equal intervals, demonstrating the validity of the
dependence on zi, with a few exceptions. Use of a constant ai25
is valid in regions where the curves are relatively flat. The
curves tend to bend downwards at higher concentrations and
the effect is more noticeable for higher valencies. The values
in Table 1 are an average over the three measurements for
which I < 0.01 mol L–1.
Curves for H + and OH – are flat, but somewhat below the
other monovalent species; this is because their chemistry is
slightly different. CO 32– appears above the other trivalent
species, somewhat near the curve for the only ion with zi = 4,
Fe (CN )64+. Also, both of these curves are not very flat, but have
a broad peak. This is an indicator that ion pairing is occurring
in the data from which these conductivities were derived. The
subtractive procedure performs even less well for modeling 2:2
compounds. We derive a reasonable ai25 for SO 42– from Na2SO4,
but if we try to model Zn 2+ and Cu 2+ (both of which are tabulated as compounds with SO 42–), the curves suggest an ai25
increasing from about 6 to 12 as ionic strengths increase from
0.0005 to 0.01. We therefore set ai25 for Cu 2+, Zn 2+, F – (for
which we have no data at 25°C), and FeCN 64+ to a typical value
for their valence, and also take the value for carbonate to be
the same as that for sulfate.
Pairing effects
To determine the pairing reduction factor parameterization
(Tables 2, 3), we compare predictions without pairing to measurements of the actual conductivities of various binary subsystems at different ionic strengths. The ratio of these two is
αjk(I ), and estimates of this from available data are shown in
Fig. 3. After some experimentation, it was found that pairing
data generally was well described by one side of a Gaussian
curve in log I . The Gaussian curve is described by three
parameters – a width, amplitude, and center. As the available
data does not constrain the center very well (curves whose
Temperature effects
Fitting polynomials for the relative temperature dependence of λi° is straightforward, and the residuals are all less than
Fig. 1. Variation of aizi with ionic strength for 24 ions with valencies of
1 to 4. Note that CO32–, Cu2+, Zn2+, and Fe(CN6)4+ are not well modeled by
this parameterization (curves are neither flat nor clustered with others of
similar valency), an indication that ion pairing effects are present in the
electrolytes from which the ion-specific data was derived.
Fig. 2. Relative error in predicted temperature dependence of KCl at a
concentration of 0.01 mol kg–1. Shown are the errors arising from ignoring the density correction, correcting for density (but without the temperature correction for ai), and finally from the full parameterization.
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Rich Pawlowicz
Conductivity of natural waters
Fig. 3. Ion pairing effects for variety of binary electrolytes at (a) 25°C and (b) 18°C. Symbols represent available data, and lines are estimated functional
forms, fit by eye.
carbonate amplitudes by the ratio of logKA, and then taking
only half of this amplitude to take into account the charge on
the pairs as discussed earlier. Alternatively, we note that association constants for the 1:2 metal bicarbonates and Na2CO3
are very similar, and use the same reduction. Both approaches
give approximately the same value, but we use the latter as
being somewhat safer. Note that the comparisons with measured river waters discussed later can be greatly improved if we
double these reduction factors, but at present, there seems to
be no justification for doing so.
Electrolytes with NO 3– and ClO 4– are also slightly prone to
pairing, and we see in Fig. 3 a decrease of a few percent at I =
0.1 mol L–1. Cl – shows signs of pairing with Zn 2+ (and hence
presumably with Cu 2+ as well), but not with any other ion. We
have limited data for F – which is known to pair strongly with
bivalent metals, but since such combinations are not limnologically important, we make no correction. No information is
known about the behavior of Br – and I – with bivalent metals.
Bivalent metal hydroxides also are known to associate reasonably strongly, but again we have no conductivity data.
peaks are rightwards and above the present ones have a virtually identical goodness-of-fit) it seemed more useful to set the
width and center parameters to empirical constants for each
set of valencies, changing only the amplitude of the curve for
different ion pairs. Amplitudes are fitted by eye, with a bias
towards matching the data at lower ionic strengths, as inaccuracies in our parameterization (16) which is used in deriving
the plotted points becomes large at higher strengths. The
largest reductions are found for 2:2 metal sulfates.
The available derived amplitudes correlate well with association constants (logKA ) tabulated in Millero (2001) allowing
us to estimate the others. We derive amplitudes for bivalent
metal carbonate αjk by scaling those for bivalent metal sulfates
by the ratio of tabulated logKA. Pairing effects of around 5% at
I = 0.1 mol L–1 are seen in 1:2 sulfates in spite of the lower data
quality at 18°C. This is about 20% of the effect seen in 1:2 carbonates, although, in this case, tabulated association constants suggest that the pairing effects should be similar, so correlation does not seem useful here. Instead, we estimate the
reduction for unknown 1:2 carbonates by setting them to the
value found for Na2CO3.
