An Investigation of Zdanovskii's Rule for Predicting the Water Activity

Article pubs.acs.org/jced

An Investigation of Zdanovskii’s Rule for Predicting the Water Activity of Multicomponent Aqueous Strong Electrolyte Solutions Darren Rowland* and Peter M. May School of Chemical and Mathematical Sciences, Murdoch University, Murdoch, WA 6150, Australia ABSTRACT: The effectiveness of Zdanovskii’s rule as a fundamental method of predicting water activities in mixed aqueous strong electrolyte solutions has been investigated. Unfortunately, when literature data available for more than 100 ternary systems (i.e., two electrolytes plus water) were examined, sufficient measurements were rarely available to test Zdanovskii’s rule conclusively. This is because few mixed systems have been studied by more than one (independent) investigator and, in these few cases, the differences between reported water activities are generally much larger than typical estimates of experimental uncertainty. Most of the electrolyte systems considered in this work are found statistically to conform with Zdanovskii’s rule but often only within a worryingly large tolerance in residuals. However, when data are taken from a single source, deviations from Zdanovskii’s rule, significant in comparison to the stated experimental precision, can often be attributed equally well to small, unrecognized, systematic errors. This dilemma can only be resolved by more experimental measurements of sufficient accuracy to achieve better agreement between independent observers. Our findings imply that (a) confidence in deviations from Zdanovskii’s rule typically depends on a subjective judgment of data quality and (b) Zdanovskii’s rule, albeit often difficult to validate, provides an impressively general method for estimating water activities in mixed aqueous strong electrolyte solutions, with a certainty that, objectively, is as good as or better than current experimental capability.

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INTRODUCTION A key concept in modeling the properties of multicomponent electrolyte solutions is to use fundamental mixing rules to predict the mixture properties based on the empirical properties of the binary systems.1 The advantages of this approach are that solution properties can be estimated when the multicomponent system has not been experimentally characterized and that relatively few parameters are needed to describe the system’s thermodynamics. Since comprehensive experimental characterization of all multicomponent mixtures over wide ranges of temperature, pressure, and concentration is an impossibly large task, good theoretical predictions are required. The literature shows that considerable effort has accordingly been directed toward the identification and validation of mixing rules. As a result, multiple mixing rules are described even for the same property. In particular, since water activity is such a pivotal solution property, a large number of different mixing rules have been proposed for its prediction.2−8 Most of these use the concentrations of the binary electrolyte solutions so that the water activity can be determined solely from the specified composition of the mixture. However, an important exception is the “iso-activity” rule, discovered independently by Zdanovskii9 and Stokes and Robinson.10 It is known in the literature as Zdanovskii’s rule,11 the Zdanovskii−Stokes− Robinson rule,12 or the iso-active/isopiestic rule.13,14 The term Zdanovskii’s rule has precedence and is used throughout this work. © 2012 American Chemical Society

Although numerous claims have been made regarding the validity of Zdanovskii’s rule,11−13,15−17 no systematic study appears ever to have been performed. This work describes such an investigation. In particular, deviations between the measured and predicted osmotic coefficients are compared with likely estimates of the experimental error.

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THEORY According to Zdanovskii’s rule, solutions having equal water activity, when mixed in any proportion, produce a solution with the same water activity.9 Stated another way, many solutions with equal water activity exhibit no net effective interaction when mixed; that is, changes of solvation between the dissolved components on mixing are apparently absent. Since interactions between the solutes and the solvent occur and can be important in the binary solutions, but interactions between the solutes in mixed solutions are not evident,10,18 the behavior is termed “semi-ideal”. Algebraically, the rule is written as m1/m10 + m2 /m20 = 1

(1)

where m1 and m2 are the molal concentrations of solutes 1 and 2 in the mixture and mi0 is the concentration of the binary solution of i at the same water activity as the mixture. Received: June 20, 2012 Accepted: August 17, 2012 Published: August 23, 2012 2589

dx.doi.org/10.1021/je3006612 | J. Chem. Eng. Data 2012, 57, 2589−2602

Journal of Chemical & Engineering Data

Article

combination of the end point electrolytes is chosen, so-called “component rules” have been described, as discussed by Zhong and Friedman.23 These assign the concentrations of the neutral electrolyte combinations uniquely, based on the total ionic concentrations, and are necessary because in any mixture containing two or more cations and two or more anions, the amount of each electrolyte solute (nMX) present cannot be determined unambiguously from the concentrations of the ions.17 A particular rule which has yielded satisfactory results24 is due to Reilly and Wood25

