J. Phys. Chem. 1994, 98, 6870-6875
6870
A Simplified Mean Spherical Approximation for the Prediction of the Osmotic Coefficients of Aqueous Electrolyte Solutions Tongfan Sun, Jean-Louis Unard, and Amyn S. Teja’ Fluid Properties Research Institute, School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0100 Received: February 16, 1994; In Final Form: May 10, 1994”
A simplified mean spherical approximation (SMSA), containing only hard-sphere and long-range interaction terms, is proposed. The capability of the model to represent the osmotic coefficients of aqueous electrolyte solutions is demonstrated. The SMSA method yields results that are comparable to the original MSA method but require much less computational effort. To allow the representation of osmotic coefficients over large ranges of temperature and composition, a versatile form of cation diameter has been introduced and shown to yield accurate results. The SMSA model has been used to correlate the osmotic coefficients of 120 electrolyte solutions at ambient conditions and the osmotic coefficients of a group of univalent alkali metal chlorides at high temperatures. In addition, the predictive capabilities of the model were successfully tested on more than 50 electrolyte solutions.
Introduction Aqueous electrolyte solutions are encountered in many industrial applications, including oil recovery and absorption refrigeration. They also exhibit appreciabledeviationsfrom ideal solution behavior, particularly at high temperatures and concentrations, so that their thermodynamic description poses a significant theoretical challenge. A knowledge of their thermodynamic properties is therefore important from both theoretical and practical considerations. The most widely used methods for thecalculations of the thermodynamic propertiesof concentrated electrolyte solutionsare thoseof Pitzer’.’ and B l ~ m . The ~ , ~Pitzer method is analogous to a virial expansion and has been applied to many electrolyte system^^^^ but requires a large number of coefficients for treating concentrated solution^.^ Blum’s method has a basis in cluster expansion theory and the mean spherical approximation (MSA) and is receiving increasing interest”I6 for practical applications. The MSA theory models the ions in an electrolyte solution as hard spheres of arbitrary sizes and charges and reduces to the Debye-Huckel theory at low concentrations. It has been shown to represent deviationsfrom ideality in many electrolyte solutions over a wide range of concentrations. However, it involves extensive computations and has generally been applied only for ambient temperature conditions. The purpose of the present work is, first, to develop a simple and accurate method for the correlation and prediction of osmotic coefficients of concentrated electrolyte solutions; second, to use this method to predict the properties of mixtures of electrolytes at room temperature and for single electrolytes at high temperatures; and finally, to correlate the available data for aqueous LiBr solutions over a large range of conditions.
The MSA Method To date, the MSA theory seems to be the most promising description of the primitive model, in which solute ions of an electrolyte solution are modeled as electrically charged hard spheres of different diameters, while the solvent is treated as a continuous dielectric medium. The potential energy of such a system is a result of long-range Coulombic attractions and shortrange hard-sphere repulsions. Using this potential energy and a mean spherical approximation, Blum3 solved the Ornstein0
Abstract published in Advance ACS Abstracts, June 15, 1994.
Zernike relation and expressed the molar osmotic coefficient of an electrolyte system as
9,- 1 = -(r3/3Vo)- (.p,)’/@PoA’)
+ $HS
(1)
where PO is the number density of the charged species in the system and ~ H isS the hard-sphere contribution. The quantities I?, A, P,,, and a are determined from the following expressions: a’ = 4?re2/(tkT)
(2)
(3) (4)
In these equations, e is the unit electron charge, c is the dielectric constant of the solvent, k is the Boltzmann constant, and Tis the temperature. The quantities pi. ui, and zI are the number density, the hard-sphere diameter, and the valence of the ionic species i, respectively. The inverse shielding length I? is determined from eq 4-6 by iteration. Although the above equations are analytic, the iterative calculations required to evaluate the quantities are cumbersome. Some studies* have suggested that it may be possible to set P, = 0 in some special cases, e.g., when the ionic diameters are similar or when the Bjerrum length Iz+z_)eZ/(tkT d-) is large as in molten salt systems. In these cases, the computations become much simpler. It is thus of interest to investigate the influence of this hybrid term P,, upon the correlative and predictive power of the MSA model. We call the version of the MSA model, obtained by arbitrarily setting the term P,, to zero, the simplified mean spherical approximation (SMSA). In the present work, we demonstrate the application of the SMSA to a variety of electrolyte solutions over an extended range of concentrations and temperatures. In the following section, the equations for the SMSA are presented, and their validity and justification are discussed. The SMSA Method With Pn = 0, the molar osmotic coefficient of electrolyte solutions reduces to thesumof two terms, a long-rangeinteraction
0022-365419412098-6870%04.50/0 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 27, 1994 6871
Osmotic Coefficients of Aqueous Electrolyte Solutions term and a hard-sphere term. The hard-sphere term is estimated here through the compressibilityequation of state for hard-sphere mixtures proposed by Mansoori et a1.l' The resulting molar osmotic coefficient is given by
where R, S,and V denote the average radius, the surface, and the volume of the hard-sphere mixture, respectively. Thus
R = O.5xx,ai
S =rZx,u?
