Aqueous solutions of the alkaline-earth metal chlorides at elevated temperatures. Isopiestic molalities and thermodynamic properties

J. Chem. Thermodynamics 1996, 28, 1325–1358

Aqueous solutions of the alkaline-earth metal chlorides at elevated temperatures. Isopiestic molalities and thermodynamic properties a H. F. Holmes and R. E. Mesmer Chemical and Analytical Sciences Division, Oak Ridge National Laboratory, P.O. Box 2008 , Oak Ridge, TN 37831 -6110 , U.S.A. Isopiestic results for SrCl2 (aq) and BaCl2 (aq) were obtained at T = 498 K and T = 523 K. As a check on previously reported results for MgCl2 (aq) (from T = 383 K to T = 473 K), new isopiestic measurements were made over the temperature range from 383 K to 523 K. The new measurements are slightly different from our previous results. Here, NaCl(aq) was used as the reference electrolyte for computation of osmotic and activity coefficients. Our published model for the thermodynamic properties of CaCl2 (aq) was used in a global least-squares treatment of the present results in combination with published results to provide comprehensive equations for the properties of these three electrolyte solutions. At elevated temperature, the available results consist primarily of excess Gibbs free energies (our isopiestic results), particularly for SrCl2 (aq) and BaCl2 (aq). Void regions in the temperature and pressure ranges of acceptable experimental results were estimated by reference to the properties of CaCl2 (aq). Trends in the thermodynamic properties of aqueous solutions of the alkaline-earth metal chlorides were uniform at all temperatures and pressures used in the present work. 7 1996 Academic Press Limited

1. Introduction A recent paper(1) from this laboratory presented a model for the thermodynamic properties of aqueous CaCl2 based on the ion-interaction treatment of Pitzer.(2) Empirical parameters of the model were adjusted by least-squares fitting of a combination of new experimental results and previously published studies. The present work is a similar treatment for the three remaining members of the common alkaline-earth metal chlorides, namely, MgCl2 (aq), SrCl2 (aq), and BaCl2 (aq). Our previous isopiestic results(3,4) for aqueous solutions of these three electrolytes terminated at T 1 473 K due to limitations of the isopiestic facility. Subsequent modification of the facility(5) permits operation through T 1 523 K and, in the present paper, we report isopiestic results for these three electrolytes in a Research sponsored by the Division of Chemical Sciences, Office of Basic Energy Sciences at Oak Ridge National Laboratory, managed by Lockheed Martin Energy Research Corporation for the U.S. Department of Energy under contract number DE-AC05-96OR22464. The submitted manuscript has been authored by a contractor of the U.S. Government under contract No. DE-AC05-96OR22464. Accordingly, the U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of the contribution, or to allow others to do so, for U.S. Government purposes.

0021–9614/96/121325 + 34 $18.00/0

7 1996 Academic Press Limited

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the temperature range from 473 K to 523 K. There have been recent suggestions(6–8) that our earlier isopiestic results for MgCl2 (aq)(3) are in error by several per cent. The earlier measurements have been repeated and the results are reported in this paper. There are previous mathematical models for the thermodynamic properties of these electrolytes at elevated temperatures, particularly for MgCl2 (aq).(7,9) Phutela et al.(10) published equations for the thermodynamic properties of MgCl2 (aq), CaCl2 (aq), and SrCl2 (aq). A similar treatment of the results for BaCl2 (aq) was recently completed by Monin.(11) With the exception of Valyashko et al.(7) who depend mostly on vapor pressure results at higher temperatures, these studies(9–11) rely heavily on our isopiestic results(3,4) at the higher temperatures. With the exception of our isopiestic measurements reported here and in previous publications,(3,4) there are very few comprehensive sets of thermodynamic results for these three solutes at temperatures above 373 K, particularly for SrCl2 (aq) and BaCl2 (aq). In the case of MgCl2 (aq), there are the heat capacity measurements of White et al.(12) which extend to T = 598 K and molality m = 2.26 mol·kg−1 and, from the same laboratory, enthalpy of dilution measurements between T = 373 K and T = 473 K by Mayrath and Wood.(13) There are also enthalpy of dilution results for MgCl2 (aq) from Oak Ridge National Laboratory(14) and Brigham Young University(15) which are applicable at the upper temperature limit of the present study. Vapor pressure measurements of MgCl2 (aq), reported by Liu and Lindsay,(16) cover the temperature range from 348 K to 573 K but have a limited range of molalities. The volumetric results of Ellis(17) extend to T = 473 K for all of the common alkaline-earth metal chlorides in aqueous solution. Quite recently, Brendler and Voigt(6) reported isopiestic results at T = 428 K which are related to the present work. There are several sets of vapor pressure results available at elevated temperatures, mostly from the Russian schools and for MgCl2 (aq), but they are generally above the upper molality limit of the present study. The solubility of the reference salt NaCl dictated the upper molality limits for MgCl2 (aq) and SrCl2 (aq), which were about 4 mol·kg−1 and 5 mol·kg−1, respectively. Results for BaCl2 (aq) were limited by the solubility of BaCl2 (s), which was about 4.1 mol·kg−1 at T = 523 K. The lower molality limit of our present results is about (0.4 to 0.5) mol·kg−1.

2. Experimental Stock solutions of the three alkaline-earth metal chlorides were prepared from the solid hydrates and deionized water. The BaCl2 was ‘‘ultrapure’’ grade from Alfa Chemicals and was used without further purification. The MgCl2 (‘‘AR’’ grade from Mallinckrodt) and SrCl2 (‘‘AR’’ grade from Mallinckrodt) were each recrystallized twice from triply distilled water. The three stock solutions were analysed gravimetrically by precipitation of the chloride as AgCl. Triplicate determinations agreed to 0.03 per cent, or better. Mass aliquots of the stock solutions were used as experimental samples. Reference solutions for the isopiestic measurements were prepared from mass aliquots (after drying overnight at

MCl2 (aq) at high temperatures

1327

T = 400 K) of the same ‘‘ultrapure’’ NaCl used for all our previous isopiestic experiments.(3) The experimental procedure and the apparatus, including the modifications necessary for operation above T = 473 K, have been described in previous publications.(3,5) A recent review contains a detailed description of the Oak Ridge National Laboratory high-temperature isopiestic facility.(18) The essence of an isopiestic experiment is allowing two or more solutions (normally aqueous) to attain equilibrium with a common vapor phase. Equilibrium molalities are determined from a suitable analysis which, in our case, is in-situ weighing with an electromagnetic balance. Water activities (and solute molalities) can be adjusted by addition or removal of water, as liquid, through a stainless-steel capillary tube which extends from the vapor phase of the pressure vessel to a valve at laboratory temperature. Measurement of the pH of the samples at the conclusion of an experimental run indicated that hydrolysis of the alkaline-earth metal chlorides was negligible. One would not expect hydrolysis to be a significant factor at these temperatures and molalities.

3. Results Equilibrium isopiestic molalities for MgCl2 (aq) and the reference electrolyte NaCl(aq) are listed in table 1 along with the corresponding osmotic coefficients. The same quantities for SrCl2 (aq) and BaCl2 (aq) are given in table 2. Tables 1 and 2 are each a combination of results from two separate loadings of the isopiestic apparatus. Osmotic coefficients f were calculated from the listed molalities of tables 1 and 2 by means of the equation: fx = 2fs ms /3mx .

(1)

Equation (1) is specific for the present charge type (2 − 1), and the subscripts s and x refer to NaCl(aq) and the test electrolyte, respectively. Since equation (1) assumes complete dissociation of both electrolytes, the resulting osmotic coefficients are stoichiometric quantities. Osmotic coefficients for the isopiestic standard NaCl(aq) were calculated from the equations of Pitzer et al.(19)

4. Treatment of results The ion-interaction treatment of Pitzer(20) continues to be the preferred method for describing the thermodynamics of aqueous solutions. His equations have been used for isothermal treatment of a single property at both normal and elevated temperatures, as well as for global descriptions of several thermodynamic properties over a range of temperatures and pressures. The ion-interaction treatment has been found to be equally applicable to aqueous solutions of mixed electrolyte systems. We used Pitzer’s treatment in all of our previous studies of aqueous solutions of electrolytes and continue to do so in the present work.

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TABLE 1. Isopiestic molalities m and the corresponding osmotic coefficients f for MgCl2 (aq). (m° = 1 mol·kg−1 ) f