Zdanovskii’s rule has a remarkably broad applicability. It was based originally on experimental observations on aqueous solutions, for which it has been claimed to hold for numerous strong electrolyte mixtures.11 Similar behavior has also been found in aqueous mixtures of nonelectrolytes,10 nonaqueous electrolyte solutions,12 and molten salt mixtures.19,20 In the latter cases, where the solvent is not water, the equivalent but more general thermodynamic property, solvent f ugacity, is conserved.21 This generality is relevant because it indicates that Zdanovskii’s rule represents a fundamental behavior of solution mixing.14,22 In contrast, mixing rules employing constancy of ionic strength (e.g., refs 4, 5, 7) do not apply readily to other solutions, even of aqueous nonelectrolytes.11 To calculate the water activity of a multicomponent solution given its composition, an algorithm is required to locate the linear mixing path defined by Zdanovskii’s rule. The details of an efficient algorithm for this purpose have been published previously.1 Stated briefly, the starting point is the specification of the solute concentrations in terms of neutral (usually electrolyte) components. Since the water activity of all binary solutions always decreases with increasing solute concentration (Figure 1), the dashed lines (Figure 2) are known not to be the

2nMnX nMX =

(

z M |z X | νM(X)νX(M)

1/2

)

∑c nczc + ∑a na|za|

(2)

where nM and zM are the number of moles and algebraic charge of the cation M, respectively, and nX and zX are the analogous quantities for anion X. The summations are over all cations c and anions a in solution. Since any electroneutral solution satisfies ∑cnczc = ∑ana|za|, this can be simplified to nMX =

nMnX gcd(z M , |z X|) ∑c nczc

(3)

where gcd() is the greatest common divisor function and the relation νM(X) = |zX|/gcd(zM, |zX|) is used to determine the number of MzM ions in the electrolyte formula, along with the similar expression for νX(M). Applying this component rule to mixtures of electrolytes, as is done in the present work, ensures that the solution water activity obtained using Zdanovskii’s rule is calculated consistently, no matter how the concentrations in the multicomponent system are specified.

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RESULTS More than 100 ternary electrolyte solutions were investigated. This was possible for mixtures at 25 °C (and 1 atm) because water activities/osmotic coefficients for many binary electrolyte solutions have been experimentally well-characterized and correlated using the Pitzer equations (with parameters taken from May et al.26). Mixtures containing electrolytes that are known to form associating species and/or are not wellcorrelated by the standard Pitzer equations,26 for example, sulfuric acid, were excluded. Residual plots for all ternary mixtures were examined initially to identify outlying or erroneous data. Some examples of inconsistencies are shown in Figure 3. The experimental water activity is plotted as the abscissa. The arrows show points which clearly lie well outside the trend of the majority of the data. Although these systems have very large average deviations from Zdanovskii’s rule, it is apparent that this cannot be taken as evidence of nonideal mixing. Unfortunately, a number of other ternary systems are similarly poorly characterized. Table 1 shows the statistical results of our analysis. Mixtures are grouped according to the type and number of ionic components. Agreement between the measured osmotic coefficients and values predicted by Zdanovskii’s rule is expressed as the root-mean-square difference (rmsd = N ∑i=1 [(ϕi − ϕzdan,i)2/N]1/2, where ϕi and ϕzdan,i are the experimental and predicted osmotic coefficient, respectively, and N is the number of data points). The number of data, rmsd, and the range of water activities (aw) are shown for each data set. The primary values for N and rmsd refer to the data whose water activity was calculated

Figure 1. Water activity of exemplar electrolyte and nonelectrolyte solutions (from top to bottom: sucrose, NaCl, Ca(NO3)2, LaCl3).

Figure 2. Mixing paths for two binary solutions of concentration m1 and m2, respectively, having equal water activity (solid line) and unequal water activities (e.g., dashed lines).

Zdanovskii mixing path because the water activities at the end points differ. However, the Zdanovskii mixing path can be found by repeatedly adjusting the binary solution concentrations to reduce the difference between the water activities of the end members. Hence, the Zdanovskii mixing path (Figure 2, solid line) can be robustly determined. To avoid determining different values for the water activity of the same multicomponent solution depending on which 2590



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Figure 3. Examples of outlying data identified by examining the residual plots for all electrolyte mixtures. Thin dark and thick light symbols represent mixtures with composition dominated by the first and second components, respectively. Left: LiCl + BaCl2,27 right: NaOH + NaCl.28