Metal bicarbonates are a limnologically important subsystem. Association constants for metal carbonates and bicarbonates are well correlated (Millero and Hawke 1992), so we
could estimate the reduction amplitude by scaling the metal
Overall accuracy for binary electrolytes
Fig. 4a–c shows the relative error in predictions of conductivity without pairing against tabulated values for 30 binary
electrolytes at 25°C. The error when pairing effects are included
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Rich Pawlowicz
Conductivity of natural waters
Fig. 4. Relative error in our parameterizations for 65 binary electrolytes. (a–e) are predictions without the pairing parameterization, (f–j) include pairing. Note change of vertical scale. (a,f) 1:1 electrolytes at 25°C. (b,g) 2:1 and 2:2 electrolytes at 25°C. (c,h) 3:1 and 4:1 electrolytes at 25°C. (d,i) 1:1
electrolytes at 18°C. (e,j) 2:1 and 2:2 electrolytes at 18°C.
this arises from disagreement in values of λi° at 18°C derived
from different sources. The comparisons in Fig. 4 are perhaps
overly accurate, as most of the data also was used in estimation
of one or more of our empirical parameters.
is shown in Fig. 4f–h and generally is less than 1% for concentrations up to 0.01 mol L–1. Three points in Fig. 4a and 4f below
the rest are for NaHCO3. Interestingly, the underestimate is
roughly constant with changing concentrations, a pattern not
generally seen when other parameterization errors arose, possibly suggesting a systematic error in tabulated data. Other than
these anomalies the error is a mix of over- and under-estimates.
We also find the prediction error for a further 34 compounds
whose conductivities at 18°C were listed in Dean (1999) (Fig.
4d–e without pairing, and 4i–j with pairing effects included). A
typical error is on the order of 2% for concentrations up to 0.01
mol L–1 in all cases, and somewhat better than that for 1:1 electrolytes, with predicted conductivities being biased downwards
at highest ionic strengths. Note that many of the curves appear
to be shifted vertically (accounting for much of the scatter);
Accuracy for natural waters
Although the performance of our algorithm for binary electrolytes is satisfactory, it is not clear if similar results can be
expected for more complex (e.g., natural) systems. The most
comprehensive measurements of natural systems are those for
seawater with salinities in the range of 0.005 to 42. Using the
standard seawater salinity/conductivity relationship and the
ionic composition of seawater (ignoring the minor constituent
B (OH )4– for which we have no chemical data), we show the relative error in our algorithm in Fig. 5. The difference between the
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Rich Pawlowicz
Conductivity of natural waters
Fig. 6. Comparison between measured and predicted κ25 for 32 North
Fig. 5. Relative difference (thick lines) between conductivities predicted
American rivers (discrete symbols, Hamilton 1978), as well as for seawater for salinities of 0.1–3 (lines). Shown are results from our full algorithm,
with and without pairing effects, and without ionic-strength-dependence,
as well as the methods proposed by W96 and SMEWW. a) All points. b)
Seawater only (note change in vertical scale).
for seawaters of different salinities and temperatures using both the algorithm developed here and standard seawater equations (see text for
details). Thin lines are contours of conductivity (μS cm–1). Conductivitybased salinities are unitless but are numerically close to g kg–1.
two is < 1% for salinities < 3 over virtually all temperatures, (i.e.
over roughly a doubling in conductivity at fixed salinity), with
largest errors at 0°C. As this bias is systematic, it may be due to
lesser accuracy and/or scarcity of λ°i data at low temperatures.
Next, the effectiveness of the algorithm at 25°C for a variety of North American waters is shown in Fig. 6. The observational dataset is not without its own problems, which we
can see when first estimating the conductivity by simply
summing up the infinite dilution conductivities for each ion
(i.e. using λi = λi° in Eq. 7). As this would be an upper bound
(it ignores all ionic interactions, and overestimates conductivity by about 16% in this dataset), it suggests that at least
two data points near κ25 ≈ 50 μS cm–1 are problematic. We
apply our algorithm without ion pairing and find it gives
results almost identical to those using the algorithm of W96
(also shown). Thus including relaxation and electrophoresis
effects reduces the overestimation to about 8%. Including the
ion pairing effects of bivalent metal sulfates reduces the error
to about 4% on average, and adding the effects of metal carbonate pairing using the full algorithm reduces this to 3%,
with a root-mean-square difference (RMSD) of 5%. The
SMEWW algorithm overestimates the conductivity by about
2% (RMSD 4%) for this particular set of freshwater data. Performance of our algorithm is not as good as the comparison
with binary electrolytes, but much of this mismatch could
occur because of uncertainties in the chemical analysis. The
relative error Ê in κ25 arising from the chemical analysis can
be estimated by normalizing the difference between anion
and cation sums (and by taking advantage of the fact that κ25
≈ 105Ceq in our units for fresh waters):
Δκ 25
Ê =
=
κ 25
∑
∑
N+
j
N+
j
z j c+j − ∑ z k ck−
N−
k
z j c+j + ∑ z k ck−
N−
(26)
k
We find Ê is about 2% here, typical for high-quality data with
κ25 > 300 μS cm–1 (SMEWW). In addition, laboratory measurements of κ25, even when done properly, typically have an
uncertainty of at least 1−2%. This could account for half of the
RMSD, suggesting that the parameterization error is only a few
percent at most.
