Calculating the conductivity of natural waters - Earth and Ocean ...

1Calculating the conductivity of natural waters2Rich Pawlowicz ∗3Sept 26 20074Running head: Conductivity of natural waters∗ Dept. of Earth and Ocean Sciences, University of British Columbia 6339 Stores Rd., Vancouver, BritishColumbia, Canada, V6T 1Z4. email: rich@eos.ubc.ca, tel: 604-822-1356

1Abstract2345678910111213141516171819An algorithm is developed to compute the conductivity of lake and dilute ocean waterfrom measured chemical composition at arbitrary temperature and pressure. The complexmixed 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 innatural freshwaters. Bounds on the accuracy of the algorithm for specific classes of binaryelectrolytes are assessed and it is estimated that the algorithm has an overall accuracy of betterthan 2% for salinities less than about 4 g L −1 . Comparison with seawater conductivities ismuch better than 1%, but predicted conductivities of some published analyses of river watersare about 3% too high. Some of this difference may be due to a lack of data on ion pairingeffects between bivalent metals and bicarbonate but may also result from uncertainties inthe measured chemical composition and measured conductivity. An iterative procedure incorporatingthis algorithm is used to compute reference conductivity at 25 ◦ C and salinityfrom in situ measurements of conductivity in waters where only relative amounts of ions areknown. 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 somewhatdifferent than that for KCl solutions. However, derived salinities are quite sensitive to thecomposition, and the ratio of ionic salinity to reference conductivity varies between 0.6 and0.9 mg L −1 (µS cm −1 ) −1 .2

1Acknowledgements234Discussions with Roger Pieters, Bernard Laval, and the comments of an anonymous reviewergreatly improved this work, which was supported by the Natural Sciences and Engineering Re-search Council of Canada through grant 194270-04.3

1Introduction234567891011121314The salinity (mass of dissolved salts per unit mass of solution) of seawater has for manyyears been operationally estimated by measurements of its conductivity, and empirical equationsexist to convert between the two (Lewis, 1980). As a practical matter this is possible because therelative ratios of the ionic constituents are similar enough in all oceans that similar conductivitiesyield similar salinities (with an error ≪ 0.1%). The situation is somewhat different in freshwatersystems. Dissolved ions are present in amounts large enough that a conductivity (in the range of 5to 7000 µS cm −1 ) can be easily measured, but as the relative ratios of these ions can vary widelya conversion to salinity is not straightforward. However, once salinity is available, density anda number of other thermodynamic properties can be computed using standard parameterizationswhich are for many purposes insensitive to exact composition (Chen and Millero, 1986). Astandard practise is to convert measurements of conductivity κ (µS cm −1 ) at a temperature T ( ◦ C)to a reference conductivity κ 25 at a fixed temperature of 25 ◦ C using the approximate temperaturedependence of a typical electrolyte:κ 25 =κ1 + 0.0191∆T(1)151617where ∆T = T − 25 is the temperature anomaly from 25 ◦ C (Clesceri et al., 1998, hereafterSMEWW). Note that most freshwater has T < 5 ◦ C. Reference conductivity is then converted toa measure of total dissolved solids (TDS, mg L −1 ) using a relationship such asTDS = 0.65κ 25 (2)4

1234567891011121314151617181920212223(Snoeyink and Jenkins, 1980), where the scale factor for different systems is known to varybetween 0.55 and 0.9 mg L −1 (µS cm −1 ) −1 in general usage (SMEWW), and as high as 1.4 inmeromictic saline lakes (Hall and Northcote, 1986). The error arising from eqn. (1) when appliedto natural waters is not known, and the error arising from eqn. (2) could be as much as 30%. Accuratedeterminations of salinity, which are required in many deep and weakly stratified systemswhere 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 forwhich empirical governing relationships can be determined. SMEWW present a semi-empiricalmethod for relating salinity and κ 25 (i.e. determining eqn. 2) by summing up infinite dilutionconductivities for all constituents and then scaling by a specified function of ionic strength. Thisprovides good results for North American fresh waters. Sorensen and Glass (1987) assumedconstancy of the product of viscosity and a power of conductivity (a modified Walden product)and developed a pH-dependent formula for adjusting conductivities for lakes in Minnesota, Wisconsin,and Michigan (i.e. to avoid eqn. 1). McManus et al. (1992) avoided both problems bydeveloping an empirical equation directly relating measured conductivity to salinity in CraterLake. This was done by fitting a polynomial to simultaneous measurements of conductivity andtemperature from two dilutions of lake water over a range of temperatures (similar to the methodused for seawater). Wüest et al. (1996), hereafter W96, presented a general parameterizationfor predicting the conductivity of lake water from measurements of ionic composition and usedthis to develop polynomial relationships between measured conductivities and salinity for LakeMalawi. A similar approach that requires some speciation information a priori was described byTalbot et al. (1990).In general little information has been presented about the general reliability and limitations5

12345678910111213of the different approaches (although they proved useful in the particular cases considered), norhave they been formally developed. The physical chemistry of limnological conductivity is stillpoorly understood (Hall and Northcote, 1986). Below we review existing theory, and then formallydevelop and assess an algorithm for predicting the conductivity of natural waters of knowncomposition at arbitrary temperature and pressure, and for determining reference conductivity,using principles of physical chemistry. The algorithm has two novel features: first, the complexelectrolyte is considered as a sum of binary electrolytes rather than a sum of ions, andsecond, the effects of ion association (as well as relaxation and electrophoresis) are includedin the parametrization of binary electrolytes. All numerical parameters (other than pressure dependence)are determined from basic chemical data. We also outline a procedure for estimatingsalinity from measured conductivities using our algorithm. Comparisons are made to previousformulations and the accuracy of the algorithm is evaluated using predictions of freshwater anddilute seawater conductivities.14Theory1516171819The problem of predicting the conductivity of dilute electrolytes is of general chemical interestand much theoretical and experimental work has been carried out (e.g., Robinson andStokes, 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 electrolyteof arbitrary concentration by:κ = Λcn e (3)6