MgCl2 m/m°

NaCl m/m°

f

MgCl2 m/m°

NaCl m/m°

f

MgCl2 m/m°

NaCl m/m°

T = 383.05 K 1.0801

1.3334

2.1830

1.0498

1.4511

2.3457

1.1310

2.0808

3.5514

0.9608 0.9719

1.7827 1.8340

2.7977 2.9001

0.9935 1.0009

0.6997 0.6974 0.7322 0.7325 0.7444 0.7494 0.7561 0.7697 0.7982

0.5309 0.5353 0.8852 0.8875 0.9812 1.0237 1.1241 1.2228 1.4160

0.6688 0.6721 1.1634 1.1670 1.3076 1.3715 1.5144 1.6703 1.9876

0.8093 0.8355 0.8464 0.9210 0.9286 1.0250 1.0279 1.0346

1.4837 1.6363 1.6980 2.0755 2.1120 2.4721 2.4849 2.5179

0.6315 0.6309 0.6300 0.6528 0.6536 0.6555 0.6559

0.4242 0.4259 0.4301 0.8695 0.8833 0.9077 0.9146

0.4968 0.4983 0.5027 1.0652 1.0834 1.1166 1.1257

0.6845 0.6850 0.7272 0.7284 0.7672 0.7712 0.8545

0.9093 0.9102 0.9186 0.9736 0.9719 0.9711 0.9715 1.0278 1.0298

0.6976 0.7003 0.7203 0.9159 0.9189 0.9332 0.9374 1.1499 1.1570

1.0163 1.0238 1.0284 1.4108 1.4127 1.4321 1.4386 1.8275 1.8411

0.8743 0.8757 0.8859 0.9212 0.9298 0.9522 0.9573 0.9867

0.7170 0.7313 0.7694 0.9892 1.0023 1.0703 1.0937 1.2251

1.0301 1.0515 1.1159 1.4673 1.4984 1.6279 1.6690 1.9040

T = 413.74 K T = 444.03 K T = 473.85 K 1.9306 3.0969 1.9633 3.1646

1.0823

2.3019

3.9034

2.1040 2.3744 2.4870 3.2266 3.3022 4.1533 4.1825 4.2569

1.0992 1.1014 1.1647 1.1668 1.2478 1.2500 1.2992 1.3023

2.7925 2.8000 3.0661 3.0782 3.4430 3.4528 3.6771 3.6800

4.9231 4.9436 5.6313 5.6610 6.6547 6.6825 7.3422 7.3636

T = 524.12 K 1.2019 1.2135 1.5353 1.5454 1.8243 1.8489 2.3234

1.5376 1.5535 2.0658 2.0819 2.5593 2.6043 3.5350

0.8566 0.9389 0.9409 1.0135 1.0135 1.2123 1.2242

2.3398 2.7038 2.7229 3.0782 3.1016 3.8852 3.9499

3.5658 4.4146 4.4515 5.3158 5.3528 7.7957 7.9999

1.0982 1.1026 1.1815 1.1850 1.1960 1.1991 1.2014 1.2058 1.2183

T = 383.11 K 1.4046 1.4125 1.6614 1.6719 1.7091 1.7198 1.7246 1.7404 1.7796

2.3202 2.3400 2.8632 2.8863 2.9659 2.9883 3.0006 3.0335 3.1191

1.3060 1.3173 1.3219 1.4664 1.4728 1.5241 1.5278 1.6465 1.6529

2.0389 2.0758 2.0873 2.4983 2.5139 2.6538 2.6634 2.9757 2.9865

3.7119 3.7952 3.8237 4.8324 4.8746 5.2381 5.2638 6.1179 6.1549

0.9971 1.0516 1.0609 1.0932 1.1066 1.1176 1.1302 1.2425

T = 413.56 K 1.2561 1.4648 1.4999 1.6088 1.6628 1.7068 1.7532 2.1310

1.9665 2.3687 2.4381 2.6638 2.7716 2.8602 2.9568 3.7935

1.2569 1.3416 1.3595 1.4972 1.5177 1.6316 1.6486

2.1795 2.4439 2.4979 2.9105 2.9687 3.2826 3.3340

3.9045 4.5429 4.6785 5.7604 5.9231 6.8445 6.9956

T = 498.72 K

1329

MCl2 (aq) at high temperatures TABLE 1—continued f

MgCl2 m/m°

NaCl m/m°

f

MgCl2 m/m°

NaCl m/m°

f

MgCl2 m/m°

NaCl m/m°

T = 443.44 K 0.8053 0.8109 0.8266 0.8343 0.8544 0.8632 0.8800 0.8953 0.9254 0.9464

0.6605 0.6890 0.7673 0.8057 0.9086 0.9517 1.0519 1.1113 1.2535 1.3377

0.8997 0.9437 1.0668 1.1281 1.2944 1.3659 1.5286 1.6353 1.8855 2.0433

0.9787 0.9956 0.9979 1.0658 1.0953 1.1127 1.1372 1.1421 1.1575 1.1921

1.4832 1.5528 1.5573 1.8351 1.9494 2.0121 2.1061 2.1229 2.1846 2.2985

2.3139 2.4489 2.4605 3.0188 3.2604 3.3982 3.6034 3.6418 3.7768 4.0474

1.2199 1.2658 1.2978 1.3381 1.3678 1.4676 1.4951 1.5883 1.6127

2.4007 2.5440 2.6580 2.7952 2.8960 3.2063 3.2951 3.5817 3.6514

4.2854 4.6483 4.9294 5.2820 5.5470 6.4309 6.6916 7.5930 7.8309

0.7444 0.7530 0.7681 0.7727 0.7820 0.7866 0.8262 0.8353 0.8442 0.8658 0.8771

0.7335 0.7347 0.7821 0.8065 0.8719 0.9007 1.1210 1.1689 1.2492 1.3360 1.3945

0.9508 0.9631 1.0438 1.0818 1.1805 1.2251 1.5833 1.6142 1.7895 1.9508 2.0546

0.9066 0.9201 0.9536 0.9682 1.0031 1.0190 1.0454 1.0598 1.0985 1.1132

T = 473.18 K 1.5414 1.6050 1.7589 1.8236 1.9785 2.0445 2.1827 2.2416 2.3963 2.4504

2.3232 2.4434 2.7425 2.8722 3.1893 3.3301 3.6082 3.7388 4.0913 4.2210

1.1628 1.1794 1.2143 1.2280 1.2988 1.3144 1.3767 1.3895 1.4540 1.4622

2.6294 2.6918 2.8277 2.8680 3.1296 3.1808 3.4033 3.4517 3.6825 3.7252

4.6635 4.8186 5.1595 5.2748 5.9800 6.1304 6.7845 6.9283 7.6625 7.7862

0.6801 0.6784 0.6853 0.6870 0.6952 0.6986 0.7053 0.7077 0.7242 0.7302

0.5631 0.5829 0.6163 0.6172 0.6615 0.6854 0.7496 0.7768 0.9238 0.9725

0.6893 0.7118 0.7603 0.7632 0.8278 0.8618 0.9509 0.9886 1.1992 1.2712

0.7476 0.7714 0.7863 0.7815 0.8192 0.8573 0.8499 0.9044 0.8997

T = 498.22 K 1.1070 1.2690 1.3241 1.3332 1.5413 1.7019 1.7110 1.9506 1.9677

1.4746 1.7331 1.8372 1.8389 2.2032 2.5164 2.5117 2.9976 3.0073

0.9761 1.0017 1.0224 1.0733 1.1042 1.1777 1.2066 1.2671 1.2991

2.2904 2.4112 2.5425 2.7256 2.8580 3.1589 3.2845 3.5487 3.6821

3.7106 3.9755 4.2433 4.7126 5.0406 5.8355 6.1766 6.9303 7.3406

THERMODYNAMICS AND BASIC EQUATIONS

The ion-interaction model assumes that the excess Gibbs free energy G E can be adequately represented by: GE/ww RTm° = f G(I/m°) + 4B G(m/m°)2 + 23/2C(m/m°)3,

(2)

G

where f is defined as: f G = −2Af heG ,

(3)

B G = b (0) + b (1)h1G + b (2)h2G ,

(4)

G

and B is given by:

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H. F. Holmes and R. E. Mesmer

TABLE 2. Isopiestic molalities m and the corresponding osmotic coefficients f for SrCl2 (aq) and BaCl2 (aq). (m° = 1 mol·kg−1 ) f

SrCl2 m/m°

f

BaCl2 m/m°

NaCl m/m°

f

SrCl2 m/m°

1.1341

2.4150

0.9432

2.9037

T = 413.45 K 3.9034

1.0347

2.4720

0.8459

3.0236

T = 443.76 K 3.8157

0.7274 0.7299 0.7300 0.7975 0.7995 0.8588 0.8615 0.9724

0.8572 0.8829 0.8888 1.4735 1.4881 1.9864 2.0076 2.9190

0.6741 0.6748 0.6750 0.7090 0.7095 0.7310 0.7319 0.7380

0.9250 0.9549 0.9613 1.6576 1.6767 2.3336 2.3632 3.8458

T = 473.68 K 1.0830 0.9722 1.1182 1.0014 1.1257 1.0035 1.9813 1.0416 2.0040 1.0430 2.7836 1.0758 2.8218 1.0756 4.3738

0.6658 0.6674 0.6653 0.6552 0.6659 0.6658 0.6740 0.6741 0.6743 0.6751 0.6928 0.6928 0.7160 0.7169 0.7553 0.7567 0.7605

0.4083 0.4081 0.4123 0.4150 0.4170 0.4180 0.7405 0.7497 0.7745 0.7858 1.0614 1.0724 1.3693 1.3789 1.7967 1.8124 1.8503

0.6401 0.6377 0.6364 0.6355 0.6347 0.6345 0.6264 0.6259 0.6250 0.6252 0.6297 0.6285 0.6351 0.6354 0.6468 0.6471 0.6483

0.4247 0.4270 0.4310 0.4344 0.4375 0.4386 0.7967 0.8074 0.8356 0.8486 1.1678 1.1821 1.5437 1.5557 2.0980 2.1192 2.1705

T = 498.27 K 0.4875 0.7621 0.4884 0.7617 0.4919 0.7634 0.4951 0.7945 0.4981 0.7949 0.4991 0.8305 0.8982 0.8316 0.9094 0.8655 0.9396 0.8666 0.9544 0.8971 1.3152 0.8986 1.3286 0.9267 1.7356 0.9279 1.7491 0.9488 2.3566 0.9497 2.3797 0.9657 2.4366 0.9660

0.6242 0.6239 0.6243 0.6238 0.6238 0.6254 0.6244 0.6221 0.6222 0.6305 0.6299 0.6473 0.6482 0.6690 0.6694 0.6932 0.6939 0.7216

0.4960 0.4964 0.4985 0.4999 0.5034 0.7637 0.7647 0.7707 0.7732 1.0128 1.0194 1.3556 1.3676 1.6903 1.6959 2.0114 2.0205 2.3764