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CASE STUDIES: INDEPENDENTLY MEASURED SYSTEMS Fortunately, some ternary systems listed in Table 1 have been independently studied. These systems have been examined to provide information about the extent to which measurements of the same property can differ between experimenters. Hence, the size of typical error in ternary solution measurements has been estimated. These systems consist mostly of the major seawater ions: sodium, magnesium, potassium, chloride, and sulfate. Since the electrolytes formed from these ions are relatively easy to handle, the measurement errors should be comparatively small. Numerical error from poorly characterized binary electrolyte properties should also be negligible in these cases. NaCl + MgCl2. Sodium chloride and magnesium chloride salts comprise the majority of electrolyte substance in natural waters. Accordingly, the properties of this system have been very extensively studied. Residual plots for each data source are shown in Figure 5. Wu et al.75 presented a small data set which is well-represented by Zdanovskii’s rule. Platford’s data76 and the large data set of Rard and Miller77 are similarly well predicted overall: at water activities greater than 0.81, Rard and Miller’s data agree with Zdanovskii’s rule within ± 0.005. However, in more concentrated solutionscorresponding to the region in which pure NaCl approaches saturationthere is a noticeable increase in the measured osmotic coefficients. While the cause of this systematic effect can be diagnosed in this instance (it is due to changing the isopiestic reference from NaCl to CaCl2), in other cases unrecognized sources of systematic error make quantification of the deviations from Zdanovskii’s rule problematic. The data points of Dinane and Mounir78 exhibit much greater random scatter than the other data. There is no significant pattern in the residuals, and most of the calculated deviations fall within the experimental precision. However, some of the data have residuals of 0.03 or more, suggesting a variation greater than 2 % in the experimental values, precluding any possibility that they can be sensibly used to assess adherence of this system to Zdanovskii’s rule. KCl + MgCl2. The available independent sources of KCl + MgCl2 data are in reasonable agreement. The observed trends indicate that the osmotic coefficients predicted by Zdanovskii’s rule are consistently larger than the experimental measurements (Figure 6).

without extrapolating beyond the upper concentration limit for the binary electrolytes;26 secondary values (obtained when all the data were analyzed) are given in parentheses. While the rmsd values provide some information about the agreement between the experimental results and Zdanovskii’s rule predictions, crucially, they do not indicate whether the deviations are systematic or random in nature. This needs to be done by graphical inspection. It is important to bear in mind that deviations in the osmotic coefficient from the Pitzer equations of 0.01 to 0.02 are common for concentrated binary electrolyte solutions.109 The data for many ternary mixtures are more or less predicted to within this limit. However, some data sets show deviations significantly larger in magnitude (as shown, for example, in Figure 4). For the mixtures Li2SO4 + Rb2SO4 and CaCl2 + Mg(NO3)2, the solutions having intermediate ionic-strength fractions have consistently larger residuals than the solutions comprised predominantly of one salt. Such a pattern of residuals provides credible evidence in favor of solute−solute interactions, and the conclusion can be drawn that the deviations from Zdanovskii’s rule are not artifactual. However, such cases are exceptional. In contrast, although the ternary mixtures HClO4 + NaClO4 and KCl + CaCl2 exhibit residuals of relatively large magnitude, their pattern implies that this is not likely to be due to a failure of Zdanovskii’s rule. Rather, since large deviations are evident even in the almost-pure solutions, it is clear that the model description of the binary electrolyte solutions is inconsistent with the binary solution properties implied by the ternary data set. The effects of poorly characterized binary solutions are observed even for systems where the principal component is described reasonably accurately. For example, when K2SO4 is mixed with Li2SO4 or Cs2SO4, much lower water activities are reported50,51 for the mixture than are achievable in pure K2SO4(aq), due to its low solubility. Hence, the Zdanovskii’s rule predictions result in large rmsd values in the extrapolated range (Table 1). These issues are discussed further below. Observed water activities may show deviations from Zdanovskii’s rule for many reasons. Assessing whether Zdanovskii’s rule is accurate enough for practical applications is made difficult because most of the systems in Table 1 have only one data source. This means that experimental errors caused by poor procedure or chemical impurities could well cause deviations indistinguishable from nonideal mixing. 2591



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Table 1. Survey of Ternary Aqueous Mixtures Tested Using Zdanovskii’s Rulea system

ref

N

rmsd

aw range

HCl + NaCl HClO4 + LiClO4 HClO4 + NaClO4 LiCl + LiNO3 LiCl + NaCl

5 29 29 30 31 32 33 34 35 31 36 29 31 28 37 38 39 40 41 42 34 43 44 37 45 37 38 46 47 48 49 50 50 50 51 52 53 50 51 52 54 55 51 56 57 58 59 60 61 62 63 58 58 64 65 66 5 67 68 69 27

