An Investigation of Harned's Rule for Predicting the Activity

Article pubs.acs.org/jced

An Investigation of Harned’s Rule for Predicting the Activity Coefficients of Strong Aqueous Electrolyte Solution Mixtures at 25 °C Darren Rowland* and Peter M. May Chemistry, School of Engineering and Information Technology, Murdoch University, Murdoch, Western Australia 6150, Australia S Supporting Information *

ABSTRACT: From a critical evaluation of the relevant literature, Harned coefficients at 25 °C and their corresponding trace activity coefficients are reported for 72 mixtures of strong aqueous electrolytes with a common ion. The work confirms the generality of Harned’s rule in its simplest form, that is, with only one (linear) term. Uncertainties in predicted mean activity coefficients are typically less than 0.01, comparable with the experimental reproducibility. Unless there are good grounds to suggest that significant changes in chemical speciation are occurring, experimental measurements that do not conform to the rule should be regarded with suspicion. We find that the estimates of errors made by authors of single potentiometric investigations are often as much as an order of magnitude too optimistic. Even the work of Harned and colleagues, which was typically of high quality, is sometimes much worse than is claimed or implied in the literature. The unfortunately common practice of introducing a quadratic term (as a second adjustable parameter) to extend Harned’s rule is unjustified, within the true current limits of experimental error. Harned’s rule with just one coefficient provides a description of aqueous strong electrolyte mixing thermodynamics which is widely applicable, satisfactorily accurate, and without sign of any systematic problem.

■

Harned and Robinson in 1968,6 many experimental studies have been reported, but there has been no comprehensive review, nor corresponding analysis and evaluation, of them. The purposes of the present work are accordingly 4-fold: to assemble the available values, to compare them against published activity coefficient data for each respective solution mixture, to perform an up-to-date critical assessment, and to determine some recommended values for practical use at convenient ionic strengths. Harned’s rule is of great importance on account of its remarkable generality and simplicity. It describes how the logarithm of the activity coefficient of one strong electrolyte, B, in a two-component solution mixture at constant ionic strength varies in direct proportion to the concentration of the other strong electrolyte, C (ref 7, p 438):

INTRODUCTION Although most systems of applied thermodynamic interest are mixtures, for obvious reasons of both logic and pragmatism, much greater theoretical and experimental attention is given to pure substances. A common belief is that physicochemical property simulation should be more troublesome with multicomponent than single-component systems, and this is certainly the case with aqueous complexing systems. However, with aqueous strong electrolytes, so-called “linear mixing rules” of various kinds have become well-established and imply the opposite; that is, as long as the pure binary (one salt plus water) systems have already been characterized, the thermodynamic behavior of mixtures is comparatively simpler and hence easier to model. Of all the mixing rules available to predict the thermodynamic properties of multicomponent strong electrolyte systems, one of the most valuable is Harned’s rule for activity coefficients, described first by Harned1,2 and subsequently explored and confirmed by many others. Harned’s rule is the method of choice whenever the equation can be parametrized from experimental measurements on mixtures.3 In contrast, Zdanovskii’s rule4,5 lacks precision when used to calculate the separate activity coefficients for each component in a mixture (although it is more universally applicable, requiring only that the pure solutions have been characterized). Unfortunately, the Harned equation parametersor so-called “Harned coefficients”are widely scattered in the literature, sometimes seriously inconsistent and often presented in different algebraic forms. Since the work of © XXXX American Chemical Society

′ mC log10(γBmix ) = log10(γBpure) − αB(C) in C

(1)

where α′B(C) is the Harned coefficient and mC is the concentration of C on the molal scale. Other forms of Harned’s equation appearing in the literature include: log10(γBmix ) = log10(γBpure) − αB(C)IC in C

(2)

log10(γBmix ) = log10(γBpure) − αB(C)yC I in C

(3)

Received: July 19, 2016 Accepted: November 3, 2016

A

DOI: 10.1021/acs.jced.6b00651 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

Article

predictive for mixtures comprising more than two electrolytes without the need for further adjustable parameters. Predictions with eq 8 have been found to agree with activity coefficients and solubility data for quaternary and higher order strong electrolyte mixtures within the estimated experimental uncertainty of the data.11−16 This means that the Harned coefficients obtained in this work from common ion mixtures can be used more generally to describe strong electrolyte systems with any number of components at 25 °C. Methodology. An extensive search of the chemical literature located more than 1400 Harned coefficients from 93 references. These data have been assembled and entered into the FIZ database of our JESS (Joint Expert Speciation System)17,18 software package. The references are listed in the Supporting Information. For the compatibility of the Harned coefficients with an existing database structure designed for other physicochemical properties, the concentrations of the mixture components in this database refer to the line having constant ionic strength connecting the end points (i.e., of the pure, binary solutions) and a concentration transform is applied when the electrolyte component is not 1:1. Concentrations can be specified in units of either “m” (mol·kg−1) or “M” (mol·dm−3). Other forms of Harned’s rule appearing in the literature are transformed automatically into eq 2, the JESS canonical form. A computer program was then developed to generate sets of activity coefficient data from each Harned parameter in the database (with activity coefficients of the pure salts calculated from the Pitzer equations) so that values calculated by it could be analyzed and compared with the corresponding activity coefficient data previously assembled from the literature.2,3,6,19−163 Database entries were assessed and weighted in the usual way.17 Significant inconsistencies between data under the same conditions were resolved in favor of the work judged to be of better standard. Further computer programs were then used to express every individual activity coefficient value in terms of both a Harned coefficient and a trace activity coefficient for all recognized aqueous strong electrolyte mixtures at a given I and having yC > 0.25. The internal consistency of the resulting data sets for given mixtures at each ionic strength could thus be examined. In this way, indisputable outliers were identified and eliminated. Averages and spreads for each Harned coefficient could finally be determined, yielding a recommended value for the trace activity coefficient under each particular condition along with a rough estimate of its uncertainty where possible. These uncertainty estimates were framed as maximum deviations, cognisant of the relatively small number of calculations being averaged in most cases.

and * y log10(γBmix ) = log10(γBpure) − αB(C) in C C

(4)

where IC is the ionic strength contribution of component C, I is the stoichiometric ionic strength, and yC = IC/I is the ionic strength f raction of component C. These equations can of course be written in terms of natural logarithms, but this is done only infrequently. Since the eqs 1−4, and those with a natural log base, all apply at a given ionic strength, they differ (at that ionic strength) just in the magnitude of their coefficients α′B(C), αB(C), and α*B(C). To deal with observed deviations from the above eqs 1−4, most notably for (a) the mixed systems {NaOH + NaCl}(aq) and {KOH + KCl}(aq), (b) systems with ions of higher charge, and (c) solutions at high ionic strength, equations which include a quadratic term have also often been described. Equation 5 is an example. ′ mC − βB(C) ′ mC2 log10(γBmix ) = log10(γBpure) − αB(C) in C

(5)

However, references in the literature to quadratic extensions such as appear in eq 5 find that the β coefficient is almost invariably small. This raises questions about its validity, as will be considered below. Finally, but of great theoretical importance, Harned’s rule leads directly to the evaluation of so-called “trace activity coefficients”, which are the extrapolated values for the activity coefficient of component B in the limit of zero concentration. In particular, from eq 2 for an infinitely small concentration of B in C such that IC = I: log10(γBtrace ) = log10(γBpure) − αB(C)I in C

(6)

As with mean activity coefficients of pure salt solutions, there is little theoretical guidance for the behavior of the trace activity coefficient at high ionic strength. However, it follows from Debye−Hückel theory that under dilute conditions the trace activity coefficient of B in a solution of C is related to the trace activity coefficient of C in a solution of B at the same ionic strength by log10(γBtrace ) in C |z B +z B −|

=

log10(γCtrace ) in B |zC +zC −|

(7)

where zB+ and zB− are the algebraic charges on the cation and anion, respectively. An excellent example of a real mixture displaying this behavior, even up to high ionic strength, is {HCl + NaCl}(aq) (ref 7, p 433). For many systems it is further observed that the trace activity coefficients fall between the activity coefficients of the pure salts at the same ionic strength. In cases where the two Harned coefficients αB(C) and αC(B) are not available from experiment, they may be estimated using other equations. For example, by Zdanovskii’s rule (see ref 8) or by the Pitzer equations (see refs 9, 10). The forms of Harned’s rule shown above apply to ternary electrolyte mixtures. A logical extension to Harned’s rule for representing the activity coefficient of an electrolyte in a multicomponent system is log10(γBmix ) = log10(γBpure) − in CD...

