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J. Phys. Chem. B 2008, 112, 15376–15381
Semi-ideal Solution Theory. 2. Extension to Conductivity of Mixed Electrolyte Solutions Yu-Feng Hu,* Xian-Ming Zhang, Ji-Guang Li, and Qian-Qing Liang High Pressure Fluid Phase BehaVior & Property Research Laboratory, China UniVersity of Petroleum, Beijing 102200, People’s Republic of China ReceiVed: June 26, 2008; ReVised Manuscript ReceiVed: August 9, 2008
Conductivities were measured for the ternary systems NaCl-LaCl3-H2O and KCl-CdCl2-H2O and their binary subsystems NaCl-H2O, KCl-H2O, CdCl2-H2O, and LaCl3-H2O at 298.15 K. The semi-ideal solution theory for thermodynamic properties of aqueous solutions of electrolyte mixtures was used together with the Eyring absolute rate theory to study conductivity of mixed electrolyte solutions. A novel simple equation for prediction of the conductivity of mixed electrolyte solutions in terms of the data of their binary solutions was established. The measured conductivities and those reported in literature were used to test the newly established equation and the generalized Young’s rule for conductivity of mixed electrolyte solutions. The comparison results show that the deviation of a ternary solution from the new conductivity equation is closely related to its isopiestic behavior and that the deviations are often within experimental uncertainty if the examined system obeys the linear isopiestic relation. While larger deviations are found in the system with large ion pairing effect, the predictions can be considerably improved by using the parameters calculated from its isopiestic results. These results imply that the previous formulation of the thermodynamic properties of aqueous solutions of electrolyte mixtures has a counterpart for transport properties. 1. Introduction One of the objectives of the theory of ionic solutions is to calculate various properties of mixed electrolyte solutions from properties for single electrolytes, plus a minimal amount of additional information that involves the interactions in the mixtures and cannot be found in binary solutions.1 An important aspect of this objective has been achieved with three empirical rules, i.e., Harned’s rule,2 Young’s rule,3 and Zdanovskii’s rule (sometimes referred to as the linear isopiestic relation).4 Young’s rule3 has been extended to transport properties.1,5 Zdanovskii’s rule was first discovered empirically by Zdanovskii for the molalities of ternary unsaturated electrolyte solutions4 and was derived for unsaturated nonelectrolyte solutions by Stokes and Robinson using the semi-ideal hydration approach.6–9 Since then, it was experimentally extended to the molalities of unsaturated aqueous solutions of electrolytes and nonelectrolytes.10,11 Recently, it has received considerable attention,12–23 as it can be coupled with the thermodynamic relations to yield the simple predictive equations for thermodynamic properties (activity coefficient of each solute in multicomponent solutions,12,13,20,24 volumetric properties,12,13,19 thermal properties,12,13,15a and surface tension15b) and thermodynamically consistent combinations of various models for electrolytes and nonelectrolytes.14 Besides, it has been extended to saturated solutions12,13 and highly supersaturated solutions related to aerosols.14,16 At the same time, efforts have also been made to explore theoretical justifications of the Zdanovskii’s rule. Mikhailov derived the rule for very dilute electrolyte solutions where the Debye-Hu¨ckel theory applies.21 However, extensive isopiestic results indicate that the rule is much more widely applicable than can be theoretically justified by Mikhailov.21 Rard also derived the rule by assuming that the osmotic coefficients of the binary and multicomponent solutions are equal under isopiestic equili* To whom correspondence should be addressed. E-mail: Huyf3581@ sina.com. Fax: 86-10-89733846. Tel: 86-10-89733846.
brium,23 which is evidently quite reasonable for the solutions of 1:1 electrolytes. Recently, a semi-ideal solution theory has been presented to describe the thermodynamic behavior of the multicomponent electrolyte solutions M1X1- · · · -MnXn(N1Y1)sat- · · · -(N{n′}Y{n′})sat-H2O at constant activities of all N1Y1, ..., N{n′}Y{n′}, and H2O, from which the new linear isopiestic relations for saturated solutions and thereby the Zdanovskii’s rule for unsaturated electrolyte solutions were theoretically derived.13 The theory proves that, under the condition of constant activities of all N1Y1, ..., N{n′}Y{n′}, and H2O, the average hydration numbers characterizing the ion-solvent interactions have the same values in the mixture as in the subsystems. It states that, for the generalized linear isopiestic relations (from which the Zdanovskii’s rule can be reproduced) to hold, the contributions from the ion-ion interactions to the solvent activity should be the same in the mixture as in the subsystems and that the effect of complex formation upon mixing the nonideal electrolyte mixtures MiXi-(N1Y1)sat-, ..., -(N{n′}Y{n′})sat-H2O (i ) 1, 2, ..., n) at constant activities of all N1Y1, ..., N{n′}Y{n′}, and H2O should be negligible compared to those of the Debye-Hu¨ckel contribution and the ion-solvent interactions. The first one has been substantiated,13 and the justification of the second one is limited to isopiestic measurements. The formation of neutral and/or ion complexes upon mixing the binary solutions would decrease the conductivity of the thusobtained solutions of electrolyte mixtures. Besides, conductivity measurements are often used to test electrolyte theories.5 Furthermore, conductivity is one of the principal transport properties of electrolyte solutions, not only for its intrinsic interest, but also for technical and industrial applications such as batteries and plating. Therefore, it is of interest to determine whether the theory for the thermodynamic properties of electrolyte mixtures can be extended to conductivity and whether the above-mentioned second statement can be justified by the conductivity measurements. In the following sections, the simple