0.5894 0.5891 0.5887 0.5879 0.5879 0.5799 0.5773 0.5737 0.5732 0.5703 0.5691 0.5689 0.5690 0.5714 0.5719 0.5746 0.5746 0.5754

0.5253 0.5257 0.5286 0.5304 0.5342 0.8236 0.8272 0.8357 0.8393 1.1197 1.1284 1.5426 1.5580 1.9787 1.9852 2.4267 2.4399 2.9803

T = 523.20 K 0.5756 0.7231 0.5758 0.7432 0.5786 0.7451 0.5798 0.7685 0.5841 0.7689 0.8942 0.7965 0.8939 0.7980 0.8976 0.8242 0.9007 0.8280 1.1959 0.8305 1.2026 0.8321 1.6351 0.8446 1.6511 0.8450 2.0874 0.8575 2.0954 0.8577 2.5442 0.8670 2.5576 0.8664 3.0832

f

BaCl2 m/m°

2.9353 3.2229 3.2459 3.6684 3.6888 4.1390 4.1509

NaCl m/m°

4.3950 4.8928 4.9319 5.6701 5.7047 6.4900 6.5056

1.8661 1.8790 1.8989 2.2111 2.2213 2.5899 2.6035 2.9743 2.9881 3.3587 3.3724 3.7582 3.7739 4.1358 4.1501 4.5249 4.5369

0.6485 0.6478 0.6482 0.6537 0.6536 0.6542 0.6540 0.6464 0.6457

2.1928 2.2095 2.2364 2.6874 2.7017 3.2878 3.3101 3.9826 4.0102

2.4608 2.4755 2.5049 2.9868 3.0006 3.5848 3.6057 4.2078 4.2297 4.8383 4.8631 5.5029 5.5294 6.1228 6.1474 6.7512 6.7697

2.3929 2.6651 2.6870 3.0070 3.0131 3.4205 3.4325 3.9076 3.9423 4.0322 4.0509 4.3260 4.3559 4.6882 4.7261 5.0709 5.1020

0.5754 0.5727 0.5722 0.5655 0.5649

3.0068 3.4584 3.4986 4.0868 4.1011

3.1087 3.5177 3.5526 4.0461 4.0548 4.6921 4.7150 5.4541 5.5206 5.6518 5.6857 6.1191 6.1597 6.6828 6.7349 7.2712 7.3098

1331

MCl2 (aq) at high temperatures TABLE 2—continued f

SrCl2 m/m°

f

BaCl2 m/m°

NaCl m/m°

f

SrCl2 m/m°

f

BaCl2 m/m°

NaCl m/m°

1.0121

1.4231

0.9255

1.5562

T = 383.05 K 2.1830

0.9752

1.5620

0.8760

1.7390

T = 413.74 K 2.3457

1.0115

2.3264

0.8406

2.7995

T = 444.03 K 3.5514

0.8622 0.8692 0.8836

1.9866 2.0507 2.1708

0.7342 0.7361 0.7398

2.3328 2.4215 2.5925

T = 473.85 K 2.7977 0.8883 2.9001 0.9406 3.0969

2.2123 2.6487

0.7404 0.7442

2.6542 3.3476

3.1646 3.9034

0.6669 0.6697 0.6683 0.6687 0.7058 0.6988 0.7094 0.7199 0.7108 0.7206 0.7330 0.7402 0.7552 0.7623 0.8082

0.5552 0.5547 0.5586 0.5870 0.9183 0.9303 1.0296 1.0656 1.1958 1.3062 1.5420 1.6222 1.8103 1.8852 2.3652

T = 498.72 K 0.6666 0.8129 0.6688 0.8581 0.6721 0.8596 0.7070 0.8615 0.1634 0.8630 1.1670 0.8670 1.3076 0.8999 1.3715 0.9010 1.5144 0.9301 1.6703 0.9310 1.9876 0.9609 2.1040 0.9620 2.3744 0.9749 2.4870 0.9755 3.2266

2.4126 2.9008 2.9236 2.9414 2.9595 3.0046 3.4110 3.4228 3.8392 3.8579 4.4709 4.4864 4.9001 4.9127

0.6548 0.6485 0.6478 0.6470 0.6469 0.6455

2.9951 3.8384 3.8799 3.9169 3.9482 4.0355

3.3022 4.0875 4.1226 4.1533 4.1825 4.2569 4.9231 4.9436 5.6313 5.6610 6.6547 6.6825 7.3422 7.3636

0.6158 0.6155 0.6154 0.6199 0.6211 0.6228 0.6242 0.6254 0.6256 0.6416 0.6421 0.6671 0.6679

0.4351 0.4365 0.4403 0.4419 0.4698 0.9114 0.9249 0.9514 0.9590 1.2823 1.2948 1.6737 1.6855

T = 524.12 K 0.4968 0.6922 0.4983 0.6947 0.5027 0.7419 0.5084 0.7440 0.5425 0.7838 1.0652 0.7845 1.0834 0.8187 1.1166 0.8186 1.1257 0.8483 1.5376 0.8563 1.5535 0.8703 2.0658 0.8724 2.0819

2.0219 2.0525 2.6763 2.6938 3.2389 3.2657 3.8108 3.8400 4.5391 4.7491 5.4117 5.5424

0.5731 0.5735 0.5709 0.5712

2.4419 2.4863 3.4777 3.5088

2.5593 2.6043 3.5350 3.5658 4.4146 4.4515 5.3158 5.3528 6.4401 6.7756 7.7957 7.9999

0.6315 0.6322

0.5882 0.5905

0.6330 0.6359 0.6398 0.6415 0.6462 0.6478 0.6548

1.3427 1.4801 1.7666 1.8719 2.1156 2.2186 2.9193

0.5861 0.5845 0.5838

0.4571 0.4597 0.4642

0.5701 0.5689 0.5681 0.5677 0.5682 0.5673 0.5708 0.5711

0.9956 1.0148 1.0475 1.0567 1.4479 1.4654 1.9558 1.9713

with ww being the mass of water, m° = 1 mol·kg−1, m the molality, I the ionic strength {I = (1/2)Si mi zi2 with zi the charge number on ion i}, and Af the Debye–Hu¨ckel osmotic-coefficient parameter. Ion-interaction parameters b(0), b (1), and C are temperature- and pressure-dependent, and are determined from a

1332

H. F. Holmes and R. E. Mesmer

least-squares fit of the equations to experimental results. It is helpful to define various functions of ionic strength: heG = (2/b)·ln{1 + b(I/m°)1/2},

(5)

u1 = a1 ·(I/m°) ,

(6)

u2 = a2 ·(I/m°)1/2,

(7)

1/2

f 1

h = exp(−u1 ),

(8)

h2f = exp(−u2 ),

(9)

h = 2{1 − (1 + u1 )h }/u ,

(10)

h2G = 2{1 − (1 + u2 )h2f }/u22 .

(11)

f 1

G 1

2 1

The quantities b and a1 are usually given the constant values of 1.2 and 2.0, respectively, while the value of a2 depends on the charge type involved. Osmotic and activity coefficients can be obtained from the excess Gibbs free energy by differentiation with respect to the amount of substance for the solvent or solute, respectively. For the osmotic coefficient f this treatment gives: f = 1 + 2f f + (4/3)(m/m°)B f + (25/2/3)(m/m°)2C,

(12)

where f f, the long-range electrostatic term, is given by: f f = −Af hef ,

(13)

and Bf, the second virial coefficient, is given by: B f = b (0) + b (1)h1f + b (2)h2f ,

(14)

hef = (I/m°)1/2/{1 + b(I/m°)1/2}.

(15)

with hef defined as: In the case of the activity coefficient g, differentiation gives an expression (excluding the non-determinable terms) for a single ionic species, which has no thermodynamic reality. Combination of the expressions for a cation and an anion gives the equation for the activity coefficient of a neutral electrolyte, a useful and real quantity. For the present case the equation is: ln g = 2f g + (4/3)(m/m°)B g + 23/2(m/m°)2C.

(16)

For the activity coefficient, the long-range electrostatic term f g is: f g = −Af (heG + hef ),

(17)

and the second virial coefficient B g is given by: B g = B G + B f.

(18)

We shall be interested in combining enthalpies and heat capacities with the excess Gibbs free energy in a comprehensive analysis. Established thermodynamic relations give equations relating these three quantities. The first derivative of the excess

MCl2 (aq) at high temperatures

1333

Gibbs free energy with respect to temperature (at constant pressure and composition) gives the apparent relative molar enthalpy as: Lf = f L − 4RT 2{(m/m°)B L + (m/m°)2C L/21/2},

(19)

where the superscript L denotes a derivative with respect to temperature, resulting in the equations: f L = (3/2)AL heG ,

(20)

where AL is the Debye–Hu¨ckel term for enthalpy, and B L = bL(0) + bL(1) h1G + bL(2) h2G + b (1)h1L + b (2)h2L ,

(21)

with bL(0) = (1b (0)/1T )p,m ,

h1L = (1h1G /1T )p,m ,

(22)

and likewise for bL(1) , bL(2) , C L, and h2L . With few exceptions (the present work being one), normal usage of the ion-interaction model sets h1L = h2L = 0. The molar integral enthalpy of solution Dsol Hm is related to Lf through the enthalpy of solution at infinite dilution Dsol Hma by: Dsol Hm = Dsol Hma + Lf .

