Ind. Eng. Chem. Res. 2010, 49, 7671–7677
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Systematic Study of the Simple Predictive Approaches for Thermodynamic and Transport Properties of Multicomponent Solutions Jia-Zhen Yang,*,† Jian-Guo Liu,‡ Jing Tong,† Wei Guan,† Da-Wei Fang,† and Chuan-Wei Yan‡ College of Chemistry, Liaoning UniVersity, Shenyang 110036, China, and State Key Laboratory for Corrosion and Protection, Institute of Metal Research, Chinese Academy of Sciences, 62 Wencui Road, Shenyang 110016, China
The densities, viscosities, and conductivities were measured for the ternary solution NaCl + urea + H2O, the quintuple solution NaCl + Na2SO4 + KCl + K2SO4 + H2O, and their binary subsystems at 298.15 K. The results together with the densities, viscosities, and conductivities of multicomponent solutions reported in the literature were used to study the predictability of the Young’s rule for density and conductivity, the rule of Patwardhan and Kumar for density, Hu’s equation for the viscosity of mixed electrolyte solutions based on the Eyring’s absolute rate theory and the rule of Patwardhan and Kumar, and the semiideal solution theory for thermodynamic and transport properties. The results show that all the tested equations can provide comparable and accurate predictions for the densities of multicomponent electrolyte solutions. The semiideal solution theory is applicable to the aqueous solutions of electrolytes and nonelectrolytes, and its predictions for the densities of the examined solutions are in nice agreement with the experimental results. The simple equation based on Eyring’s absolute rate theory and the rule of Patwardhan and Kumar and the semiideal solution theory can provide nice predictions for the viscosity of the tested electrolyte solutions. The predictions for the viscosity of the ternary solution NaCl + urea + H2O by the semiideal solution theory are also in accordance with the measured viscosities. The semiideal solution theory can provide better predictions for the conductivities of the tested electrolyte solutions than the extended Young’s rule, and their predictions are both in good agreement with the experimental results. The advantages of the semiideal solution theory are briefly discussed and reviewed. 1. Introduction The thermodynamic and transport properties of aqueous electrolyte solutions are very important to a variety of fields, including chemistry and chemical engineering, separation process, wastewater treatment, pollution control, oil recovery, and so on. The solutions encountered in chemical industrial processes appear to be often concentrated electrolyte solutions. However, while extensive data have been reported in the literature for the thermodynamic and transport properties of binary aqueous solutions, relatively few measurements have been made on multicomponent solutions. The thermodynamic and transport properties of multicomponent nonelectrolyte solutions are also very important. For example, sugars are key ingredients for several areas of food production, such as the confectionery,1 the ice cream production,2 and the bakery industries.3 In addition, monosaccharides with the same molecular weight display different thermodynamic and transport properties. This feature can be fruitfully used by the manufacturers to obtain a multicomponent solution having the desired sweetening ability or viscosity by mixing different simple and complex sugars.4 To reduce trials, it will be desired to develop a predictive theory or equation that is able to predict the thermodynamic and transport properties of the multicomponent solution in terms of the properties of their binary solutions.4 On the other hand, one of the objectives of the theory of solutions is to calculate various properties of multicomponent solutions from properties for single solute, plus a minimal amount of additional information that involves the interactions in the mixtures that cannot be found in the binary solutions.5 * To whom correspondence should be addressed. † Liaoning University. ‡ Chinese Academy of Sciences.