10 (10) 15 (15) 36 (69) 30 (30) 36 (36) 39 (44) 32 (45) 37 (42) 19 (21) 26 (26) 26 (30) 30 (33) 16 (26) 28 (47) 8 (8) 13 (13) 51 (85) 40 (47) 60 (60) 110 (110) 35 (40) 21 (21) 28 (28) 12 (12) 31 (35) 12 (12) 8 (8) 44 (44) 14 (25) 45 (60) 30 (30) 27 (27) 21 (61) 25 (41) 43 (46) 24 (28) 18 (18) 24 (72) 54 (72) 28 (28) 18 (22) 23 (23) 11 (22) 26 (29) 117 (117) 9 (17) 16 (26) 24 (30) 72 (72) 36 (36) 39 (39) 12 (15) 17 (23) 20 (82) 20 (30) 32 (32) 12 (12) 45 (60) 54 (60) 34 (43) 19 (28)

0.11 (0.11) 0.006 (0.006) 0.013 (0.028) 0.032 (0.032) 0.008 (0.008) 0.017 (0.017) 0.022 (0.027) 0.052 (0.055) 0.040 (0.047) 0.013 (0.013) 0.026 (0.028) 0.012 (0.014) 0.006 (0.020) 0.16 (0.17) 0.005 (0.005) 0.01 (0.01) 0.008 (0.01) 0.017 (0.016) 0.026 (0.026) 0.031 (0.031) 0.020 (0.022) 0.011 (0.011) 0.030 (0.030) 0.013 (0.013) 0.054 (0.070) 0.002 (0.002) 0.003 (0.003) 0.010 (0.010) 0.011 (0.050) 0.004 (0.027) 0.018 (0.018) 0.009 (0.009) 0.035 (0.070) 0.059 (0.074) 0.084 (0.090) 0.014 (0.016) 0.009 (0.009) 0.021 (0.023) 0.026 (0.074) 0.010 (0.010) 0.014 (0.022) 0.012 (0.012) 0.009 (0.14) 0.004 (0.004) 0.047 (0.047) 0.027 (0.036) 0.045 (0.068) 0.012 (0.013) 0.007 (0.007) 0.058 (0.058) 0.098 (0.098) 0.026 (0.026) 0.01 (0.022) 0.075 (0.076) 0.027 (0.060) 0.016 (0.016) 0.078 (0.078) 0.006 (0.009) 0.009 (0.022) 0.004 (0.009) 0.53 (0.64)

0.9650−0.7910 0.9317−0.7609 0.9805−0.4444 0.9932−0.5179 0.9826−0.7604 0.9900−0.7113 0.9901−0.7311 0.9900−0.5560 0.9900−0.7220 0.9841−0.7627 0.9934−0.5724 0.9723−0.7715 0.9846−0.7551 0.9680−0.4452 0.8946−0.8343 0.9670−0.8442 0.9835−0.7553 0.9935−0.7750 0.9845−0.7723 0.9771−0.7718 0.9903−0.7021 0.9900−0.7770 0.9494−0.8438 0.9629−0.8526 0.9877−0.7049 0.9342−0.8580 0.9675−0.9430 0.9966−0.9348 0.9878−0.7732 0.9904−0.6291 0.9935−0.7831 0.9604−0.8602 0.9636−0.8507 0.9709−0.9110 0.9710−0.8185 0.9919−0.8180 0.9880−0.9607 0.9704−0.8431 0.9653−0.7555 0.9920−0.8511 0.9962−0.9248 0.9800−0.9424 0.9653−0.8363 0.9921−0.9090 0.9910−0.9025 0.9720−0.5301 0.8601−0.2575 0.9015−0.6172 0.9853−0.6114 0.9900−0.6030 0.9275−0.4075 0.9771−0.6410 0.9744−0.5182 0.8063−0.4285 0.9906−0.2690 0.9961−0.8780 0.9750−0.9150 0.9792−0.5264 0.9799−0.3316 0.9925−0.8118 0.9886−0.7084

LiCl + KCl LiCl + CsCl LiCl + NH4Cl LiNO3 + NaNO3 LiNO3 + NH4NO3 LiClO4 + NaClO4 Li(C2H3O2) + Na(C2H3O2) NaOH + NaCl NaCl + NaBr NaCl + NaNO3 NaCl + KCl NaCl + CsCl

NaCl + NH4Cl NaCl + Pr4NCl NaBr + KBr NaNO3 + NH4NO3 KCl + KBr KCl + KNO3 KNO3 + NH4NO3 CsCl + NH4Cl NH4Cl + NH4NO3 Li2SO4 + Na2SO4 Li2SO4 + K2SO4 Li2SO4 + Rb2SO4 Li2SO4 + Cs2SO4 Li2SO4 + (NH4)2SO4 Na2SO4 + K2SO4 Na2SO4 + Rb2SO4 Na2SO4 + Cs2SO4 Na2SO4 + (NH4)2SO4 Na2HPO4 + (NH4)2HPO4 K2SO4 + Rb2SO4 K2SO4 + Cs2SO4 K2SO4 + (NH4)2SO4 BaCl2 + ZnCl2 CaCl2 + Ca(NO3)2 CaCl2 + Ca(ClO4)2 CaCl2 + CoCl2 CaCl2 + MgCl2