∑ X = C,D,...

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RESULTS AND DISCUSSION In line with the majority of experimental investigations, we chose to examine the available data at 25 °C and at rounded ionic strengths of 0.1, 0.5, 1.0, 2.0, and 3.0 mol·kg−1. No consideration has been given, and hence no conclusions can properly be drawn, for ionic strengths above 3.0 mol·kg−1. A total of 72 ternary common ion mixtures were found with sufficient data to permit evaluation of the Harned coefficient for at least one rounded ionic strength. Of these 72 mixtures, there were 23 “complementary salt pairs”, in which the activity coefficient of electrolyte 1 had been characterized (as B) in the presence of electrolyte 2 (as C) and vice versa. Over 500 Harned coefficients and trace activity coefficients were obtained

αB(X)IX (8)

where parameter αB(X) is specific to component B in the presence of component X.11 As αB(X) can be derived from measurements on ternary electrolyte solutions, eq 8 is B

DOI: 10.1021/acs.jced.6b00651 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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This may be due to the type of ion selective electrode used. Similar issues were found with the data for {NaF + NaClO4}(aq) at lower ionic strengths from Hernández-Luis et al.165 who also used an ion selective electrode, but in this case we attribute the waywardness to anomalies which these authors themselves identified. Linear or Quadratic Dependence? Following what appears to be the mistaken lead by Harned and Cook,166 many reports in the literature concerned with the thermodynamics of mixing aqueous strong electrolytes reach a categorical conclusion in favor of the existence of the quadratic/β term in eq 5. A list of such statements is given in the Supporting Information. However, it is found in almost all of such cases, and often acknowledged, that the observed curvature is minor: for example, 0.02. Given that trace activity coefficients are obtained from extrapolations of the target ion to zero concentration, the outcome summarized in Table 1 constitutes a robust validation of the generality of Harned’s rule. Several aspects of the results shown in Table 1 are noteworthy, the initial three having fundamental implications. First, Harned’s rule works for strong electrolytes regardless of their charge type; that is, it is not confined to 1:1 electrolytes. Second, the trace activity coefficients change systematically with ionic strength, and they tend to change only slowly above I = 1.0 mol·kg−1. Third, in more than half of the complementary salt pairs, the trace activity coefficients of electrolyte 1 and electrolyte 2 (in mixtures with the other electrolyte) are in close agreement with eq 7. Fourth, the high percentages shown in Table 1 for some mixtures, for example, Mg+2|K+1/Cl−1 and K+1|Mg+2/Cl−1, are no cause for concern: they arise simply because (unusually) the activity coefficient curves of the two neat, binary electrolytes intersect; this happens at about the ionic strength in question (so, in calculating the percentage, the divisor tends to zero, making any change whatsoever large in fractional terms). The results in Table 1 provide a useful set of parameters for the prediction of activity coefficients in mixtures of aqueous strong electrolytes. The overall sense of orderly behavior, despite the diversity of systems and sources, supports the notion that Harned’s rule is widely applicable. The random nature of the worst-fitting results suggests that experimental error is the main cause of deviations from linear behavior. A number of systems where the data were poorly fitting were examined in detail. With the system {NaBr + LiBr}(aq) at various ionic strengths, the activity coefficients of NaBr from Yan133 do not accord with Harned’s rule: the corresponding α parameter values vary systematically at given ionic strengths. C

DOI: 10.1021/acs.jced.6b00651 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

0.0383 0.786(−) 2%

2

0.00189 0.754(0.1) 0%

D

24

H+1|Ca+2/Cl−1

16

8

7

H+1|Ca+2/Br−1

4 0.0575 0.793(0.002) 5% 11 0.0390 0.786(0.008) 4%

3

5

Ca+2|H+1/Br−1

0.0159 0.613(−) 17%

2 0.0176 0.474(0.001) 25% 2 −0.0952 0.570(0.01) 25% 6 0.0664 0.735(0.02) 17% 10 0.0505 0.713(0.01) 13%

18

7

1

5

1

Cl−1|Ca+2/NO3−1

3

3

12

Br−1|Ca+2/Cl−1

12

4

H+1|Bu4N+1/Cl−1

0.278 0.576(−) 63% 8 0.0896 0.681(−) 62%

4

6

6

2

H+1|Bu4N+1/Br−1

0.330 0.745(−) 53% 2 0.102 0.775(−) 47%

4

Na+1|Ba+2/ClO4−1 1

6

0.0258 0.662(−) 7%

Ba+2|Na+1/ClO4−1

4

29

6

7

Na+1|Ba+2/Cl−1

7

59

5

5

5 −0.0632 0.792(0.006) −6%

12 0.0616 0.704(0.02) 14% 4 −0.00537 0.454(0.02) 2% 21 −0.0141 0.660(0.02) −4% 5 −0.0262 0.717(0.01) −8%

Ba+2|Na+1/Cl−1

7

26

K+1|Ba+2/Cl−1

Na+1|Ba+2/Br−1

4

17

Ba+2|K+1/Cl−1

19 0.0734 0.780(0.02) 6%

9

3

54

30

2 −0.0851 0.527(0.01) 19% 7 0.0767 0.726(0.009) 17%

H+1|Ba+2/Cl−1

9

3

12

9

H+1|Ba+2/Br−1

2 −0.0571 0.621(0.005) 5% 7 0.0847 0.788(0.004) 7%

2

Ba+2|H+1/Cl−1

3

Ba+2|H+1/Br−1

H+1_NH3|Ba+2/Cl−1

1

14

1

H+1|Al+3/Cl−1

0.5 6

0.1

2 −0.0936 0.550(0.01) 31% 7 0.0768 0.737(0.007) 26% 8 −0.0728 0.480(0.003) 24% 37 0.0642 0.698(0.007) 21% 4 0.00212 0.404(0.02) −1% 46 −0.0104 0.619(0.01) −6% 5 −0.00578 0.698(0.008) −3% 3 −0.0428 0.448(0.02) 20% 20 0.00823 0.646(0.02) 4% 4 0.0120 0.448(0.003) −9% 2 −0.0253 0.668(0.005) −19% 4 0.297 0.444(−) 80% 8 0.0870 0.663(−) 76% 1 0.100 0.389(−) 273% 2 0.0211 0.429(0.001) 33% 1 −0.103 0.621(−) 40% 5 0.0648 0.758(0.008) 26% 12 0.0524 0.718(0.01) 21%

4

−0.0636 0.375(−) 16% 8 0.0611 0.704(0.001) 15%

1.0

ionic strength (mol·kg−1)

Al+3|H+1/Cl−1

electrolyte mixture

Table 1. Harned Coefficient Analysis at Different Ionic Strengthsa

17

8

2

5

6

6

6

23

5

7

33

5

36

8

9

3

13

6

2 0.0285 0.399(0.002) 50% 2 −0.121 0.906(0.009) 68% 6 0.0652 0.872(0.008) 37% 12 0.0547 0.788(0.02) 31%

4 0.0926 0.662(−) 104%

2 −0.100 0.699(0.01) 47% 7 0.0777 0.823(0.009) 36% 6 −0.0844 0.567(0.03) 40% 25 0.0667 0.746(0.009) 32% 4 0.00717 0.372(0.01) −8% 26 −0.00417 0.585(0.01) −5% 5 0.00339 0.725(0.005) 3% 3 −0.0332 0.448(0.02) 27% 15 0.0108 0.639(0.02) 9% 4 0.00316 0.450(0.001) −5% 4 −0.0174 0.660(0.009) −28%