We also show on this figure the error in predictions of conductivity for dilute seawater at 25°C with salinities of 0.1 to 3.
As discussed earlier, agreement is very good (to better than
0.2% for κ25 < 3000 μS cm–1). However, the limnological “tuning” of the SMEWW algorithm for freshwaters is shown clearly
as it underestimates the conductivity of seawaters by up to
20%. Pairing related effects are small in seawater (a reduction
of about 0.3%), but still improve the results substantially. The
error for our algorithm curves downwards with increasing
conductivity (which is roughly proportional to ionic
strength), in the same fashion as the curves in Fig. 4, whereas
that of W96 curves upwards at similar conductivity.
To illustrate the wider utility of this algorithm, we consider
results for typical fresh waters. Several tables describing the
ionic composition of world river waters are given in Wetzel
(2001). Table 10-1 of that reference provides a breakdown by
continent, and table 10-2 provides a breakdown by the dominant rock type in the drainage. In addition, we take ionic composition of several lakes from other papers. From this database, we compute both ionic salinity and reference
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Rich Pawlowicz
Conductivity of natural waters
Fig. 7. (a) κ25 for κ5 = 100 μS cm–1. At top are shown results for two specific lakes (McManus et al. 1992; W96) as well as a dilute seawater and a 1
mmol L–1 KCl solution. Next are world average and averages by river type, followed by world average and averages by continent (Wetzel 2001). At bottom is the result assuming Eq. 1. Both the full algorithm described here as well as that of W96 and SMEWW are used. (b) κ25 for same ionic compositions. (c) Ratio of ionic salinity to κ25 for all water types.
μS cm–1, but in most situations careful calibration would be
required to differentiate between the two. Repeating the procedure for a measured conductivity of 1000 μS cm–1 results in
a slightly larger variation of about 3%. It appears that the
major advantage of our algorithm is not in computing reference conductivity itself (except as a quality-control measure
for chemical analyses), but rather in computing the salinity
(or at least the numerical factor in Eq. 2). By combining the
parameterization of W96 for computing conductivity with our
iterative procedure for determining reference conductivity, we
can test the apparent difference between the two parameterizations sensitively. Their algorithm gives values about 1 μS
cm–1 higher in all situations except for the KCl solution (which
is well-modeled). The constant difference suggests a systematic bias between the two parameterizations, which is likely
related to the form (16) as ion pairing corrections do not seem
to affect the reference conductivity calculation to any great
extent. Although the calculation in SMEWW is not claimed to
be valid for temperatures more than a few degrees away from
25°C we also can apply it in this case. As they also take temperature dependence to be constant for all constituents, not
surprisingly, the predicted reference conductivity is independent of chemical composition; it is also somewhat high.
conductivity (for this comparison, we have neglected the nonionic components of salinity which can be substantial and
must be accounted for in practical situations). Figure 7b shows
that computed reference conductivity varies greatly from very
small values in granitic basins to large values in carbonate
basins. Differences in the computed κ25 values using this algorithm and that of W96 can be as large as 5%, but this is too
small to be easily apparent in this presentation. The differences are more apparent when we show the ratio of salinity to
reference conductivity (Fig. 7c); this is equivalent to determining the numerical coefficient in Eq. 2. This ratio varies
widely from a low of 0.65 for African rivers and sandstone
drainages to a maximum of 0.9 for carbonate drainages. The
generally lower conductivities using our algorithm result in
ratios higher by 1-5% than those found using W96, but
slightly smaller than those using SMEWW. The ratios for
dilute seawater and KCl solutions are around 0.5, somewhat
smaller than that for most lake waters.