Ca(NO3)2 + Mg(NO3)2 MgCl2 + Mg(NO3)2

MgSO4 + MnSO4 HCl + BaCl2 HClO4 + Ba(ClO4)2 HClO4 + UO2(ClO4)2 LiCl + BaCl2

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Table 1. continued system LiCl + CaCl2 LiCl + MgCl2 LiCl + NiCl2 LiCl + Li2SO4 LiNO3 + Li2SO4 NaCl + BaCl2 NaCl + CaCl2 NaCl + CoCl2 NaCl + MgCl2

NaCl + MnCl2 NaCl + SrCl2 NaCl + FeCl3 NaCl + Na2SO4 NaClO4 + Ba(ClO4)2 NaClO4 + UO2(ClO4)2 Na2SO4 + MgSO4

KCl + CaCl2 KCl + MgCl2

KCl + SrCl2 KCl + FeCl3 KCl + K2SO4 KCl + K2HPO4 KNO3 + Mg(NO3)2 K2SO4 + MgSO4

CsCl + BaCl2 NH4Cl + BaCl2 NH4Cl + MgCl2 NH4Cl + (NH4)2SO4 NH4NO3 + Ba(NO3)2 NH4NO3 + Ca(NO3)2 NH4NO3 + Mg(NO3)2 NH4NO3 + (NH4)2SO4 (NH4)2SO4 + MgSO4 Me4NCl + NiCl2 MgCl2 + MgSO4

Mg(NO3)2 + MgSO4 ZnCl2 + ZnSO4 HCl + NaClO4 HCl + Ba(ClO4)2 LiCl + Na2SO4 NaCl + KBr NaCl + KNO3 NaCl + K2SO4 NaCl + MgSO4

ref

N

rmsd

aw range

27 27 70 30 30 71 72 73 74 60 75 76 77 78 62 79 80 81 82 75 83 68 75 84 85 86 87 88 89 90 82 53 91 92 88 93 94 69 95 96 97 98 99 99 99 100 101 70 75 64 102 103 64 104 5 5 105 37 38 53 106

34 (43) 32 (34) 8 (8) 19 (23) 25 (30) 48 (48) 19 (30) 53 (53) 31 (33) 27 (27) 15 (15) 86 (87) 261 (287) 19 (19) 31 (31) 90 (90) 45 (45) 88 (130) 88 (88) 15 (15) 55 (88) 36 (60) 15 (15) 14 (16) 24 (24) 57 (57) 27 (27) 75 (120) 20 (36) 72 (72) 89 (89) 19 (19) 47 (55) 40 (40) 72 (87) 40 (40) 22 (22) 49 (51) 12 (12) 31 (36) 41 (51) 59 (64) 12 (17) 41 (44) 37 (41) 26 (40) 37 (37) 8 (8) 14 (14) 0 (62) 44 (44) 40 (40) 0 (57) 39 (39) 5 (7) 4 (12) 17 (17) 8 (8) 16 (16) 16 (19) 18 (18)

0.083 (0.55) 0.072 (0.14) 0.001 (0.001) 0.021 (0.025) 0.015 (0.014) 0.003 (0.003) 0.022 (0.039) 0.005 (0.005) 0.079 (0.081) 0.010 (0.010) 0.003 (0.003) 0.008 (0.008) 0.006 (0.007) 0.017 (0.017) 0.007 (0.007) 0.016 (0.016) 0.002 (0.002) 0.006 (0.017) 0.008 (0.008) 0.016 (0.016) 0.017 (0.024) 0.009 (0.024) 0.008 (0.008) 0.014 (0.015) 0.011 (0.011) 0.027 (0.027) 0.019 (0.019) 0.020 (0.028) 0.017 (0.041) 0.029 (0.029) 0.034 (0.034) 0.003 (0.003) 0.01 (0.015) 0.009 (0.009) 0.055 (0.057) 0.028 (0.028) 0.013 (0.013) 0.027 (0.029) 0.003 (0.003) 0.042 (0.052) 0.007 (0.010) 0.008 (0.008) 0.013 (0.015) 0.010 (0.011) 0.029 (0.028) 0.032 (0.10) 0.006 (0.006) 0.043 (0.043) 0.015 (0.015) 0.0 (0.26) 0.015 (0.015) 0.027 (0.027) 0.0 (0.24) 0.050 (0.050) 0.066 (0.096) 0.20 (0.18) 0.015 (0.015) 0.014 (0.014) 0.020 (0.020) 0.005 (0.005) 0.011 (0.011)