V = (r/6)Cx,u,'
(8)
The packing fraction r] and the molar fraction xi are defined as follows:
(9)
The shielding parameter r is obtained from the implicit equation
The summations are carried out over all positive and negative ions. In the spirit of perturbation theory, the ion diameters are allowed to vary with temperature for each ionic species. For simplicity, however, the adjustment is made only for cations while the anion size is kept constant and close to its Pauling crystal diameter. This treatment is based on a previous finding1ss18 showing that the cation size increases more than the anion size, a behavior interpretable in term of preferential hydration. An effectivecation diameter is introduced and expressed as a function of the cation concentration as follows: u+(c+)
= uo + UC+
+ bc;
(12)
where UO, a, and b are coefficients and c+ is the molar cation concentration. The coefficientsof eq 12 can be determined for each electrolyte by a least-squares analysis of experimental data, e.g., data on molal osmotic coefficients &,,ap as a function of salt molality m. For T = 298 K,the parameters are evaluated as follows: 1. We start by setting the anion diameter equal to the Pauling diameter and by assuming initial values for the coefficients UO, a, and b. Typically, uo = 4 A, a = -0.1, and b = 0.01. 2. The experimental variable m is converted to the corresponding molar concentration c using the following equationlg involving the solution density do, c = mdo/(l
+ 0.001mw)
(13)
where w is the molecular weight of the solute. Thenumber density can then be obtained. 3. I' is then evaluated using eq 11 and a Newton-Raphson algorithm with initial value ro = 4 2 . 4. Equations 7-1 0 are solved for the molar osmotic coefficient &,colsr and the value is converted to the molal scale using eq 13 and
pi
where Vis the solution volume and V,,,,, is the partial molar volume of the solvent.
5. The deviations between &,,& and & a p are minimized to obtain the parameters UO, a, and b for each electrolyte.
Results for Sigle Electrolytes We have applied both the MSA and SMSA methods described aboveto 120singleelectrolytesolutions. Theambient temperature data used in our calculations were obtained from the publication by Hamer and Wu20 or the compilation of Loboz1and covered the concentration scale from the lowest concentrations to the saturation limit, at 298 K. The solution density data needed to convert from the molal to the molar scale were obtained from the handbook of Sbhnel and Novotny.z2 Average absolute deviations (AAD) between calculated and experimentalosmoticcoefficients were computed and are summarized in Tables 1-3. The tables also contain all the parameters used in the calculation, including diameters of the anions (in the caption of Table 1). The results show that both methods are able to correlate experimental data with almost the same degree of accuracy, with differences being of the order of 0.01%. A detailed inspection of Table 1 indicates that theSMSAmethod yieldsbetterresults than theMSAmethod for 68 salts, the same results for 37 salts, and slightly poorer results for 11 salts. It is not surprising that the two methods yield almost the same results for systems in which the cation and anion size are close to each other, because the P,, term is negligible for such systems. However, the good agreement between the two methods is surprising when the size difference between the cation and anion is significant. In order to understand the success achieved with the SMSA method, it is instructive to review the intrinsic deficiencies of the MSA model. A study of modern liquid theories by Barker and Henderson23has revealed that the density expansion coefficients of the MSA cannot be expressed in terms of integrals involving the Mayer f function, and as a result, the MSA does not become exact in the limit of low density nor does it give an exact value of the second virial coefficient. This is due to the inherent limitations of the radial distribution function of the MSA. By comparing the MSA with computer simulations, Andersen et al.24have shown that the radial distribution function of the MSA at contact for a 1-1 electrolyte is qualitatively incorrect even at low concentrations. Also, Lee* has shown that the radial distribution functions of cation-cation and anion-anion interactions obtained from the MSA become negative at high densities, which is not physically realistic. More recentiy, Gering et al.15 showed that the short-range term (the P,, term) of the MSA becomes negative at moderate concentrations and is of opposite sign to the analogous term in other successful electrolyte theories such as that of Pitzer. To remedy these inherent inadequacies, modifications of the MSA have been proposed by Anderson and Chandler,zS by Ste11,26and by Gering et al.lS A successful modification in which the short-range term is replaced by an exponential term (the EXP-MSA approach) was proposed by Ste11.26 We have performed EXP-MSA calculations for the 120 electrolytesolutions studied in this work and compared the results with our SMSA results. The fitting accuracy of the EXP-MSA method was found to be better for 60 electrolytes, about the same for 30 electrolytes,and worse for 30 electrolytes, which typically contain complex anions such as S042-or multi-chlorides. It is therefore likely that the short-range term is not always positive in sign, especially in complex systems. It is also instructive to compare the three parts of eq 1 (MSA) with the two parts of eq 7 (SMSA) for a particular salt, e.g., LiBr. These comparison are shown in Table 2 where it can be seen that the P,, term of eq 1 constitutes about 0.1% of the total contribution to the osmotic coefficientin the concentration range 0.8-2.5 M. For the remaining concentrations (below 0.8 M and about 2.5 M up to 20 M), the P,, term is negligible (contributing less than 0.01% to the total value). It would therefore appear that setting P, = 0 makes little difference in this system.