−0.0990 0.501(−) 39% 9 0.0633 0.758(0.1) 25% 4

2.0

13

5

6

31

3

27

6

40

19

2

3

1

9

20

11 0.0543 0.909(0.005) 38%

2 0.0362 0.387(0.008) 64%

4 0.0972 0.676(−) 126%

−0.0320 0.487(−) 36% 21 0.0137 0.653(0.02) 16% 1

13 −0.104 0.711(0.07) 54% 5 0.0646 0.846(0.19) 33% 1 −0.00974 0.602(−) −18% 2 −0.105 0.977(0.007) 58% 2 0.0784 0.957(0.01) 43% 15 −0.0875 0.715(0.03) 50% 31 0.0675 0.830(0.01) 38% 4 0.00976 0.365(0.003) −18% 22 −0.00036 0.571(0.02) −1%

3.0

Journal of Chemical & Engineering Data Article

DOI: 10.1021/acs.jced.6b00651 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

E

1

0.274 0.750(−) 99%

3

3 −0.191 0.783(0.02) −23%

4 0.0408 0.650(0.003) 39%

6

6

Na+1|Cs+1/Cl−1

3 −0.0930 0.673(0.01) −54% 6

3

Cs+1|Na+1/Cl−1

Cs+1|Mg+2/Cl−1

12

Li+1|Cs+1/Cl−1

12

38

24

2

18

19

8

6

K+1|Cs+1/Cl−1

13 −0.0708 0.656(0.003) 37% 17 0.105 0.669(0.01) 54% 3 −0.0741 0.658(0.02) 119% 4 0.00374 0.647(0.006) 6%

17 −0.0268 0.670(0.004) −11% 15 −0.0213 0.698(0.003) −7%

24

Cs+1|Li+1/Cl−1

3

Cs+1|K+1/Cl−1

25

H+1|Cs+1/Cl−1

5 0.127 0.770(0.006) 52%

19

Cs+1|H+1/Cl−1 8

18

Na+1|Co+2/Cl−1

OH−1|Cs+1/Cl−1

19

K+1|Co+2/Cl−1

Cl−1|Co+2/NO3−1

1

18

21

15

8

64

12

18

Na+1|Ca+2/Cl−1

28

17 −0.0324 0.674(0.003) −13% 2 −0.107 0.578(0.005) 40% 5 −0.0142 0.707(0.02) −5% 23 0.0264 0.469(0.02) −9% 12 0.0110 0.673(−) 4%

6

H+1|Ce+3/Cl−1

32

Ca+2|Na+1/Cl−1

7

4

19

0.5

6

17

Na+1|Ca+2/Br−1

3 −0.286 0.669(0.01) 30% 13 −0.0754 0.794(0.01) −8% 28 0.132 0.597(0.10) −13% 6 0.0451 0.769(0.02) 4%

0.1

−0.0280 0.582(−) 34% 4 0.0424 0.597(0.001) 52% 4

8 0.0129 0.587(0.01) 29% 4 −0.0541 0.618(0.01) 36% 8 0.0986 0.615(0.007) 65%

16 −0.0250 0.640(0.003) −21% 15 −0.00868 0.672(0.004) −6% 2 0.0939 0.628(0.007) 61% 16 −0.0602 0.627(0.001) 35% 26 0.101 0.642(0.01) 59%

4 0.0359 0.415(0.01) −28% 50 −0.0249 0.640(0.02) −20% 6 −0.0537 0.555(0.02) 36% 11 −0.00729 0.700(0.02) −5% 18 −0.0169 0.469(0.01) 10% 17 −0.00546 0.667(0.02) −3% 4 −0.0177 0.303(−) 4% 8 0.0848 0.666(0.02) 19%

1.0

ionic strength (mol·kg−1)

Ce+3|H+1/Cl−1

5

Ca+2|Na+1/Br−1

K+1|Ca+2/Cl−1

Ca+2|K+1/Cl−1

electrolyte mixture

Table 1. continued

12

11

18

12

12

6

43

29

18

19

6

12

17

12

6

7

30

6

7 −0.0148 0.535(0.007) 23% 8 0.0423 0.552(0.01) 66%

19 −0.0455 0.616(0.006) 30% 29 0.0996 0.641(0.01) 65% 4 −0.0169 0.540(0.001) 56% 8 0.0105 0.547(0.02) 35% 8 −0.0259 0.563(0.01) 19% 10 0.110 0.555(0.03) 83%

5 −0.00193 0.743(0.02) −3% 4 −0.0196 0.498(0.008) 23% 8 −0.00595 0.690(0.02) −7% 4 −0.0426 0.319(−) 15% 8 0.0908 0.667(−) 31% 4 0.0328 0.412(−) 273% 17 −0.0209 0.632(0.005) −54% 15 −0.00231 0.679(0.008) −3%

4 0.0223 0.410(0.009) −44% 24 −0.0182 0.624(0.009) −36%

2.0

6

6

9

12

5

6

20

17

24

19

6

6

20

27

10

30

5

4 −0.0108 0.521(0.007) 19% 4 0.0395 0.546(0.02) 69%

8 −0.0396 0.635(0.006) 27% 14 0.0985 0.670(0.007) 68% 4 −0.0112 0.522(0.001) 47% 3 0.00685 0.543(0.006) 29% 8 −0.0170 0.543(0.02) 13% 5 0.115 0.524(0.03) 91%

6 −0.0209 0.574(0.01) 39% 18 −0.00295 0.733(0.04) −6% 4 −0.0510 0.374(−) 22% 4 0.0918 0.701(−) 39% 4 0.0343 0.424(−) 253% 17 −0.0170 0.641(0.02) −198% 19 0.00489 0.694(0.02) 12%

3 0.0212 0.429(0.01) −107% 24 −0.0147 0.631(0.02) −74%

3.0

Journal of Chemical & Engineering Data Article

DOI: 10.1021/acs.jced.6b00651 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

F

18

H+1|K+1/Cl−1

4 2

11

La+3|H+1/Cl−1

H+1|Li+1/Br−1

H+1|Li+1/Cl−1

H+1|Li+1/ClO4−1

Li+1|H+1/Cl−1

12

7

H+1|La+3/Cl−1

6

28

1

5

12

23

1

K+1|H+1/Cl−1 5 0.0961 0.776(0.02) 5% 2 0.248 0.464(0.001) −12% 2 0.00398 0.803(0.004) 6% 9 0.00875 0.792(0.01) 64%

1

5

18

37

1

6

12

24

50

1

7

H+1|K+1/Br−1

7

7

12 0.0653 0.701(0.01) 50% 5 −0.0567 0.693(0.004) 43% 1 0.0874 0.717(−) 54% 16 0.0582 0.706(0.02) 44% 8 −0.0740 0.707(−) 56% 8 0.106 0.668(−) 15% 3 −0.00200 0.344(0.004) 0% 1 0.00318 0.790(−) 7% 20 0.00534 0.751(0.008) 28% 4 −0.0140 0.751(−) 72%

H+1_NH3|H+1/Cl−1

16

17

7 0.0613 0.782(0.009) 43% 5 −0.0768 0.781(0.002) 54% 1 0.114 0.783(−) 63% 13 0.0802 0.779(0.01) 57%

11

4

H+1|H+1_NH3/Cl−1

4

12

12

6

5

4

6

6

4 −0.0861 0.809(0.004) 4968%

0.258 0.589(−) 65%

−0.0569 0.806(−) −8% 1 −0.00150 0.657(−) 3% 3 0.0234 0.663(0.003) 56% 4 0.0868 0.717(−) 53% 8 0.0160 0.778(−) 38% 8 −0.0170 0.770(−) 40% 4 −0.0236 0.776(0.02) 1517% 4

4

0.5

0.0770 0.678(−) 18% 1 −0.00626 0.624(−) 13% 3 0.0191 0.630(0.005) 51% 4 0.0800 0.732(−) 51% 8 0.0160 0.848(−) 44% 8 −0.0130 0.834(−) 36% 4 −0.0168 0.842(0.02) 518% 4 −0.0480 0.816(−) 100% 13 0.0677 0.728(0.02) 58% 5 −0.0521 0.682(0.009) 41% 1 0.0824 0.728(−) 54% 38 0.0562 0.712(0.02) 44% 16 −0.0716 0.713(0.007) 56% 8 0.0991 0.645(−) 23% 4 −0.0589 0.341(0.006) 14% 1 0.00703 0.866(−) 19% 27 0.00491 0.801(0.007) 24% 12 −0.0131 0.796(0.004) 63% 4 −0.0228 0.860(0.007) 55%