0.9930−0.2490 0.9810−0.4609 0.9785−0.9730 0.9864−0.7717 0.9864−0.7728 0.9834−0.9042 0.9900−0.8191 0.9824−0.7677 0.9920−0.6821 0.9690−0.8161 0.9699−0.8229 0.9963−0.7530 0.9910−0.6592 0.9900−0.7960 0.9904−0.7993 0.9621−0.7703 0.9801−0.8513 0.8662−0.6588 0.9700−0.8615 0.9699−0.8229 0.9842−0.4877 0.9772−0.4815 0.9862−0.9379 0.9939−0.8424 0.9904−0.8770 0.9759−0.8491 0.9787−0.8497 0.9677−0.5535 0.9839−0.7130 0.9839−0.8526 0.9700−0.8615 0.9878−0.9505 0.9913−0.7788 0.9713−0.9449 0.9820−0.9017 0.9909−0.9810 0.9902−0.9380 0.9925−0.8781 0.9462−0.9144 0.9900−0.6469 0.9865−0.7661 0.9900−0.7438 0.9975−0.9578 0.9967−0.5917 0.9967−0.5266 0.9869−0.7983 0.9905−0.9405 0.9785−0.9730 0.9894−0.9297 0.8090−0.4285 0.9930−0.8737 0.9801−0.9396 0.8086−0.4284 0.9783−0.9011 0.9640−0.5250 0.9380−0.7200 0.9889−0.8754 0.8945−0.8479 0.9645−0.9289 0.9867−0.9266 0.9818−0.8767

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Table 1. continued system NaBr + KCl NaNO3 + KCl Na2SO4 + KCl Na2SO4 + CsCl Na2SO4 + MgCl2 KCl + MgSO4 K2SO4 + Mg(NO3)2 CaCl2 + Mg(NO3)2 Ca(NO3)2 + MgCl2

N

ref 37 38 53 105 106 107 108 58 58

12 12 14 16 17 17 35 22 18

(12) (24) (20) (17) (17) (43) (35) (35) (24)

rmsd 0.010 0.004 0.002 0.014 0.015 0.020 0.066 0.025 0.036

(0.010) (0.011) (0.002) (0.014) (0.015) (0.036) (0.066) (0.054) (0.042)

aw range 0.9336−0.8616 0.9768−0.8494 0.9693−0.9112 0.9914−0.8248 0.9862−0.8767 0.9788−0.8900 0.9894−0.9647 0.9730−0.5180 0.9678−0.5280

a

If N is given as 10 (20), then 20 total data were processed, but only 10 were within the valid range of the Pitzer parameterization of the binary electrolytes.26

Figure 4. Examples of ternary mixtures with large, systematic deviations from the osmotic coefficient values predicted by Zdanovskii’s rule. Thin dark and thick light symbols represent mixtures with composition dominated by the first and second components respectively. Top left: Li2SO4 + Rb2SO4,50 top right: CaCl2 + Mg(NO3)2,58 bottom left: HClO4 + NaClO4,29 bottom right: KCl + CaCl2.86

Rard and Miller42 measured the system at only two different ionic-strength fractions. The mixture compositions did not correspond exactly to any of those used by Robinson, so the osmotic coefficients cannot be directly compared. While the pattern and magnitude of the residuals follow the trend of Robinson’s data, Rard and Miller’s analysis concludes that the osmotic coefficients determined by Robinson are around (0.5 to 1.0) % too low.42 The data set of El Guendouzi et al.34 has lower rmsd values than the other literature sources. However, the osmotic coefficient values are more internally scattered and, therefore, appear to be less reliable. Uncertainties of such magnitude between the results of independent observers preclude a meaningful objective assessment of Zdanovskii’s rule. CaCl2 + MgCl2. The calcium chloride plus magnesium chloride system has been measured by three groups. The data

Since the few experimental osmotic coefficient data reported at nearly identical concentrations tended to differ by 0.01 or less, it is possible that some of the deviation exhibited from Zdanovskii’s rule is due to nonideal mixing. However, the likelihood of this remains unclear: a significant number of the data lie outside the region where KCl(aq) could be characterized by the Pitzer equations due to solubility constraints, which might contribute to the size of the residuals at low water activities. NaCl + CsCl. Residual plots for the mixture of sodium chloride and cesium chloride are shown in Figure 7. Robinson41 measured the system at six different ionic-strength fractions. Nonideal mixing is seemingly evident: the solutions with equal moles of each salt have consistently larger residuals than other compositions at the same total molality. 2594



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Figure 5. Residual plots for the ternary system NaCl + MgCl2. Thin dark and thick light symbols represent mixtures with composition dominated by NaCl and MgCl2, respectively. Top left: ref 75, top right: ref 76, bottom left: ref 77, bottom right: ref 78.