6812 The Journal of Physical Chemistry, Vol. 98, No. 27, 1994
Sun et ai.
TABLE 1: Correlation of the Osmotic Coeffkients of 1-1,l-51-3,1-4, and 2-2 Electrolytes at 298 I(. coefficients of q 12 max. m
00
a
b
coefficients of q 12
lOOAAD (9%)
SMSA MSA
max. m
uo
a
b 0.191 0.0203 -0.154 0.0490 0.0263 0.0504 0.247 0.247 -0.0721 0.0215 0.0793 0.0655 0.0182 -0.00939 -0.0441 -0.0383 -0.0363 -0.0392 -0.0343 0.113 -0.0269 -0.0441 -0,0381 -0.0377 -0.0460 -0.0430 -0.129 -0.03 12 -0.0320 0.153 0.00536 -0.0745 0.139 -0.0543 -0.0584 -0.0761 -0.0833 -0.0593 -0.0786 -0.0539 -0.0761 -0.0652 -0.0621 0.0714 0.179 0.0581 0.532 -0.00838 -0.150 0.0500 0.198 0.162 0.0199 0.0520 0.461 0.201 0.221
lOOAAD (9%) SMSA MSA
4.0119 0.13 0.14 2.6 5.46 -1 -06 0.64 0.64 CSCl 11.0 2.33 0.185 0.43 5.8 5.33 -0.287 0.00120 0.40 0.62 0.63 16.0 4.94 -0.107 HCl 0.0503 0.18 0.18 1.07 7.8 3.16 KCI 5.0 3.42 -0.327 2.85 2.85 5.0 5.76 -0.446 0.76 0.77 -0.00320 0.60 0.62 19.2 4.59 -0.0268 LiCl 0.27 5.0 5.61 -0.414 0.27 0.0114 NH4CI 7.4 3.38 -0.160 0.56 0.57 0.34 0.35 5.5 5.76 -0.504 0.0246 NaCl 6.1 3.77 -0.190 0.88 0.90 0.25 0.24 0.00740 2.0 1.71 -0.951 7.8 3.01 -0.0575 0.20 0.27 RbCl 4.0 3.89 -1.45 0.11 0.11 CsBr 5.0 1.97 0.186 -0.00857 0.84 0.84 0.24 0.0148 5.5 6.68 -0.345 0.23 HBr 6.0 5.29 -0.218 0.50 0.53 7.1 5.58 -0.342 0.21 0.21 0.0390 KBr 5.5 3.33 -0.273 0.68 0.70 0.76 0.76 -0.00386 5.5 4.74 -0.814 LiBr 20.0 4.65 -0.0101 0.71 0.71 3.6 6.03 -0.542 0.40 0.40 NaBr 0.000855 9.0 3.75 -0.0376 0.55 0.55 4.0 6.06 -0.414 0.09 0.09 0.0403 RbBr 5.0 2.88 -0.241 0.97 0.98 0.06 6.0 5.15 -0.291 0.0769 -0.0326 0.06 3.0 1.91 0.49 0.49 CSI 0.96 3.6 7.25 -0.202 0.00558 0.95 10.0 5.39 -0.152 0.36 0.43 HI 0.25 0.0392 0.25 3.8 7.26 -0.209 4.5 3.63 -0.298 0.38 0.43 KI 0.34 0.185 0.33 3.6 7.15 -0.228 LiI 3.0 5.77 -0.834 0.41 0.47 0.17 0.202 0.17 3.6 7.19 -0.220 1.1 3.62 -0,892 0.36 0.42 NH4I 0.58 0.58 3.7 7.30 -0.240 0.41 0.48 12.0 4.01 -0.00575 -0.00123 NaI 0.12 0.0283 2.0 7.23 -0.593 0.12 5.0 2.63 -0.0997 0.42 0.47 RbI 0.24 0.0360 4.1 7.27 -0.243 0.25 3.5 4.71 -0.300 0.46 0.49 CsF 3.9 6.93 -0.143 0.56 17.5 3.31 0.0431 -0.00367 0.55 KF 0.48 0.51 3.9 7.00 -0.193 0.04 0.754 1.0 3.31 -1.91 0.04 NaF 0.42 0.48 0.04 0.7 17 0.05 3.6 7.06 -0.202 3.5 4.60 -0.594 0.42 0.47 RbF 3.6 7.23 -0.200 0.51 0.000161 0.51 20.0 4.10 -0.0562 0.34 0.39 KOH 1.6 8.44 -0.700 0.29 0.167 -0.0571 0.29 4.0 1.22 LiOH 0.68 0.71 0.249 3.9 1.46 1.32 0.0699 -0.00459 1.32 29.0 2.87 NaOH 0.22 0.25 0.79 0.79 4.0 7.32 -0.242 0.00676 20.0 3.71 -0.175 HNO3 0.26 0.31 0.31 0.31 4.0 7.26 -0.222 0.00308 13.5 4.06 -0.158 0.38 0.41 LiNO3 6.5 1.85 1.79 5.33 -1.14 0.138 10.8 2.50 -1.08 1.74 1.76 NaNO3 0.32 0.00741 0.31 5.0 6.12 -1.17 4.6 3.74 -0.0628 HCIO, 1.29 1.30 0.28 0.0523 0.28 4.6 6.74 -0.293 LiC104 4.5 3.67 -0.520 0.91 0.91 1.3 5.31 -1.66 0.23 0.0919 0.22 6.0 2.18 -0.728 NaClO4 1.15 1.15 0.47 0.0980 0.48 4.6 6.69 -0.349 NaH2POd 6.5 1.35 -0,323 0.99 1.00 0.09 0.436 0.09 4.5 6.76 -0.339 1.8 1.14 -1 -27 1.18 1.20 KH2m4 0.18 0.19 4.5 6.49 -0.218 0.0390 BaC12 1.8 5.53 -0.536 1.15 1.16 0.67 -0.0201 0.62 4.6 6.73 -0,256 7.5 5.78 -0,0107 1.03 1.04 CaC12 0.41 -0.0148 0.37 4.9 6.45 -0.256 4.1 6.22 -0.262 1.19 1.20 COCl2 0.23 0.0570 0.23 4.7 6.38 -0.181 5.8 5.73 -0.779 cuc12 1.14 1.14 0.27 4.7 6.57 -0,301 0.24 0.0194 FCCI~ 2.1 6.10 -0.3 18 1.17 1.18 