J. Chem. Thermodynamics 1998, 3, 312 Partial molar volumes at infinite dilution in aqueous solutions of NaCl, LiCl, NaBr, and CsBr at temperatures from 55 K to 725 K Josef Sedlbauer, Department of Chemistry, Technical Uniersity Liberec, Halkoa 6, 461 17 Liberec, Czech Republic Eric M. Yezdimer, and Robert H. Wood a Department of Chemistry and Biochemistry and Center for Molecular Engineering Thermodynamics, Uniersity of Delaware, Newark, DE 19716, U.S.A. Partial molar volumes at infinite dilution provide a convenient test of theoretical models of aqueous solutions. In this communication, previously published experimental results of the apparent molar volumes for NaCl, NaBr, LiCl, and CsBr at near-critical conditions were extrapolated to infinite dilution. In the temperature range included in this study Ž55 to 725. K, ionic association processes must be considered. Using recently proposed equilibrium constants for alkali halides included in this study, the extrapolations were corrected for ion-association effects. Partial molar volumes at infinite dilution for free ions and ion pairs are reported for each electrolyte. 1998 Academic Press Limited KEYWORDS: volumes; aqueous; salts; high temperature 1. Introduction Measurements of the apparent molar volumes in aqueous solutions of several alkali halides at near-critical conditions were performed in the laboratory at the University of Delaware in previous years and the experimental results were Ž1 3. published in this Journal. For many purposes, namely tests of predictions of theoretical models, it is convenient to process these results and calculate the partial molar volumes at infinite dilution. However, at temperatures above T 55 K, the extrapolations cannot be accurate without consideration of ionassociation reactions which occur in the solutions. Independent conductance measurements are therefore required as a source from which the equilibrium constants of association reactions may be evaluated. Except for NaClŽ aq., Ž47. conductance measurements were not thought to be extensive enough to warrant their use in an extrapolation, until the recent results of Zimmerman et al. Ž8. and Gruszkiewicz and Wood Ž9. were obtained. The latter authors have recently proposed a To whom correspondence should be addressed E-mail: RWOOD@udel.edu. 219614983 $25.ct97262 1998 Academic Press Limited
4 J. Sedlbauer, E. M. Yezdimer, and R. H. Wood equations for calculating the equilibrium constants of association reactions for several 11 electrolytes. The purpose of this communication is to provide estimates of the partial molar volumes at infinite dilution for four alkali halides. The results for NaCl are also compared with the earlier extrapolations reported by Majer and Wood. Ž3. 2. Results Association reactions which occur in the solutions of 11 electrolytes at high temperatures above T 55 K are represented by the following equations: A Ž aq. B Ž aq. ABŽ aq., Ž 1. mž A. mž B. m Ž 1. m, Ž 2.. mž AB.. m, Ž 3. where m is stoichiometric molality and is the degree of association of the electrolyte in the solution, which can be calculated from the equation for the equilibrium constant of reaction Ž. 1 : K. 1. m.. 4 2 2 Because there is no reliable model for the activity coefficient of the ion pairs available, was considered to be equal to unity at all experimental conditions and the mean activity coefficient was estimated from the extended DebyeHuckel formula: Ž.. 12. 12.. 12 ln A I Ž 11.2 I. Ž 21.2. lnž 1 1.2 I., Ž 5. 4 where A is the osmotic slope in the DebyeHuckel limiting law, calculated from Ž11. Archer and Wang, and the ionic strength I Ž 1.. Ž mm., with m. 1 1 mol kg. The empirical correlation proposed by Gruszkiewicz and Wood Ž9. for representation of the equilibrium constant of reaction Ž. 1 for alkali halides was used in calculations referred to in this study. The six-parameter formula is given by:. 3 lg K a 1.2. Ž T T. Ž a.. Ž a.. 1 2 3 4... 3 3 a4 exp a5 TTc T a6, 6. where T 1K, Tc647.15 K, 1kg m, is the density of pure water,. 8 a31291., a41.7768, a5.37829, a64.417, and the adjustable parameters a1 and a2 are listed in table 1. The density range used to evaluate.. these parameters was 65 kg m to 2 kg m for NaClŽ aq. and CsBrŽ aq., and.. 65 kg m to 25 kg m for LiClŽ aq. and NaBrŽ aq.. In accordance with the assumptions made for activity coefficients Žideal behavior