4

1.0

ionic strength (mol·kg−1)

NO3−1|H+1/ClO4−1

4

12

Cl−1|H+1/Br−1

Cl−1|H+1/ClO4−1

12

0.139 0.779(−) 73%

Br−1|H+1/Cl−1

1

6

2

4

H+1|H+1_NH3/Br−1

2

H+1_NH3|Na+1/Br−1

6

6

4

3

H+1|Gd+3/Cl−1

−0.0320 0.799(−) −2% 2 0.123 0.771(−) 6% 1 0.00924 0.768(−) −16% 2

0.1

Na+1|H+1_NH3/Cl−1

3

H+1|Eu+3/Cl−1

H+1|Et4N+1/Br−1

electrolyte mixture

Table 1. continued

6

6

26

1

6

15

38

1

7

7

1

6

4

4

6

−0.0197 1.11(−) 253%

4 −0.0922 0.414(0.008) 32% 1 0.00614 1.14(−) 20% 18 0.00510 0.990(0.008) 25% 4 −0.0140 0.984(−) 68% 4 −0.0112 1.11(0.009) 46%

5 0.0718 0.728(0.02) 58% 5 −0.0490 0.717(0.02) 39% 1 0.0811 0.810(−) 55% 27 0.0584 0.775(0.01) 47% 11 −0.0621 0.764(0.01) 50%

1

0.0752 0.717(−) 27% 1 −0.0109 0.620(−) 23% 3 0.0225 0.605(0.007) 65% 4 0.0807 0.811(−) 54%

4

2.0

6

8

3

6

34

1

6

4

−0.0138 0.644(−) 28%

−0.101 0.547(−) 44%

6 0.00408 1.29(0.003) 22% 4 −0.0140 1.28(−) 74%

1

4 −0.0468 0.777(0.03) 38% 1 0.0832 0.926(−) 56% 26 0.0630 0.856(0.01) 52% 4 −0.0540 0.827(−) 44%

1

3.0

Journal of Chemical & Engineering Data Article

DOI: 10.1021/acs.jced.6b00651 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

G

2 3 9 3

Pr4N+1|H+1/Br−1

H+1|Sm+3/Cl−1

H+1|Sr+2/Br−1

Sr+2|H+1/Br−1

3

15

6

6

3

3

Ni+2|H+1/Cl−1

3

9

6

6

3

9

6

9

5 0.000991 0.793(0.01) 0% 2 −0.155 0.635(0.006) 14% 1 −0.221 0.727(−) 34% 2 0.0260 0.788(−) 1% 7 0.0779 0.790(0.002) 7% 2 −0.0120 0.623(0.001) 1%

4 0.0608 0.704(0.009) 9% 4 0.0550 0.744(−) 15% 6 0.0433 0.719(0.004) 11% 2 −0.125 0.557(0.01) 32% 4 −0.113 0.548(−) 26% 4 0.0746 0.693(−) 11% 11 0.0680 0.733(0.01) 17% 2 −0.0657 0.536(0.003) 16%

19

7

7

26

H+1|Ni+2/Cl−1

2

Na+1|H+1/Cl−1

8

1

6

11

H+1|Na+1/Cl−1

1

5

H+1|Ni+2/Br−1

1

H+1|Na+1/Br−1

4

12

6

4

Mn+2|H+1/Cl−1

11

6

6

17

H+1|Mn+2/Cl−1

6

6

H+1|Nd+3/Cl−1

6

Mg+2|H+1/Cl−1

19

3

9

6

26

H+1|Mg+2/Cl−1

3

6 0.0563 0.743(0.001) 15% 2 −0.125 0.599(0.005) 34% 13 0.0374 0.723(0.008) 10% 4 −0.0685 0.537(0.003) 19% 7 0.0489 0.714(0.009) 12% 4 −0.116 0.545(0.001) 29% 1 0.0414 0.756(−) 37% 6 0.0350 0.725(0.006) 39% 5 −0.0569 0.728(0.001) 64%

Na+1|H+1/ClO4−1

3

Mg+2|H+1/Br−1

9

0.5

5

6

H+1|Mg+2/Br−1

0.194 0.759(−) 66% 3 0.0662 0.792(0.001) 6% 2 −0.141 0.647(0.003) 13% 14 0.0127 0.791(0.02) 1% 4 −0.0651 0.629(0.001) 6% 11 0.0376 0.786(0.01) 3% 3 −0.126 0.630(0.001) 11% 1 0.0701 0.791(−) 54% 9 0.0494 0.784(0.007) 57% 1 −0.0550 0.787(−) 63%

1

0.1

6 0.0552 0.775(0.002) 24% 2 −0.119 0.676(0.02) 51% 4 0.0430 0.734(0.002) 18% 4 −0.0688 0.554(0.001) 29% 8 0.0534 0.716(0.01) 20% 4 −0.109 0.571(0.003) 42% 1 0.0409 0.801(−) 38% 18 0.0318 0.753(0.002) 35% 13 −0.0580 0.752(0.001) 64% 3 0.0474 0.732(0.003) 42% 4 −0.0703 0.741(0.007) 63% 4 0.0665 0.695(0.01) 15% 4 0.0621 0.763(−) 26% 6 0.0478 0.725(0.004) 19% 2 −0.114 0.593(0.01) 46% 4 −0.0672 0.470(−) 20% 4 0.0711 0.687(−) 16% 6 0.0709 0.748(0.005) 26% 2 −0.0813 0.567(0.002) 30%

1.0

ionic strength (mol·kg−1)

H+1|Na+1/ClO4−1

2

H+1|Me4N+1/Cl−1

Li+1|H+1/ClO4−1

electrolyte mixture

Table 1. continued

3

15

6

6

2

9

6

6

6

4

7

14

1

4

12

6

13

1

9

6 0.0566 0.907(0.002) 37% 1 −0.114 0.983(−) 74% 9 0.0464 0.819(0.02) 30% 4 −0.0700 0.689(0.02) 45% 9 0.0578 0.777(0.01) 32% 3 −0.101 0.709(0.007) 57% 1 0.0395 0.981(−) 39% 10 0.0311 0.878(0.004) 35% 5 −0.0579 0.876(0.004) 65% 2 0.0593 0.800(0.009) 50% 4 −0.0680 0.833(0.004) 57% 4 0.0664 0.747(0.02) 23% 4 0.0562 0.908(−) 36% 6 0.0492 0.944(0.004) 37% 1 −0.108 0.783(−) 66% 4 −0.0236 0.397(−) 9% 4 0.0735 0.723(−) 26% 10 0.0700 0.852(0.01) 36% 2 −0.0944 0.749(0.002) 49%

2.0

3

6

6

3

9

5

8

1

5

9

10

6

3

0.0731 0.798(−) 32% 4 0.0717 1.00(−) 44% 2 −0.102 1.10(0.002) 64%

4

6 0.0488 0.944(0.004) 37% 2 −0.102 1.09(0.02) 78%

3 −0.0662 0.969(0.01) 53%

7 0.0601 0.873(0.02) 41% 3 −0.0992 0.949(0.02) 67% 1 0.0417 1.23(−) 42% 6 0.0311 1.07(0.005) 35%

9 0.0458 0.964(0.01) 38%

4 0.0572 1.11(0.001) 47%

3 0.00200 1.58(0.03) 14%

3.0

Journal of Chemical & Engineering Data Article

DOI: 10.1021/acs.jced.6b00651 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