10.1021/jp805833g CCC: $40.75 2008 American Chemical Society Published on Web 11/07/2008
Semi-ideal Solution Theory
J. Phys. Chem. B, Vol. 112, No. 48, 2008 15377
equation for prediction of conductivity of mixed electrolyte solutions in terms of the data of its binary solutions was established on the basis of the semi-ideal solution theory.13 The conductivities of the ternary systems NaCl-LaCl3-H2O and KCl-CdCl2-H2O and their binary subsystems NaCl-H2O, KCl-H2O, CdCl2-H2O, and LaCl3-H2O were measured at 298.15 K. The measured conductivities, together with the conductivities reported in the literature, were used to test the derived equations and the assumption of the semi-ideal solution theory. 2. New Predictive Equation for Conductivity 2.1. Semi-ideal Solution Theory. It is assumed that all electrolyte solutes (MiXi) are completely dissociated and produce νMi cations of charges zMi and νXi anions of charges zXi, respectively. Note that the cations and anions are denoted by + Mi and Xi instead of Mzi Mi and Xzi Xi.13 13 In semi-ideal solution theory, the ion-ion interactions in the semi-ideal mixture M1X1-M2X2-H2O are treated with Debye-Hu¨ckel theory, and the ion-solvent interactions are described by the stepwise hydration equilibrium.6–8 That is, the Gibbs energy of the above system is composed of two terms, namely G ) GDH + GHy, where GDH is the Debye-Hu¨ckel contribution, and GHy describes the semi-ideal mixture of the resulting species based on mole fraction x (ln a ) ln aDH + ln aHy, where aHy ) x). The stepwise hydration equilibrium can be symbolized by
U · H2O(l-1) + H2O a U · H2O(l)
(
mo1
+
m2 mo2
)1
aw ) constant and 0 e
mi mio
)
e1
( )
with
∆G* TR
]
xHo,ideal ) aH2O 2O,i
(5)
ln σi ) (1 - xHo,ideal ln σi · H2O(Li) 2O,i )
(6)
and
For eq 5 to hold, it is convenient to introduce an electrolytespecific parameter ci and express xHo,ideal as 2O,i
xHo,ideal ) 2O,i
55.51 - ciνimio 55.51 - ciνimio + νimio
(7)
where νi ) νMi + νXi. The parameter ci relies on aH2O. The conductivity of the pseudo liquid i · H2O(Li), σi · H2O(Li), depends only on the value of L (i.e., depends only on water activity). A combination of eqs 5 and 7 gives
ci )
aH2O 55.51 1 - aH2O νimio
(8)
As mentioned above, the average hydration numbers have the same values in the mixture as in the subsystems of the same water activity and, therefore, the species i · H2O(Li) (i ) 1 or 2) are the same in ternary solution M1X1-M2X2-H2O as in the subsystems MiXi-H2O of the same water activity. Furthermore, the process of mixing the solutions MiXi-H2O (i ) 1 or 2) at constant water activity is as simple as that of mixing the ideal solutions MiXi-H2O (i ) 1 or 2) of equal water mole fraction. Accordingly, the ternary solution M1X1-M2X2-H2O may also be described by an ideal solution 1 · H2O(L1)-2 · H2O(L2)-H2O that satisfies
(2)
(3)
(4)
where σmixture and σideal are the conductivities of a mixed solution and an ideal solution, respectively. Superscript “pure” together with subscript “i” represents the property of pure substance. ∆G+ is the molar excess activation free energy of flow. The summation in eq 4 is carried out over all constituents of the solution (solvent and solutes). 2.3. New Equation for Conductivity. The actual species in the i-th solution MiXi-H2O is free water molecules and Mi · H2O(l) and Ni · H2O(l’) (in the following sections, they are denoted by i · H2O(Li) for convenience). The species i · H2O(Li) are the charge carriers and thus the conductivity of the solution relies on their number and their mobilities. The species i · H2O(Li) has a constant composition at a given water activity. This couples well with the fact that σH2O ) 0 allows us to describe the conductivity (σi) of the solution MiXi-H2O by invoking an o,ideal ideal solution that is composed of xio,ideal · H2O(Li)(1 - xH2O,i ) pseudo o,ideal liquid i · H2O(Li) and xH2O,i free water molecules and satisfies
xHideal ) aH2O 2O
2.2. Eyring Absolute Rate Theory for Conductivity. The Eyring absolute rate theory25 can be used to describe conductivity:
σmixture ) σideal exp
xi ln σipure
i
(1)
where U ) Mi and Xi with i ) 1 or 2 and l ) 2, ..., n. It is shown that, under the condition of constant water activity, the average hydration numbers characterizing the ion-solvent interactions have the same values in the mixture as in the subsystems. It has also been shown that, if there is no complex formation, then the ion-ion Coulombic interactions in mixed-electrolyte solutions can be adequately accounted for in terms of those existing in their component binary solutions of equal water activity. Under the above conditions, the process of mixing the nonideal electrolyte solutions MiXi-H2O (i ) 1 or 2) at constant water activity is as simple as that of mixing the ideal mixtures MiXi-H2O (i ) 1 or 2) of equal water mole fraction, such that the changes in the thermodynamic properties accompanying the process of mixing the MiXi-H2O (i ) 1 or 2) nonideal solutions obey the same linear relations as mixing the classical ideal solutions. The molalities of MiXi (i ) 1 or 2) in M1X1M2X2-H2O, mi, are related to their values (moi ) in the binary solutions MiXi-H2O (i ) 1 or 2) by the Zdanovskii’s rule:
m1
[∑
σideal ) exp
(9)
ideal ln σ ) x1ideal · H2O(L1) ln σ1 · H2O(L1) + x2 · H2O(L2) ln σ2 · H2O(L2) (10)
with
xHideal ) 2O
55.51 - c(ν1m1 + ν2m2) 55.51 - c(ν1m1 + ν2m2) + ν1m1 + ν2m2
(11)
ideal ideal ideal and x1ideal · H2O(L1) + x2 · H2O(L2) + xH2O ) 1, where xi · H2O(Li) ) νimi/ (55.51 - c(ν1m1 + ν2m2) + ν1m1 + ν2m2). From eqs 2 -11, we obtain
15378 J. Phys. Chem. B, Vol. 112, No. 48, 2008
c)
c1ν1m1 + c2ν2m2 ν1m1 + ν2m2
Hu et al.