12345where c is the concentration (mol L −1 ) of the electrolyte, and n e = ν 1 z 1 = ν 2 z 2 . ν i are the molesof the ith constituent ion formed from each mole of electrolyte, and z i the algebraic valency(absolute value of charge) on that ion. At low concentrations the equivalent conductivity of adissociated electrolyte can be represented by a limiting Debye-Hückel-Onsager equation of theformΛ = Λ ◦ − (AΛ ◦ + B) √ I. (4)67where Λ ◦ is the infinite dilution equivalent conductivity, different for every electrolyte, and I(mol L −1 ) the stoichiometric ionic strength:I = c 2∑N ciν i z 2 i = 1 2∑N cic i z 2 i (5)891011121314151617Here c i = ν i c are the ionic concentrations, and N c = 2 the number of ionic constituents. Thefirst form for I is useful when dealing with simple electrolytes; the second will be useful whendealing with natural waters (i.e. when N c > 2). A, B are constants that can be developed fromtheory 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 thesolution rearranges behind the moving ions, and an electrophoretic effect from the viscous dragof neighbouring 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 expansionsare of terms such as 1/(1 + ka ′ ) where a ′ is an “ionic diameter” which varies by less than afactor 2 and k can be expressed as a function of other fundamental parameters. Numerically, ka ′is of order z i√I (∼ z2i√ c) which must formally be ≪ 1. Validity is thus marginal in lakewaters187

12345678for ionic constituents with a valency greater than 1 for any order of expansion. More complextheoretical approaches have also been applied to less dilute systems (e.g., Anderko and Lencka,1997). As a practical matter many different quasi-empirical forms parameterizing finite dilutioneffects have been proposed (see, e.g., Robinson and Stokes (1970), Horvath (1985), or Barthelet al. (1998) for summaries) to provide useful predictions at greater concentrations than thosemodelled by eqn. (4). All of these approaches are generally limited to simple binary electrolytesthat 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 sumof infinite dilution ionic equivalent conductivities for the cation (λ ◦ + ) and anion (λ◦ − ):9Λ ◦ = λ ◦ + + λ◦ − (6)10and so at small concentrations a modification of (3):∑N cκ = c i z i λ i (I) (7)i11121314151617with N c = 2 is taken as a starting point for determining the conductivity of dilute systemsin general, where the ionic equivalent conductivities λ i are each dependent on ionic strengthand can be modelled by an equation which has a limiting form (4). An obvious generalizationproposed by W96 and Talbot et al. (1990) for natural waters is to use (7) with N c > 2 (i.e. to addtogether the effects of all ions present). A problem with this approach is that the conductivityof some systems (notably 2:2 electrolytes like MgSO 4 but also a number of 1:1 electrolytesinvolving NO3 − or ClO4 − ) is somewhat less than this sum at even very small concentrations. Ion8

12345pairing, a mechanism not included in (4), provides a self-consistent explanation for this effect.Ion pairing or association occurs when oppositely charged ions are closer than a certain criticaldistance. They then act as a neutral molecule (in symmetrical electrolytes) and hence donot contribute to conductivity. The standard way of dealing with this situation is to presume adissociated fraction ˜α ≤ 1, so thatκ = ˜α2∑c i z i λ i (˜αI) (8)i67Parameterization of ˜α involves both an association “constant” K A for the particular ion pairinvolved through an equilibrium of the formMg 2+ + SO 2−4 ⇄ MgSO 48withK A = 1 − ˜α˜α 2 cf 2 (9)91011121314where f ≤ 1 is an activity coefficient for the ions parametrizing the deviations from ideal thermodynamicbehavior of the solution (Davies, 1962). In fact both K A and f are also functions oftemperature and ionic strength. In very dilute solutions ˜α decreases with ionic strength, but thisdecrease can slow or even (for ions with higher valencies) pass through a minimum at an ionicstrength of around 0.1–1 mol L −1 and then increase. Note that the conductivity parameterizationdescribed by SMEWW also uses the activity, but applies it somewhat inappropriately as a9

1conductivity reduction factor for finite concentrations:∑N cκ 25 = f 2ic i z i λ ◦25i (10)234567with f(I) parameterized using the ion-independent monovalent Davies equation at 25 ◦ C (Davies,1962) and λ ◦25iinfinite dilution equivalent conductivities at 25 ◦ C. As we shall see, this semiem-pirical procedure performs surprisingly well for lake waters, but badly for seawater of the sameconductivity.If the electrolyte is a 2:1 system, say Mg(HCO 3 ) 2 , then ion pairs may also form but the pairhas a charge and can therefore contribute to conductivity:Mg 2+ + 2HCO − 3 ⇄ MgHCO + 3 + HCO − 38910The 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 activi-ties have been estimated for the constituents of natural waters (Millero, 2001), other necessarydata, e.g., infinite dilution ionic equivalent conductivities for forms like MgHCO 3 + , are virtually11nonexistent. However, noting that most ions other than H + and OH − have similar λ ◦2512i(see, e.g.1314151617Table 1) we can crudely estimate that the conductivity of the right hand side of this equilibrium isabout 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, whichis required for investigations into chemistry but, we believe, is overly complicated for the more10

12practical matter of predicting conductivity to reasonable accuracy at the current state of knowl-edge, our approach will therefore be to writeκ = α(I)2∑c i z i λ i (I) (11)i3456where α(I) ≤ 1 is now defined as an ionic-strength-dependent reduction factor for pairing andspeciation effects in a binary electrolyte. Its functional form (discussed below) will be assumedconstant for particular valency combinations, with an amplitude that is correlated with the availableassociation constants when not directly available from data.7Materials and Procedures89101112In order to take into account ion pairing we generalize the existing theory for binary electrolytesto one for complex electrolytes by considering the system as a weighted sum of binaryelectrolytes formed by partitioning ions into all combinations of available anions and cations,rather than a sum of ions as in eqn. (7). The weighting of the partition is by ionic equivalentfractions, which was found by Anderko and Lencka (1997) to be effective in modelling ternary13electrolytes. We have:N + N∑∑−κ =jkc + j z jc − k z kC eqΛ jk µS cm −1 (12)14where the equivalent concentration C eq (mol L −1 ) is defined asN∑ +C eq = c + j z jj(N −)∑= c − k z k , = 1 ∑N cc i z i2ki(13)11