V of NaClŽ aq., LiClŽ aq., NaBrŽ aq., and CsBrŽ aq. 2 5 Ž. TABLE 1. Parameters of equation 6 for different alkali halides 4. Electrolyte a a 1 2 NaClŽ aq. 1.589 4.3583 LiClŽ aq. 1.3889 1.3583 NaBrŽ aq. 1.2289.6417 CsBrŽ aq. 1.1189 1.6417. of ion pairs and no interaction between ion pairs and free ions, we can express the apparent molar volumes of electrolytes by: 4 exp. V Ž 1. V Ž A. V Ž B.. V Ž AB., Ž 7. where the apparent molar volumes of ion pairs are considered to be constant and equal to partial molar volumes at infinite dilution V : AB V Ž AB. V. Ž 8. The molality dependence of the apparent molar volumes of free ions is represented by a simple form of the Pitzer ion-interaction model: Ž. 12. ½. 5.. V V V A V B V A 1.2 ln 1 1.2 II 2 RT m B, where AV is the DebyeHuckel slope for volume, calculated again from Archer Ž11.. 1 and Wang, I 1 mol kg, and BV is the ion-interaction parameter. Equations Ž. 8 and Ž. 9 substituted into equation Ž. 7 yield the final formula for extrapolating the experimental results of the apparent molar volumes. A weighted least-squares procedure was used for the calculations with weights equal to 1 2, where is the estimated experimental uncertainty presented in the original papers. Ž1 3. Our simple model could not be used for the description of experimental results at higher molalities, where the ionic association is very high, or re-dissociation processes occur, because the interaction between strongly polar ion pairs and free ions are presumably very important under these conditions. We followed Majer and Wood Ž3. and used only experimental results up to the target molality m. 1.5 mol kg for extrapolations at T 65 K, and results up to the molality. 1 m.1 mol kg at T 65 K. It should be noted that some of the conditions reported by Majer and Wood Ž3. for certain alkali halides do not contain a suitable number of experimental points Ž at least four. or a solid distinguishable trend over the molality range examined to obtain a reliable extrapolation. Therefore, the results for those conditions are not reported in this study. With the limitations of molality range outlined above, all three adjustable parameters V, B, and V, were found to be numerically correlated in some V AB cases, and the accuracy of the results of interest V and VAB might be reduced by AB Ž 9.
6 J. Sedlbauer, E. M. Yezdimer, and R. H. Wood TABLE 2. Comparison of extrapolations for NaClŽ aq. using different equations for ionic association constants, where V VAB V, V is the infinite dilution volume of free ions, and VAB is the infinite dilution volume of ion pairs a b a b 3 1 3 1 3 1 3 1 T p V V V V K MPa cm. mol cm. mol cm. mol cm. mol 548 11 4 58 53 548 25 88 32 8 7 597 21 171 96 167 156 64 19 211 133 223 217 64 27 165 9 152 148 651 28 721 687 59 54 665 28 5756 531 7713 73 67 28 12 933 1631 13467 651 33 44 344 577 567 669 33 14 94 1671 1654 681 33 3655 3327 576 4787 687 33 55 4754 7595 6848 691 33 6542 5571 8921 7889 548 37 81 27 27 597 39 125 57 95 89 64 37 138 68 112 8 651 38 299 23 4 4 673 38 65 62 15 691 38 1859 1828 2779 2732 73 38 358 3221 4991 4648 7 38 4288 3784 5865 5334 a Results from Reference 3. b Results of this study. this effect. In order to stablize the extrapolation and to allow for the use of a minimum number of adjustable parameters, we used the thermodynamic relation: V V RT Ž ln K p. RT Ž ln K. Ž p. AB T T T RT Ž ln K ln., where T is the coefficient of isothermal compressibility of pure water. Applying this relation, equation Ž. 6 yields:.. 2 2 V V V RT ln a a T Ž AB T 2 3.. T 4 3 3 2 2 3a4 exp a5 TTc T a6 a6. 11 In table 2, the new results for V and V in NaClŽ aq. solutions are compared with the older calculations of Majer and Wood. Ž3. Agreement between both sets is very good, considering the difficulty of extrapolation, and gives us confidence in our extrapolations. It is surprising that equation Ž 11. yields V Ž LiCl. V Ž NaCl.; the difference is small and probably reflects the difficulty of calculating.