of Robinson and Bower61 have an rmsd = 0.007 (Table 1), which is satisfactory. In contrast, the rmsd values of the data sets of El Guendouzi et al.62 and Christov63 are almost 0.06 and 0.1, respectively. Residual plots for the data sets are shown in Figure 8. The Robinson and Bower61 data are predicted extremely well by Zdanovskii’s rule for water activities greater than 0.75. In this region, all of the residuals are within ± 0.01. At lower water activities, the MgCl2-dominated mixtures are less wellpredicted, with deviations up to 0.025 in magnitude. However, the CaCl2-dominated solutions are well-represented across the entire range of water activity. On the other hand, many of the residuals for the other data sets are greater than ± 0.05 and display large scatter. Regarding the Christov data,63 poor agreement with the author’s own model suggests that the measurements are subject to considerable random error. It can be concluded that this system conforms with Zdanovskii’s rule and that reported experimental data deviating from Zdanovskii’s rule can be regarded as less reliable than the rule itself. K2SO4 + MgSO4. The system potassium sulfate plus magnesium sulfate has received attention from three different groups. Predicting the data sets using Zdanovskii’s rule resulted in average deviations of > 0.05,88 0.028,93 and 0.013.94 The residual plots for the three data sets are shown in Figure 9. The pattern of residuals from Kuschel and Seidel’s88 data suggest that their measurements are internally consistent. However, the residuals have large magnitudes, and neither the K2SO4- or MgSO4-dominated solutions are particularly wellpredicted.

The data from Todorović and Ninković93 fall within a small range of water activity values, corresponding to reasonably dilute solutions. It is evident that the solutions with K2SO4 as the principal component are apparently not in good agreement with Zdanovskii’s rule. However, the solutions with greater amounts of MgSO4 are increasingly well-predicted. This, and the differences between the two data sets,88,93 suggest the deviations from Zdanovskii’s rule may well be artifactual. On the other hand, the data set of Mounir et al.94 is more internally scattered but has smaller residuals than the other literature sources. Among these data sets, several measurements have been reported for solutions with virtually identical compositions. For example, the osmotic coefficient of 0.4 m K2SO4 + 1.2 m MgSO4 (a, Figure 9) is given as 0.56188 and 0.650.94 Similarly, 0.45 m K2SO4 + 1.35 m MgSO4 (b, Figure 9) has 0.57388 and 0.668,94 and the mixture with 0.55 m K2SO4 + 1.65 m MgSO4 (c, Figure 9) has 0.60588 and 0.718.94 In each of these cases, the difference in the measured osmotic coefficients is around 0.1 (i.e., greater than 10 %), precluding any sensible assessment of Zdanovskii’s rule. The situation must evidently remain unresolved until more experimental results of sufficient accuracy become available to provide better agreement between independent observers.

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DISCUSSION There is a broad body of evidence in the literature supporting the general validity of Zdanovskii’s rule.22 For example, Hu and co-workers have successfully used Zdanovskii’s rule to predict a variety of thermodynamically related but distinct physicochemical properties such as density110 and viscosity.111 To all this 2595



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Figure 6. Residual plots for the ternary system KCl + MgCl2. Thin dark and thick light symbols represent mixtures with composition dominated by KCl and MgCl2, respectively. Top left: ref 87, top right: ref 88, bottom left: ref 89.

can be added the findings of the present work shown in Table 1 where the rmsd is, say, less than 0.01. More pointedly, this paper demonstrates that there are few mixed electrolyte systems for which Zdanovskii’s rule conclusively fails. That Zdanovskii’s rule works well with so many systems, and often with such disparate electrolytes, is indeed remarkable. So, it only remains to consider those cases where there is significantly greater disagreement between the experimental reported osmotic coefficients and the Zdanovskii’s rule predictions. To draw meaningful conclusions in these cases it is necessary to make a careful assessment of the source and magnitude of the relevant experimental errors. According to Rard and Platford,112 measurements of osmotic coefficients using the isopiestic method should be reproducible to better than 0.3 %, that is, approximately 0.003 in ϕ. Other experimental techniques are generally less accurate: the hygrometric method used by El Guendouzi and co-workers yields results which are only reproducible to approximately 2 % in moderately concentrated solutions,27 for example. In practice, osmotic coefficients for many ternary solutions have internal inconsistencies as great as 5 %. Published data on the systems NaOH + NaCl,28 CaCl2 + MgCl2,63 and those by Teruya et al.5 have an internal scatter of 10 % or more. This means not only that far fewer systems can be evaluated for conformance with Zdanovskii’s rule than might be expected, but also that considerable caution is necessary before concluding that Zdanovskii’s rule is not being obeyed. While single studies using the isopiestic method can achieve osmotic coefficients to a precision of 0.3 %, independent measurements tend to exhibit deviations that are often considerably larger than this,112 casting doubt on the level of