(12)
and
ln σ ) z1 ln σ1 + z2 ln σ2
(13)
where zi ) ratio of the mole fraction of i · H2O(Li) (1 or 2) in the ternary ideal solution 1 · H2O(L1) - 2 · H2O(L2) - H2O to the mole fraction of i · H2O(Li) in the binary ideal solution i · H2O(Li) o,ideal H2O, i.e., zi ) xiideal · H2O(Li) ⁄ xi · H2O(Li). The σi represents the conductivity of the binary solution of salt i and water, MiXi-H2O (i ) 1 or 2), having the same water activity as that of the ternary solution M1X1-M2X2-H2O. Note that Good has systematically investigated the different effects on fluidity of various electrolyte solutions.26 Their results, together with that of polarographic studies,27 show that the free energies of activation of electrolyte solutions are principally determined by the ion-solvent interactions, and ion-ion interactions are negligible by comparison. As mentioned above, the ion-ion Coulombic interactions in the semi-ideal solutions of electrolyte mixtures can be adequately accounted for in terms of those existing in their component binary solutions of equal water activity. These results, together with the following relationship, indicate that the ∆G+ is also principally determined by the ion-solvent interactions:
σ)
yF2 [(ζ r )-1 + (ζcrc)-1] 6πNAMφ a a
water. NaCl and KCl were dried under vacuum over CaCl2 for 7 days at 423 K immediately prior to their use.13 The stock solutions of CdCl2 or LaCl3 were prepared from the triplicaterecrystallized salts. The molalities of CdCl2 stock solutions were analyzed by the titration (of Cl- with AgNO3) method.28 The molalities of LaCl3 stock solutions ware analyzed by EDTA and the titration (of Cl- with AgNO3) methods.13 The NaCl and KCl solutions were prepared by mass using double-distilled deionized water and NaCl or KCl with a precision of ( 5 × 10-5 g (the conductivity of the doubledistilled deionized water was less than 0.050 µS cm-1). All masses were corrected to vacuum. The CdCl2 and LaCl3 solutions were made by diluting the stock solution of CdCl2 or LaCl3 with their concentrations determined above and were kept in airtight vessels under nitrogen. The aqueous solutions of NaCl + LaCl3 or KCl + CdCl2 were prepared by mixing the binary solutions with known concentrations by mass. All solutions were prepared immediately before use, and the uncertainty was ( 5 × 10-5 mol kg-1. The conductivity measurements were carried out with a METLER TOLEDO SevenEasy conductivity meter calibrated with the standard aqueous potassium chloride solutions. The temperature of the cell was kept constant to within ( 0.005 K by circulating thermostatted liquid and the temperature was measured with a calibrated calorimeter thermometer ((0.006 K).
(14)
where 0 e y e 1 is the degree of dissociation. ζa and ζc are the “correction” factors taking into account the specific interactions between the mobile ions in the melt, ra and rc are the anion and cation hydrodynamic radii. The variables F, NA, M, and φ are the Faraday constant, the Avogadro number, molar mass, and kinematic viscosity, respectively. 3. Experimental Section All of the chemicals used were of the highest purity obtainable commercially and recrystallized twice from doubly distilled
4. Results and Discussions 4.1. Comparisons of the Measured Conductivities with the Values Reported in Literature. Table 1 and Figure 1 compare the measured conductivities of the binary solutions NaCl-H2O, KCl-H2O, CdCl2-H2O, and LaCl3-H2O at 298.15 K with the reported values.28–30 It is seen that the agreements are good. 4.2. Test Procedure. Equation 13 was tested by comparing its results with the measured/reported conductivities. The procedure is briefly described as follows: (1) Represent the measured/reported conductivities of all the binary solutions by the following polynomial equations
TABLE 1: Measured Conductivities for the Binary Systems MiXisH2O at 298.15 K mNaCl (mol kg-1) 0.2075 0.3959 0.5997 0.7817 1.0102 1.1987 1.5018 1.7945 2.4085 2.9996 3.5952 4.1857 4.8002
mKCl (mol kg-1) 0.2003 0.4004 0.4996 0.9998 1.5034 2.0012 2.5027 3.0039 3.5024
a
σNaCl (mS cm-1) σLaCl3 (mS cm-1) σNaCl (mS cm-1) σLaCl3 (mS cm-1) exp mLaCl3 (mol kg-1) expa exp mNaCl (mol kg-1) expa ref 30 mLaCl3 (mol kg-1) ref 29 20.92 37.60 54.42 68.46 85.02 97.80 116.82 133.68 164.62 188.85 208.12 223.80 235.02
0.2 0.5 1.0 2.0 3.0 4.0
20.23 46.34 84.29 144.69 188.78 219.28
20.21 46.42 84.27 144.88 189.35 220.39
0.0996 0.1997 0.2997 0.3994 0.4984 0.6000 0.6981 0.7983 1.0000 1.2011
26.33 47.36 66.80 84.22 98.98 112.36 123.85 134.18 151.48 163.36
0.10 0.50 1.00
26.41 99.17 151.43
26.36 99.36 151.60
σKCl (mS cm-1) σCdCl2 (mS cm-1) σKCl (mS cm-1) σCdCl2 (mS cm-1) exp mKCl (mol kg-1) expa exp mCdCl2 (mol kg-1) expa ref 30 mCdCl2 (mol kg-1) ref 28 24.56 46.92 57.60 108.68 156.05 199.24 239.40 276.41 310.33
0.2 0.5 1.0 2.0 3.0
Calculated from the equations shown in Table 2.