12and can be computed from the sum of either cation or anion ionic equivalent concentrations (c + j z jor c − k z k respectively) which are in principle identical in a neutral electrolyte. Here j subscriptsrefer to cations with concentrations c + j (mol L−1 ), k subscripts to anions with concentrations c − k ,34 Λ jk is the equivalent conductivity of a binary subsystem formed from the combination of one of5678the N + cations and one of the N − anions, and N c = N + + N − .As we typically have N + , N − ∼ 5 in lakes using (12) implies knowing the equivalent conductivityof ∼ 25 compounds over a range of temperature, concentration, and pressure. Suchdata is not available. Instead we write:Λ jk = (λ j + λ k )α jk Γ(p) (14)91011so that we must only parameterize the equivalent conductivities for ∼ 10 individual ions, addingan ion pairing reduction factor α jk for those combinations that require it, and using a generalpressure dependence. Note that this formulation (eqns. 12-14) can be rewritten as:∑N cκ = c i z i λ i (I)α i Γ(p) (15)i12131415where α i ≤ 1 is an equivalent-fraction-weighted average of the α jk associated with all the othercations for an anion and all the other anions for a cation. This shows the relationship betweenthe procedure developed here with expansions of the form (7).There are many possible parameterizations for the ionic equivalent conductivities. After12

1testing several, we find that a simple representation of the formλ i =λ ◦ i1 + a i z i√I(16)is sufficiently accurate. Temperature and pressure dependence in situations other than T = 25 ◦ Cand p = 0 is parameterized by multiplicative terms:λ ◦ i = λ ◦25i (1 + f i (∆T)) µS cm −1 (mol L −1 ) −1 (17)a i = a 25i (1 + β∆T) (mol L −1 ) −1/2 (18)Γ(p) = 1 + γ p (p)(1 + γ T (∆T)) (−) (19)2345678910At constant temperature and pressure there is only one parameter (a i ), usually slightly less than1 in these units, that must be determined empirically for each ion, but having this term in the denominatorgives it a slightly wider range of validity than a form like (4). This form is simpler thanthe 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 mustestimate the unknown coefficients for some important ions (like CO3 − ) without sufficient data itis more robust. f i are polynomials that express approximate doubling in λ ◦ i over temperaturesbetween 0 ◦ C and 35 ◦ C, and β a change of about 10% in a i over the same range. Pressure-relatedchanges are relatively small (of order 0.1% over 100 m) but are included for completeness. One11advantage of this parameterization is that the numerical parameters other than λ ◦25iare gener-1213ally very similar. This facilitates the approximation of effects arising from more exotic ions notincluded here, if desired.13

1We parameterize the pairing reduction using a gaussian curve:α jk = 1 − A jk e −w jk(log √ I−log √ I 0 jk )2 (20)2where the center I 0 jk and width w jk are taken to be constant for particular combinations of valen-cies, so that only an amplitude A jk must be specified for each ion pair. The center Ijk 0 3than typical ionic strengths of interest (i.e. we generally need only a monotonically increasingis larger45segment of this functional form). Temperature dependence is again of the formA jk = A 25jk (1 + β∆T) (21)67891011121314151617Numerical values for all ionic parameters are listed in Table 1, and those that are taken assimilar for all ions in Table 2. Interaction amplitudes are given in Table 3. In addition, lim, atoolbox 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 importantones in order to better assess the algorithm, and to facilitate application in waters withexotic chemistry.Equations (12-21) can be used to determine the conductivity of a sample with known chemicalcomposition at specified temperature and pressure. Salinity is found merely by summingthe various constituents. Alternatively, it may be useful to use measurements of conductivityto estimate the salinity in a particular system. Rather than develop a second set of equationsspecifically for this purpose we can perform this calculation by iteration. Note that salinity deter-14

f i = F 1 ∆T + F 2 ∆T 2 + F 3 ∆T 3Ion z i λ ◦25i a i F 1 F 2 F 3- mS cm −1 (mol L −1 ) −1 (mol L −1 ) −1/2 ( ◦ C) −1 ( ◦ C) −2 ( ◦ C) −3Ca 2+ 2 59.47 0.85 2.213 × 10 −2 9.854 × 10 −5 −1.027 × 10 −6Mg 2+ 2 53.00 0.90 1.901 × 10 −2 2.096 × 10 −5 -Na + 1 50.08 0.80 2.189 × 10 −2 9.159 × 10 −5 −1.127 × 10 −6NH 4 + 1 73.50 0.62 1.931 × 10 −2 1.113 × 10 −4 2.546 × 10 −6K + 1 73.48 0.62 1.941 × 10 −2 5.786 × 10 −5 −1.593 × 10 −7Cl − 1 76.31 0.62 2.012 × 10 −2 6.217 × 10 −5 −4.921 × 10 −8HCO3 − 1 44.50 0.91 1.909 × 10 −2 1.573 × 10 −5 −1.320 × 10 −6NO3 − 1 71.42 0.74 1.874 × 10 −2 6.764 × 10 −5 8.948 × 10 −7F − 1 55.40 0.80 2.089 × 10 −2 - -CO3 2− 2 69.30 0.74 1.772 × 10 −2 −6.001 × 10 −5 -SO4 2− 2 80.00 0.74 2.119 × 10 −2 6.746 × 10 −5 -H + 1 349.65 0.30 1.388 × 10 −2 −3.152 × 10 −5 −6.372 × 10 −7OH − 1 198.00 0.31 1.644 × 10 −2 1.403 × 10 −5 4.500 × 10 −6Ba 2+ 2 63.60 0.83 2.084 × 10 −2 8.885 × 10 −5 -Sr 2+ 2 59.40 0.84 2.096 × 10 −2 7.322 × 10 −5 -Ag + 1 61.90 0.62 1.969 × 10 −2 4.308 × 10 −5 -Li + 1 38.66 0.96 2.294 × 10 −2 1.313 × 10 −4 4.451 × 10 −7Br − 1 78.10 0.60 1.960 × 10 −2 6.678 × 10 −5 4.371 × 10 −7I − 1 76.80 0.61 1.950 × 10 −2 6.370 × 10 −5 7.755 × 10 −7ClO4 − 1 67.30 0.66 1.724 × 10 −2 −2.359 × 10 −5 -Zn 2+ 2 52.80 0.80 2.200 × 10 −2 1.287 × 10 −4 -Cu 2+ 2 53.60 0.80 2.329 × 10 −2 1.676 × 10 −4 -La 3+ 3 69.70 0.81 2.191 × 10 −2 8.119 × 10 −5 -Fe(CN 6 ) 4+ 4 110.4 0.62 - - -Table 1: Ion-dependent numerical parameters. Limnologically important ions are listed at thetop.15