Therefore, it is both practically and theoretically important to develop the theories and models to predict the properties of multicomponent solutions in terms of the available information on the binary solutions. Many simple predictive equations are available in literature for the thermodynamic properties of aqueous solutions.6-19 Recently, the simple predictive approaches also have been reported for the transport properties.5,20-26 However, the systematic comparisons of these equations with the experimental results are still lacking in the literature. Therefore in this study the densities, viscosities, and conductivities were measured for the ternary solution NaCl + urea + H2O, the quintuple solution NaCl + Na2SO4 + KCl + K2SO4 + H2O, and their binary subsystems at 298.15 K. The results together with the densities, viscosities, and conductivities of multicomponent solutions reported in the literature were used to check the predictability of the well-known approaches, such as Young’s rule27,28 for density and conductivity, the rule of Patwardhan and Kumar9-11 for density, Hu’s22 equation for the viscosity of mixed electrolyte solutions based on Eyring’s absolute rate theory29 and the rule of Patwardhan and Kumar,9-11 and the semiideal solution theory23,26,30,31 for thermodynamic and transport properties. 2. Experimental Section All of the used chemicals are the highest purity obtainable commercially. They were recrystallized twice from doubly distilled water. NaCl and KCl were dried under vacuum over CaCl2 for 7 days at 423 K.30 Na2SO4 and K2SO4 were dried under vacuum over CaCl2 for 3 days at 573 K.32 Urea was dehydrated at room temperature under vacuum over CaCl2 to constant weight.30
10.1021/ie100752w 2010 American Chemical Society Published on Web 07/12/2010
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The solutions were prepared by mass using double-distilled deionized water with a precision of (5 × 10-5 g. The conductivity of the double-distilled deionized water was less than 0.050 µS cm-1. All masses were corrected to vacuum. The multicomponent aqueous solutions were prepared by mixing the binary solutions with known concentrations by mass. All solutions were prepared immediately before use. The uncertainty was (5 × 10-5 mol kg-1.33 The densities of solutions were measured with an Anton Parr model DMA 4500 oscillating U-tube densitometer thermostatted to better than (0.01 K. The temperature in the measuring cell was monitored with a digital thermometer. The densimeter was calibrated by double-distilled water. The density of water at 298.15 K was taken as 0.99704 g cm-3.34 In all the measured variables the uncertainty in densities was (2 × 10-5g cm-3. The viscosities were measured using a modified CannonUbbelohde suspended level capillary viscometer. A thoroughly cleaned and perfect dried viscometer filled with liquid was placed vertically in a glass-sided water thermostat. The temperature was maintained to 298.15 ( 0.01 K. After attaining thermal equilibrium the efflux times of the flow of the liquids were recorded with a digital stop watch with a precision of (0.01 s. Triplicate measurements were performed at each composition. At 298.15 K, the viscosity of water is 0.8903 mPa · s,35 and the uncertainty in viscosity was δη,water ) (ηexpt,water - 0.8903)/0.8903 ) 0.02%, where ηexpt,water is the viscosity of water measured in this study. The viscosity of solution is given by η ) ηo
Fτ Foτo
(1)
where ηo is the viscosity of water. F and Fo are the densities of the solution and water, respectively. τ and τo are the flow times of the solution and water, respectively. The conductivity measurements were carried out with a METTLER 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 water, and the temperature was measured with a calibrated calorimeter thermometer ((0.006 K). 3. Comparisons of the Measured Properties for the Binary Solutions with the Data Reported in the Literature Tables 1-3 are listed in the Supporting Information and compare the measured densities, viscosities, and conductivities for the binary solutions NaCl + H2O, KCl + H2O, Na2SO4 + H2O, K2SO4 + H2O, and urea + H2O at 298.15 K with the reported values.23,36-39 It is seen that the agreements are good. 4. Comparisons with the Measured Thermodynamic and Transport Properties 4.1. Comparisons with the Measured Thermodynamic Properties. Young’s cross-square rule27,28 was originally formulated to relate the heats of mixing at constant ionic strength I for the six pairs of electrolytes.27,28,40,41 It has been extended to other thermodynamic properties, and the extensions are in accord with the experimental data. The extended Young’s rule27,28 for the density of the multicomponent electrolyte solutions can be expressed as