24.53 57.64 108.70 199.14 276.13
24.61 57.78 108.85 199.24 277.04
0.1235 0.3763 0.5546 0.8347 1.0725 1.2906 1.4756 1.7249 2.4922 2.9842 3.4428
10.56 20.15 23.86 27.26 28.58 29.12 29.08 28.70 25.82 23.26 20.85
0.15000 0.30000 0.44993 0.59994 0.80000 1.0000 2.0000 3.0000
11.91 17.92 21.90 24.61 26.94 28.29 27.89 23.20
11.940 17.896 21.856 24.532 26.896 28.270 27.899 23.115
Semi-ideal Solution Theory
J. Phys. Chem. B, Vol. 112, No. 48, 2008 15379
∆eqi ) σi(eqi) - σi(exp)
n
σi(cal) )
∑ Al(mio)l⁄2
(15)
l)0
where σi(cal) and moi represent the conductivity and molality of the binary aqueous solution MiXi-H2O. The optimum fit was obtained by varying l until the value of δσ,i)∑Ni)1|σi(exp) - σi(cal)| × σi(exp)-1/N is less than a few parts in 10-4. The thus obtained equations and δσ,i’s for the examined binary solutions are shown in Table 2. (2) Determine the compositions (moi ) of the binary solutions having the same water activity as that of the ternary solution of given molalities mi (i ) 1 or 2) using the osmotic coefficients (φ) of the binary solutions MiXi-H2O (i ) 1 or 2)31 and eq 2. (3) Determine the values of ci using eq 8 and the water activities of the binary solutions. (4) Insert the values of σi(cal) from eq 15 into eq 13 and eq 16 (Young’s rule)1,3,5 to yield the predictions for the ternary solutions of given mi (i ) 1 and 2), which were then compared with the corresponding experimental data:
σI ) y1σ1,I + y2σ2,I
(16)
with yi ) Ii/(I1 + I2), where I is ionic strength. σI, σ1,I, and σ2,I are the conductivities of the ternary solution M1X1-M2X2-H2O and its binary subsystems MiXi-H2O (i ) 1 or 2) of equal ionic strength. In this paper, the differences between predicted and measured conductivities were defined by
Figure 1. Variation of the δ′σ,i ()σi(exp) - σi(ref j)) value with the molality for the binary systems NaCl + H2O, KCl + H2O, CdCl2 + H2O, and LaCl3 + H2O at 298.15 K.
TABLE 2: Parameters for the Measured Conductivities of the Binary Systems MiXi-H2O at 298.15 K MiXisH2O NaClsH2O KClsH2O CdCl2sH2O LaCl3sH2O
a
equation σ ) 1.0901 - 2.2842m0.5 + 112.3929m - 25.0216m1.5 2.1133m2 + 0.2240m2.5 σ ) -52.6662 + 230.3676m0.25 329.3049m0.5 + 285.3005m0.85 24.9945m1.5 σ ) 44.7048 - 214.1430m0.25 + 332.6784m0.5 - 145.3099m0.85 + 10.3556m1.5 σ ) 54.611 - 436.925m0.5 + 1631.169m - 2062.008m1.5 + 1317.141m2 - 352.555m2.5
N δσ,i ) ∑i)1 |σi(exp)-σi(cal)| × σi(exp)-1/N.