(23)

An additional derivative with respect to temperature, following thermodynamic definitions, yields an equation for the apparent molar heat capacity Cp,f as: Cp, f = Cp,af + (3/2)AJ heG − 4RT 2{(m/m°)B J + (m/m°)2C J/21/2},

(24)

where Cp,af is the apparent molar heat capacity at infinite dilution, AJ is the Debye–Hu¨ckel coefficient for the heat capacity, and B J is obtained from: BJ = (12B G/1T 2 )p,m + (2/T )(1B G/1T )p,m ,

(25)

and in a like manner for C J. If experimental results are known as a function of temperature, a useful equation relating Dsol Hma and Cp,af is: Cp,af = C°p,2 (s) + (1Dsol Hma /1T )p ,

(26)

where C°p,2 (s) is the molar heat capacity of the pure solid solute. The pressure dependence for the thermodynamic functions is given by the volumetric properties which are the pressure derivatives of the Gibbs free energy. The pertinent equation for the apparent molar volume Vf is: Vf = Vfa + (3/2)AV heG + 4RT{(m/m°)B V + (m/m°)2C V/21/2 }.

(27)

Here, Vfa is the apparent molar volume at infinite dilution, Av is the Debye–Hu¨ckel coefficient for volume, and the superscript V denotes a derivative with respect to pressure. In terms of the pressure derivatives of the ion-interaction parameters, B V is given by: B V = bV(0) + bV(1) h1G + bV(2) h2G .

(28)

1334

H. F. Holmes and R. E. Mesmer

Our third virial coefficient for the excess Gibbs free energy, the quantity C in equations (2), (12), and (16), is the ion-interaction parameter C f usually tabulated from analyses of experimental results.(21) The above equations are specific for an electrolyte of the (2 − 1) charge type. PREVIOUS EXPERIMENTAL RESULTS

All of the thermodynamic results selected and used in the overall least-squares fit are listed in table 3. Table 3 is not intended to be an exhaustive listing of the available thermodynamic results for MgCl2 (aq), SrCl2 (aq), and BaCl2 (aq). Priority was given to covering the broadest possible range of temperature, pressure, and molality, and also to have a wide variety of types of experimental results. For sources using NaCl(aq) as the isopiestic standard, the results were recalculated using the Pitzer et al.(19) formulation for NaCl(aq). Except as noted, studies using other isopiestic standards were used without recalculation. In order to weight properly the various types of experimental results, we assigned an estimated standard error to each set of results, or where it was deemed advisable, to individual experimental points. In those cases in table 3 where a range of standard errors is indicated, the large errors usually apply to only a few results, normally outliers or points at the upper molality limit of the simple ion-interaction model (see section on results of the least-squares analysis). The three factors considered in making the error assignments were the reported experimental uncertainty, internal consistency of the particular set of results as measured by an isothermal fit with the ion-interaction model, and compatibility with other results. For a few points, it was obviously necessary, from the least-squares results, to assign standard errors larger than those listed in table 3. ION-INTERACTION MODEL

Equations for the thermodynamic properties of these electrolyte solutions were given in a previous section. In order that equations be useful, one must be able to calculate the required ion-interaction parameters at the desired temperatures and pressure. The selected empirical function of temperature and pressure f(T, p) is: f(T, p) = F0 + F1 (p/p°) + F2 (p/p°)2,

(29)

where p° = 0.1 MPa, and F0 , F1 , and F2 are functions of temperature only, and are given by: F0 = p1 + (1/2)p2 (T/T0 ) + (1/6)p3 (T/T0 )2 + (1/12)p4 (T/T0 )3 + (1/6)p5 (T/T0 )2{ln(T/T0 ) − (5/6)} + p6 {(T/2T0 ) + 3T 22 /2TT0 + (T2Tx /T )ln T } + p7 {2(TyT0 /T ) + 1}ln Ty ;

(30)

F1 = p8 + p9 (T0 /T ) + p10 (T/T0 ) + p11 (T/T0 )2 + T x− 1 p12 + T y− 1 p13 ;

(31)

F2 = p14 + p15 (T0 /T ) + p16 (T/T0 ) + p17 (T/T0 )2,

(32)

and

1335

MCl2 (aq) at high temperatures TABLE 3. Thermodynamic results for MCl2 (aq).a (m° = 1 mol·kg−1 ) Property b

Method c

T/K

f

iso

f

iso

383 523 298.15

f

iso

298.15

f

iso

298.15

f

iso

298.15

f

iso

298.15

f

iso

298.15

f

iso

298.15

f

iso

298.15

f

iso

298.15

f

iso

298.15

f

iso

298.15

f

iso

373

f

f.t.

Lf , Ddil Hm

cal

Ddil Hm

fcal

255 273 298.15 353 473 523 523

Ddil Hm

fcal

Ddil Hm

fcal

Ddil Hm

cal

273 473 298.15

Ddil Hm

cal

298.15

Ddil Hm

cal

298.15

Ddil Hm

cal

298.15

Dsol Hm

cal

298.15

Cp, f

fcal

Cp, f

cal

CP,f

cal

349 499 353 453 298.15

CP,f

cal

283 403

p/MPa

m/m°

M = Mg 0.42 sat f 3.95 0.1 0.11 0.13 0.1 1.41 2.96 0.1 0.39 2.09 0.1 0.08 2.76 0.1 1.14 5.92 0.1 1.10 2.42 0.1 0.43 2.00 0.1 0.11 2.02 0.1 0.46 5.11 0.1 0.31 3.80 0.1 0.23 1.15 sat 0.92 4.04 0.1 0.03 2.03 0.1 0.003 1.99 7.0 0.02 3.52 10.3 0.03 1.04 sat 0.002 5.43 0.1 0.19 5.90 0.1 0.19 1.21 0.1 0.0001 0.10 0.1 0.005 5.70 0.1 0.001 0.14 2.3 0.03 17.9 2.26 sat 0.22 0.90 0.1 0.46 5.11 sat 0.38 0.95

ss d

Reference

sf e

0.003 0.010 0.003

This work

1.051

22

0.360

0.003 0.020 0.003 0.010 0.003 0.025 0.003 0.010 0.003 0.010 0.003 0.015 0.003 0.010 0.003 0.030 0.003 0.010 0.003

23

0.908

24

1.230

25

0.757

26

0.951

27

1.074

28

1.128

29

1.073

30

0.943

31

1.089

32

0.434

0.003 0.080 0.005 0.016

33

1.271

34

0.737

h

35

0.774

h

14

1.030

h

15

1.134

h

36

0.963

h

37

0.836

h

38

0.802

h

39

1.177

h

40

0.903

h

40

0.750

4i 25 i 4i 25 i 4i 15 i 4i 35 i

12

0.901

41

0.708

30

0.932

42

0.930

1336

H. F. Holmes and R. E. Mesmer TABLE 3—continued

Property b

Method c

T/K

CP,f

fcal

0.6

CP,f

fcal

298.15 373 298.15

CP,f

fcal

298.15

0.1

Vf

v.t.

0.6

Vf

v.t.

297 372 298.15

Vf

v.t.

298.15

0.1

Vf

v.t.

0.1

Vf

m.f.

Vf

v.t.

Vf

v.t.

278 318 273 323 308 368 298.15

Vf

dil.

Vf

dil.

Vf

pycn.

f

iso

f

iso

f

iso

383 524 383 474 298.15

f

iso

298.15

f

iso

298.15

f

iso

298.15

ln g

e.m.f.

ln(g/gR )

e.m.f.

283 343 298.15

Dsol Hm

cal

Ddil Hm

cal

276 313 303

Cp, f

fcal

298.15

Cp, f

cal

298.15

Cp, f

fcal

298.15 373

298.15 348 323 473 298.15

p/MPa

0.1

0.1

0.1 100 0.1 0.1 40.3 0.1 2.03 0.1

m/m°

ss d

Reference

sf e

0.11 0.53 0.14 5.03 0.013 0.34 0.11 0.53 0.14 5.03 0.004 0.34 0.01 5.28 0.009 0.315 0.35 4.61 0.03 2.95 0.055 6.17 0.1 1.0 0.001 0.703

4i 15 i 4i 9i 5i 15 i 0.2 j

43

1.059

44

1.163

45

0.672

43

0.651

0.2 i

44

0.166

0.2 j 0.3 j 0.2 j 0.6 j 0.2 j 2.1 j 0.2 j

45

0.370

46

0.343

47

0.878

48

0.293

0.2 j 0.5 j 0.2 j 2.0 j 0.2 j 2.0 j 0.2 j 0.5 j

49

0.756

50

1.012

17

1.012

51

1.172

g

0.711

4

0.781

52

0.997

53

1.100

54

0.631

55

0.966

56

1.123

57

0.616

h

58

0.627

h

59

0.954

5i

45

0.555

5i 12 i 5i 20 i

60

0.837

43

0.965

M = Sr sat 0.41 5.54 sat 0.47 4.22 0.1 0.70 3.13 0.1 0.11 2.19 0.1 2.68 3.84 0.1 0.08 4.04 0.1 0.01 0.30 0.1 0.03 3.02 0.1 0.03 3.15 0.1 0.23 1.01 0.1 0.013 0.329 0.1 0.75 2.76 0.6 0.03 1.05

0.003 0.025 0.003 0.010 0.003 0.012 0.003 0.007 0.003 0.010 0.003 0.018 0.01 0.07 0.02 0.05

1337

MCl2 (aq) at high temperatures TABLE 3—continued Property b

Method c

T/K

Vf

v.t.

Vf

v.t.

297 372 298.15

f

iso

f

iso

f

iso

383 524 383 474 298.15

f

iso

298.15

f

iso

298.15

f

iso

373

f

iso

394

f

iso

318

f

iso

353

f

iso

298.15

f

iso

298.15

f

iso

298.15

f

v.p.

f

f.t.

f

v.p.

298.15 318 268 273 298.15

ln g

e.m.f.

ln g

e.m.f.

ln(g/gR )

e.m.f.