n
F)
∑yF
o,I i i
(2)
i)1
n with yi ) Ii/(∑j)1 Ij) being the ionic strength fraction. F and Fio,I are the densities of the multicomponent solution and its binary subsystems i + H2O (i ) 1, 2, ..., n) of equal ionic strength. Rule of Patwardhan and Kumar. On the basis of the assumption that single-electrolyte solutions of equal ionic strength mix ideally, Patwardhan and Kumar presented a set of predictive equations for the thermodynamic properties of electrolyte solutions, including water activity, activity coefficient of either solute in mixed electrolyte solutions, vapor pressure, density, adiabatic compressibility, enthalpy of mixing, heat capacity, expansibility, freezing point depression, and Gibbs energy.9-11 These equations can yield good predictions for the mixed electrolyte solutions using only the properties of the binary solutions. No binary interaction coefficients are required. The established predictive equation for density of mixed electrolyte solutions can be expressed as
n
F)
n
∑ Y / ∑ (Y /F i
i)1
i
o,I i )
(3)
i)1
with Yi ) yi + miMi, where yi is the ionic strength fraction, mi is molality, Mi is molar mass, and Fio,I is the densities of the binary solutions having the same ionic strength as that of the multicomponent solution. It is worthwhile to note that the agreement between the predictions of eq 3 and the experimental data for a number of aqueous electrolyte solutions seems to be very impressive. Semiideal Solution Theory. Hu has proposed the simple predictive equations for water activity, density, adiabatic compressibility, enthalpy of mixing, heat capacity, expansibility, freezing point depression, and surface tension on the basis of the thermodynamic relations12-19 and the Zdanovskii rule. Hu and Lee12,21 have established the simple predictive equation for surface tension and viscosity by joint use of the Patwardhan and Kumar rule (or the linear isopiestic relation) and the fundamental Butler equations (or the Eyring absolute rate theory29). Because the Zdanovskii rule was discovered empirically for the molalities of ternary unsaturated electrolyte solutions,42 these simple equations are also empirical. However, recently Hu et al.23,26,30,31 proposed a semiideal solution theory to describe the process of mixing the binary nonideal solutions at constant H2O activity.23,26,30,31 In this theory, the ion-ion interactions in the semiideal mixture are treated with DebyeHu¨ckel theory and the interactions between solvent and solute (ion or nonelectrolyte solute) are described by the stepwise hydration equilibrium: U · H2O(l-1) + H2O a U · H2O(l)
(4)
where U ) Ci, Ai, or nonelectrolyte solute with i ) 1 or 2 and l ) 2, ..., n; Ci and Ai denote cation and anion. That is, the Gibbs energy of the tested system is expressed as G ) GDH + GHy, where GDH is the Debye-Hu¨ckel contribution and GHy describes the semiideal mixture of the resulting species based on mole fraction x. It is shown that, under the condition of constant water activity, the average hydration numbers characterizing the solvent-solute (the ion or the nonelectrolyte solute) interactions have the same values in the mixture as in its subsystems. The authors have shown that if there is no complex formation during the mixing process, then the solute-solute interactions in multicomponent solutions can be adequately
Ind. Eng. Chem. Res., Vol. 49, No. 16, 2010
accounted for in terms of those already existing in their component binary solutions of equal water activity. Under these conditions, the process of mixing the nonideal solutions i + H2O (i ) 1, 2, ..., n) at constant water activity is as simple as that of mixing the ideal mixtures i + H2O (i ) 1, 2, ..., n) of equal H2O mole fraction. Then, they derived a series of novel linear equations for prediction of thermodynamic properties of multicomponent solutions from the properties of their binary subsystems, including chemical potential, water activity, activity of either solute, Gibbs energy, enthalpy, entropy, thermal properties, and volumetric properties.30,31 From the novel linear isopiestic equations for the molalities of multicomponent saturated solutions, the widely applied empirical Zdanovskii’s rule is first derived theoretically for concentrated electrolyte solutions,42,43 mi