0.42 0.37 0.00448 4.6 6.69 -0.280 5.9 6.41 -0.214 0.79 0.79 MgCh 0.23 0.0205 0.24 4.6 6.73 -0.31 1 MnC12 7.7 6.15 -0.491 1.05 1.06 0.35 -0.0143 0.32 4.6 6.75 -0,330 NiC12 5.7 6.16 -0.196 1.13 1.14 0.30 0.0300 0.28 4.0 5.29 -0.298 4.0 5.74 -0.221 1.32 1.34 SrCl2 0.0361 0.58 3.2 7.08 -0.641 0.55 3.5 4.58 -0.61 1 UOZC12 1.72 1.74 0.202 2.02 1.6 4.02 -0.465 6.0 4.97 -1.28 2.04 ZnCl2 0.05 0.05 0.136 0.13 1.4 4.92 -1.33 0.12 2.3 6.00 -0.694 BaBr2 0.39 0.38 1.22 1.27 9.2 6.03 -0.003 14 -0.0194 10.0 2.76 0.171 CaBr2 2.67 2.67 0.0164 0.31 0.35 0.7 2.79 5.6 6.87 -0.340 0.0258 0.06 0.06 MgBn 0.000922 0.22 0.24 3.0 4.25 -0.423 MnBr2 5.6 6.71 -0.398 0.90 0.91 0.00683 0.37 0.42 3.6 5.24 -0.737 4.7 6.71 -0.347 NiBr2 1.77 1.80 0.253 0.32 0.34 4.0 4.79 -0.606 2.1 6.39 -0.826 SrBr2 2.68 2.77 0.128 3.25 3.30 10.0 6.18 -1.16 ZnBr2 5.5 2.61 -0.134 0.80 0.80 0.317 0.26 0.25 4.5 2.94 -0.423 2.0 6.94 -1.02 BaI2 1.12 1.12 0.157 0.27 0.29 2.5 4.99 -1.30 1.9 7.00 -0.688 CaIz 1.55 1.57 0.27 0.32 1.8 4.12 -1.06 0.0233 5.0 7.22 -0.399 0.42 0.42 MgI2 0.216 0.35 0.37 2.0 6.84 -0.760 3.5 4.81 -0.724 SrI2 2.04 2.06 0.280 1.43 1.37 11.9 8.25 -2.20 ZnI2 0.0681 0.79 0.79 6.0 4.28 -0.716 Ca(NOd2 aThe anion diameters are18 3.056 A for C1, 3.374 A for Br, 3.604 A for I, 3.378 A for OH,2.934 A for F, 2.501 A for HzP04, 3.774 A for NO3, 4.79 A for clod, and 2.981 A for SO+
Prediction for Mixed Electrolytes Since all pairwise interactions involving the ions are accounted for in the SMSA summations without any limitation as to the number of pairs, the SMSA approach can be extended to mixed electrolyte solutions in a straightforward manner. The calculations involve the parameters obtained by a fit of single electrolyte solution data and require no additional adjustable parameters. The SMSA method is therefore predictive for mixed electrolyte solutions. The predictive capability of the SMSA was demonstrated by calculatingosmotic coefficients of 50 mixed electrolytes solutions consisting of a wide variety of ions. The mixtures included 1-1, 1-2, 1-3, 2-1, and 2-2 electrolytes with common anions and cations and without any common ions. All experi-
mental data were at 298 K,and we covered the concentration range up to the limit of the experimental conditions. In most cases, experimentalosmotic coefficients of mixtureswere obtained from measured isopiestic ratios reported in the literature. The solution densities required in the conversion were estimated by averaging densities of solutions of single electrolytes according to the molar fraction of the electrolyte.'* The coefficients a, b, and c in eq 12 were taken from the single salt results, and the molar cation concentration c+ was obtained from the sum of the cation concentrations. The AADs between calculated and experimental values for the 50 mixtures studied are summarized in Table 3. Also shown in Table 3 are the predictions using the MSA method. Agreement between predicted and experimental
Osmotic Coefficients of Aqueous Electrolyte Solutions
TABLE 2 Comparison of the Various Contributions to 0, for the Aqueous LiBr System Using the MSA Model (41) molality, m 0.001 0.005 0.05
0.2 0.5 0.6 0.7 0.8 0.9 1.o
1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5 .O
5.5 6.0 7.O 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.o