2

Cl−1|K+1/NO3−1

H

6

4 −0.0226 0.667(0.007) 54% 4 0.0191 0.667(0.004) 46% 6

Na+1|K+1/Cl−1

18

18

12 −0.0121 0.621(0.008) 32% 12 0.0231 0.624(0.004) 62%

23

24

6

K+1|Mn+2/Cl−1

K+1|Na+1/Cl−1

1

Mg+2|K+1/SO4−2

6

6

16

20

16

4

8

20

6

24

4

8

1

5

Mg+2|K+1/Cl−1

4 −0.0881 0.740(0.02) 106% 28 0.0290 0.565(0.01) 9% 6 −0.0356 0.671(0.02) 31% 7 0.0792 0.673(0.03) 68% 6 −0.0416 0.665(0.02) 39% 6 0.0641 0.666(0.01) 60% 2 −0.00142 0.606(0.002) −3% 36 −0.0308 0.649(0.006) −29% 4 0.0204 0.451(0.003) −19%

3 −0.0155 0.640(0.006) −5% 8 −0.0290 0.646(0.01) 104% 1 −0.0534 0.563(−) 63% 19 0.0551 0.532(0.002) 40%

9 0.0564 0.711(0.007) 21%

1.0

K+1|Mg+2/SO4−2

48

4

K+1|Me4N+1/Cl−1

K+1|Mg+2/Cl−1

10

Li+1|K+1/Cl−1 18

10

12

K+1|Li+1/Cl−1

20

12 0.0125 0.640(0.002) 16%

19

Li+1|K+1/Br−1

12

19

K+1|Li+1/Br−1

20

38

24

1

12

4

14

5

8 0.0335 0.625(0.01) 7% 14 −0.0682 0.712(0.01) 59% 15 0.0428 0.716(0.01) 37%

−0.0591 0.622(−) 61% 19 0.0607 0.606(0.02) 39% 1

3 −0.0187 0.673(0.003) −4%

2 0.0561 0.708(0.001) 14%

0.5

Cl−1|K+1/SO4−2

13

24

1

4

2

6 3 0.0856 0.753(0.004) 7% 15 −0.0949 0.788(0.002) 81% 15 0.0115 0.790(0.003) 10%

−0.0777 0.759(−) 60% 1 0.317 0.714(−) 160% 1

6 0.0522 0.784(0.008) 5%

0.1

ionic strength (mol·kg−1)

Cl−1|K+1/OH−1

NO3−1|K+1/Cl−1

1

12

H+1_CO3−2|K+1/Cl−1

Cl−1|K+1/F−1

Br−1|K+1/SO4−2

Sr+2|H+1/Cl−1

H+1|Sr+2/Cl−1

electrolyte mixture

Table 1. continued

11 0.0207 0.522(0.02) 10% 2 0.0544 0.462(0.01) −46% 2 0.174 0.460(0.01) 147% 10 −0.0335 0.670(0.02) 33% 12 0.0669 0.678(0.02) 65% 10 −0.0125 0.608(0.002) −54% 4 −0.0265 0.648(0.003) −87% 4 0.0169 0.461(0.002) −56% 1 −0.0549 0.297(−) −22% 1 0.00767 0.0718(−) −3% 4 0.0274 0.506(−) 50% 16 −0.00898 0.598(0.008) 26% 15 0.0231 0.604(0.008) 68%

19 0.0501 0.456(0.03) 42% 4 −0.0572 0.431(−) 48%

3 −0.00972 0.621(0.02) −5%

6 0.0584 0.775(0.004) 31%

2.0

18

17

6

2

2

6

6

17

10

10

6

20

4

12

18

6 −0.0342 0.721(0.02) 33% 6 0.0749 0.692(0.01) 73% 11 −0.0192 0.651(0.003) −175% 4 −0.0239 0.672(0.008) 11485% 2 0.0196 0.498(0.001) 9440% 2 −0.0359 0.255(0.001) −21% 1 0.00180 0.0600(−) −1% 4 0.0188 0.501(−) 74% 11 −0.00896 0.606(0.009) 27% 12 0.0236 0.610(0.01) 71%

17 0.0459 0.415(0.004) 42% 4 −0.0511 0.381(−) 47%

16 0.0588 0.881(0.01) 38% 7 −0.0915 0.857(0.03) 59% 3 −0.00118 0.600(0.003) −1%

3.0

Journal of Chemical & Engineering Data Article

DOI: 10.1021/acs.jced.6b00651 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

0.311 0.725(−) 23%

16 −0.0513 0.802(0.006) −77% 14 −0.129 0.804(0.01) 194%

I

5

10 7 2

Br−1|Na+1/acetic−1

Cl−1|Na+1/acetic−1

Br−1|Na+1/ClO4−1

5

acetic−1|Na+1/Br−1

Na+1|Mg+2/SO4−2

Na+1|Mg+2/Cl−1

Mg+2|Na+1/Cl−1

4 0.00363 0.792(0.002) 5% 6 −0.0490 0.789(0.01) 71% 5 −0.0411 0.785(0.001) 49% 1 −0.0415 0.788(−) −146%

3 −0.0873 0.793(0.001) −9%

6

25

9

7

51

12

5 −0.00507 0.741(0.003) −10% 5 0.00229 0.694(0.008) −5% 21 −0.0294 0.705(0.001) 43% 4 0.00330 0.693(−) 9%

8 0.00677 0.492(0.009) −2% 39 −0.0188 0.696(0.01) −7%

7

Na+1|Mg+2/Br−1

6

32

10

7

6

79

42

7

16

5 0.0899 0.447(0.01) 9%

5

Cl−1|Mg+2/SO4−2

11

6

Na+1|Li+1/ClO4−1 0.270 0.582(−) 8% 5 −0.0929 0.797(0.005) −10%

2

Li+1|Na+1/ClO4−1

1

12

14

12

Na+1|Li+1/Cl−1

20

15 −0.0186 0.769(0.002) −28% 15 −0.0919 0.774(0.009) 136% 12

20

Na+1|Li+1/Br−1

20

Li+1|Na+1/Cl−1

20

Li+1|Na+1/Br−1

8 0.00440 0.799(0.01) 6% 9 −0.0701 0.809(0.02) 102% 8 0.0223 0.733(0.005) 32% 8 −0.0459 0.732(0.003) 66% 2 0.0807 0.745(0.005) 53% 4 −0.0728 0.746(0.001) 47% 8 0.0790 0.394(0.01) 12% 5 −0.0171 0.716(0.01) −13% 28 −0.00492 0.478(0.02) 3% 59 −0.0141 0.680(0.02) −10% 4 −0.0342 0.338(0.002) −7% 4 −0.00916 0.773(0.001) −22% 6 0.0127 0.669(0.007) −31% 26 −0.0235 0.695(0.007) 39% 4 0.00510 0.681(−) 13% 6

19

5

7

6

56

11

7

22

6

3

37

17

9

8

6

26

8

4

6

73

26

16

5

2

7

6

18

18

18

21

NO3−1|Li+1/Cl−1

11 0.00275 0.911(0.002) 13% 11 −0.0193 0.916(0.01) 93% 5 0.0385 0.857(0.03) 54% 4 −0.0345 0.863(0.03) 48% 11 0.0288 0.808(0.02) 42% 24 −0.0378 0.799(0.02) 55% 3 0.0751 0.831(0.01) 53% 4 −0.0662 0.827(0.01) 47% 12 0.0696 0.354(0.02) 21% 5 −0.00147 0.741(0.02) −3% 7 −0.0110 0.525(0.008) 17% 42 −0.0102 0.704(0.02) −16% 4 −0.0320 0.278(0.001) −13% 5 −0.0151 0.913(0.002) −47% 5 0.0206 0.669(0.03) −65% 17 −0.0176 0.728(0.01) 34% 4 0.00850 0.708(−) 21%

11 0.0224 0.518(0.02) 56% 1 0.0953 0.556(−) 33% 15 −0.0122 0.607(0.005) −19%

2.0

18

18

1

13

17

14 −0.0185 0.631(0.001) −13%

13 0.0245 0.571(0.01) 20%

1.0

Cl−1|Li+1/NO3−1

18

15

6

18

11 0.0428 0.618(0.03) 17% 1 0.0607 0.662(−) 11% 14 −0.0250 0.668(0.003) −9%

0.5

ionic strength (mol·kg−1)