δσ,ia 4.5 × 10-4 1.0 × 10-4 7.8 × 10-4 7.5 × 10-4
(17)
4.3. Test of Equation 13. Table 3 compares measured30,32 and predicted conductivities for aqueous solutions of (1:1 + 1:1) electrolyte mixtures at 298.15 K, including NaClKCl-H2O, NaBr-KBr-H2O, and NaI-KI-H2O. It is seen that the values of δσ,eq 13 (δσ,eq i ) ∑jN) 1|σj(exp)-σj(eq i)| × σj(exp) -1/N) for the three ternary solutions are smaller than those of δσ,eq 16. Table 4 contains the comparisons of “experimental” conductivities33 with predicted values at 298.15 K for the ternary solution of (1:1 + 1:2) electrolyte mixture, NaCl-MgCl2-H2O. The TABLE 3: Comparisons of Measured30,32 and Predicted Conductivities for Aqueous Solutions of (1:1 + 1:1) Electrolyte Mixtures at 298.15 K mB
mC
σ (mS cm-1)
(mol kg-1) (mol kg-1) exp eq 13 eq 16
∆σ (mS cm-1) ∆eq 13
∆eq 16
0.20 0.30 0.20 0.40 0.60 0.80 0.40 0.80 1.20 1.60 0.80 1.60 2.40 3.20
0.30 0.20 0.80 0.60 0.40 0.20 1.60 1.20 0.80 0.40 3.20 2.40 1.60 0.80
NaCl(B)sKCl(C)sH2O 53.10 52.95 53.24 50.83 50.67 50.96 103.80 103.55 103.93 98.82 98.41 99.02 93.85 93.46 94.10 89.06 88.78 89.19 187.99 187.27 188.37 176.58 175.85 177.50 165.79 164.97 166.62 155.41 154.65 155.75 315.35 314.58 317.21 290.28 288.90 293.01 265.97 264.65 268.80 242.66 241.83 244.60 δσ,eq ia
0.0671 0.1685 0.3397 0.6910 1.4371 2.2576 3.1778 0.0503 0.1264 0.2550 0.5192 1.0820 1.7015 2.4082 0.0336 0.0843 0.1700 0.3466 0.7244 1.1418 1.6212
0.0335 0.0843 0.1699 0.3455 0.7186 1.1288 1.5889 0.0503 0.1264 0.2550 0.5192 1.0820 1.7015 2.4082 0.0671 0.1686 0.3401 0.6932 1.4488 2.2837 3.2423
NaBr(B)sKBr(C)sH2O 11.78 11.70 11.74 -0.08 -0.04 27.71 27.65 27.77 -0.06 0.06 52.46 52.62 52.88 0.16 0.42 99.04 98.66 99.26 -0.38 0.22 180.56 179.71 181.07 -0.85 0.51 248.01 247.99 250.22 -0.02 2.21 302.08 302.54 305.79 0.46 3.71 12.08 12.04 12.08 -0.04 0.00 28.67 28.55 28.68 -0.12 0.01 54.45 54.57 54.87 0.12 0.42 103.30 102.99 103.66 -0.31 0.36 190.52 189.53 191.09 -0.99 0.57 264.90 263.95 266.48 -0.95 1.58 326.52 325.88 329.59 -0.64 3.07 12.42 12.39 12.43 -0.03 0.01 29.68 29.49 29.60 -0.19 -0.08 56.55 56.56 56.82 0.01 0.27 107.88 107.43 108.02 -0.45 0.14 201.04 199.87 201.25 -1.17 0.21 282.09 280.95 283.22 -1.14 1.13 351.80 350.42 353.74 -1.38 1.94 -3 a 3.5 × 10 4.3 × 10-3 δσ,eq i
0.0670 0.1691 0.3420 0.7017 1.4891 2.3940 3.4874 0.0503 0.1269 0.2568 0.5267 1.1206 1.8093 2.6503 0.0335 0.0846 0.1713 0.3517 0.7499 1.2137 1.7864
0.0335 0.0846 0.1710 0.3508 0.7446 1.1970 1.7437 0.0503 0.1269 0.2568 0.5267 1.1206 1.8093 2.6503 0.0670 0.1692 0.3425 0.7034 1.4997 2.4274 3.5728
NaI(B)sKI(C)sH2O 11.66 11.62 11.67 27.86 27.79 27.91 53.10 53.20 53.45 100.68 100.62 101.16 186.04 185.75 186.92 258.00 258.52 260.30 317.44 315.84 318.61 12.02 11.98 12.04 28.74 28.69 28.84 55.06 55.12 55.42 104.90 104.79 105.38 196.20 195.29 196.64 275.10 274.57 276.61 340.80 338.89 341.95 12.35 12.38 12.41 29.83 29.64 29.76 57.06 57.11 57.37 109.55 109.13 109.67 206.30 205.31 206.49 291.90 291.09 292.87 362.80 362.68 365.39 δσ,eq ia
a
N δσ,eq i)∑j)1 |σj(exp) - σj(eq i)| × σj(exp)-1/N.
-0.15 -0.16 -0.25 -0.41 -0.39 -0.28 -0.72 -0.73 -0.82 -0.76 -0.77 -1.38 -1.32 -0.83 3.8 × 10-3
0.14 0.13 0.13 0.20 0.25 0.13 0.38 0.92 0.83 0.34 1.86 2.73 2.83 1.94 4.4 × 10-3
-0.04 0.01 -0.07 0.05 0.10 0.35 -0.06 0.48 -0.29 0.88 0.52 2.30 -1.60 1.17 -0.04 0.02 -0.05 0.10 0.06 0.36 -0.11 0.48 -0.91 0.44 -0.53 1.51 -1.91 1.15 0.03 0.06 -0.19 -0.07 0.05 0.31 -0.42 0.12 -0.99 0.19 -0.81 0.97 -0.12 2.59 -3 2.8 × 10 4.0 × 10-3
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Hu et al.