Finally, in Fig. 7a, we illustrate the effect of differing chemical composition by computing the reference conductivity for
a measured conductivity of 100 μS cm–1 at T = 5°C for each
case. The computed κ25 vary between 160 and 161.6 μS cm–1
for all compositions, i.e. we can consider the correction to be
composition-independent to within about ±1%. Equation 1
performs admirably in spite of its crudeness, predicting a
numerical correction within 1% of our composition-dependent corrections. Taking the composition to be pure KCl is less
useful, as the computed reference conductivity is less than 159
Discussion
The conductivity of natural waters depends on chemical
composition. Conductivity is reduced by the effects of relaxation and electrophoresis at finite concentrations, and by ion
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Conductivity of natural waters
solids in freshwater and estuarine systems with a TDS of less than
around 4 g L–1, of arbitrary composition. A reliable determination
of the accuracy of our algorithm is limited by uncertainties in
routine chemical analysis, but it appears to be accurate to at least
within a few percent on average, and perhaps much better.
Even if more traditional techniques for salinity determination
are utilized, this algorithm is useful as a check on the results.
association between certain pairs of ions. Temperature
dependence is strong over the limnological range, but is somewhat independent of the ionic composition. Pressure dependence is smaller. The formal assessment of our algorithm clearly
indicates the shortcomings of existing conductivity parameterizations and provides guidance about the uncertainties.
We find that the simple temperature correction in Eq. 1 is
sufficient to determine reference conductivity with an error of
around 1% for all water types considered. Sorensen and Glass
(1987) found previously that accounting for ionic composition changed the correction to reference conductivity by an
average of 0.3% for lakes in Minnesota, Wisconsin, and Michigan. Conversion to salinity does require a knowledge of the
ionic composition.
Electrophoresis and relaxation effects were accounted for in
parameterizations described by W96 and Talbot et al. (1990).
However, our formulation, which requires only one empirically-derived parameter per ion to model ionic-strengthdependence, appears to provide results as good as those of the
former when ion association does not occur, and is valid to
slightly greater concentrations than the more formal theoretical expansions utilized in the latter.
The strategy of considering natural waters as a mix of (possibly associated) binary electrolytes, rather than a simple sum
of ions as in all other parameterizations described, gives much
better results in lake waters because it naturally allows ion
pairing effects to be included, and the weighting by equivalent
fractions reduces to a sum of ions when no pairing effects
occur. The numerical results indicate that ion association
effects are not modeled in W96, and that accounting for these
effects leads to a significant improvement. Although our ion
association parameterization is relatively crude, it is sufficient
to show that ion association is important in fresh waters when
κ25 is greater than about 50 μS cm–1, but rather less so in dilute
seawaters of even the same (approximate) salinity. Also, it is
found that pairing effects in bivalent metal sulfates generally
are more important than those in bivalent metal carbonates in
their effect on reductions in conductivity. Talbot et al. (1990)
accounts for changes in conductivity due to speciation in
metal bicarbonates, but requires speciation information a priori, and does not include the more important effects arising
from bivalent metal sulfates. An empirical reduction factor
that presumably accounts for all three effects, described in
SMEWW, may perform slightly better than our algorithm for
some waters, but it clearly underperforms in dilute seawaters
and hence is not necessarily trustworthy.
Both W96 and Talbot et al. (1990) include a correction for
temperature assuming the separability of temperature and
concentration dependencies, but for more precise results a
small joint dependence also must be accounted for, as we do
here (SMEWW is valid only near 25°C).
In addition to evaluating previously described methods of
computing conductivity, the algorithms described here should
have wide utility in providing better estimates of the dissolved
Comments and recommendations
The largest uncertainty in the development of the conductivity algorithm is related to the effects of ion pairing with
nonsymmetrical (i.e. 1:2) electrolytes. Not only are the available theoretical guidelines very complex, but accurate laboratory measurements of conductivity (especially for the important carbonate and bicarbonate subsystems) are lacking.
For practical reasons, we have largely avoided speciation
issues in developing the algorithm, with the exception of the
carbonic acid system. Carbonate and bicarbonate ions each
are accounted for explicitly in the conductivity calculation.
However, chemical analyses often characterize this system
using measurements of total alkalinity and pH. To convert to
ionic concentrations, dissociation constants are required;
Millero (1995) provides the required equations valid over a
range of temperatures at ocean salinities of 0-5. Approximate
freshwater salinities (e.g. estimated via Eq. 2), appropriately
scaled to the nondimensional ocean salinity scale, may be substituted. A limnological equation of state (Chen and Millero
1986) should be used to convert from units of mol kg–1 solution to the units of mol L–1 that are used here. A further issue
is that the carbonate and bicarbonate ionic concentrations are
assumed to remain constant with changes in temperature in
the reference conductivity and salinity calculations. In fact,
the pH of a closed system will increase by order 0.1 for a 10°C
decrease in temperature and this can change the ionic concentrations slightly. The dependence may be worth investigating when highest precision is required.
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Submitted 14 February 2007
Revised 5 October 2007
Accepted 7 February 2008
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