Na+1|La+3/Cl−1

K+1|Sr+2/Cl−1

1

1

OH−1|K+1/SO4−2

4

13

0.1

K+1|Ni+2/Cl−1

electrolyte mixture

Table 1. continued

16 −0.0132 0.625(0.03) 40% 54 −0.00794 0.758(0.03) −24% 4 −0.0283 0.247(0.002) −16% 2 −0.0201 1.13(0.001) −77% 5 0.0226 0.703(0.02) −87% 22 −0.0138 0.789(0.02) 30% 4 0.0109 0.762(−) 26%

4 0.0302 0.942(0.007) 43% 4 −0.0379 0.933(0.02) 54% 2 0.0749 0.952(0.01) 54% 3 −0.0623 0.943(0.02) 45% 8 0.0693 0.354(0.02) 21%

15 −0.00872 0.605(0.01) −27% 4 0.00770 0.681(0.02) 5% 9 0.0106 1.08(0.005) 40% 8 −0.0223 1.13(0.02) 84%

17 0.00876 0.536(0.02) 104%

3.0

Journal of Chemical & Engineering Data Article

DOI: 10.1021/acs.jced.6b00651 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

−0.0833 0.764(−) 52%

J

Sr+2|Na+1/Cl−1

Na+1|Sr+2/Cl−1

7

1

OH−1|Na+1/SO4−2 5 −0.111 0.800(0.002) −11%

25

Na+1|Ni+2/Cl−1

7

6

Cl−1|Na+1/CO3−2

Na+1|Sr+2/Br−1

6

F−1|Na+1/ClO4−1

SO4−2|Na+1/Cl−1 −0.0400 0.661(−) 80% 4 0.0480 0.645(−) 11% 19 −0.0115 0.691(0.003) −4% 1 0.0646 0.638(−) 13% 5 −0.0318 0.722(0.006) −11% 4

12

7

25

6

30

41

14 0.0576 0.638(0.008) 12%

Cl−1|Na+1/SO4−2

22

11

OH−1|Na+1/Cl−1

56

1

9

14

11

6

1

8 0.00436 0.678(0.01) −34% 4 0.0500 0.643(−) 46% 1 −0.0583 0.643(−) 54% 4 0.0410 0.650(−) 50%

6

18

Cl−1|Na+1/OH−1

8 0.0674 0.766(0.005) 5%

1

9

14

2

22

12

16

1

3 −0.00652 0.779(0.009) 23%

2 −0.469 0.869(0.02) −37% 4 −0.0207 0.736(−) 83%

16 0.0694 0.643(0.005) 69% 4 −0.0600 0.664(−) 59% 1 −0.104 0.785(−) −21%

0.5

5 −0.00604 0.698(0.009) −4% 9 0.000488 0.658(0.004) 0%

0.0490 0.588(−) 17% 20 −0.00818 0.671(0.006) −5%

4

12 −0.00042 0.659(0.008) −2% 4 −0.0228 0.665(−) 122% 9 0.00570 0.650(0.01) −109% 4 0.0450 0.594(−) 52% 1 −0.0473 0.601(−) 54% 39 0.0364 0.605(0.009) 47% 8 −0.0371 0.600(0.002) 48% 8 0.0165 0.634(0.06) −175% 8 0.0186 0.645(0.04) 198% 27 0.0494 0.588(0.01) 15% 20 −0.0366 0.340(0.007) 11%

17 0.0423 0.625(0.01) 44%

1.0

ionic strength (mol·kg−1)

NO3−1|Na+1/Cl−1

Cl−1|Na+1/NO3−1

H+1_CO3−2|Na+1/Cl−1

Cl−1|Na+1/H+1_CO3−2

Cl−1|Na+1/formic−1

6

6

Cl−1|Na+1/ClO4−1

ClO4−1|Na+1/Cl−1

2

6

NO3−1|Na+1/Br−1

Br−1|Na+1/SO4−2

21

0.1

Br−1|Na+1/NO3−1

electrolyte mixture

Table 1. continued

19

12

34

11

11

6

6

14

29

16 −0.00414 0.684(0.003) −6%

4 0.0356 0.570(0.003) 49% 4 −0.0340 0.562(0.007) 47% 7 0.0182 0.617(0.11) −152% 9 0.0328 0.610(0.14) 273% 22 0.0485 0.537(0.02) 22% 8 −0.0387 0.287(0.004) 17%

10 0.0110 0.638(0.02) 459%

23 0.0557 0.569(0.02) 60%

2.0

6

12

25

6

12

40

6

6

12

45

14

6

12

6

26

9 0.00267 0.705(0.02) 4% 4 −0.0177 0.515(−) 27%

0.0440 0.530(−) 26% 19 0.00239 0.706(0.01) 6%

4

30 0.0365 0.558(0.02) 51% 8 −0.0336 0.553(0.01) 47% 4 0.0285 0.590(0.05) −202% 5 0.0343 0.625(0.03) 243% 27 0.0478 0.516(0.02) 26% 8 −0.0397 0.267(0.005) 22%

8 0.00366 0.700(0.02) 16% 4 −0.0208 0.708(−) 91% 9 0.0138 0.653(0.02) 192%

21 0.0554 0.560(0.02) 61% 4 −0.0750 0.736(−) 83%

3.0

Journal of Chemical & Engineering Data Article

DOI: 10.1021/acs.jced.6b00651 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Cl−1|Rb+1/SO4−2

Electrolyte mixtures are specified in JESS notation as described in the text. At each ionic strength, the top row gives the number of points in database for the mixture, number of points with yB > 0.25 used to calculate the Harned coefficient, and the Harned coefficient; the second row gives the corresponding trace activity coefficient, the error estimate as a maximum deviation (in parentheses when more than one value is calculated), and the change in ln(activity coefficient) of the trace value from that of the pure binary solution relative to the other pure binary solution: (ln(gam-mix) − ln(gamB)) × 100/ (ln(gamC) − ln(gamB). The mixtures are in alphabetical order with cations placed first and anions placed second, with each complementary salt pair being tabulated together where applicable.

6

5 3 0.0603 0.506(0.003) 46% 5 3 0.0697 0.583(0.003) 47% 5 Cl−1|Rb+1/NO3−1

0.1 electrolyte mixture

Cl−1|Ni+2/SO4−2

Table 1. continued

Article

−1 Figure 1. log10(γmix and t = 25 °C with HCl in MCl2) at I = 1.0 mol·kg M = {Mg, Ca, Sr, Ba, Mn, Ni} (from top to bottom). Each series is displaced downward 0.01 for clarity. Data sources: Mg;72 Ca;71,90 Sr;32,35,164 Ba;30,34,75,108,164 Mn;65,85 Ni.76,110 Slopes of the straight lines correspond to Harned coefficients.

Figure 2. Results from Case Study 1natural logarithms of the activity coefficients for the system {NaOH + NaCl}(aq), showing activity coefficient curves for pure NaOH(aq), ln(γpure NaOH), as the upper black line, for pure NaCl(aq), ln(γpure NaCl), as the middle blue line, and for trace the trace activity coefficient of NaCl(aq), ln(γNaCl in NaOH ), as 141 as the lower two red straight line determined by Falciola et al. mix segments. Values for ln(γNaOH in NaCl ) (open symbols) and lnmix (γNaCl in NaOH) (filled symbols) at I = 1.0 mol·kg−1166 and I = 2.0 and 3.0 mol·kg−1103 are displaced to either side by 0.05 mol·kg−1 for clarity of representation.

same ionic strength, making this system anomalous among strong electrolyte mixtures. Exact quantification of the trace activity coefficients is made difficult because the measured activity coefficients (in terms of natural logs) are discrepant by as much as 0.01 at I = 1 mol·kg−1 and 0.05 at I = 3 mol·kg−1. This belies the implied precision of activity coefficients in the literature, which is typically reported as three or four decimal placessee ref 94 for instance. In fact these discrepancies in the activity coefficients of ternary mixtures between independent investigators are similar in size to the differences between critically assessed values in JESS for the binary salts from independent workers, as can be seen in Figure 2 scattered around the (topmost) curve for pure NaOH(aq). This is not the first time Harned’s results have been questioned. Pitzer and Kim (ref 169, p 5707) commented that

a

6

4

0.0727 0.341(−) 17% 3 0.0536 0.425(0.004) 46% 4 −0.112 0.913(−) −63% 6

1.0 0.5

ionic strength (mol·kg−1)