Parameter Valueβ 1/300 ( ◦ C) −1γ p (p) 9.0718 × 10 −6 p − 2.8013 × 10 −10 p 2 (-)γ T (∆T) −2.5751 × 10 −2 ∆T + 8.46082 × 10 −4 ∆T 2 (-)1:1 Ijk 0 , w jk 0.6 (mol L −1 ), 0.24 (-)1:2 Ijk 0 , w jk 0.1 (mol L −1 ), 0.32 (-)2:1 Ijk 0 , w jk 0.1 (mol L −1 ), 0.32 (-)2:2 Ijk 0 , w jk 0.2 (mol L −1 ), 0.28 (-)1:4 Ijk 0 , w jk 0.1 (mol L −1 ), 0.54 (-)Table 2: Numerical parameters taken as similar for all ions. See text for details.A 25jk (-) Li + Na + K + Ag + NH 4 + Mg 2+ Ca 2+ Sr 2+ Ba 2+ Zn 2+ Cu 2+F − 0 0.00- 0.00- 0 0 0 0 0 0 0 0Cl − 0.00- 0.00+ 0.00+ 0 0.00+ 0.00+ 0.00+ 0.00+ 0.00+ 0.04- 0.04Br − 0.00- 0 0.00+ 0 0 0 0 0 0 0 0I − 0 0.00+ 0.00+ 0 0 0 0 0 0 0 0NO3 − 0.00- 0.00- 0.00+ .010+ 0.01- 0.00- 0.01- 0.03 0.05- 0 0ClO4 − 0.00+ 0.00+ 0.02+ 0 0 0 0 0 0 0 0HCO3 − 0 0.00+ 0.00+ 0 0 0.10 0.10 0.10 0.10 0.20 0.30SO4 2− .015- 0.01+ 0.02- 0.06- 0.04- 0.27- 0.32- 0.13 0.30 0.33+ 0.36+CO3 2− 0.11 0.11+ 0.12+ 0.11 0.11 0.35 0.45 0.20 0.42 0.66 1.00Fe(CN 6 ) 4− 0 0.04- 0.10+ 0 0 0 0 0 0 0 0Table 3: Numerical parameters for ion pairing interactions. Numbers with 2 decimal placeshave been estimated in some way: trailing ‘+’ are estimates from data at 25 ◦ C, trailing ‘-’ arefrom lower quality data mostly at 18 ◦ C, and remaining values were estimated by correlationand/or similarity. Single digit numbers are values set to zero in the absence of any quantifiableinformation, even through some, e.g. combinations of F − and bivalent metals, are known to pairstrongly.16

12345mination is not possible without making additional assumptions about the chemical compositionof the system. In the simplest case, we assume that the system is composed of dilutions or concentrationsof a known chemical composition, i.e. we have concentrations ĉ i (mol L −1 ) for eachconstituent (both ionic and nonionic) in a standard composition, so that at any actual location theconcentrations are c i = dĉ i where d is an unknown scale factor. The true salinity S ′ (in units of6g L −1 of solution) is:S ′ =N∑c+N niN∑c+N nc i M i = diĉ i M i (22)78with M i the molar mass of a particular constituent. The sum is over both the N c ionic and N nnonionic constituents. We note that in situ measured conductivity:∑N c∑N cκ = λ i (I)α i (I)c i z i = d λ i (I(d))α i (I(d))ĉ i z i (23)ii91011121314151617depends sensitively on d through a multiplicative term on the RHS, and weakly through a dependenceof λ i and α i on ionic strength I which in turn depends on d. To determine d we takean initial guess d 0 , compute I and thence the summation on the RHS of (23), dividing the measuredconductivity κ by this sum to generate a new estimate d 1 , and then repeat until valuesd 0 , d 1 , d 2 , ..., d N , ... converge, which tends to happen within a few iterations. The reference conductivityκ 25 is not actually needed to determine salinity, but if required it can then be computedby setting T = 25 ◦ C in (23) once d is known.Conversion of the salinity S ′ to salinity S (g kg −1 solution) requires a determination of thedensity ρ which itself has a salinity dependence (Chen and Millero, 1986),S = S ′ /ρ(S, T, p) (24)17

1234567Again this can be done iteratively, substituting numerical values of S ′ for S on the RHS, computinga new S and then resubstituting on the RHS until convergence occurs.In particular situations other assumptions and/or knowledge of the system may apply. Forexample, the system may be a mixture of two different standard compositions. The above procedurecould be modified to estimate the mixing faction. In addition, computation of density (orother derived parameters such as static stability) to a useful accuracy may sometimes depend ondetails of the chemical composition rather than on salinity alone (e.g., W96).8Data sources910The parameterization (12-21) is carefully written to allow us to determine the unknown nu-merical parameters in a straightforward way. First, we parameterize the relationships at T =1125 ◦ C and p = 0 dbar using data available from numerous sources. Values for λ ◦25iare readily121314available for a large variety of ions and can generally be taken from a single modern source; weuse Lide (2007). To determine a 25i data is required for λ 25i at different concentrations, sourceswere Lide (2007), except for carbonate data from Dean (1999). As such data is provided only forbinary electrolytes and not individual ions (i.e. for Λ 2515must be made. We take a 25jk = λ25j + λ 25 ) an arbitrary initial choicek16ito be identical for Cl − and K + . Thereafter the ionic equivalent1718192021conductivity 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 reductiona small number of points appeared to show aberrant behavior compared to trends throughneighbouring points. These were deleted from the analysis. An inconsistency arose with tabulationsof K 2 CO 3 and Na 2 CO 3 which were listed as being measured at 18 ◦ C in both Dean (1999),18