TABLE 4: Comparisons of Measured33 and Predicted Conductivities for Aqueous Solution of (1:1 + 1:2) Electrolyte Mixture NaCl(B)sMgCl2(C)sH2O at 298.15 K mB (mol
kg-1
0.3000 0.3750 0.6000 0.7500 0.9000 0.2500 0.5000 0.7500 0.1250 0.2500 1.5000 0.4000 0.5000 a
σ (mS cm-1)
mC -1
) (mol kg ) exp eq 13 0.1000 0.1250 0.2000 0.2500 0.3000 0.2500 0.5000 0.7500 0.3750 0.7500 0.5000 1.2000 1.5000
43.73 53.00 78.49 93.73 107.70 59.25 101.83 133.74 65.07 108.66 152.65 143.47 157.89
43.76 53.05 78.49 93.70 107.74 59.32 101.80 134.09 65.17 108.56 153.57 143.66 159.00
eq 16
∆σ (mS cm-1) ∆eq 13
43.93 0.03 53.25 0.05 78.63 0.00 93.71 -0.03 107.51 0.04 59.53 0.07 101.64 -0.03 132.83 0.35 65.30 0.10 108.33 -0.10 151.68 0.92 142.32 0.19 156.66 1.11 δσ,eq i a 1.8 × 10-3
∆eq 16 0.20 0.25 0.14 -0.02 -0.19 0.28 -0.19 -0.91 0.23 -0.33 -0.97 -1.15 -1.23 4.2 × 10-3
N δσ,eq i ) ∑j)1 |σj(exp) - σj(eq i)| × σj(exp)-1/N.
“experimental” conductivities were determined by the following procedure. (1) The densities of NaCl and MgCl2 reported by Zhang and Han34 and Toshlakl29 were used to calculate the molalities moi (in mol kg-1) of the two salts in the binary solutions from their volumetric concentrations Coi (in mol L-1); they were also used together with Hu’s equation19 to calculate the densities of the ternary solution at given compositions mNaCl and mMgCl2. (2) The calculated densities of the ternary solution were used to determine the volumetric concentrations Ci of the two salts in the ternary solution from their molalities mi. (3) The conductivities at given compositions mNaCl and mMgCl2 were calculated by substituting the Ci values to the polynomial equation σ ) f(CNaCl, CMgCl2) that was obtained by fitting to the reported conductivities. We see that the predictions of eq 13 agree well with those of eq 16. Table 5 compares measured and predicted conductivities for the ternary solution of (1:1 + 1:3) electrolyte mixture, NaCl-LaCl3-H2O, at 298.15 K. It is observed that eq 13 gives smaller deviations. The above results also show that eqs 13 and 16 are obeyed within the experimental error for all the examined cases where mixing occurred at constant water activity and at constant ionic strength, respectively. Table 6 compares measured conductivities with predicted values at 298.15 K for the ternary solution with large deviations from the linear isopiestic relation (eq 2), KCl-CdCl2-H2O. The change in P in mixing MiXi-H2O (i ) 1 or 2) at constant I is1
∆mP ) PI - y1P1,I - (1 - y1)P2,I
(18)
where P might be the free energy of the solution per kilogram of solvent, any of its thermodynamic derivatives, or a transport coefficient such as the specific conductivity. PI and Pi,I are the properties of the ternary solution M1X1-M2X2-H2O and its binary subsystems MiXi-H2O (i ) 1 or 2) of equal ionic strength. A natural and general form for the y dependence of ∆mPI is35
∆mP ) y(1 - y)
TABLE 5: Comparisons of Measured and Predicted Conductivities for Aqueous Solution of (1:1 + 1:3) Electrolyte Mixture NaCl(B)sLaCl3(C)sH2O at 298.15 K
∑ (1 - 2y)nPn(I)
(19)
ng0
where the Pn does not depend on y. The generalized Harned’s rule may be stated as eqs 18 and 19, with the condition that the terms in Pn are negligible.1 The deviations from the Harned’s rule for thermodynamic properties are often no larger than the experimental uncertainty. However, Wood and co-workers have shown that larger deviations, typically about 10× that of the experimental uncertainty, are found in some systems with large ion-pairing effects.36 Similarly, the results from Table 6
mB (mol
kg-1
0.0306 0.0681 0.1325 0.2706 0.0586 0.1312 0.2661 0.0869 0.1901 0.3898 0.7867 0.1212 0.2613 0.5230 1.0507 0.1391 0.3331 0.6485 1.2936 0.1672 0.3804 0.7709 1.5503 0.1891 0.4483 0.8962 1.8036 0.2226 0.5080 1.0150
σ (mS cm-1)
mC -1
) (mol kg ) exp 0.0945 0.0883 0.0776 0.0547 0.1900 0.1779 0.1554 0.2852 0.2679 0.2346 0.1683 0.3793 0.3560 0.3126 0.2251 0.4753 0.4431 0.3907 0.2835 0.5721 0.5365 0.4713 0.3413 0.6666 0.6233 0.5486 0.3973 0.7612 0.7137 0.6292
27.30 28.98 31.97 38.55 49.80 52.93 58.65 69.82 73.86 81.52 96.96 87.85 92.89 102.11 121.36 102.65 109.29 119.86 140.72 117.16 123.36 135.28 156.81 128.82 135.95 147.61 169.55 139.66 146.72 159.01
eq 13
eq 16
27.39 29.01 32.10 38.74 49.89 53.02 58.77 70.18 74.05 81.71 97.34 88.13 93.03 102.44 121.61 103.34 109.71 120.34 141.31 117.40 123.97 136.04 157.90 129.28 136.80 149.27 170.75 140.28 147.84 160.35
27.77 29.52 32.54 39.00 49.81 52.86 58.54 70.16 74.01 81.43 96.20 88.11 92.80 101.55 119.18 103.12 108.94 118.39 137.72 116.86 122.55 132.98 153.79 128.39 134.56 145.23 166.84 138.96 144.96 155.61 δσ,eq ia
a
∆σ (mS cm-1) ∆eq 13
∆eq 16
0.09 0.03 0.13 0.19 0.09 0.09 0.12 0.36 0.19 0.19 0.38 0.28 0.14 0.33 0.25 0.69 0.42 0.48 0.59 0.24 0.61 0.76 1.09 0.46 0.85 1.66 1.20 0.62 1.12 1.34 4.3 × 10-3
0.47 0.54 0.57 0.45 0.01 -0.07 -0.11 0.34 0.15 -0.09 -0.76 0.26 -0.09 -0.56 -2.18 0.47 -0.35 -1.47 -3.00 -0.30 -0.81 -2.30 -3.02 -0.43 -1.39 -2.38 -2.71 -0.70 -1.76 -3.40 9.4 × 10-3
N δσ,eq i ) ∑j)1 |σj(exp) - σj(eq i)| × σj(exp)-1/N.