5

3.0

4

6 2.0

0.0740 0.322(−) 22% 3 0.0502 0.377(0.005) 47% 4 −0.0998 1.06(−) −73%

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Figure 4. Results from Case Study 2activity coefficient change in mixtures of {KBr + K2SO4}(aq) as reported by Zhang et al.158 for ionic strengths (top to bottom) I = 0.01, 0.1, 0.2, 0.5, 1.0, 2.0, and 3.0 mol· kg−1, with lines indicating fits calculated from parameters reported by Zhang et al.158

Figure 3. Additional results from Case Study 1natural logarithms of the activity coefficients for the system {NaOH + NaCl}(aq), showing linear fits to three independent sets of experimental data, together with the scatter of measured points. Data at I = 3.0 mol·kg−1 come from Harned and Harris167 (diamonds), Falciola et al.141 (squares), and Mironov and Sokolova103 (circles). Data in the top two sets come from Harned and Cook166 (diamonds), Falciola et al.141 (squares), and Mironov and Sokolova103 (circles).

doubtful given that at 0.2 the slope is negative. Accordingly, Harned’s rule with one (linear) parameter describes these measurements more than adequately. Moreover, the effect of ionic strength on mixing behavior is negligible or close to it, in line with much evidence that at higher concentrations Harned coefficients become less dependent on concentration.174 We also note, incidentally, that the experimental data at 0.1 are not matched using the author’s tabulated parameters and that their −1 value of γpure KBr (in their Table 1) is 0.934 at 0.0052 mol·kg compared to literature values of 0.926 or 0.927.175−178 Case Study 3: {CaBr2 + CaCl2}(aq). The system {CaBr2 + CaCl2}(aq) has been studied by Tialowska-Mocharla and Atkinson94 with Ca2+ and Br− ion-selective electrodes. The work appears to have been carefully performed, but regrettably, several aspects of it are problematic. First, the potential of the cell was reported to be stable to within 0.5 mV, corresponding to an uncertainty of ±0.005 in the activity coefficients, even though Tialowska-Mocharla and Atkinson94 give four significant figures in the logarithms of their activity coefficients. Second, analysis of their data with the equations of Pitzer169 leads to ternary interaction parameters that vary randomly with ionic strength, and with the (Cl−,Br−) interaction parameters tending to be large and positive, in contrast to literature reports that they should be independent of ionic strength and zerovalued169,179 or small and negative.180 Finally, regarding the significance of the β parameters reported by TialowskaMocharla and Atkinson,94 consider Figure 5. The α values change systematically but the β values are evidently scattered. It would be difficult to interpolate or extrapolate these β values to other ionic strengths and be confident of achieving anything meaningful. Case Study 4: {CsCl + MgCl2}(aq). As a last example, the system {CsCl + MgCl2}(aq) has been studied by Hu et al.152 with the Cs+ ion-selective electrode. They concluded that Harned’s rule with quadratic term fitted the experimental activity coefficients very well. Their Harned coefficients are shown in Figure 6. Although the α values are subject to some scatter, they at least display a plausible trend with ionic strength. However, the β values are small and flip randomly between positive and negative values. This strongly suggests that the β values are not

Harned and Gancy’s38 results for {HCl + KCl}(aq) may be subject to larger error than they assumed. This conclusion was corroborated by citing several sources: Lietzke and O’Brien170 found that both activity coefficients in the mixture obeyed Harned’s rule in linear form; mixing in the common ion system {HBr + KBr}(aq)171 was more consistent with the careful work of Guntelberg;172 and Bates et al.173 have shown that the Ag−AgCl electrode of the kind used by Harned and Gancy38 is more erratic than had been thought at the time. All of these factors lead us to reject Harned and Cook’s166 data from our analysis. We have also favored the results of Falciola et al.141 over those of Mironov and Sokolova103 since, in the same study by the latter authors, the data for the corresponding K+ system show some worrying inconsistencies. Case Study 2: {KBr + K2SO4}(aq). Careful consideration of the recent work by Zhang et al.,158 depicted in Figure 4, provides a graphic illustration of the overfitting to be found in this area of research. The authors used a “battery cell without liquid junction”, designated as K-ISE|KBr(m1),K2SO4(m2)|Br-ISE, to measure the activity coefficients in mixtures of KBr(aq) and K2SO4(aq) with a precision of 0.1 mV, typical of such ion-selective electrodes. The data at each ionic strength were fitted to the equation: mix pure ln(γKBr ) = ln(γKBr ) − αKBr(K 2SO4 )yK SO in K SO 2

4

2

−

4

βKBr(K SO )yK2 SO 2 4 2 4

i.e., a function with two adjustable parameters, αKBr(K2SO4) and βKBr(K2SO4), describing the change in ln(γmix KBr in K2SO4) away from the activity coefficient of the neat solution, ln(γmix KBr), as a function of the ionic strength fraction of K2SO4, yK2SO4. These fits show no sign whatsoever of a systematic effect that would warrant a second adjustable parameter. By eye, the lines look as straight as could be expected within the scatter of small random effects. There is some indication, one might conclude, of a positive slope at lower ionic strengths, but even this seems L

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curvature of the activity coefficients with respect to concentration has to be rejected. There is a strong tendency in the literature, within the area of solution chemistry at least, to pick out the data of highest quality (as critically assessed) and then to treat such data as if they were error-free. However, later independent investigations of similar or better quality often yield results that differ significantly from the earlier selected values, as illustrated in Figure 2. We have reached similar conclusions in other contexts, even for properties like density that can nowadays be measured routinely to high precision.181 The problem is that no matter how carefully experimental work is performed it is impossible to avoid systematic errors. Such errors cannot be quantified by internal scrutiny. In other words, experts tend to overemphasize the accuracy of individual work (often, but not always, their own). As a common consequence it becomes difficult for them, and others following them, to “see the forest for the trees”. A broader perspective in the context of Harned’s rule makes it clear that using the β parameters in eq 5 to fit small observed effects is not justified and has been a major distraction in the literature. Apart from enticing subsequent investigators to overfit their experimental data, theoretical frameworks with an abundance of empirical parameters (such as occur in the Pitzer equations) can accommodate these spurious effects and, hence, suffer from various deleterious consequences associated with too many degrees of freedom. Once it is accepted that aqueous strong electrolytes obey Harned’s rule (with one parameter) up to I = 3 mol·kg−1, it becomes much easier to identify and to reject anomalous experimental measurements. This is the great advantage of having a simple theoretical equation with a minimum of flexibility. Uncertainties in mean activity coefficients predicted with Harned’s rule are typically less than 0.01. Therefore, unless there are good grounds to suggest significant changes in chemical speciation, experimental measurements of electrolyte solution mixing not conforming to Harned’s rule should be regarded with suspicion. It may one day become possible to characterize more subtle real effects, but considerably improved experimental techniques, much superior to those available today, will be needed. Until then, it is safer to conclude that fits involving an extended Harned equation with a quadratic term are merely following systematic errors. We have found that the estimates of errors made by authors of single potentiometric investigations are typically as much as an order of magnitude too optimistic. Although the experimental work of Harned and colleagues was usually of high quality, particularly given the technological limitations of their time, the accuracy often claimed in the literature for their results is worse, and sometimes much worse, than stated. We are not at all suggesting that hypothetical models should take precedence over observation but rather depend on the axiom that scientific knowledge comes from agreement between experiment and theory. When, and only when, such agreement is reached is it sensible to assess the overall result in terms of the simplicity of the model’s premises, the patterns evident in the data, and the likely magnitude of experimental error. Judged by these criteria, this work shows that Harned’s rule with only one coefficient provides a description of aqueous−strong-electrolyte mixing thermodynamics which is widely applicable, satisfactorily accurate, and without sign of any systematic problem. On the other hand, there is an undeniable weakness in using the Harned coefficients reported here which arises because they