$, - 1 -0.01 1 -0.023 -0.052 -0.058 -0.026 -0.012 0.002 0.018 0.034
0.050
0.083 0.118 0.154 0.190 0.227 0.323 0.422 0.524 0.630 0.738 0.848 0.961 1.075 1.308 1.542 1.776 2.004 2.224 2.430 2.620 2.791 2.941 3.067 3.170 3.249 3.305 3.339
-r3/3rp0
$HS
-0.012 -0.024 -0.060 -0,092 -0.112 -0.116 -0.119 -0.121 -0.123 -0.125 -0.128 -0.130 -0.132 -0.133 -0,134 -0.137 -0.139 -0.140 -0.141 -0.142 -0.143 -0.144 -0.144 -0.146 -0.148 -0.149 4.151 4.153 -0.155 -0.157 -0.159 -0,161 -0.164 -0.166 -0.169 -0.172 -0.175
0.000 0.001 0.006 0.034 0.087 0.104 0.122 0.140 0.158 0.176 0.212 0.249 0.287 0.325 0.363 0.461 0.562 0.665 0.772 0.881 0.992 1.105 1.220 1.454 1.691 1.926 2.156 2.377 2.585 2.777 2.950 3.102 3.231 3.336 3.418 3.477 3.515
-(aPn)2/8poA2 0.0000 O.oo00 -0.0002 -0,0005 -0.0008 -0.0008 -0.0009 -0.0010 -0.0010 -0.0010 -0.001 1 -0.001 1 -0.0012 -0.0012 -0.0012 -0.0012 -0.0012 -0.0012 -0.0012 -0.001 1 -0.001 1 -0.0010 -0.0010 -0.0008 -0.0007 -0.0006 -0.0005 -0.0003 -0.0002 -0.OOO1 -0.0001 0.0000 0.0000 0.0000
-0.0001 -0.0002 -0.0003
values was found to be comparable for the MSA and SMSA methods and generally better than 2%for both methods. A typical plot of experimental and calculated osmotic coefficients as a function of molality is shown in Figure 1.
Correlation and Prediction of Osmotic Coefficients at High Temperatures As noted in the Introduction,a knowledgeof osmoticcoefficients of concentrated electrolyte solutions at high temperatures is essential for the design of a variety of process equipment. Since experimentalhigh-temperaturedata are scarce, an accurate model which permits the smoothing and extrapolation of data to high temperatures is, therefore, of great practical importance. The ability of the SMSA method to predict these properties over a wide range of experimentalconditions is examined in the following section. A group of electrolyte solutions containing the alkali metal chlorides-LiCl, NaCl, KCl, and CsC1-was chosen for this study. This group is the most common family of metal salts used in industry, which makes the present study important from a practical point of view. Also, experimental data for these electrolytes are available over a wide range of conditions, and in addition, the SMSA theory should be applicable to the alkali metal chlorides because of the simple molecular shape of their constitutive ions. Five isotherms (at 273,298,323,348, and 373 K)were selected for each electrolyte. The experimental osmotic coefficientswere obtianed from the workof Holmes and M ~ m e r ! ~ The diameters of the ions were treated as adjustable parameters in the calculations. Although the temperature variation of ionic diameters has not been fully studied previously within the framework of the MSA method, the temperature dependence of hard-sphere diameters has been extensively examined in other liquid theories for simple liquid systems and widely reported in the literat~re.23.a.~~ These studies have concluded that the temperature variation of effective hard-sphere diameters of