TABLE 6: Comparisons of Measured and Predicted Conductivities for the Ternary Solution KCl(B)sCdCl2(C)sH2O at 298.15 K mB
mC
σ (mS cm-1)
(mol kg-1) (mol kg-1) exp eq 13 eq 16 0.1010 0.2010 0.3111 0.3996 0.1995 0.4003 0.6003 0.8002 0.2991 0.6005 0.8990 0.4035 0.8000 1.1970
0.3135 0.2501 0.1818 0.0944 0.7296 0.6046 0.4550 0.2712 1.1795 1.0258 0.8238 1.6446 1.4921 1.2672
26.63 33.66 41.26 49.06 39.51 52.90 68.68 86.97 45.18 63.92 86.46 47.28 69.15 98.91
25.42 31.81 40.31 47.92 37.05 49.74 66.04 86.68 42.36 60.32 84.77 44.60 66.35 97.61
28.40 36.51 45.42 51.31 43.91 60.76 77.56 94.17 52.65 76.78 101.55
δσ,eq ia a
∆σ (mS cm-1) ∆eq 13
∆eq 16
-1.21 1.77 -1.85 2.85 -0.95 4.16 -1.14 2.25 -2.46 4.40 -3.16 7.86 -2.64 8.88 -0.29 7.20 -2.82 7.47 -3.60 12.86 -1.69 15.09 -2.68 -2.80 -1.30 4.0 × 10-2 1.2 × 10-1
N δσ,eq i ) ∑j)1 |σj(exp) - σj(eq i)| × σj(exp)-1/N.
show that eq 16 cannot provide good predictions for the system showing appreciable ion complexation, KCl-CdCl2-H2O. These comparisons may confirm that the Reilly-Wood analysis of the thermodynamic properties of mixed electrolyte solutions37 has a counterpart for conductivity in mixed electrolyte solutions. Alternatively, the preliminary calculations show that eq 13 also cannot give good predictions for the system KCl-CdCl2-H2O, while the isopiestic measurements38 at 298.15 K show that this system exhibits large deviations from eq 2. The isopiestic results of KCl-CdCl2-H2O can be described by the relationship mKCl/ o o 39 mKCl + mCdCl2/mCdCl 2 ) 1 + 0.4400mKClmCdCl2/(mKCl + mCdCl2), which was used together with the osmotic coefficients of the binary
Semi-ideal Solution Theory solutions KCl-H2O and CdCl2-H2O31 to determine the compositions (moi ) of these binary solutions having the same water activity as that of the ternary solution of given molalities mi (i ) 1 or 2). The values of ci and c were calculated from eqs 8 and 12 and then were used together with eqs 11 and 13 to yield predictions for the ternary system. The results are shown in the sixth column of Table 6. It is clear that the predictions are reasonably good, implying that the isopiestic behavior of electrolyte mixtures also have their counterparts for conductivities in electrolyte mixtures. The above results suggest that eq 13 will hold well for the conductivity of the multicomponent electrolyte solutions obeying the linear isopiestic relation and that the isopiestic results for the largely deviating systems can be used to improve the predictions for their conductivities. Because extensive isopiestic measurements have been reported in the literature, this inference is of great importance. Equation 13 is a dynamical analogue of our formulation of the thermodynamics of aqueous solutions of single-electrolyte and their mixtures that provides a simple and practical way to predict the thermodynamic properties of mixed electrolyte solutions from those of their binary subsystems.12,13,15,19 The deviations from these simple equations provide information about the complex formation upon mixing the aqueous solutions of single electrolyte at a fixed water activity. The inference that can be made is that the semiideal solution theory can also be extended to study the viscosity of multicomponent electrolyte solutions. Work addressing this opportunity is now underway in this laboratory. 5. Conclusions The conductivities were measured for the ternary systems NaCl-LaCl3-H2O and KCl-CdCl2-H2O and their binary subsystems NaCl-H2O, KCl-H2O, CdCl2-H2O, and LaCl3-H2O at 298.15 K. A novel, simple equation for the prediction of conductivity of mixed electrolyte solutions in terms of the data of its binary solutions was established based on the semi-ideal solution theory and the Eyring absolute rate theory. The measured conductivities and those reported in the literature were used to test the applicability of the established equation and the generalized Young’s rule for the conductivities of the mixed electrolyte solutions. The comparison results show that the deviations from the new equation are often within the experimental uncertainty if the examined systems obey the linear isopiestic relation. While larger deviations are found in the system with a large ion-pairing effect, the predictions can be considerably improved by using the parameter c calculated from its isopiestic results. These simple equations can yield good predictions for the conductivities of the ternary electrolyte solutions from the data of the binary solutions, indicating that they can make use of the information on the binary solutions, avoid much of the complexity involved in calculation of multicomponent conductivities, and provide good predictions for the mixed electrolyte solutions.