1234567891011121314151617181920Dobos (1975), and Milazzo (1963), but whose trend using (4) was more consistent with Λ o at25 ◦ C from those and other sources.Coefficients for the temperature dependence f i are obtained by fitting polynomials to dataavailable in older sources for a far smaller number of ions in the range 0−35 ◦ C. Tabled values ofλ i for a given ion are sometimes numerically different in different sources, likely by scale factorsdue to changes in the definitions of standard physical constants and/or experimental issues, so itis more satisfying to use them only to determine relative changes. For most ions we take valuesat 7 temperatures from Robinson and Stokes (1970). Values at 3 temperatures for CO 2−3 , HCO− 3and 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 a i . Exam-ination of this joint dependency is more speculative as tabulated data is scarce. However, ontheoretical grounds we expect the temperature-dependency in ka ′ terms to be somewhat inde-pendent of the particular ions so we take a best fit for tabulated KCl data (Lide, 2007) as beinggenerally applicable.Ion pairing for bivalent metal sulphates was parameterized using data published by Tomšičet 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 wereavailable only at a temperature of 18 ◦ in Dean (1999); the general quality (accuracy, number ofvalues) of this data appeared lower than that found in Lide (2007). When no conductivity datawas available we used correlations between fitted A 25jkand association constants in electrolytes21of like valency, tabulated in Smith and Martell (1976), Millero (2001), and Millero and Hawke22(1992), to estimate other A 25jk .23 19

123456789101112131415The pressure dependence was determined from measurements of the conductivity of seawaterwith salinities in the range of 2 to 35, temperatures from 0 to 30 ◦ C and pressures from 0 to 4000dbar (Bradshaw and Schleicher, 1980). As salinity dependence is weak over the limnologicalrange and including it would complicate the method we ignore it by extrapolating data to asalinity of zero.An independent data set for binary compounds at a temperature of 18 ◦ was taken from Dean(1999) for verification of predictions. Practical accuracy was determined by comparison withmeasured conductivities and chemical compositions of lake and river waters listed in Wetzel(2001) and Hamilton (1978). For seawater comparisons we computed a salinity using equationsdetermined by Perkin and Lewis (1980) with the low-salinity correction of Hill et al. (1986) fora given conductivity and temperature. We then used a salinity-dependent chemical compositiondescribed 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 conductivityas a function of salinity (Poisson, 1980) due to a lack of low-salinity data in derivation ofthe latter.16Assessment1718192021We first assess the degree to which the the available data shows the decomposition into ionicconductivities (eqn. 7) to be valid when pairing does not occur, irrespective of the parametrizations,by finding sets of 4 binary electrolytes whose equivalent conductivities should sum to zeroat 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 to20

a i25 zi87654← Cu 2+ and Zn 2+z =1 i234Fe(CN ) 4+ 63CO 32−21H + ,OH − →010 −3 10 −2 10 −1Ionic Strength (mol L −1 )Figure 1: Variation of a i z i with ionic strength for 24 ions with valencies of 1 to 4. Note thatCO3 2− , Cu 2+ , Zn 2+ and Fe(CN 6 ) 4+ are not well modelled by this parameterization (curves areneither flat nor clustered with others of similar valency), an indication that ion pairing effects arepresent in the electrolytes from which the ion-specific data was derived.12340.1 mol L −1 , but for some other quads the residual was less than 0.4% only for concentrationsless than 0.01 mol L −1 . This would then represent a maximum achievable accuracy. We use suchcomparisons to identify binary electrolytes with weakest pairing effects, and then add/subtractthem to estimate equivalent conductivities for particular ions at tabulated concentrations.21

1Ionic equivalent conductivities at finite concentrations2Having estimated values for λ 25i , we rearrange (16) to form an expression for a 25i z i as a3function of ionic strength:a 25i z i = √ 1 ( λ◦25iIλ 25i)− 1(25)45Results are shown in Figure 1 for all ions considered. Note the general clustering of lines for ionsof similar valencies, with clusters at equal intervals, demonstrating the validity of the dependence6on z i , with a few exceptions. Use of a constant a 25iis valid in regions where the curves are7891011relatively flat. The curves tend to bend downwards at higher concentrations and the effect is morenoticeable for higher valencies. The values in Table 1 are an average over the 3 measurementsfor which I < 0.01 mol L −1 .Curves for H + and OH − are flat but somewhat below the other monovalent species; thisis because their chemistry is slightly different. CO 2−3 appears above the other trivalent species,somewhat near the curve for the only ion with z i = 4, Fe(CN) 4+6 . Both of these curves are12also not very flat but have a broad peak. This is an indicator that ion pairing is occurring in1314the data from which these conductivities were derived. The subtractive procedure performs even15less well for modelling 2:2 compounds. We derive a reasonable a 25ifor SO 2−4 from Na 2 SO 4 ,1617181920but then if we try to model Zn 2+ and Cu 2+ (both of which are tabulated as compounds withSO4 2− ) the curves suggest an a25 i increasing from about 6 to 12 as ionic strengths increase from0.0005 to 0.01. We therefore set a 25i for Cu 2+ , Zn 2+ , F − (for which we have no data at 25 ◦ C)and FeCN 4+6 to a typical value for their valence, and also take the value for carbonate to be thesame as that for sulphate.22

0.0150.01Without density compensationconstant a itemperature−compensated a irelative error in Λ0.0050−0.005−0.010 10 20 30 40 50temperature ( o C)Figure 2: Relative error in predicted temperature dependence of KCl at a concentration of 0.01mol kg −1 . Shown are the errors arising from ignoring the density correction, correcting fordensity but without the temperature correction for a i , and finally from the full parameterization.1Temperature effects2345678Fitting polynomials for the relative temperature dependence of λ ◦ i is straightforward andthe residuals are all less than 0.3%, although this likely represents a slight overfitting (i.e. thepolynomial order is higher than required), which may mask inaccuracies in the data. Differencesin the order of the predicting polynomial arise because some ions are tabulated more extensivelythan others so 3rd order fits are not always possible.The effect of joint temperature and concentration dependence is small but systematic. Figure2 shows the relative error in predicted conductivity at a variety of temperatures for a 0.0123