288 318 283 343 298.15

Dsol Hm

cal

Ddil Hm

cal

273 368 303

Ddil Hm

cal

298.15

Ddil Hm

cal

298.15

Cp, f

fcal

298.15

Cp, f

cal

298.15

p/MPa 0.6 0.1

m/m°

ss d

Reference

sf e

0.03 1.99 0.004 0.33

0.2 j 2.3 j 0.2 j 0.4 j

43

0.870

45

0.385

g

0.930

4

0.585

61

0.964

62

0.586

M = Ba sat 0.42 4.10 sat 0.49 3.62 0.1 0.09 1.78 0.1 0.85 1.79 0.1 0.05 1.80 sat 0.71 2.22 sat 0.50 3.00 0.1 0.50 1.60 0.1 0.50 2.21 0.1 0.36 1.76 0.1 0.53 1.70 0.1 0.50 1.62 sat 0.87 1.60 0.1 0.033 0.991 sat 0.31 1.79 0.1 0.01 1.80 0.1 0.04 0.30 0.1 0.03 1.65 0.1 0.001 0.012 0.1 0.20 1.09 0.1 0.0001 0.096 0.1 0.023 1.50 0.1 0.027 0.387 0.1 0.50 1.62

0.003 0.017 0.003 0.007 0.003 0.008 0.003 0.003 0.010 0.010

63

0.653

64

0.965

0.003

65

0.875

0.003 0.013 0.003 0.007 0.003 0.010 0.003 0.006 0.003 0.010 0.003 0.015 0.003 0.015 0.005 0.035 0.01 0.03 0.01 0.05 0.02

66

1.247

67

0.603

68

0.910

69

0.923

30

0.820

70

1.079

71

0.921

72

0.981

73

0.876

74

0.602

57

0.774

h

76

0.728

h

59

0.766

h

39

0.854

h

76

1.136

12 i 50 i 5i 12 i

45

0.337

30

0.328

1338

H. F. Holmes and R. E. Mesmer TABLE 3—continued

Property b

Method c

T/K

Cp, f

cal

298.15

0.1

Cp,af

cal

0.1

Vf

v.t.

273 373 298.15

0.1

298.15

0.1

Vf Vf

v.t.

288 413

p/MPa

0.1 20

ss d

m/m° 0.20 1.59 0.0 0.004 0.387 0.50 1.62 0.01 1.58

5i 25 i 10 i 0.2 j 0.6 j 0.8 j 1.5 j 0.2 j 2.0 j

Reference

sf e

77

0.430

75

0.268

45

0.887

30

1.016

78

0.827

a Two entries indicate a range of values. b Thermodynamic property explicitly used in the overall least-squares fit. c Experimental method: iso, isopiestic; e.m.f., electrochemical cell potential; v.p., vapor pressure; cal, calorimetric; fcal, flow calorimetric; v.t., vibrating tube; pycn., pycnometric; f.t., freezing temperature; dil., dilatometric; m.f., magnetic float. d Assigned standard error. e Standard deviation of fit from overall least-squares analysis. This is a dimensionless quantity since it is a ratio of the deviation and the assigned standard error. f Vapor pressure of pure water. g This work. h 0.02·Ddil Hm or 20 J·mol−1, whichever is greater. i J·K−1·mol−1. j cm3 ·mol−1

(where T1 = 647 K, T2 = 227 K, T0 = 1 K); Tx = (T − T2 )/T0 ;

(33)

and Ty = (T1 − T )/T0 .

(34)

Equations (29) through (34) are used for calculating values of b , b , and C. The quantity Vfa of equation (27) is defined as: (0)

(1)

Vfa = V1 + V2 (p/p°) + V3 (p/p°)2,

(35)

V(i = 1, 2, 3) = p18 + p19 (T0 /T ) + p20 (T/T0 ) + p21 (T/T0 )2 + T x− 1 p22 + T y− 1 p23 .

(36)

where a p, f

In a similar manner C

of equation (24) is defined as:

Cp,af = J0 − J1 (p/p°) − J2 (p/p°)2 − J3 (p/p°)3,

(37)

with J0 = p24 + p25 (T0 /T ) + p26 ln(T/T0 ) + p27 (T/T0 ) + p28 (T/T0 )2 + T x− 1 p29 + T y− 1 p30 ,

(38)

and J(i = 1, 2, 3) = 2p19 (T0 /T )2 + 2p21 (T/T0 ) + 2p22 (T/T0T x3 ) + 2p23 (T/T0T y3 ).

(39)

Equations (29) to (39) are consistent with the thermodynamic requirements relating the temperature and pressure derivatives of the various properties, and provide the basic temperature and pressure dependence necessary for the application of the

MCl2 (aq) at high temperatures

1339

ion-interaction model in describing the thermodynamic properties of MgCl2 (aq), SrCl2 (aq), and BaCl2 (aq). Refinements such as the use of b (2) and a2 , and the pressure and temperature dependence of a1 , a2 , and b(2), were determined during the least-squares calculations, and are described in the following section. RESULTS OF THE LEAST-SQUARES ANALYSIS

Isothermal and isobaric fitting of the ion-interaction model to each type of result was done at every temperature and pressure at which sufficient results were available. The isothermal fits gave a general indication of the temperature and pressure dependence of the ion-interaction parameters, provided a check on the suitability of the assigned standard errors, and indicated the precision of the results being examined. There is a limit to the range of m for which an acceptable fit of experimental results can be obtained from the simple ion-interaction model. The majority of the high-temperature results are our isopiestic measurements where the molality is limited by the solubility of the reference salt, NaCl. Previous work(1) has demonstrated that the ion-interaction model is quite adequate over this range of molalities. For those few cases, where necessary, our analysis has been limited to the molality range where the simple ion-interaction model gives an adequate fit. The osmotic coefficients of Rard and Miller for MgCl2 (aq)(23) and SrCl2 (aq)(54) at T = 298.15 K are examples where the molality range was limited. Rard and Miller(23,54) also found that it was not possible to apply the ion-interaction model to their entire set of results—an analysis which required a seven-term power series in molality. One can extend the range of the simple model by adding terms in higher powers of the molality, such as was done by Ananthaswamy and Atkinson(79) for CaCl2 (aq). We have chosen not to add terms, primarily because of the lack of suitable results at high molalities and elevated temperatures. Isothermal fits of the osmotic coefficients became noticeably less satisfactory with increasing temperature when using the normal value of 2.0 for a1 . Allowing a1 to assume an optimum value gave an acceptable fit of the isopiestic results at all temperatures and molalities used in the present work. Isopiestic results for CaCl2 (aq)(1) behaved in a similar manner when fit with the standard ion-interaction model. The simple linear dependence of a1 on temperature: a1 = 2 − 18.1·10−4·{(T/K) − 298.15},

(40)

used for CaCl2 (aq) was found to be adequate for representation of all of the thermodynamic results used in this paper. The derivative of a1 with respect to temperature must, of course, be included in equations (21) and (25). Enthalpy of dilution results for CaCl2 (aq) at the higher temperatures could not be fit satisfactorily with the standard ion-interaction model, particularly the results at low molalities.(1) This result had been anticipated,(80) with the behavior at low molalities attributed to ion association. The ion-interaction parameter b (2) was introduced by Pitzer and Mayorga(81) to describe systems which exhibit a moderate amount of ion association. For CaCl2 (aq),(1) the use of bL(2) {=(1b (2)/1T )p,m } gave

1340

H. F. Holmes and R. E. Mesmer

an acceptable fit of the high-temperature, low-molality enthalpy of dilution results. Pitzer and Mayorga(81) related the ion-interaction parameter b (2) to the association constant for the ion-pair complex. Frantz and Marshall(82) obtained, from their electrical conductivity results, equations for the ionization constants K of both MgCl+(aq) and CaCl+(aq) as a function of temperature and pressure. By using the inverse of their equation to define b (2) according to the relationship of Pitzer and Mayorga,(81) we obtained a good fit of the high-temperature, low-molality enthalpy of dilution results for CaCl2 (aq). However, an abbreviated expression: b (2) = −0.5/[exp{−16.5 + 7150(T0 /T )}],

(41) (82)

was found to work about as well as the equation of Frantz and Marshall for lg K as a function of temperature. Pitzer and Mayorga(81) also related a2 (necessary if one uses b(2) ) to the Debye–Hu¨ckel parameter Af . Since b(2) is important only at the upper end of our temperature range, we were able to obtain an acceptable fit of the CaCl2 (aq) results with a constant value a2 = 12, which was adopted for the present work along with equation (41) for b (2). The temperature- and pressure-independent parameters of the model equations were determined from a fit of the model to the thermodynamic results listed in table 3 using a general least-squares computer program.(83) The selected results listed in table 3 constitute, for each of the three solutes, a package of experimental points of various types, all of which were considered in the same computational cycle. Debye–Hu¨ckel limiting slopes were calculated as defined by Bradley and Pitzer,(84) which was also our source for the dielectric properties of water. The other necessary properties of water were calculated from the steam tables of Haar et al.(85) Final values of the adjustable parameters for equations (30) through (39) are listed in table 4 along with their associated standard fractional errors. The constants of table 4 were determined from a global least-squares fit of the model to the experimental results. Those parameters which had a standard fractional error q1 were set to zero. With the parameters of table 4, the overall standard deviation of fit to the experimental results is quite acceptable in every case when the results are weighted as 1/s 2, where s is the assigned standard error from table 3. As is obvious from table 3, not all types of results are available at elevated temperatures and pressures, and the scarcity is more noticeable in the cases of SrCl2 (aq) and BaCl2 (aq). Where necessary, the void regions have been estimated from our prior work(1) for CaCl2 (aq). The estimation was accomplished by setting missing parameters to the values obtained in the CaCl2 (aq) analysis as well as varying the minimum number necessary to obtain an acceptable fit of the available results. This procedure effectively fixed the temperature and pressure dependence of the excess Gibbs free energy at, or near, that of CaCl2 (aq). The fixed parameters are identified in table 4.