V of NaClŽ aq., LiClŽ aq., NaBrŽ aq., and CsBrŽ aq. 2 7 TABLE 3. Extraplated partial molar volumes at infinite dilution for NaClŽ aq., LiClŽ aq., NaBrŽ aq., and CsBr aq. V VAB V, where V is the infinite dilution volume of free ions, VAB is the infinite dilution volume of ion pairs, and B is the ion-interaction parameter in equation Ž. 9 V T p V V VAB BV K MPa cm. mol cm. mol cm. mol kg. MPa. mol 3 1 3 1 3 1 1 1 NaClŽ aq. 548.5.86 4 53 13 6.93. 548.5 24.66 32 7 5 3.. 597.45 2.65 96 156 6 1.77. 64.41 18.51 133 217 84 2.53. 64.41 27.44 9 148 58 1.92. 651.11 28. 687 54 67.9 665.39 28.1 531 73 272 4.1 669.94 28. 933 13467 4137 22.2 673.19 28. 52 14351 4299 99.7 651.11 33. 344 567 223 4.16. 668.86 33. 94 1654 56.237 681. 33. 3327 4787 146 2.36 686.55 33.1 4754 6848 294.2 691.19 33. 5571 7889 2318 34.7 696.66 33. 599 835 2315 1.74. 548.5 37.13 27 27 5.97. 597.45 38.68 57 89 2 1.25. 64.42 37.41 68 8 4 1.41. 651.11 37.99 23 4 17 3.77. 673.19 38. 62 8.149 691.2 38.1 1828 2732 94 1.9 73.12 38. 3221 4648 1427 6.8 79.55 38. 3784 5334 155 18.7 716.72 38.1 4158 5774 1616 1.16. 725.5 38.1 4393 675 1682 1.2. LiClŽ aq. 64.41 18.49 121 226 5 3.16. 64.41 27.48 82 158 76 3.1. 651. 28. 647 83 436.169 665.39 28. 4847 7 2153 3.71 669.94 28. 993 1324 4111 17.9 673.19 28. 9868 13994 4126 56.7 651.11 33. 321 582 261 5.5. 668.86 33. 39 1664 625.311 681.11 33. 3216 4713 1497 2.22 686.56 33.1 4633 6659 226 7.36 691.2 33. 5461 7662 221 18.8 64.42 37.4 61 117 56 1.84. 651. 38. 213 46 193 3.3. 673.18 38. 585 4 449.257 691.19 38.1 1758 2698 94 1.4 73.12 38. 314 4512 1372 4.88 79.55 38. 378 5177 1469 13.5 716.72 38.2 487 5534 1447 27.1 4 4 2 2 5 2 2 3 2 2
8 J. Sedlbauer, E. M. Yezdimer, and R. H. Wood TABLE 3continued T p V V VAB BV K MPa cm. mol cm. mol cm. mol kg. MPa. mol 3 1 3 1 3 1 1 1 NaBrŽ aq. 64.42 18.5 113 24 91 3.26. 64.41 27.48 76 134 58 1.84. 651.11 28.1 618 6 418.112 665.39 28. 4714 694 2226 3.11 669.94 28. 8922 1334 4382 15.2 673.19 28. 9756 14232 4476 54.7 651. 33.1 35 567 262 7.81. 668.86 32.99 3 1643 64.248 681.11 33. 3141 472 1561 1.77 686.56 33.1 4548 671 2153 6.5 691.2 33. 5383 7827 2444 2.3 64.42 37.49 56 95 9 1.28. 651.11 38. 22 84 182 3.71. 673.19 38. 562 979 417.135 691.21 38. 1715 2686 971.83 73.12 38. 387 4529 1442 3.31 79.56 38. 3656 526 164 11.3 716.71 38. 448 573 1655 47.2 CsBrŽ aq. 64.41 18.47 1 172 62 2.77. 64.42 27.46 73 1.3. 651. 28.1 64 8 44.134 665.39 28. 4667 683 2163 2.14 669.94 27.99 8892 13749 4857 14.4 673.19 28. 9698 14325 4627 42.4 651.11 33.1 297 55 28 2.36. 668.87 32.99 985 1534 549 6.37. 681.11 33. 31 4657 1556 1.4 686.56 33.1 4511 6691 218 4.57 691.2 33. 5344 7966 2622 2.5 64.43 37.46 54 65 11 1.23. 651.11 38. 197 4 143 1.78. 673.19 38. 55 934 84.117 691.21 38. 1692 2665 973.719 73.12 38. 36 4568 158 3.23 79.55 38. 363 5359 1729 11.7 716.72 38. 426 5853 1827 63.3 2 2 2 2 2 V from the pressure derivative of ln K. At the highest temperatures at every isobar, the values of V of Majer and Wood tend to be lower than the new calculations. We repeated the calculations of Majer and Wood and found the same systematic difference between V from their equation for the association constants and V from the equation of Gruszkiewicz and Wood used in this work. It should be noted that the lowest densities used in our extrapolations are
V of NaClŽ aq., LiClŽ aq., NaBrŽ aq., and CsBrŽ aq. 2 9 still within the limits of the density range used for adjusting the parameters in equation Ž. 6, while this is not true in the case of Majer s equation for lg K. The complete sets of calculated V, B V, VAB, and V for all alkali halides are reported in table 3. 3. Discussion As discussed previously, Ž3. the present extrapolation method should fail if the measurements are too close to the critical point Ž12,13. because critical effects become larger than the DebyeHuckel limiting law on the critical isotherm and isobar. However, the previous extrapolations of the NaCl results Ž3. and the present calculations do not yield any evidence for critical point effects. More recently, Gruszkiewicz and Wood Ž9. used a DebyeHuckelBjerrum activity coefficient in treating conductance results near the critical density and at 2.5 K above the critical temperature, and also found no evidence of critical point effects. FIGURE 1. Plots of parameter D V Ž RT. against Ž H O. for free ions. a, Na Ž aq. 12 T 2 Cl Ž aq.; b, Li Ž aq. Cl Ž aq.; c, Na Ž aq. Br Ž aq.; d, Cs Ž aq. Br Ž aq.., p 38 MPa;, p 33 MPa;, p 28 MPa.