1−α ij0.450.40.350.30.250.2CuSO 4ZnSO 4MgSO 4K CO 2 3Na 2CO 3K 4FeCN 6Na FeCN 4 6KClO 4CuSO 4ZnSO 4CaSO 4MgSO 4Ag 2SO 4(NH 4) 2SO 4K 2SO 4Ba(NO 3) 2AgNO 3AgNO 310 −4 10 −3 10 −2 10 −10.150.10.05010 −4 10 −3 10 −2 10 −1Ionic Strength (mol L −1 )Ionic Strength (mol L −1 )Figure 3: Ion pairing effects for variety of binary electrolytes at a) 25 ◦ C and b) 18 ◦ C. Symbolsrepresent available data, and lines are estimated functional forms, fit by eye.123456mol kg −1 KCl solution, with and without a β = 1/300 ( ◦ C) −1 dependence. In order to correctlyaccount for this effect it is necessary to convert to concentrations into units of mol L −1 usingdensity computed according to Chen and Millero (1986) as the systematic error from neglect-ing this is larger than the joint dependence. The remaining underestimate of about 0.2% arisesfrom the a 25iparameterization. We find that an identical parameterization adequately reduces thetemperature dependence of pairing effects in metal sulphates and thus assume it to be general.7Pairing effects89To determine the pairing reduction factor parameterization (Tables 2 and 3) we comparepredictions without pairing to measurements of the actual conductivities of various binary sub-24

12345678910111213141516171819systems at different ionic strengths. The ratio of these two is α jk (I), and estimates of this fromavailable data are shown in Fig. 3. After some experimentation it was found that pairing datawas generally well described by one side of a gaussian curve in log √ I. The gaussian curve isdescribed by 3 parameters - a width, amplitude, and center. As the available data does not constrainthe center very well (curves whose peaks are rightwards and above the present ones havea virtually identical goodness-of-fit) it seemed more useful to set the width and center parametersto empirical constants for each set of valencies, changing only the amplitude of the curvefor different ion pairs. Amplitudes are fitted by eye, with a bias towards matching the data atlower ionic strengths as inaccuracies in our parameterization (16) which is used in deriving theplotted points becomes large at higher strengths. The largest reductions are found for 2:2 metalsulphates.The available derived amplitudes correlate well with association constants (log K A ) tabulatedin Millero (2001) allowing us to estimate the others. We derive amplitudes for bivalent metalcarbonate α jk by scaling those for bivalent metal sulphates by the ratio of tabulated log K A . 1:2sulphates show pairing effects of around 5% at I = 0.1 mol L −1 in spite of the lower data qualityat 18 ◦ C. This is about 20% of the effect seen in 1:2 carbonates, although in this case tabulatedassociation constants suggest that the pairing effects should be similar so correlation does notseem useful here. Instead we estimate the reduction for unknown 1:2 carbonates by setting themto the value found for Na 2 CO 3 .20Metal bicarbonates are a limnologically important subsystem.Association constants for212223metal carbonates and bicarbonates are well correlated (Millero and Hawke, 1992), so we couldestimate the reduction amplitude by scaling the metal carbonate amplitudes by the ratio oflog K A , and then taking only half of this amplitude to take into account the charge on the pairs as25

123456789101112discussed earlier. Alternatively we note that association constants for the 1:2 metal bicarbonatesand Na 2 CO 3 are very similar, and use the same reduction. Both approaches give approximatelythe same value but we use the latter as being somewhat safer. Note that the comparisons withmeasured river waters discussed later can be greatly improved if we double these reduction factorsbut at present there seems to be no justification for doing so.Electrolytes with NO3 − and ClO4 − are also slightly prone to pairing, and we see in Figure 3 adecrease of a few percent at I = 0.1 mol L −1 . Cl − shows signs of pairing with Zn 2+ (and hencepresumably with Cu 2+ as well), but not with any other ion. We have limited data for F − whichis known to pair strongly with bivalent metals but since such combinations are not limnologicallyimportant we make no correction. No information is known about the behavior of Br − and I −with bivalent metals. Bivalent metal hydroxides are also known to associate reasonably stronglybut again we have no conductivity data.13Overall accuracy for binary electrolytes141516171819202122Figure 4a-c shows the relative error in predictions of conductivity without pairing againsttabulated values for 30 binary electrolytes at 25 ◦ C. The error when pairing effects are includedis shown in Fig. 4f-h and is generally less than 1% for concentrations up to 0.01 mol L −1 . Threepoints in Fig. 4a and 4f below the rest are for NaHCO 3 . Interestingly the underestimate isroughly constant with changing concentrations, a pattern not generally seen when other parameterizationerrors arose, possibly suggesting a systematic error in tabulated data. Other than theseanomalies the error is a mix of over- and under-estimates. We also find the prediction error fora further 34 compounds whose conductivities at 18 ◦ C were listed in Dean (1999) (Fig. 4d-ewithout pairing, and 4i-j with pairing effects included). A typical error is on the order of 2% for26

1:12:1/2:23:1/4:11:1 (18 ° C)2:1/2:2 (18 ° C)relative error in Λ0.10.050−0.05abcde0.041:12:1/2:23:1/4:11:1 (18 ° C)2:1/2:2 (18 ° C)relative error in Λ0.020−0.02−0.04−0.06f10 −3 10 −2 10 −1ghi10 −3 10 −2 10 −110 −3 10 −2 10 −110 −3 10 −2 10 −1Concentration (mol L −1 )j10 −3 10 −2 10 −1Figure 4: Relative error in our parameterizations for 65 binary electrolytes. (a)-(e) are predictionswithout 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 at25 ◦ C. (d,i) 1:1 electrolytes at 18 ◦ C. (e,j) 2:1 and 2:2 electrolytes at 18 ◦ C.27