J. Phys. Chem. B, Vol. 112, No. 48, 2008 15381 Acknowledgment. The authors thank the National Natural Science Foundation of China (Grant Nos. 40673043, 20576073, and 20490207), CNPC Innovation Fund (04E7031), and the Program for New Century Excellent Talents in University of Ministry of Education of China (NCET-06-0088) for financial support. Valuable comments from the anonymous referees are also gratefully acknowledged. References and Notes (1) Wu, Y. C.; Koch, W. F.; Zhong, E. C.; Friedman, H. L. J. Phys. Chem. 1988, 92, 1692. (2) Harned, H. S. J. Am. Chem. Soc. 1935, 57, 1865. (3) Young, T. F.; Wu, Y. C.; Krawet, A. A. Discuss. Faraday Soc. 1957, 24, 37. (4) Zdanovskii, A. B. Tr. Solyanoi Lab., Vses. Inst. Galurgii, Akad. Nauk SSSR 1936. (5) Miller, D. G. J. Phys. Chem. 1996, 100, 1220. (6) Stokes, R. H.; Robinson, R. A. J. Phys. Chem. 1966, 70, 2126. (7) Scatchard, G. J. Am. Chem. Soc. 1921, 43, 2387. (8) Scatchard, G. J. Am. Chem. Soc. 1921, 43, 2406. (9) Stokes, R. H.; Robinson, R. A. J. Am. Chem. Soc. 1948, 70, 1870. (10) Robinson, R. A.; Stokes, R. H. J. Phys. Chem. 1962, 66, 506. (11) Okubo, T.; Ise, N. J. Phys. Chem. 1970, 74, 4284. (12) Hu, Y. F. J. Phys. Chem. B 2003, 107, 13168. (13) Hu, Y. F.; Fan, S. S.; Liang, D. Q. J. Phys. Chem. A 2006, 110, 4276. (14) Clegg, S. L.; Seinfeld, J. H. J. Phys. Chem. A 2004, 108, 1008. (15) (a) Hu, Y. F. Bull. Chem. Soc. Jpn. 2001, 74, 47. (b) Hu, Y. F.; Lee, H. J. Colloid Interface Sci. 2004, 269, 442. (16) Clegg, S. L.; Seinfeld, J. H.; Edney, E. O. J. Aerosol Sci. 2003, 34, 667. (17) May, P. M. Mar. Chem. 2006, 99, 62. (18) Chan, C. K.; Ha, Z. J. Geophys. Res. 1999, 104, 30193. (19) Hu, Y. F. Phys. Chem. Chem. Phys. 2000, 2, 2380. (20) Hu, Y. F. J. Chem. Soc., Faraday Trans. 1998, 94, 913. (21) Mikhailov, V. A. Russ. J. Phys. Chem. 1968, 42, 1414. (22) Rush, R. M.; Johnson, J. S. J. Phys. Chem. 1968, 72, 767. (23) Rard, J. A. J. Chem. Thermodyn. 1989, 21, 539. (24) Majima, H.; Awakura, Y. Metal. Trans 1988, B 19, 349. (25) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Proces; McGraw-Hill: New York, 1941. (26) Good, W. Electrochim. Acta 1965, 10, 1. (27) Giegtenberg, D.; Van Riesenbeck, G.; Nurnberg, H. W. Z. Anal. Chem. 1962, 186, 102. (28) (a) Indaratna, K.; McQuillan, A. J.; Matheson, R. A. J. Chem. Soc., Faraday Trans. I 1986, 82, 2755. (b) Mcquillan, A. J. J. Chem. Soc., Faraday Trans. I 1974, 70, 1558. (29) Toshlakl, I. J. Chem. Eng. Data 1984, 29, 45. (30) Ruby, C. E.; Kawai, J. J. Am. Chem. Soc. 1926, 48, 1119. (31) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd (revised) ed; Butterworth: London, 1965. (32) Stearn, A. E. J. Am. Chem. Soc. 1922, 44, 670. (33) Bianchi, H.; Corti, H. R.; Fernandez-Prini, R. J. Solution Chem. 1989, 18, 485. (34) Zhang, H. L.; Han, S. J. J. Chem. Eng. Data 1996, 41, 516. (35) Friedman, H. L. J. Chem. Phys. 1960, 32, 1351. (36) Srna, R. F.; Wood, R. H. J. Phys. Chem. 1975, 79, 1535. (37) Reilly, P. J.; Wood, R. H. J. Phys. Chem. 1969, 73, 4292. (38) Filippov, V. K.; Yakimov, M. A.; Makarevskii, V. M.; Luking, L. G. Russ. J. Inorg. Chem. 1971, 16, 1653. (39) Chen, H.; Sangster, J.; Teng, T. T.; Lenzi, F. Can. J. Chem. Eng. 1973, 51, 234.
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