J. Sedlbauer, E. M. Yezdimer, and R. H. Wood There are several sources which contribute to the uncertainties of the extrapolations reported in table 3. Experimental uncertainties in the values of apparent molar volumes are unavoidable and we tried to reduce their effect on the extrapolated values by using the weighted least-squares procedure. Uncertainties in representing the association constants and their pressure derivatives are other sources of error in extrapolations. To estimate the effect of inaccuracies in calculations of the degree of association and V is a difficult task: we can only judge their contributions from the differences between our extrapolations and those of Majer and Wood Ž see table 2., because different equations for ln K were used in both sets of calculations. Disagreement becomes apparent at high temperatures, where it is almost per cent Ž 2 per cent in one case.. Another source of error arises from our assumptions and the method used for extrapolation. In order to summarize all the effects mentioned above and to estimate the FIGURE 2. Plots of parameter D V Ž RT. against Ž H O. for ion pairs. a, NaClŽ aq. 12 AB T 2 ; b, LiClŽ aq.; c, NaBrŽ aq.; d, CsBrŽ aq.., p 38 MPa;, p 33 MPa;, p 28 MPa.
V of NaClŽ aq., LiClŽ aq., NaBrŽ aq., and CsBrŽ aq. 2 11 TABLE 4. Parameters of equation 12 for different alkali halides, free ions and ion pairs. The average relative error of the correlation is given by, and is the maximum relative error. a. a. a. a.. 2 4 7 2 2 1 2 3 NaClŽ aq. Free ions 3.26 18.274 6.36 24.451 4.3 11.3 Ion pair 8.1361 4.291 7.9752 5.596 4.6 12.4 LiClŽ aq. Free ions 26.472 13.656 25.956 16.851 3.8 9.9 Ion pair 6.9149 3.5438 6.8524 4.4183 4.4 11.7 NaBrŽ aq. Free ions 27.747 14.45 27.386 17.784 3.7.7 Ion pair 8.1557 4.2348 8.148 5.254 3.8.8 CsBrŽ aq. Free ions 28.721 14.962 28.467 18.576 4.1 8.6 Ion pair 9.1988 4.8289 9.3178 6.88 4.9 13.7 accuracy of tabulated results, we made an independent check of our calculations. It has been our experience, Ž3,14,15,18. which has some background in the theory of near-critical phenomena Ž16. that, near the critical point plots, of the functions 4 V V HO RT or V Ž RT. m 2 T T against density, or some function of density, produce smooth curves on which the points at the same densities are very close to each other, regardless of pressure. Plots of D V Ž RT. 12 T and D V Ž RT. 12 AB T against, are shown in figures 1 and 2. For T 65 K, the new extrapolations are in excellent agreement with the V values of Archer. Ž17. At T 65 K, we found that a third-order polynomial in density provided good correlation of D for free ions and also ion pairs of all solutes: 12 D a a. a. a., 12 2 3 12 1 Ž. 2 Ž. 3 Ž.. where 1 kg m. Parameters of equation Ž 12. and calculated average relative errors and maximum relative errors are summarized in table 4. The accuracy of this description of D12 allows us to estimate the uncertainty of our extrapolations due to random experimental errors and the choice of the method to be about 5 per cent of the absolute values of V and VAB.. The values of V predicted from equation 12 at 4 kg m follow the. order: LiCl NaCl NaBr CsBr, and the values of VAB at 35 kg m increase in the opposite way: CsBr NaBr NaCl LiCl. Both orders are in agreement with the arguments put forward by Majer and Wood Ž3. about the influence of free ions and ion pairs of different sizes on the properties of solutions. However, at lower densities, the sequence for free ions is changed to: NaCl CsBr NaBr LiCl, but the differences are not much larger than our estimated uncertainties. Similarly, the sequence for ion pairs is changed at higher densities to: LiCl NaBr CsBr NaCl. It should be noted that at higher densities the