0.0143.54000−0.005−0.0170000360000.005salinity2.5230005000−0.0051.51200000.50.00510000 5 10 15 20 25 30temperature ( o C)Figure 5: Relative difference (thick lines) between conductivities predicted for seawaters of differentsalinities and temperatures using both the algorithm developed here and standard seawaterequations (see text for details). Thin lines are contours of conductivity (µS cm −1 ). Conductivitybasedsalinities are unitless but are numerically close to g kg −1 .123456concentrations 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 manyof the curves appear to be shifted vertically (accounting for much of the scatter); this arises fromdisagreement in values of λ ◦ i at 18◦ C derived from different sources. The comparisons in Fig. 4are perhaps overly accurate as most of the data was also used in estimation of one or more of ourempirical parameters.28

1Accuracy for natural waters23456789Although the performance of our algorithm for binary electrolytes is satisfactory it is notclear if similar results can be expected for more complex (e.g., natural) systems. The mostcomprehensive measurements of natural systems are those for seawater with salinities in therange of 0.005 to 42. Using the standard seawater salinity/conductivity relationship and theionic composition of seawater (ignoring the minor constituent B(OH) − 4 for which we have nochemical data) we show the relative error in our algorithm in Figure 5. The difference betweenthe two is < 1% for salinities less than 3 over virtually all temperatures, (i.e. over roughly adoubling in conductivity at fixed salinity), with largest errors at 0 ◦ C. As this bias is systematic itmay be due to lesser accuracy and/or scarcity of λ ◦ idata at low temperatures.10Next, the effectiveness of the algorithm at 25 ◦ 11C for a variety of North American waters is1213141516171819202122shown in Figure 6. The observational dataset is not without its own problems, which we can seewhen first estimating the conductivity by simply summing up the infinite dilution conductivitiesfor each ion (i.e. using λ i = λ ◦ i in eqn. 7). As this would be an upper bound (it ignoresall ionic interactions, and overestimates conductivity by about 16% in this dataset) it suggeststhat at least two data points near κ 25 ≈ 50 µS cm −1 are problematic. We apply our algorithmwithout 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 overestimationto about 8%. Including the ion pairing effects of bivalent metal sulphates reduces the error toabout 4% on average, and adding the effects of metal carbonate pairing using the full algorithmreduces this to 3%, with a root-mean-square difference (RMSD) of 5%. The SMEWW algorithmoverestimates the conductivity by about 2% (RMSD 4%) for this particular set of freshwater data.29

0.25a↑ (0.47)b0.0150.2FreshwaterRelative error in predicted κ 250.150.10.050−0.05−0.10.010.0050−0.005I=0W96No−pairingFullSMEWWSeawaterI=0W96No−pairingFullSMEWW10 1 10 2 10 3−0.0110 2 10 3Measured κ 25(µS cm −1 )Figure 6: Comparison between measured and predicted κ 25 for 32 North American rivers (discretesymbols, Hamilton, 1978), as well as for seawater for salinities of 0.1-3 (lines). Shownare results from our full algorithm, with and without pairing effects, and without ionic-strengthdependence,as well as the methods proposed by W96 and SMEWW. a) All points. b) Seawateronly (note change in vertical scale)30

12345Performance of our algorithm is not as good as the comparison with binary electrolytes but muchof this mismatch could occur because of uncertainties in the chemical analysis. The relativeerror Ê in κ 25 arising from the chemical analysis can be estimated by normalizing the differencebetween anion and cation sums (and by taking advantage of the fact that κ 25 ≈ 10 5 C eq in ourunits for fresh waters):Ê = ∆κ 25= | ∑ N +∑ jκ N+25jz j c + j − ∑ N −kz k c − k |z j c + j + ∑ N −kz k c − k(26)67891011121314151617181920We find Ê is about 2% here, typical for high-quality data with κ 25 > 300 µS cm −1 (SMEWW).In addition, laboratory measurements of κ 25 , even when done properly, typically have an uncertaintyof at least 1 − 2%. This could account for half of the RMSD, suggesting that theparametrization error is at most only a few percent.We also show on this figure the error in predictions of conductivity for dilute seawater at25 ◦ C with salinities of 0.1 to 3. As discussed earlier, agreement is very good (to better than 0.2%for κ 25 < 3000 µS cm −1 ). However, the limnological “tuning” of the SMEWW algorithm forfreshwaters 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 substantiallyimprove the results. 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 break-31

Full W96 SMEWWLake MalawiCrater LakeKCLSeawater S=0.1World (2)CarbonateVolcanicGraniteShaleGneissSandstoneWorld (1)AsiaEuropeSouth AmericaAustraliaNorth AmericaAfricaeqn. (1)abc158 160 162 164 166κ 25for κ 5=10010 1 10 2κ 250.5 0.6 0.7 0.8 0.9S/κ 25Figure 7: (a) κ 25 for κ 5 = 100 µS cm −1 . At top are shown results for two specific lakes (Mc-Manus et al., 1992; Wüest et al., 1996) as well as a dilute seawater and a 1 mmol L −1 KClsolution. Next are world average and averages by river type, followed by world average andaverages by continent (Wetzel, 2001). At bottom is the result assuming eqn. (1). Both the fullalgorithm described here as well as that of W96 and SMEWW are used. (b) κ 25 for same ioniccompositions. (c) Ratio of ionic salinity to κ 25 for all water types.32

1234567891011121314151617181920212223down by the dominant rock type in the drainage. In addition, we take ionic composition of severallakes from other papers. From this database we compute both ionic salinity and reference conductivity(for this comparison we have neglected the nonionic components of salinity which canbe substantial and must be accounted for in practical situations). Figure 7b shows that computedreference conductivity varies greatly from very small values in granitic basins to large values incarbonate basins. Differences in the computed κ 25 values using this algorithm and that of W96can be as large as 5% but this is too small to be easily apparent in this presentation. The differencesare more apparent when we show the ratio of salinity to reference conductivity (Fig. 7c);this is equivalent to determining the numerical coefficient in eqn. (2). This ratio varies widelyfrom a low of 0.65 for African rivers and sandstone drainages to a maximum of 0.9 for carbonatedrainages. The generally lower conductivities using our algorithm results in ratios higher by1-5% than those found using W96, but slightly smaller than those using SMEWW. The ratiosfor dilute seawater and KCl solutions are around 0.5, somewhat smaller than that for most lakewaters.Finally, in Figure 7a we illustrate the effect of differing chemical composition by computingthe reference conductivity for a measured conductivity of 100 µS cm −1 at T = 5 ◦ C for eachcase. The computed κ 25 vary between 160 and 161.6 µS cm −1 for all compositions, i.e. wecan consider the correction to be composition-independent to within about ±1%. Eqn. (1) performsadmirably in spite of its crudeness, predicting a numerical correction within 1% of ourcomposition-dependent corrections. Taking the composition to be pure KCl is less useful as thecomputed reference conductivity is less than 159 µS cm −1 , but in most situations careful calibrationwould be required to differentiate between the two. Repeating the procedure for a measuredconductiviy of 1000 µS cm −1 results in a slightly larger variation of about 3%. It appears that33