Figure 5. Harned coefficients (αCaBr2(CaCl2), open diamond; βCaBr2(CaCl2), filled diamond; αCaCl2(CaBr2), open triangle; βCaCl2(CaBr2), filled triangle) for the system {CaBr2 + CaCl2}(aq) reported by Tialowska-Mocharla mix = log10 γpure and Atkinson94 according with log10 γCaBr CaBr2 − αCaBr2(CaCl2) 2 2 mix yCaCl2 − βCaBr2(CaCl2)yCaCl2 and log10 γCaCl2 = log10 γpure CaCl2 − αCaCl2(CaBr2) 2 yCaBr2 − βCaCl2(CaBr2)yCaBr . 2

Figure 6. Harned coefficients (αCsCl(MgCl2), open diamond; βCsCl(MgCl2), filled diamond) for the system {CsCl + MgCl2}(aq) reported by Hu et al.152 according with ln γCsCl = ln γpure CsCl + αCsCl(MgCl2)yMgCl2 + βCsCl(MgCl2)y2MgCl2.

significant and that the curvature in the activity coefficients described by Hu et al.152 is more likely due to a systematic error, probably of experimental origin. These results again illustrate the widespread overfitting in the literature that we have identified in general.

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CONCLUSION In spite of frequent usage of the quadratic extension to Harned’s rule in the literature (see examples in the Supporting Information), the resulting β parameters often display unrealistic characteristics. In particular, the β values do not tend to differ from zero by statistically significant amounts, nor do they vary smoothly with the ionic strength of the system. We consider the second point to be particularly damning. Scattered β values are unquestionably contrary to what one would expect if the underlying data possessed significant/real quadratic dependence. Accordingly, in such cases the reported M

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(9) Khoo, K. H.; Lim, T.-K.; Chan, C.-Y. Activity Coefficients for the System HCl + CoCl2 + H2O at 298.15 K. J. Chem. Soc., Faraday Trans. 1 1978, 74, 2037−2044. (10) Esteso, M. A.; Fernandez-Merida, L.; Hernandez-Luis, F. F.; Gonzalez-Diaz, O. M. Activity Coefficients in the System Sodium Chloride + Disodium Succinate + Water at 25 °C. Ber. Bunsenges. Phys. Chem. 1989, 93, 213−217. (11) Khoo, K. H.; Chan, C.-Y. Activity Coefficient of Hydrochloric Acid in the System Hydrochloric Acid-Ammonium ChloridePotassium Chloride-Water at Constant Total Molality 0.5 mol kg−1 at 298.15 K. J. Chem. Eng. Data 1979, 24, 28−30. (12) Akerlöf, G. The Calculation of the Composition of an Aqueous Solution Saturated with an Arbitrary Number of Highly Soluble Strong Electrolytes. J. Am. Chem. Soc. 1934, 56, 1439−1443. (13) Akerlöf, G. A Study of the Composition of the Liquid Phase in Aqueous Systems Containing Strong Electrolytes of Higher Valence Types as Solid Phases. J. Phys. Chem. 1937, 41, 1053−1076. (14) Gieskes, J. M. T. M. Activity Coefficients of Sodium Chloride in Mixed Electrolyte Solutions at 25 °C. Z. Phys. Chem. 1966, 50, 78−90. (15) Wood, J. R. Thermodynamics of Brine-salt Equilibria - I. The Systems NaCl-KCl-MgCl2-CaCl2-H2O and NaCl-MgSO4-H2O at 25 °C. Geochim. Cosmochim. Acta 1975, 39, 1147−1163. (16) Jiang, C. Activity Coefficients of Hydrochloric Acid in Concentrated Electrolyte Solutions. 2. HCl + BaCl2 + KCl + H2O, HCl + LiCl + KCl + H2O, and HCl + NaCl + KCl + H2O at 298.15 K. J. Chem. Eng. Data 1996, 41, 117−120. (17) May, P. M.; Rowland, D.; Königsberger, E.; Hefter, G. JESS, a Joint Expert Speciation System - IV: A Large Database of Aqueous Solution Physicochemical Properties with an Automatic Means of Achieving Thermodynamic Consistency. Talanta 2010, 81, 142−148. (18) May, P. M. JESS at Thirty: Strengths, Weaknesses and Future Needs in the Modelling of Chemical Speciation. Appl. Geochem. 2015, 55, 3−16. (19) Akerlöf, G. Investigations of Sulfate Solutions. Experimental Methods and Results on Cells without Liquid Junction. J. Am. Chem. Soc. 1926, 48, 1160−1188. (20) Randall, M.; Breckenridge, G. F. The Activity Coefficient of Hydrogen Chloride in Aqueous Solutions with Barium and Lanthanum Chlorides at 25 °C. J. Am. Chem. Soc. 1927, 49, 1435− 1445. (21) Harned, H. S.; Schupp, O. E., Jr. Activity Coefficients and Dissociation of Water in Cesium Chloride Solutions. J. Am. Chem. Soc. 1930, 52, 3892−3900. (22) Harned, H. S.; Mason, C. M. The Activity Coefficient of Hydrochloric Acid in Aluminum Chloride Solutions. J. Am. Chem. Soc. 1931, 53, 3377−3380. (23) Hawkins, J. E. The Activity Coefficients of Hydrochloric Acid in Uni-univalent Chloride Solutions at Constant Total Molality. J. Am. Chem. Soc. 1932, 54, 4480−4487. (24) Harned, H. S.; Copson, H. R. The Dissociation of Water in Lithium Chloride Solutions. J. Am. Chem. Soc. 1933, 55, 2206−2215. (25) Murdoch, P. G.; Barton, R. C. The Activity Coefficients of Hydrochloric Acid in Aqueous Solutions Containing Either Sodium Dithionate or Perchloric Acid. J. Am. Chem. Soc. 1933, 55, 4074−4079. (26) Vance, J. E. The Dissociation of Water in Lithium Bromide Solutions at 25 °C. J. Am. Chem. Soc. 1933, 55, 4518−4521. (27) Vance, J. E. The Dissociation of Water in Strontium Chloride Solutions at 25 °C. J. Am. Chem. Soc. 1933, 55, 2729−2733. (28) Mason, C. M.; Kellam, D. B. The Activity Coefficient of Hydrochloric Acid in Cerous Chloride Solutions at 25 °C. J. Phys. Chem. 1933, 38, 689−692. (29) Owen, B. B.; Cooke, T. F., Jr. The Thermodynamics of Aqueous Hydrochloric Acid-Hydrobromic Acid Mixtures at Constant Total Molality. J. Am. Chem. Soc. 1937, 59, 2277−2279. (30) Glueckauf, E. Activity Coefficients in Concentrated Solutions Containing Several Electrolytes. Nature (London, U. K.) 1949, 163, 414−415. (31) Davidson, A. W.; Argersinger, W. J., Jr. Equilibrium Constants of Cation Exchange Processes. Ann. N. Y. Acad. Sci. 1953, 57, 105−115.

each refer only to a particular ionic strength and to only one complementary salt pair. This is especially worrying because these values are, of course, obtained by extrapolation and often by thermodynamic calculation rather than direct measurement. The key issue in understanding the physicochemical behavior of strong electrolytes remains: there is currently no sound fundamental model for their specific ion interactions.18,182 Such a model would accurately predict and smooth Harned coefficients as a thermodynamically consistent function of ionic strength. Until then, the application of Harned’s rule with one linear parameter to endorse experimentally measured data will be the outcome from this work of most benefit.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.6b00651. Examples of categoric statements regarding β and the quadratic term in eq 5 and literature references found with Harned coefficients (PDF)

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +61 8 9360 1393. Fax: +61 8 9360 6498. E-mail: [email protected]. Present Address

D.R.: Centre for Energy, School of Mechanical and Chemical Engineering, The University of Western Australia, Crawley, Western Australia, 6009. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Sandra Blair for entering much of the Harned coefficient data into the FIZ database of JESS.

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REFERENCES

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DOI: 10.1021/acs.jced.6b00651 J. Chem. Eng. Data XXXX, XXX, XXX−XXX