12 J. Sedlbauer, E. M. Yezdimer, and R. H. Wood uncertainties in evaluating VAB are high, because the ions are only weakly associated under these conditions. Similarly, for V at lower densities, the uncertainties are higher, because at these conditions and at molalities included in this study, most of the ions are associated. On the other hand, the regularity of these unexpected features suggests that random experimental errors are not responsible for the effect and that some other systematic error might be involved: perhaps, for example, a systematic effect arising from our neglect of iondipole and dipoledipole interactions in calculating the activity coefficients in equation Ž. 4. We would expect to observe a minimum of V near the maximum of T ; however, the experimental data do not extend far enough beyond this maximum to enable us to discern the predicted minimum within the uncertainties of our extrapolations. Based on the above findings, we conclude that the extrapolated values of the partial molar volumes at infinite dilution of free ions reported in table 3 are reliable, with uncertainties varying from 5 per cent to per cent as the temperature increases. The absolute values of the partial molar volumes at infinite dilution of ion pairs are lower and, therefore, the relative uncertainties are higher; we estimate them to vary from 7 per cent to 2 per cent as the temperature decreases. The authors would like to thank John O Connell and Vladimir Majer for their helpful discussions and comments. This work was supported by the Department of Energy Ž DOE. under grant number DEFG2-89ER-148 and by the National Science Foundation under grant CHE9416564. Such support does not constitute endorsement by the DOE or NSF of the views expressed in this article. REFERENCES 1. Majer, V.; Hui, L.; Crovetto, R.; Wood, R. H. J. Chem. Thermodynamics 1991, 23, 213229. 2. Majer, V.; Hui, L.; Crovetto, R.; Wood, R. H. J. Chem. Thermodynamics 1991, 23, 36578. 3. Majer, V.; Wood, R. H. J. Chem. Thermodynamics 1994, 26, 11431166. 4. Fogo, J. K.; Benson, S. W. J. Am. Chem. Soc. 1954, 22, 212216. 5. Pearson, D.; Copeland, C. S.; Benson, S. W. J. Am. Chem. Soc. 1963, 85, 4749. 6. Quist, A. S.; Marshall, W. L. J. Phys. Chem. 1968, 72, 68473. 7. Lukashov, Yu. M.; Komissarov, K. B.; Golubev, B. P.; Smirnov, S. N.; Svistunov, E. P. Teploenergetika 1975, 22, 7881. 8. Zimmerman, G. H.; Gruszkiewicz, M. S.; Wood, R. H. J. Phys. Chem. 1995, 99, 1161211625. 9. Gruszkiewicz, M. S.; Wood, R. H. J. Chem. Thermodynamics 1997, submitted.. Pitzer, K. S. Actiity Coefficients in Electrolyte Solutions. 2nd edition. Pitzer, K.S.: editor. CRC Press: Boca Raton, FL. 1991. 11. Archer, D. G.; Wang, P. J. J. Phys. Chem. Ref. Data 1991, 19, 371411. 12. Levelt Sengers, J. M. H.; Everhart, C. M.; Morrison, G.; Pitzer, K. S. Chem. Eng. Commun. 1986, 47, 31528. 13. Levelt Sengers, J. M. H.; Harvey, A. H.; Crovetto, R.; Gallagher, J. S. Fluid Phase Equilib. 1992, 81, 857. 14. O Connell, J. P.; Sharygin, A. V.; Wood, R. H. Ind. Eng. Chem. Res. 1996, 35, 2882812. 15. Hnedkovsky, L.; Wood, R. H.; Majer, V. J. Chem. Thermodynamics 1996, 28, 125142. 16. Harvey, A. H.; Levelt Sengers, J. M. H.; Tanger, J. C., IV. J. Phys. Chem. 1991, 95, 932937. 17. Archer, D. G. J. Phys. Chem. Ref. Data 1992, 21, 793829. 18. Cooney, W. R.; O Connell, J. P. Chem. Eng. Commun. 1987, 56, 34149. O-661 ( ) Receied 17 January 1997; in final form 19 May 1997