12345678910111213the major advantage of our algorithm is not in computing reference conductivity itself (except asa quality-control measure for chemical analyses), but rather in computing the salinity (or at leastthe numerical factor in eqn. 2). By combining the parameterization of W96 for computing conductivitywith our iterative procedure for determining reference conductivity we can sensitivelytest the apparent difference between the two parameterizations. Their algorithm gives valuesabout 1 µS cm −1 higher in all situations except for the KCl solution (which is well-modelled).The constant difference suggests a systematic bias between the two parameterizations which islikely related to the form (16) as ion pairing corrections do not seem to affect the reference conductivitycalculation to any great extent. Although the calculation in SMEWW is not claimedto be valid for temperatures more than a few degrees away from 25 ◦ C we can also apply it inthis case. As they also take temperature dependence to be constant for all constituents not surprisinglythe predicted reference conductivity is independent of chemical composition; it is alsosomewhat high.14Discussion15161718192021The conductivity of natural waters depends on chemical composition. Conductivity is reducedby the effects of relaxation and electrophoresis at finite concentrations, and by ion associationbetween certain pairs of ions. Temperature dependence is strong over the limnologicalrange but is somewhat independent of the ionic composition. Pressure dependence is smaller.The formal assessment of our algorithm clearly indicates the shortcomings of existing conductivityparameterizations and provides guidance about the uncertainties.We find that the simple temperature correction (1) is sufficient to determine reference con-34

1234567891011121314151617181920212223ductivity with an error of around 1% for all water types considered. Sorensen and Glass (1987)previously found that accounting for ionic composition changed the correction to reference conductivityby an average of 0.3% for lakes in Minnesota, Wisconsin, and Michigan. Conversionto salinity does require a knowledge of the ionic composition.Electrophoresis and relaxation effects were accounted for in parameterizations described byW96 and Talbot et al. (1990). However, our formulation, which requires only one empiricallyderivedparameter per ion to model ionic-strength-dependence, appears to provide results asgood as those of the former when ion association does not occur, and is valid to slightly greaterconcentrations 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 muchbetter results in lake waters because it naturally allows ions pairing effects to be included, andthe 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 modelled in W96, and thataccounting for these effects leads to a significant improvement. Although our ion associationparameterization is relatively crude, it is sufficient to show that ion association is important infresh waters when κ 25 is greater than about 50 µS cm −1 , but rather less so in dilute seawatersof even the same (approximate) salinity. Also, it is found that pairing effects in bivalent metalsulphates are generally more important than those in bivalent metal carbonates in their effect onreductions in conductivity. Talbot et al. (1990) accounts for changes in conductivity due to speciationin metal bicarbonates, but requires speciation information a priori, and does not includethe more important effects arising from bivalent metal sulphates. An empirical reduction factorthat presumably accounts for all three effects, described in SMEWW, may perform slightly better35

123456789101112than our algorithm for some waters, but it clearly underperforms in dilute seawaters and hence isnot necessarily trustworthy.Both W96 and Talbot et al. (1990) include a correction for temperature assuming the separabilityof temperature and concentration dependencies but for more precise results a small jointdependence must also 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 algorithmsdescribed here should have wide utility in providing better estimates of the dissolvedsolids in freshwater and estuarine systems with a TDS of less than around 4 g L −1 , of arbitrarycomposition. A reliable determination of the accuracy of our algorithm is limited by uncertaintiesin routine chemical analysis, but it appears to be accurate to at least within a few percent onaverage, and perhaps much better. Even if more traditional techniques for salinity determinationare utilized this algorithm is useful as a check on the results.13Comments and Recommendations1415161718192021The largest uncertainty in the development of the conductivity algorithm is related to theeffects of ion pairing with nonsymmetrical (i.e. 1:2) electrolytes. Not only are the availabletheoretical guidelines very complex, but accurate laboratory measurements of conductivity (especiallyfor 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 are each accountedfor explicitly in the conductivity calculation. However, chemical analyses often characterizethis system using measurements of total alkalinity and pH. To convert to ionic concen-36

12345678910trations dissociation constants are required; Millero (1995) provides the required equations validover a range of temperatures at ocean salinities of 0-5. Approximate freshwater salinities (e.g.estimated via eqn. 2), appropriately scaled to the nondimensional ocean salinity scale, may besubstituted. A limnological equation of state (Chen and Millero, 1986) should be used to convertfrom units of mol kg −1 solution to the units of mol L −1 that are used here. A further issue is thatthe carbonate and bicarbonate ionic concentrations are assumed to remain constant with changesin temperature in the reference conductivity and salinity calculations. In fact the pH of a closedsystem will increase by order 0.1 for a 10 ◦ C decrease in temperature and this can change the ionicconcentrations slightly. The dependence may be worth investigating when highest precision isrequired.11References121314Anderko, A. and M. M. Lencka (1997). Computation of electrical conductivity of multicompo-nent aqueous systems in wide concentration and temperature ranges. Ind. Eng. Chem. Res. 36,1932–1943.1516Barthel, J. M. G., H. Krienke, and W. Kunz (1998). Physical Chemistry of electrolyte solutions:modern aspects, Volume 5 of Topics in physical chemistry. New York: Springer.171819Bešter-Rogač, M., V. Babič, T. M. Perger, R. Neueder, and J. Barthel (2005). Conductometricstudy of ion association of divalent symmetric electrolytes: I. CoSO 4 , NiSO 4 , CuSO 4 andZnSO 4 in water. J. Mol. Liquids 118, 111–118.37

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