The Journal of Physical Chemistry, Vol. 98, NO. 27, 1994 6873 TABLE 3 Prediction of the Osmotic Coefficients of Mixed Electrolyte Solutions at 298 K l00AAD (9%) system ref max. P SMSA MSA KCl + LiCl 5.0 27 2.2 2.3 0.92 0.88 LiNO3 + LiCl 6.0 27 1.1 NaCl + LiCl 6.0 27 1.o CsCl + KCl 27 0.52 5.0 0.54 5.5 6.0 CsCl + LiCl 27 5.5 0.35 LiCL + NaCl 5.8 0.34 28 0.90 5.9 0.75 LiNO3 + NaNOo 28 1.7 6.6 KCl + NaCl 1.7 29 1.1 3.7 KCI + NaCl 1.1 30 KBr + NaCl 0.68 4.6 0.66 30 4.2 NaBr + KCl 1.6 1.6 30 KBr + NaBr 1.4 4.5 1.4 30 0.48 0.49 KBr + KCl 4.4 30 4.4 0.64 0.61 NaBr + NaCl 30 1.6 1.6 KNO3 + NaNO3 3.0 31 6.2 0.62 0.66 NaNO3 + NaCl 31 0.61 0.61 KNO3 + NaCl 31 3.7 KNO3 + KCl 1.4 1.5 31 3.7 3.4 5.9 NaN03 + KCI 3.3 31 1.8 CsCl + NaCl 6.0 1.8 32 2.1 KH2P04 + KCl 33 3.O 3.0 33 4.3 4.9 NaH2P04 + NaCl 4.3 33 3.0 1.8 KH2PO4 + NaCl 3.0 3.1 5.0 NaH2PO4 + KCl 3.1 33 2.8 KH2PO4 + NaH2P04 33 2.8 4.0 34 0.38 2.5 0.38 Na2SO4 + NaCl 0.71 K2SO4 + KCI 34 0.71 2.3 0.93 34 0.93 3.6 K2sO4 + Na2SO4 0.96 3.6 0.96 34 K2S04 + NaCl 1.8 34 1.8 Na2SO4 + KCI 4.4 0.80 35 0.81 4.3 BaCl2 + LiCl 1.2 35 1.3 4.1 BaC12 + CsCl 9.4 1.5 36 1.5 Na2S04 + NaCl 1.6 36 1.6 8.8 MgSO4 + Na2S04 1.4 36 1.5 6.8 MgC12 + MgSO4 NaCl + MgSO4 5.9 36 0.82 0.86 37 1.6 1.6 CaCl2 + NaCl 8.6 1.1 38 1.1 5.3 BaC12 + NaCl 9.8 39 1.5 MnCl2 + NaCl 1.5 40 1.3 CaClz + KC1 1.3 6.4 5.1 41 0.68 0.70 BaC12 + KCl 11.4 42 1.4 1.4 15.2 1.4 1.4 43 23.4 2.4 2.4 43 18.1 43 3.4 3.3 19.0 43 3.1 3.1 7.3 44 1.3 1.3 12.4 44 1.8 1.8 a I = ionic strength = 0.5E2,2ml. molecular fluids is always negative and typically in the range 6u/6T=-1 X l w t o - 5 X 10-4 A/K,correspondingtoadecrease of a few percent in the diameter for an increase of 100 K in temperature. Unfortunately,this behavior is not even qualitatively followed by the electrolyte systems studied in this work. It was found that, of the four chloride solutions, only LiCl solutions follow the same trend, i.e. exhibit negative values of 6u/6T, the remaining chlorides exhibiting positive values of 6u/6T. This can be seen in Figure 2 in which the 273 K isotherm is at the top of the 4,,, versus m plots, because the diameter at 273 K is largest at the specified density (molal concentration)and thus is associated with the largest value of For the other three chlorides, the hard-sphere diameter did not exhibit a monotonous increase with temperature. Rather, these effective diameters attained their maximumvalues at temperatures between 273 and 373 K (Figure 3). This complex behavior may be due to the fact that the SMSA model allows for two types of hard spheres, with unequal sizes and different degrees of softness. It may also be due to the temperature dependence of the electrostatic fields generated by the charged hard spheres. These findings concerning the alkali metal chlorides indicate that at least a parabolic temperature
6874 The Journal of Physical Chemistry, Vol. 98, No. 27, 1994 1.8
I
I
I
I
Sun et al.
I
KCI
0.6 0
I
I
I
I
1
2
3
4
I
5
I
I
6
7
o
298 K
e
323 K
+
373 K
0.8 0
1
2
Salt Molality m
3
4
5
6
7
Salt molality m
Figure 1. Osmotic coefficients of aqueous NaCl-LiCl solutions at 298 K as a function of the solution molality and of the molar fraction y of NaCl in the mixed salt NaCl-LiCl. The experimentaldata2' are denoted by the squares, and the values calculated using the SMSA model are denoted by solid lines.
Figure 3. Osmotic coefficients of aqueous KCI solutions as a function of the solution molality and temperature. The experimental data" are represented by markers and the calculatedvalues using the SMSA model by solid lines.