∑m
o i
i
(
)1
aw ) constant and 0 e
mi moi
e1
)
(5)
where aw is water activity, mi and mio are the molalities of the ith solute in multicomponent solution 1 + 2 + ... + n + H2O and in the binary solutions i + H2O (i ) 1, 2, ..., n) of equal water activity. Equation 5 was derived for unsaturated nonelectrolyte solutions by Stokes and Robinson44 using the hydration equilibrium.44-46 Mikhailov derived eq 5 for very dilute electrolyte solutions where the Debye-Hu¨ckel theory applies.47 Rard43 derived the rule for the solutions of 1:1 electrolytes by assuming that the osmotic coefficients of the binary and multicomponent solutions are equal under isopiestic equilibrium. In addition, the semiideal solution theory31 also shows that the k parameter in the Mckay and Perring equation48 under isopiestic equilibrium must be set equal to the stoichiometric coefficient ν of the electrolyte solute. It is notable that Hu has successfully applied the Mckay and Perring equation to the activity coefficients of the solutes in four electroly + nonelectrolyte + H2O solutions.13 The simplicity or the ideality of the process described by the semiideal solution theory may be more understandable by invoking the physical meaning of activity and by comparing the process of mixing the nonideal solutions at constant water activity with the process of mixing the ideal solutions at constant water mole fraction. In fact, the authors have shown that their equations are the same as those equations describing the process of mixing the binary ideal solutions at constant water mole fraction. One of the advantages of the semiideal solution theory is that its simple equations are applicable to aqueous solutions of electrolyte mixtures, nonelectrolyte mixtures, and (electrolyte + nonelectrolyte) mixtures. By comparison, Young’s rule and the rule of Patwardhan and Kumar are not applicable to the mixtures involving nonelectrolyte solutes, as they are based on the ionic strength fraction, which is not defined for nonelectrolyte solutes. According to the semiideal solution theory, the specific heat capacity and the density of a multicomponent solution are related to those of its constituent binary solutions of equal water activity by30,31 n
n
F)
∑ Y / ∑ (Y /F ) i
i)1
i
i)1
o i
(6)
cP )
mi
∑m c
o o P,i i
i
7673
(7)
with Yi ) mi/mio + miMi, where cP and F are the specific heat capacity and the density of the multicomponent systems. coP,i and Fio are the specific heat capacity and density of the binary subsystems of equal water activity. M denotes molar mass. Hu16 has tested eq 7 by comparing with the experimental specific heat capacities of the ternary systems NaCl + NaNO3 + H2O and NaCl + NaClO4 + H2O at 298.15 K and of the reciprocal system Na + K + Cl + SO4 + H2O at 373.15 K, respectively. The agreement is excellent over the experimental composition n calc expt range, with δn ) 0.0004 (δn ) ∑i)1 |cexpt P,i - cP,i |/N, with cP,i calc is the experimental quantity, cP,i is the calculated values, N is the number of experimental data points), 0.0011, and 0.003 J g-1 K-1, respectively (the experimental uncertainty is (0.02 J g-1 K-1).49 Equation 6 has been compared with the density measurements for seven ternary electrolyte solutions, and the agreement is very impressive.15 Test Procedure. The test procedure is briefly summarized as follows: (1) Represent the measured densities, conductivities, and viscosities of the binary solutions by the following polynomial equations n
Fi(calc) )
∑ A (m ) l
l/2
i
(8)
l)0 n
ηi(calc) )
∑ B (m )
l/2
l
i
(9)
l)0 n
σi(calc) )
∑ C (m )
l/2
l
i
(10)
l)0
where Fi(calc), ηi(calc), σi(calc), and mi denote the density, viscosity, conductivity, and molality of the binary aqueous solution i + H2O. The optimum fit was obtained by variation of l with l < n 3 until the values of δQ,i ) ∑j)1 [(|Qi(expt) - Qi(calc)|)/Qi(expt)]/N (Q ) F, η, and σ) are less than a few parts in 10-4. The values of Al, Bl, Cl, and δQ,i obtained for the binary solutions are shown in Table 4 in the Supporting Information. (2) Determine the compositions (mio) of the binary solutions having the same water activity as that of the multicomponent solution of given molalities mi (i ) 1, 2, ..., n) using the osmotic coefficients of the binary solutions50,51 and eq 5. (3) Determine the compositions (mo,I i ) of the binary solutions having the same ionic strength as that of the multicomponent solution of given molalities mi (i ) 1, 2, ..., n). (4) Insert the values of Xoi and Xo,I i (X ) F, η, and σ) calculated from eqs 8-10 into eqs 2, 3, 6, and 13-18 to yield the predictions for the multicomponent solutions of given mi (i ) 1, 2, ..., n). In this paper, the differences between the predicted and measured densities (∆F), viscosities (∆η), and conductivities (∆σ) over the entire experimental composition range of the multicomponent solution are defined by25,26,33 ∆Q,eqi ) Xeqi - Xexpt