5. Discussion Extending the analysis to T q 523 K is beyond the scope of the present work. There are several reasons for this, an obvious one being the lack of experimental

1341

MCl2 (aq) at high temperatures

TABLE 4. Constants for calculating ion-interaction parameters and infinite dilution quantities as a function of temperature and pressure. Numbers in parentheses are the standard fractional errors of the associated constants F0 for b (0)

p1 p2 p3 p4 p5 p6 p7

3.8802·10−1 (0.038) 0 6.6587·10−5 (0.261) 5.9554·10−8 (0.209) −1.5733·10−5 (0.238) 0 −8.6449·10−4 (0.109)

p1 p2 p3 p4 p5 p6 p7

−9.4522·10−1 (0.033) 1.8955·10−2 (0.026) −1.3677·10−4 (0.027) 2.0617·10−7 (0.029) 0 0 −1.2527·10−3 (0.062)

p1 p2 p3 p4 p5 p6 p7

−3.8177·100 (0.089) 8.1964·10−2 (0.085) −1.2002·10−3 (0.054) 4.7629·10−8 a 1.6558·10−4 (0.052) −3.7158·10−3 (0.151) −3.5549·10−4 a F1 for b (0)

F0 for b (1) MgCl2 −4.5081·100 (0.200) 0 1.1284·10−2 (0.155) 4.7561·10−6 (0.142) −2.1698·10−3 (0.153) −2.4087·10−2 (0.160) 1.1906·10−2 (0.075) SrCl2 0 −1.3574·10−1 (0.006) 1.6459·10−2 (0.001) 6.5701·10−6 (0.004) −3.1030·10−3 a −2.0329·10−2 a 0 BaCl2 1.2605·10 (0.276) −7.3195·10−1 (0.106) 5.3197·10−2 (0.070) 1.8326·10−5 (0.068) −9.7531·10−3 (0.070) −4.3707·10−2 (0.175) 0 F1 for b (1)

F0 for C

−5.4822·10−2 (0.282) −4.8095·10−3 (0.153) 6.7948·10−4 (0.096) 2.3311·10−7 (0.096) −1.2578·10−4 (0.095) −1.1675·10−3 (0.074) 3.0632·10−4 (0.103) −1.3455·10−1 a 6.3445·10−3 (0.021) −3.0104·10−4 (0.034) −9.7650·10−8 (0.047) 5.4355·10−5 (0.036) −8.4515·10−5 a 0 −1.3455·10−1 a 0 2.6782·10−4 (0.031) 9.5156·10−8 (0.020) −5.0243·10−5 (0.028) −6.9843·10−4 (0.051) 0 F1 for C

p8 p9 p10 p11 p12 p13

1.1021·10−3 a −1.3361·10−1 (0.006) −3.0111·10−6 (0.007) 3.2621·10−9 (0.013) 2.3285·10−3 a −2.1508·10−2 a

MgCl2 0 0 1.0935·10−6 a −4.0084·10−9 a 0 0

−1.3801·10−6 (0.027) 0 0 5.9573·10−12 a 0 0

p8 p9 p10 p11 p12 p13

1.1021·10−3 (0.007) −1.3924·10−1 a −2.8663·10−6 a 2.9609·10−9 a 2.3285·10−3 a −2.1508·10−2 a

SrCl2 −1.4450·10−4 (0.254) 0 1.0935·10−6 a −4.0084·10−9 a 0 0

−5.9202·10−6 (0.193) 0 0 5.9573·10−12 a 0 0

p8 p9 p10 p11 p12 p13

1.1021·10−3 (0.003) −1.3924·10−1 a −2.8663·10−6 a 2.9609·10−9 a 2.3285·10−3 a −2.1508·10−2 t

BaCl2 1.8299·10−4 (0.076) 0 1.0935·10−6 a −4.0084·10−9 a 0 0

−5.4441·10−6 (0.199) 0 0 5.9573·10−12 a 0 0

1342

H. F. Holmes and R. E. Mesmer TABLE 4—continued F2 for b (0)

F2 for b (1) MgCl2 0 −1.8894·10−5 a 0 0

F2 for C

−5.5630·10−9 a 1.7685·10−6 a 0 −3.1060·10−14 (0.138)

p14 p15 p16 p17

0 0 −1.2534·10−10 a 5.2986·10−13 (0.035)

p14 p15 p16 p17

0 0 −1.2534·10−10 a 3.5462·10−13 a

p14 p15 p16 p17

0 0 −1.2534·10−10 a 3.5462·10−13 a

0 0 0 0

−5.5630·10−9 a 1.7685·10−6 a 0 0

J1 and V1

J2 and V2

J3 and V3

SrCl2 0 0 0 0

−5.5630·10−9 a 1.7685·10−6 a 0 0

BaCl2

p18 p19 p20 p21 p22 p23

4.1746·10 (0.032) −2.8044·103 (0.055) −1.0141·10−1 (0.038) 1.6676·10−4 (0.022) −7.4036·10 a −5.0495·103 a

MgCl2 −7.0869·10−2 (0.139) 7.4730·100 (0.125) 1.9279·10−4 (0.165) −2.3336·10−7 (0.169) 0 3.3394·100 (0.091)

p18 p19 p20 p21 p22 p23

5.4690·10 (0.003) −4.5899·103 a −1.3179·10−1 (0.004) 1.9530·10−4 a −7.4036·10 a −5.0495·103 a

−1.0185·10−1 b 1.0200·10 (0.005) 2.9008·10−4 a −3.8255·10−7 a 0 5.5942·100 (0.005)

0 0 1.8092·10−8 a 0 0 −1.9119·10−3 a

p18 p19 p20 p21 p22 p23

5.5273·10 (0.000) −4.5899·103 a −1.3385·10−1 (0.000) 1.9996·10−4 a −7.4036·10 a −5.0495·103 a

BaCl2 −1.0200·10−1 a 1.0027·10 a 2.9008·10−4 a −3.8255·10−7 a 0 5.7134·100 a

0 0 1.8092·10−8 a 0 0 −1.9119·10−3 a

J0 for MgCl2

J for SrCl2

J0 for BaCl2

−1.5852647·106 (0.000) 3.3506539·107 (0.001) 2.9894419·105 a −9.0334756·102 a 4.7696174·10−1 a −2.3914394·104 (0.063) −1.2043980·106 (0.002)

−1.5854892·106 (0.000) 3.3518997·107 (0.000) 2.9894419·105 a −9.0334756·102 a 4.7696174·10−1 a −2.3810424·104 a −1.1493557·106 (0.003)

−3.5139136·106 (0.002) 7.5242728·107 (0.013) 6.6033436·105 a −1.9616482·103 (0.010) 1.0038592·100 (0.019) −5.8230384·104 (0.136) −1.6408666·106 (0.037)

0 0 2.6766·10−9 a 0 0 −3.0824·10−4 a

SrCl2

p24 p25 p26 p27 p28 p29 p30

1343

MCl2 (aq) at high temperatures TABLE 4—continued

a b

H°s for MgCl2

H°s for SrCl2

H°s for BaCl2

−1.0798581·108 (0.001)

−1.0770419·108 (0.000)

−2.3693766·108 (0.015)

Set at value for CaCl2 from previous paper.(1) Set at this value.

results, particularly in the case of SrCl2 (aq) and BaCl2 (aq). Also, as we pointed out in the case of CaCl2 (aq),(1) speciation models will probably be required at temperatures above 523 K. Such models will require association constants and experimental results for very low molalities. It may be possible to derive such models from enthalpy of dilution studies, but they are not available at the present time. EXCESS GIBBS FREE ENERGY

Excess Gibbs free energies used in the present work consist of osmotic coefficients derived from isopiestic, vapor pressure, and freezing-temperature measurements, and, to a lesser extent, solute activities from measurements of electrochemical cell potentials. Based on the standard deviations of fit listed in table 3, one concludes that the model fit all of the excess Gibbs free energies essentially within the assigned standard errors. Osmotic coefficients from the high-temperature isopiestic results were fit about as well as the low-temperature results. Osmotic coefficients from our experimental isopiestic molalities at temperatures about 498 K and 523 K are plotted in figures 1, 2(a), and 2(b) for MgCl2 (aq), SrCl2 (aq), and BaCl2 (aq), respectively. Obviously, the model does an excellent job of reproducing the experimental results. Also, it is quite clear that the normal value of a1 = 2.0 does not produce an acceptable fit. This behavior is a common feature of our isopiestic results for the four alkaline-earth metal chlorides. The isopiestic results of Fangha¨nel and Grjotheim(33) for MgCl2 (aq) agree quite well with our measurements after recalculation of their results {to conform to our CaCl2 (aq) model(1)} which was necessary because they used the model of Ananthaswamy and Atkinson(79) for CaCl2 (aq) as an isopiestic standard. In the case of SrCl2 (aq), there are no excess Gibbs free energy results at temperatures between our isopiestic measurements and T = 298.15 K except the cell potential results of Longhi et al.,(56) which are for quite low molalities. There are some quite useful isopiestic results for BaCl2 (aq) at intermediate temperatures which include the early work of Soldano and his collaborators at temperatures about 373 K(64) and 394 K.(65) Patterson and his students have published two quite helpful isopiestic studies of BaCl2 (aq)(66,67) at temperatures of approximately 318 K and 353 K. The Italian school also has some cell potential results for BaCl2 (aq)(74) at low molalities and temperatures to about 343 K. The vapor pressure measurements of Liu and Lindsay(16) for MgCl2 (aq) at elevated temperatures could not be fit with the present model and an acceptable error of fit—a discrepancy which was noted in our earlier work.(3)

1344

H. F. Holmes and R. E. Mesmer

1.4

1.3

1.2

φ

1.1

1.0

0.9

0.8

0.7

0.6

0

1

2

3

4

–1

m/(mol·kg ) FIGURE 1. Osmotic coefficients f of MgCl2 (aq) against molality m. Upper plot at T = 498.22 K and T = 498.72 K; lower plot at T = 524.12 K. Lines are least squares fits of the ion-interaction model to experimental results (points). – – –, a = 2.0; ——, a = optimum value.