3.5
298K
-
4 '
3
*E
le'
-
c.
.-c @ Q)
8
2.5
-
373 K
-
3 '
.-
413 K
0)
.Y
em 8 .8E a
2
? 0i
8
233K 3.5
1.5
2.5
453K
'
4BK
2 1.5
-
'
-
'
-
1 Y . l
0
0.5 0
4
8
12
Salt molality
16
20
m
Figure 2. Osmotic coefficients of aqueous LiCl solutions as a function of the solution molality and temperature. The experimentaldata21are represented by squares and the SMSA values by solid lines. dependence needs to be introduced in order to describe the variation of the effective hard-sphere diameter. In an exploratory study of temperature effects in the MSA model, Watanasiri et al.9 introduced six parameters to account for the variation in ionic diameters. Thus u+ = u + b / T c/ P and u- = d e / T + f / P . These two equations, however, did not consider the concentration effect on the ionic diameters and are only applicable at low concentrations. Moreover, since u+ and u- are not independent of each other, independennt values of the coefficients a-f cannot be determined during the fitting procedure. This often makes optimum values of the individual parameters difficult to attain. In the present work, we have assumed the anion diameters to be independent of temperature and the cation diameters to be of the form
+
+
where T = T- 273. In this equation the coefficients ~ 0 are 3 used to account for the complex temperature effect, and 01, a2, bl, and b2 are coefficients which correspond to a and b in eq 12 required at intermediateand highconcentrations. With this modification, the predictive capability of the SMSA method is significantly improved in terms of temperature extrapolation.
5
10
15
20
25
Saltmolality m Figure 4. Osmotic coefficients of aqueous LiBr solutions as a function of the solution molality and temperature. The experimental data are represented by squaresand the calculatedvalues using the SMSA model by solid lines.
TABLE 4: Study of the Temperature Interpolation and Extrapolation Capabilities of the SMSA Model lOOAAD (4%)
system max m fit ia fit iib fit iiic LiCl 18.0 0.8 1 0.8 1 0.81 NaCl 6.0 0.65 0.65 0.75 KCl 6.0 0.68 0.68 0.73 CSCl 6.0 0.95 0.95 1.13 Fit i = fit data for all five isotherms simultaneously. * Fit ii = fit data at 273, 323, and 373 K only. Fit iii = fit data at 273, 298, and 323 K only. E
In order to demonstrate the robustness of the proposed approach, the parameters of eq 16 were obtained by independently fitting three sets of experimentaldata for each alkali metal chloride solution: (i) data as 273,298, 323,348, and 373 K (Le., all five isotherms simultaneously); (ii) data at 273,323, and 373 K, (iii) data at 273, 298, and 323 K. Average absolute deviations (AADs) between experimental and calculatedvalues of the osmoticcoefficient ,$,I for all isotherms are given in Table 4. Agreement between the first two sets of AADs was found to be excellent, indicating the success of the SMSA method for interpolating purposes. (Data at 298 and 348 K were not used to obtain the second set of parameters.) The
Osmotic Coefficients of Aqueous Electrolyte Solutions
TABLE 5
Coefficient#of Eq 16 for Four Alkali Metal Chlorides and LiBr LiCl
an I -. 002 003
a1 a2
bi b2
The Journal of Physical Chemistry, Vol. 98, No. 27, 1994 6875
0.620X -0.890 X -0.900 x -0,805X 0.209 X -0.145 X -0.224 X
KCI
NaCl 10'
1P2 10-5 10-1 10-2
10-'
10-3
0.428 X -0.440 X -0.769 X 0.349X 0.135X 0.256X -0.378 X
10'
l(r 10-I
10-I 10-'
1b2
AADs obtained with the third set of parameters were almost the same as the others for LiCl but increased slightly for the other three salts. Nevertheless, this shows that reliable extrapolation in temperature (here from 273-323 to 373 K) is possible using the SMSA model. A detailed study also revealed that accurate extrapolation in temperature requires data for at least three isotherms, with data at one temperature spanning the complete rangeof concentrationsfromdilutesolution to saturatedsolution.
Correlation of Aqueous LiBr Solution Osmotic Coefficient In this section, we demonstrate the application of the SMSA method to aqueous solutions of lithium bromide, because such solutions are widely used as working fluids in absorption refrigeration and a good representation of their thermodynamic properties is important for engineering design. A critical evaluation of some published data sets pertaining to the determination of the osmotic coefficient of lithium bromide is available e l s e ~ h e r e .For ~ ~ this work, the experimental osmotic coefficients were taken from the data of Hamer and Wu,Zo of LCnard et al.48 and of M ~ N e e l y .The ~ ~ Hamer and Wu data cover the complete range of molalities from 0.1 to 21 m at 298 K, whereas the data of LCnard et al. cover the high temperature range from 393 to 483 K with solution molalities between 9 and 21 m. In the intermediate temperature range from 313 to 373 K,we used the data of McNeely at molalities between 13 and 21 m but not his data below 13 m since they exhibit considerable scatter. These three sets of data comprise a total of 1 1 isotherms. Solution densities were obtained from the recent measurements of Lee et a1.30 for molalities higher than 3 m and temperatures higher than 373 K. The correlation of Sbhnel and Novotny2*was employed below 3 m and 373 K. Coefficientsof eq 16 were determined by fitting data along the 1 1 isotherms and are given in Table 5. The AAD for the 138 data points is 0.52%. The maximum absolute deviation (MAD) is 2.2%. The overall fit is shown in Figure 4.