accuracy. The comparison of measurements from independent investigators examined in this work suggests that the agreement between different sources generally varies from (1 to 10) %, intimating that they are generally an order of magnitude worse than investigators’ own estimates of precision. Insofar as the applicability of Zdanovskii’s rule to a given system can only be validated by measurements on that system,16 the difficulties in accurately determining osmotic coefficients of ternary electrolyte solutions are problematic. Another issue is the paucity of available experimental data. It is therefore particularly unfortunate that so many instances occur in the literature where data which could be used to test Zdanovskii’s rule have not actually been included in the paper (e.g., refs 113omitted data later published in ref 114and 115−116). The situation is even worse at temperatures other than 25 °C. Osmotic coefficient data for mixtures at elevated temperatures are extremely limited112 (examples are those of Oak Ridge, 117−125 Voigt and co-workers 126−131 and others132−134), and independent measurements at high temperatures are almost nonexistent. The need for, and value of, Zdanovskii’s rule to predict water activities under superambient conditions is therefore all the greater.

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CONCLUDING REMARKS The results of this work have importance beyond the specific contexts of water activities and Zdanovskii’s rule. Similar outcomes (unpublished) have been found in our study of other physicochemical properties at nonambient temperatures and pressures fitted by the Pitzer equations, and they are also mirrored in any insightful analysis of errors in reaction parameters (especially equilibrium constants), where there is 2596



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Figure 7. Residual plots for the ternary system NaCl + CsCl. Thin dark and thick light symbols represent mixtures with composition dominated by NaCl and CsCl, respectively. Top left: ref 41, top right: ref 42, bottom left: ref 34.

Figure 8. Residual plots for the ternary system CaCl2 + MgCl2. Thin dark and thick light symbols represent mixtures with composition dominated by CaCl2 and MgCl2, respectively. Top left: ref 61, top right: ref 62, bottom left: ref 63. 2597



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Figure 9. Residual plots for the ternary system K2SO4 + MgSO4. Thin dark and thick light symbols represent mixtures with composition dominated by K2SO4 and MgSO4, respectively. Top left: ref 88, top right: ref 93, bottom left: ref 94. The points denoted a, b, and c show measurements on ternary solutions with equal composition (see text).

any unconstrained fitting of empirical functions tends to be ruinous for modeling purposes because such empirical functions slavishly follow the effects of error both random and systematic. Better models based on sound fundamentals are therefore urgently needed. It is in this respect that the general and fundamental validity of Zdanovskii’s rule is potentially so valuable in solution chemistry. On the other hand, it is important to bear in mind that Zdanovskii’s rule is not followed by all solution mixtures of strong electrolytes. Since there is no discernible pattern to the proven exceptions, applying the general rule to specific cases without supporting experimental evidence warrants caution. We are also aware of another concern regarding the application of Zdanovskii’s rule when it is used to calculate activity coefficients (especially trace activity coefficients) using the Gibbs−Duhem equation. Since these calculations implicitly take differences between relatively large numbers, such calculations are prone to error which can be problematic.

a greater wealth of pertinent evidence.135 In all of these cases, it is clear that agreement between independently determined thermodynamic properties is strikingly worse than the precision achieved in the individual programs of measurement. While this itself is not surprisingnamely, that collective accuracy is inferior to the precision of high quality experimentsit is salutary to note the magnitude of the differences which typically occur. For example, an order of magnitude for the difference between experimental precision and likely accuracy has been suggested as a rule of thumb for equilibrium constants.136 An order of magnitude is consistent with the present work and with our other recent findings. Moreover, there is often a pronounced lack of agreement even between the best independent investigations, when such data are available (e.g., the NaCl + CsCl case study above; see also p 251, Rard and Platford112). Systematic errors are evidently much greater than is recognized in most experimental studies. The implications of this for modeling aqueous solution thermodynamics are profound and several. First, scientific understanding can be claimed only when theory is in reasonable agreement with multiple experiments performed for the same system under the same conditions by unconnected investigators. (Theories are obviously untrustworthy when they are not in accord with experiment, but equally so, measurements made in only one laboratory warrant greater caution than is commonly applied, no matter how good the reputation of the experimentalist.) Second, a better approach to the parametrization of thermodynamic functions is evidently required, one that is specifically focused on suppressing the ill-effects of numerical correlation and error propagation. Third,

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Some constructive comments from anonymous reviewers and Professor Glenn Hefter during the preparation of this manuscript are appreciated. 2598



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An Investigation of Zdanovskii's Rule for Predicting the Water Activity

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