In figure 3 we compare experimental osmotic coefficients from our previous study(3) of MgCl2 (aq) and calculated values from the present model for the same temperatures. Quite obviously, there are systematic differences between the previous values, which are for a single sample, and the present model, which combines results

1345

MCl2 (aq) at high temperatures

1.0 (a)

(b)

0.9

φ

0.8

0.7

0.6

0.5

0

1

2

3

4

5

0

1

2

3

4

–1

m/(mol·kg ) FIGURE 2. Osmotic coefficients f of: (a) SrCl2 (aq); and (b) BaCl2 (aq) aqainst molality m. Upper plots at T = 498.22 K and T = 498.72 K; lower plots at T = 524.20 K and T = 524.12 K. Lines are least-squares fits of the ion-interaction model to the experimental results (points). – – –, a = 2.0; ——, a = optimum value.

for several samples. The differences are not large, amounting to about 3 per cent at most, and increase with increasing molality. We cannot account for the differences, but place more confidence in our new results for the above reason. We wish to emphasize that the previous results(3) were not included in the present least-squares analysis. It is instructive to consider the recent isopiestic molalities measured by Brendler and Voigt(6) at T = 428.7 K in which they used MgCl2 (aq) as an isopiestic standard; more specifically, they used the model of Valyashko et al.(7) to generate osmotic coefficients for MgCl2 (aq). Their results were not considered in the present work, primarily because there is very little overlap of the molality ranges used in the two investigations. In table 5, we have listed osmotic coefficients for those molalities which are approximately common to the two studies. The osmotic coefficients of table 5 were calculated from the two models for MgCl2 (aq), and also from isopiestic molalities of CaCl2 (aq), LiCl(aq), and CsCl(aq) using our models(1,5) as isopiestic standards for the latter three electrolytes. It is quite clear that the osmotic coefficients generated from the model of Valyashko et al.(7) are substantially larger in all cases. Additional information concerning this point can be seen in figure 3 which contains

1346

H. F. Holmes and R. E. Mesmer

2.0

1.8

1.6

φ

1.4

1.2

1.0

0.8

1.0

1.5

2.5

2.0

3.0

3.5

–1

m/(mol·kg ) FIGURE 3. Osmotic coefficients f of MgCl2 (aq) aqainst molality m. Symbols are for previously published(3) experimental results: w, T = 382.0 K; q, T = 413.8 K; r, T = 445.4 K; t, T = 474.0 K. The solid and dashed lines were calculated from the models of the present work and that of Valyashko et al.,(7) respectively, for the experimental temperatures and pressures.

f against molality curves generated from the model of Valyashko et al.(7) for the four temperatures used in our earlier study of MgCl2 (aq). At the lowest experimental temperature, there is agreement between the two models although they appear to be diverging at the low and high molalities. At the other three temperatures, values

1347

MCl2 (aq) at high temperatures

TABLE 5. Osmotic coefficients f of MgCl2 (aq) at T = 428.67 K for isopiestic molalities from Brendler and Voigt.(6) (m° = 1 mol·kg−1 ) m/m°

fa

fb

fc

2.7563 2.9509 3.1304 3.3342 3.3784 2.8572 4.1167

1.450 1.517 1.579 1.650 1.665 1.831 1.921

1.389 1.450 1.508 1.574 1.588 1.745 1.831

1.377

fd

fe 1.453 1.486

1.567 1.583 1.839

1.889

1.697 1.762

Calculated from model of Valyashko et al.(7) Calculated from present model. c From isopiestic molalities using CaCl2 (aq)(1) as a standard. d From isopiestic molalities using LiCl(aq)(5) as a standard. e From isopiestic molalities using CsCl(aq)(5) as a standard. a

b

from their model are much higher than those from the present work, with the difference approaching 10 per cent for the higher molalities. Valyashko et al.(7) used the isopiestic results of Rard and Miller(23) at T = 298.15 K in their analysis to obtain a model for MgCl2 (aq). At moderate molalities, their model gives a good fit of the Rard and Miller results, but is not so good for the higher molalities reported by Rard and Miller.(23) A possible explanation is that, at the higher temperatures, Valyashko et al.(7) relied heavily on vapor pressure measurements at quite high molalities. They used the basic three-parameter ion-interaction model for ionic strengths exceeding 36 mol·kg−1, which is a somewhat surprising application, but not without precedent.(8) In figure 4, we compare the osmotic coefficients of aqueous solutions of the four alkaline-earth metal chlorides as a function of m 1/2 for three temperatures and two pressures. {The curves for CaCl2 (aq) in figures 4 to 8 were calculated from our model in the preceding paper.(1)} As is true for all of the electrolytes we have studied with the isopiestic technique, the order is not a function of temperature, i.e. Mg q Ca q Sr q Ba for all temperatures. The curves of figure 4 are constrained quite well with experimental results over the whole range of temperature at saturation pressure or at p = 0.1 MPa for T Q 373.15 K. The only exception is for SrCl2 (aq), where the only excess Gibbs free energies below T = 298.15 K are from some cell potentials for dilute solutions at T = 283 K.(56) The osmotic coefficient is not very dependent on pressure, but the pressure dependence does increase noticeably at higher temperatures. Experimentally, the pressure coefficient of f is rather well determined for MgCl2 (aq) and CaCl2 (aq) at all temperatures, whereas, for SrCl2 (aq) and BaCl2 (aq), it must be estimated at temperatures above 373 K and 413 K, respectively. Figure 5 is quite similar to figure 4 except that the thermodynamic function involved is the activity coefficient (as ln g). The same trends and comments also apply to figure 5, as it is essentially a Gibbs–Duhem integration of figure 4, since there were very few activity coefficient results considered in the least-squares analysis,

1348

H. F. Holmes and R. E. Mesmer

2.5

(a)

(b)

(c)

(d)

2.0

1.5

φ

1.0

0.5 2.5

2.0

1.5

1.0

0.5 0.0

0.5

1.0

1.5

0.0

0.5

1.0

1.5

2.0

{m/(mol·kg–1)}1/2 FIGURE 4. Calculated osmotic coefficients f of (a) MgCl2 (aq), (b) CaCl2 (aq), (c) SrCl2 (aq), and (d) BaCl2 (aq) against m 1/2. Points are for identification only. W, Q, R: p = 40 MPa. w, q, r: at saturation pressure. From top to bottom the temperatures are 300 K, 400 K, and 500 K, respectively.

and most of those were for quite dilute solutions. One slight difference between figures 4 and 5 is that the pressure coefficient of ln g is noticeably larger than the same quantity for f.

1349

MCl2 (aq) at high temperatures

2 (a)

(b)

5.0

1

2.2

1.0

0

0.5 –1 0.2 –2

–3

γ

ln γ

0.1

(d)

(c)

5.0

1 2.2

0

1.0 0.5

–1 0.2 –2 0.1 –3 0.0

0.5

1.0

1.5

0.0

0.5

1.0

1.5

2.0

{m/(mol·kg–1)}1/2 FIGURE 5. Calculated osmotic coefficients (as ln g) of (a) MgCl2 (aq), (b) CaCl2 (aq), (c) SrCl2 (aq), and (d) BaCl2 (aq) against m 1/2. Points are for identification only. W, Q, R: p = 40 MPa. w, q, r: at saturation pressure. From top to bottom the temperatures are 300 K, 400 K, and 500 K, respectively.

1350

H. F. Holmes and R. E. Mesmer

100 (a)

(b)

(c)

(d)

80

60

40

L φ /(kJ·mol–1)

20

0

80

60

40

20

0 0.0

0.5

1.0

1.5

0.0

0.5

1.0

1.5

2.0

{m/(mol·kg–1)}1/2 FIGURE 6. Calculated apparent molar enthalpy Lf of (a) MgCl2 (aq), (b) CaCl2 (aq), (c) SrCl2 (aq), and (d) BaCl2 (aq) against m 1/2. Points are for identification only. W, Q, R: p = 40 MPa. w, q, r: at saturation pressure. From bottom to top the temperatures are 300 K, 400 K, and 500 K, respectively.

1351

MCl2 (aq) at high temperatures

100 (a)

(b)

(c)

(d)

0

–100

–200

Cp,φ /(J·K –1·mol–1)

–300

–400

–500

0

–100

–200

–300

–400

–500 0.0

0.5

1.0

1.5

0.0

0.5

1.0

1.5

2.0

{m/(mol·kg–1)}1/2 FIGURE 7. Calculated apparent molar heat capacity Cp,f of (a) MgCl2 (aq), (b) CaCl2 (aq), (c) SrCl2 (aq), and (d) BaCl2 (aq) against m 1/2. Points are for identification only. W, Q, R: p = 40 MPa. w, q, r: at saturation pressure. w and W, T = 300 K; q and Q, T = 400 K; r and R, T = 500 K.

1352

H. F. Holmes and R. E. Mesmer

60 (a)

(b)

(c)

(d)

40

20

0

–20

V φ /(cm3·mol–1)

–40

–60

–80

40

20

0

–20

–40

–60

–80 0.0

0.5

1.0

1.5

0.0

0.5

1.0

1.5

2.0

{m/(mol·kg–1)}1/2 FIGURE 8. Calculated apparent molar volume Vf of (a) MgCl2 (aq), (b) CaCl2 (aq), (c) SrCl2 (aq), and (d) BaCl2 (aq) against m 1/2. Points are for identification only. W, Q, R: p = 40 MPa. w, q, r: at saturation pressure. w and W, T = 300 K; q and Q, T = 400 K; r and R, T = 500 K.