Concision The applicability of the SMSA and MSA methods was demonstrated for aqueous solutionsof 122single electrolytes and 50 binary electrolytes. It is clear from the results that the SMSA method is able to correlate osmotic coefficient data for these solutions as well as the MSA method but has the advantage of simplicity over the MSA method. A new cation diameter relation (eq 16) was proposed and was shown to be useful in accurately interpolating and extrapolating osmotic coefficients of simple concentrated electrolyte solutions over a wide range of temperatures and concentrations. This was demonstrated in detail for the LiBr system over an extended range of conditions.
Acknowledgment. This research was supportedby a grant from ASHRAE under Research Project 526-RP and by Fluid Properties Research Inc., a consortium of companies interested in the thermophysical properties of fluids. The authors also thank Dr. Lloyd Lee of the University of Oklahoma for providing a copy of the Ph.D. thesis of L. H. Landis and for helpful comments on the EXP-MSA theory. References and Notes (1) Pitzer, K.S.J . Phys. Chem. 1973,77,268.
(2) Pitzer, K.S.;Mayorga, G. J . Phys. Chem. 1973,77,2300.
0.366 X -0,126X -0.864 X -0.119 X 0.241 X 0.469 X -0.636 X
CsCl 10' 10-1
l(r
100 10-' 10-I 10-2
0.214 X -0.109 X -0.148 X 0.487X 0.324 X -0.101 x -0.845 X
10' 10-1
IO-' 100 10-1
loo IO-*
LiBr 0.467 X 10' -0.132 . .. - - X -10-3 -0.132x 10-7 -0.115 x 10-1 -0.364 X -0.309 X 10-2 0.693 X 10-5
(3) Blum, L. Mol. Phys. 1975,30, 1529. (4) Blum, L.; Hoye, J. S. J. Phys. Chem. 1977.81, 131 1. (5) Greenberg, J. P.; Moller, N . Geochim. Cosmochim. Acta 1989.53, 2503. (6) Archer, D. G.J . Phys. Chem. Ref.Data 1991, 20,509. (7) Anstiss, R. G.; Pitzer, K. S.J . Solution Chem. 1991, 20,849. (8) Lee, L. L. J . Phys. Chem. 1983,78,5270. (9) Watanasiri, S.;Brule, M. R.; Lee, L. L.J . Phys. Chem. 1982,86, 292. (10) Triolo, R.; Blum, L.; Floriano, M. A. J . Chem. Phys. 1977,67.5956; J. Phys. Chem. 1976,82,1368. (11) Triolo, R.; Grigera, J. R.; Blum, L. J. Phys. Chem. 1976,80,1858. (12) Copeman, T. W. Fluid Phase Equilib. 1986,30,237. (13) Planche, H.; Renon, H. J . Phys. Chem. 1981,85,3924. (14) Ball, F.; Planche, H.; Furst, W.; Renon, H. AIChE J.1985,31,1233. (15) Gering, K.L.; Lee,L. L.;Landis, L. H.; Savidge, J. L. Fluid Phase Equilib. 1989,48, 1 1 1. (16) Gering, K.L.; Lee, L. L.Fluid Phase Euuilib. 1989. 53. 199. (17) Manskri, G. A.; Carnahan, N. F.; Starling, K. E.; Leland, T. W. J . Chem. Phys. 1971,54, 1523. (18) Landis, L. H. MixedSalt ElectrolvteSolutions: AccurateCorrelation for Osmotic Coefficients Based on Molechar Distribution Functions. Ph.D. Thesis, University of Oklahoma, 1985. (19) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworths: London, 1959. (20) Hamer, W. J.; Wu, Y. C. J . Phys. Chem. Ref.Data 1972,I , 1047. (21) Lob0,V. M. M. HandbookofElectrolyteSolutions,PhysicalScience Data 41; Elsevier Science Publishers B. V.: Amsterdam, 1989. (22)Sdhnel, 0.; Novotny, P. Densities ofAqucousSolutions oflnorganic Substances: Elsevier Science: Amsterdam, 1985. (23) Barker, J. A.; Henderson, D. Rev. Mod. Phys. 1976,48, 587. (24) Andersen, H. C.; Chandler, D.; Weeks, J. D.; J . Chem. Phys. 1972,
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