(11)
The average relative differences between the predicted and measured conductivities, densities, and viscosities over the entire experimental composition range of the multicomponent solution are defined by25,26,33
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Ind. Eng. Chem. Res., Vol. 49, No. 16, 2010 N
δQ,i )
∑
|Qi(expt), j - Qi(calc), j |/(NQi(expt), j)
(12)
j)1
with i ) 1, 2, ..., n. Table 5 is listed in Supporting Information and compares measured and predicted densities for the multicomponent solutions NaCl + Na2SO4 + KCl + K2SO4 + H2O and NaCl + urea + H2O at 298.15 K. For the multicomponent electrolyte solutions, the average relative differences between the measured and predicted densities are δF,eq 2/δF,eq 3/δF,eq 6 ) 1.5 × 10-4/ 1.5 × 10-4/0.7 × 10-4 (Imax ) 1.6 mol kg-1). For the ternary solution NaCl + urea + H2O (mmax ) 5.6 mol kg-1), the result is δF,eq 6 ) 0.1 × 10-4. Equations 2,3, and 6 were also checked by comparisons with the reported densities25,36,37,52 for aqueous solution of the (1: 1), (1:2), (1:3), and (1:4) electrolyte mixtures. The average relative difference (δF,eq 2/δF,eq 3/δF,eq 6) between the predicted and the measured densities are 1.6 × 10-4/2.1 × 10-4/1.9 × 10-4 for NaCl + KCl + H2O (Imax ) 4.4 mol kg-1), 1.2 × 10-4/1.9 × 10-4/2.4 × 10-4 for NaCl + CaCl2 + H2O (Imax ) 6.0 mol kg-1), 2.8 × 10-4/1.3 × 10-4/1.8 × 10-4 for KCl + CaCl2 + H2O (Imax ) 4.8 mol kg-1), 6.3 × 10-4/6.2 × 10-4/ 6.6 × 10-4 for NaCl + LaCl3 + H2O (Imax ) 4.8 mol kg-1), and 8.7 × 10-3/1.6 × 10-3/1.4 × 10-3 for HNO3 + Th(NO3)4 + H2O (Imax ) 24.5 mol kg-1). It is clear the three equations give comparable results and that their predictions are in accordance with the experimental results. However, eq 6 is also applicable to the aqueous solutions of nonelectrolyte mixtures and the mixtures of electrolytes and nonelectrolytes. Further comparisons using the reported densities33 at 298.15 K show that the values of δF,eq 6 are 5.0 × 10-4 for mannitol + sorbitol + H2O (mmax ) 1.1 mol kg-1), 3.8 × 10-4 for mannitol + sucrose + H2O (mmax ) 1.1 mol kg-1), and 6.3 × 10-4 for sorbitol + sucrose + H2O (mmax ) 2.7 mol kg-1), in which mmax is the maximum molality. It is clear that the agreements are also impressive. 4.2. Comparisons with the Measured Viscosities. According to the semiideal solution theory,23,26,30,31 the viscosity of a multicomponent solution is related to those of its constituent binary solutions of equal water activity by26 ln η )
∑x i
)
o ni-H 2O
n
)
o i-H2O
o ln ηi-H 2O
∑yη
o i i,I
∑m
+ 55.51
i
∑ (m /m ) i
o i
i
o where ηi-H 2O is the viscosity of the ith bulk liquid water (i.e., the viscosity of the binary solution i + H2O having the same water activity as that of the multicomponent solution; the binary o solution i + H2O is treated as a pseudocomponent). xi-H 2O is the mole fraction of the binary solution i + H2O (i ) 1, 2, ..., n) in the multicomponent solution. Hu’s equation for the viscosity of mixed electrolyte solutions based on Eyring’s absolute rate theory and the rule of Patwardhan and Kumar can be expressed as22
n
xi
∑x i)1
o, I i
ln ηo,I i
(15)
i
o with yi ) Ii/∑i Ii, where η and ηi,I are the viscosities of the multicomponent electrolyte solution and its binary subsystems of equal ionic strength. Table 6 is listed in the Supporting Information and compares measured and predicted viscosities for the multicomponent solution NaCl + Na2SO4 + KCl + K2SO4 + H2O and the ternary solution NaCl + urea + H2O at 298.15 K. For the multicomponent electrolyte solution NaCl + Na2SO4 + KCl + K2SO4 + H2O, the δη,eq 13/δη,eq 14/δη,eq 15 is 3.2 × 10-3/3.5 × 10-3/4.3 × 10-3. For the mixture of electrolyte and nonelectrolyte, NaCl + urea + H2O (mmax ) 5.6 mol kg-1), the δη,eq -3 13 is 4.2 × 10 . Table 7 is listed in the Supporting Information and compares the predicted viscosities with the experimental results reported in literature.26,33,36,37,53-55 For the mixed electrolyte solutions, the predictions of eqs 13 and 14 are in good agreements with the measured values, except for the system with complex formation, KCl + CdCl2 + H2O. Equation 13 usually provides the best predictions, and, when coupled with the isopiestic results (that were extensively reported in the literature), the predictions for the ternary solution KCl + CdCl2 + H2O were significantly improved. Only eq 13 is applicable to the mixed nonelectrolyte solutions and its predictions for the tested systems agree well with the measured values. 