Partial molar volumes at infinite dilution in aqueous solutions of NaCl, LiCl, NaBr, and CsBr at temperatures from 550 K to 725 K

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1 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, 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 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 RWOOD@udel.edu $25.ct Academic Press Limited

2 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 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 ln A I Ž 11.2 I. Ž lnž 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 a4 exp a5 TTc T a6, 6. where T 1K, Tc K, 1kg m, is the density of pure water,. 8 a31291., a , a , 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

3 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 LiClŽ aq NaBrŽ aq CsBrŽ aq 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 II 2 RT m B, where AV is the DebyeHuckel slope for volume, calculated again from Archer Ž 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.

4 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 T p V V V V K MPa cm. mol cm. mol cm. mol cm. mol 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: V V V RT ln a a T Ž AB T T a4 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.

5 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 NaClŽ aq LiClŽ aq

6 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 NaBrŽ aq CsBrŽ aq 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

7 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 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.

8 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.

9 V of NaClŽ aq., LiClŽ aq., NaBrŽ aq., and CsBrŽ aq 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 NaClŽ aq. Free ions Ion pair LiClŽ aq. Free ions Ion pair NaBrŽ aq. Free ions Ion pair CsBrŽ aq. Free ions Ion pair 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., 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

10 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 CHE 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, Majer, V.; Hui, L.; Crovetto, R.; Wood, R. H. J. Chem. Thermodynamics 1991, 23, Majer, V.; Wood, R. H. J. Chem. Thermodynamics 1994, 26, Fogo, J. K.; Benson, S. W. J. Am. Chem. Soc. 1954, 22, Pearson, D.; Copeland, C. S.; Benson, S. W. J. Am. Chem. Soc. 1963, 85, Quist, A. S.; Marshall, W. L. J. Phys. Chem. 1968, 72, Lukashov, Yu. M.; Komissarov, K. B.; Golubev, B. P.; Smirnov, S. N.; Svistunov, E. P. Teploenergetika 1975, 22, Zimmerman, G. H.; Gruszkiewicz, M. S.; Wood, R. H. J. Phys. Chem. 1995, 99, 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 Archer, D. G.; Wang, P. J. J. Phys. Chem. Ref. Data 1991, 19, Levelt Sengers, J. M. H.; Everhart, C. M.; Morrison, G.; Pitzer, K. S. Chem. Eng. Commun. 1986, 47, Levelt Sengers, J. M. H.; Harvey, A. H.; Crovetto, R.; Gallagher, J. S. Fluid Phase Equilib. 1992, 81, O Connell, J. P.; Sharygin, A. V.; Wood, R. H. Ind. Eng. Chem. Res. 1996, 35, Hnedkovsky, L.; Wood, R. H.; Majer, V. J. Chem. Thermodynamics 1996, 28, Harvey, A. H.; Levelt Sengers, J. M. H.; Tanger, J. C., IV. J. Phys. Chem. 1991, 95, Archer, D. G. J. Phys. Chem. Ref. Data 1992, 21, Cooney, W. R.; O Connell, J. P. Chem. Eng. Commun. 1987, 56, O-661 ( ) Receied 17 January 1997; in final form 19 May 1997

Henry s Law Constants of Methane and Acid Gases at Pressures above the Saturation Line of Water

Henry s Law Constants of Methane and Acid Gases at Pressures above the Saturation Line of Water Josef Sedlbauer and Vladimir Majer 2* Department of Chemistry, Technical University of Liberec, 46 7 Liberec,