4.3. Comparisons with the Measured Conductivities. The semiideal solution theory23,26,30,31 has also been used to establish the novel simple predictive equation for conductivity of mixed electrolyte solutions in terms of the properties of its binary solutions:23
n
ln σ )
xi
∑x
o i
ln σoi
(16)
(13)
mi + 55.51(mi /moi ) i
ln η )
η)
i)1
with o xi-H 2O
where xio,I and ηio,I are the mole fraction and the viscosity the binary solution i + H2O (i ) 1, 2, ..., n) having the same ionic strength as that of the multicomponent solution. The extended Young’s rule27,28 for the viscosities can be expressed as
(14)
with xio ) (νimio)/(55.51 - ciνimio + νimio) and xCiAi ) (νimi)/ (55.51 - c(ν1m1 + ν2m2) + ν1m1 + ν2m2), where ci ) (55.51/ νimio) - (aH2O/(1 - aH2O)) and c ) (c1ν1m1 + c2ν2m2)/(ν1m1 + ν2m2). mi and mio are the molalities of the ith solute in multicomponent aqueous solution and its binary subsystems i + H2O (i ) 1, 2, ..., n) of equal water activity. ν is the salt stoichiometric coefficient, and νi ) νMi + νXi. a is the activity. σoi represents the conductivity of the binary solution of i + H2O (i ) 1, 2, ..., n), having the same water activity as that of the multicomponent solution. The generalized Young’s rule5,20 for the conductivity of the multicomponent electrolyte solutions can be expressed as n
σ)
∑yσ
o,I i i
(17)
i)1
n with yi ) Ii/(∑j)1 Ij), where I is ionic strength. σ and σ io,I are the conductivities of the multicomponent solution and its binary subsystems of equal ionic strength. The equation for the conductivity of mixed electrolyte solutions, which is analogous to Hu’s equation22 for viscosity
Ind. Eng. Chem. Res., Vol. 49, No. 16, 2010 29
based on Eyring’s absolute rate theory and the rule of Patwardhan and Kumar,9-11 can be expressed as n
ln σI )
xi
∑x i)1
o,I i
ln σo,I i
(18)
with xio ) mio/(55.51 + mio) and xi ) mi/(55.51 + m1 + m2), where σio,I is the conductivity of the binary solution of i + H2O (i ) 1, 2, ..., n) having the same ionic strength as that of the multicomponent solution. Table 8 is listed in the Supporting Information and compares measured and predicted conductivities for the multicomponent solutions NaCl + Na2SO4 + KCl + K2SO4 + H2O at 298.15 K. The δσ,eq 16/δσ,eq 17/δσ,eq 18 is 5.6 × 10-3/7.6 × 10-3/3.7 × 10-2 (Imax ) 1.6 mol kg-1). Table 9 is listed in the Supporting Information and compares the predicted conductivities with the experimental results reported in literature.23,24,53,56,57 The predictions of eqs 16 and 17 agree well with the measured values, except for the system KCl + CdCl2 + H2O. Equation 16 generally gives the best predictions, and, when compared with the isopiestic results, its predictions are considerably improved. In addition, the osmotic coefficients of the binary solutions measured at 298.15 K can be used to determine the mio at 293.15 and 303.15 K, which can be used to provide good predictions for the conductivities of the multicomponent solutions at 293.15 and 303.15 K. 5. Conclusions The predictions of Young’s rule,27,28 the rule of Patwardhan and Kumar,9-11 and the semiideal solution theory23,26,30,31 for the densities of the tested electrolyte solutions are comparable and are in nice agreement with the experimental results. The predictions of the semiideal solution theory23,26,30,31 for the densities of the ternary solution NaCl + urea + H2O also agree well with the measured values. Hu’s equation22 for the viscosity of mixed electrolyte solutions based on Eyring’s absolute rate theory29 and the rule of Patwardhan and Kumar9-11 and the semiideal solution theory26,30,31 can provide nice predictions for the viscosities of the tested electrolyte solutions. The predictions for the viscosities of the ternary solution NaCl + urea + H2O by the semiideal solution theory are also in accordance with the measured viscosities. The semiideal solution theory23,30,31 can provide better predictions for the conductivity of the tested electrolyte solutions than the extended Young’s rule and, their predictions are both in good agreement with the experimental results. Another advantage of the semiideal solution theory is that it can be coupled with the widely reported isopiestic results to improve the predictions for the systems of complex formation. Both the thermodynamic and transport properties are dominated by, or at least strongly related to, the solvation structures.58 The semiideal solution theory23,26,30,31 for the thermodynamic and transport properties is focused on the properties of the solvation properties and therefore applies well to the both properties. Hu and coauthors first derived the simple predictive equations for the thermodynamic and transport properties of multicomponent solutions by joint use of the Zdanovskii rule42,43 and the thermodynamic relations or the Eyring absolute rate theory.29 However, the predictions of the thermodynamic properties and the transport properties by the semiideal solution theory in terms of the properties of the binary solutions do not necessarily require invoking the Zdanovskii rule (e.g., see ref 26). The semiideal solution theory essentially states that if there is no significant complex formation during the process of mixing the binary nonideal solutions, then the nonideality arising from
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mixing the binary solutions can be sufficiently absorbed in the nonidealities upon mixing water and each of the solutes, which have been accounted for by the use of water activity instead of water mole fraction. Anyway, this theory first extends the linear predictive equations for the thermodynamic and transport properties of the classical ideal solutions to the properties of the multicomponent nonideal solutions. Therefore, it is a significant contribution to the development of the classical ideal solution theory. Acknowledgment This project was supported by NSFC (Grants 20773056 and 20903053) and the Education Bureau of Liaoning Province (Grant LS2010069), People’s Republic of China. Supporting Information Available: Tables listing measured densitites, viscosities, and conductivities of the binary system i + H2O, parameters of those measured values, comparisons of measured and predicted densities, viscosities, and conductivities of the quinary aqueous solutions, and average absolute differences between measured and predicted viscosities and conductivities. This information is available free of charge via the Internet at http://pubs.acs.org/. Glossary Al ) the parameter in eq 8 aw ) water activity Bl ) the parameter in eq 9 Cl ) the parameter in eq 10 cP ) the specific-heat capacity of the multicomponent systems o cP,i ) the specific-heat capacity of the binary subsystems of equal water activity expt cP,i ) experimental quantity of the specific-heat capacity calc cP,i ) the calculated values of the specific-heat capacity G ) Gibbs energy GDH ) Debye-Hu¨ckel contribution, G ) GDH + GHy GHy ) the Gibbs energy describing the semi-ideal mixture of the resulting species based on mole fraction, x I ) ionic strength k ) the parameter in the Mckay and Perring equation mi ) the molality of the ith solute in multicomponent solution 1 + 2 + ... + n + H2O mio ) the molality of the ith solute in the binary solutions i + H2O (i ) 1, 2, ..., n) having the same water activity as that of the multicomponent solution Mi ) molar mass N ) the number of experimental data points x ) mole fraction n yi ) ionic strength fraction, yi ) Ii/(∑j)1 Ij) Yi ) yi + miMi Greek Letters ∆Q,eqi ) the relative difference between predicted and measured values, ∆Q,eqi ) Xeqi - Xexpt n expt calc δn ) relative deviation defined by δn ) ∑i)1 |cP,i - cP,i |/N ηexpt,water ) the viscosity of water measured in this study ηo ) the viscosity of water ν ) the stoichiometric coefficient of electrolyte solute F ) the density of the solution Fo ) the density of water Fio,I ) the density of the binary solutions, i + H2O (i ) 1, 2, ..., n), having the same ionic strength as that of the multicomponent solution
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σ ) conductivity τ ) the flow times of the solution τo ) the flow times of water
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ReceiVed for reView March 30, 2010 ReVised manuscript receiVed June 14, 2010 Accepted June 21, 2010 IE100752W