MCl2 (aq) at high temperatures

1353

ENTHALPY

In the case of MgCl2 (aq) there are several sets of enthalpy of dilution results which, as can be seen in table 3, cover nearly the entire range of temperature, pressure, and molality. Most of the enthalpy results show acceptable agreement with each other and with other functions, although a few individual results were assigned quite large standard errors. Enthalpy of dilution results for SrCl2 (aq) and BaCl2 (aq) are quite sparse and clustered near T = 300 K. There is one set of enthalpy of solution results listed in table 3 for each of these three electrolytes. However, with the exception of the Mischenko and Stagis(58) study of SrCl2 (aq), the molalities are so dilute that the results are useful primarily for determining functions at infinite dilution. Apparent molar enthalpies of aqueous solutions of the four alkaline-earth metal chlorides are compared in figure 6 at three temperatures and two pressures. There are two obvious features of the plots in figure 6. The apparent molar enthalpies of these four salts are quite similar, having roughly the same molality dependence and value. Secondly, the pressure coefficient of Lf , which is almost insignificant at T = 300 K, becomes quite large (about 15 per cent) at T = 500 K. One should keep in mind that there are no high-temperature enthalpy results for SrCl2 (aq) and BaCl2 (aq), and that these regions of the plots in figure 6 have been estimated from the CaCl2 (aq) model.(1) HEAT CAPACITY

The results for MgCl2 (aq) from the University of Delaware(12) form a comprehensive set covering a range of molalities, pressures, and temperatures with the higher temperatures being above the upper limit of the present work. Unfortunately, no such set exists for either SrCl2 (aq) or BaCl2 (aq), with the overwhelming majority of heat capacity results concentrated at, or near, T = 298.15 K and p 1 0.1 MPa. The heat capacity measurements of Saluja and LeBlanc,(43) which are for MgCl2 (aq) and SrCl2 (aq), cover the temperature range from 298.15 K to 373 K. Although Saluja and LeBlanc’s upper molality limit is only about 1 mol·kg−1, their results are especially useful for SrCl2 (aq). Figure 7 is a comparison of the apparent molar heat capacities of aqueous solutions of these four electrolytes for the same molalities, temperatures, and pressures used in figures 4 to 6. The extrapolations to Cp,af are not shown for the plots at T = 500 K. In contrast to the excess Gibbs free energy and the apparent enthalpy, the pressure coefficient of Cp, f is not small at any temperature. The pressure dependence does increase with temperature, but not as much proportionally as shown in figures 4 to 6. In general, the molality dependence of Cp, f increases with temperature, but this is probably related to the fact that the plots at high temperatures start at much lower values of Cp,af . Except for the region near T = 300 K and p = 0.1 MPa, the plots for BaCl2 (aq) are entirely estimated, although they are constrained somewhat by the temperature dependence of f. In addition, the values of Cp,af for T = 273 K to T = 373 K from Criss and Cobble(75) were used in the least-squares analysis.

1354

H. F. Holmes and R. E. Mesmer

VOLUME

In contrast to the situation for CaCl2 (aq), there are no extensive sets of volumetric results for these three members of the alkaline-earth family for the entire temperature range of the present study. The most extensive set of results, from Puchalska and Atkinson’s study of BaCl2 (aq),(78) covers the temperature range of only 303 K to 413 K with reasonably complete coverage of pressure up to, and including, 20 MPa. Millero and his co-workers(46–48) obtained comprehensive results for the (p, Vm , T ) properties of MgCl2 (aq) over the temperature range of 273 K to 373 K for pressures up to 100 MPa. Ellis(17) published apparent molar volumes for aqueous solutions of all the alkaline-earth metal chlorides at temperatures through 473 K. Unfortunately, Ellis’s results for SrCl2 (aq) and BaCl2 (aq) do not mesh acceptably with the other thermodynamic results listed in table 3. Saluja and LeBlanc’s results for the apparent molar volumes through T = 372 K are the only acceptable volume results for SrCl2 (aq) at temperatures other than T = 298.15 K. Calculated apparent molar volumes for aqueous solutions of the four common alkaline-earth metal chlorides are plotted in figure 8 for the conditions and molalities used in figures 4 to 7. The apparent molar volumes have some traits in common with the apparent molar heat capacities, i.e. the pressure coefficient is not negligible at any temperature, and increases with increasing temperature as does the dependence

TABLE 6. Activity coefficients (as ln g) of MgCl2 (aq). (m° = 1 mol·kg−1 ) T/K:

273.15

298.15

323.15

373.15

423.15

473.15

523.15

m/m° 0.1 0.2 0.5 1.0 2.0 3.0 4.0

−0.6091 −0.6819 −0.6839 −0.4822 0.1957 1.0495 2.0083

−0.6407 −0.7217 −0.7403 −0.5647 0.0537 0.8348 1.7065

−0.6829 −0.7753 −0.8158 −0.6713 −0.1194 0.5846 1.3661

ln g (p = psat ) −0.7913 −0.9120 −1.0059 −0.9317 −0.5117 0.0572 0.6946

−0.9355 −1.0916 −1.2503 −1.2573 −0.9756 −0.5320 −0.0141

−0.1336 −1.3339 −1.5707 −1.6722 −1.5467 −1.2390 −0.8456

−1.4326 −1.6900 −2.0233 −2.2362 −2.2880 −2.1281 −1.8647

0.1 0.2 0.5 1.0 2.0 3.0 4.0

−0.6008 −0.6704 −0.6652 −0.4539 0.2398 1.1068 2.0770

−0.6326 −0.7109 −0.7237 −0.5405 0.0892 0.8789 1.7571

ln g −0.6744 −0.7642 −0.7791 −0.6475 −0.0849 0.6266 1.4134

(p = 20 MPa) −0.7805 −0.9190 −0.8984 −1.0713 −0.9861 −1.2218 −0.9036 −1.2177 −0.4712 −0.9193 0.1062 −0.4638 0.7495 0.0625

−1.1046 −1.2984 −1.5226 −1.6084 −1.4597 −1.1354 −0.7302

−1.3775 −1.6223 −1.9342 −2.1244 −2.1466 −1.9677 −1.6915

0.1 0.2 0.5 1.0 2.0 3.0 4.0

−0.5931 −0.6597 −0.6474 −0.4264 0.2832 1.1630 2.1436

−0.6252 −0.7009 −0.7077 −0.5166 0.1249 0.9226 1.8056

ln g −0.6666 −0.7539 −0.7829 −0.6233 −0.0494 0.6690 1.4589

(p = 40 MPa) −0.7705 −0.9038 −0.8856 −1.0522 −0.9662 −1.1932 −0.8739 −1.1755 −0.4272 −0.8567 0.1585 −0.3884 0.8050 0.1441

−1.0778 −1.2651 −1.4750 −1.5411 −1.3631 −1.0197 −0.6037

−1.3229 −1.5549 −1.8430 −2.0048 −1.9875 −1.7840 −1.4948

1355

MCl2 (aq) at high temperatures TABLE 7. Activity coefficients (as ln g) of SrCl2 (aq). (m° = 1 mol·kg−1 ) 273.15

298.15

323.15

0.1 0.2 0.5 1.0 2.0 3.0 4.0

−0.6527 −0.7594 −0.8506 −0.7914 −0.4106 0.1229 0.7375

−0.6739 −0.7802 −0.8655 −0.7984 −0.4173 0.0937 0.6617

−0.7082 −0.8197 −0.9108 −0.8509 −0.4948 −0.0275 0.4759

−0.8090 −0.9426 −1.0711 −1.0594 −0.8045 −0.4595 −0.1054

0.1 0.2 0.5 1.0 2.0 3.0 4.0

−0.6452 −0.7480 −0.8286 −0.7536 −0.3473 0.2046 0.8312

−0.6664 −0.7694 −0.8454 −0.7645 −0.3618 0.1636 0.7392

ln g −0.7003 −0.8086 −0.8907 −0.8174 −0.4404 0.0404 0.5506

0.1 0.2 0.5 1.0 2.0 3.0 4.0

−0.6379 −0.7371 −0.8076 −0.7176 −0.2867 0.2831 0.9213

−0.6594 −0.7592 −0.8264 −0.7323 −0.3090 0.2298 0.8126

−0.6930 −0.7982 −0.8717 −0.7855 −0.3885 0.1049 0.6210

T/K:

373.15

423.15

473.15

523.15

−0.9530 −1.1213 −1.3119 −1.3778 −1.2615 −1.0539 −0.8434

−1.1522 −1.3651 −1.6334 −1.7913 −1.8244 −1.7453 −1.6522

−1.4458 −1.7121 −2.0676 −2.3220 −2.5001 −2.5317 −2.5262

(p = 20 MPa) −0.7988 −0.9370 −0.9290 −1.1012 −1.0482 −1.2812 −1.0225 −1.3314 −0.7458 −1.1912 −0.3868 −0.9679 −0.0256 −0.7491

−1.1238 −1.3301 −1.5845 −1.7238 −1.7297 −1.6331 −1.5302

−1.3913 −1.6452 −1.9791 −2.2104 −2.3588 −2.3732 −2.3594

ln g (p = 40 MPa) −0.7894 −0.9227 −0.9165 −1.0829 −1.0267 −1.2525 −0.9871 −1.2866 −0.6889 −1.1214 −0.3165 −0.8820 0.0509 −0.6549

−1.0980 −1.2982 −1.5387 −1.6582 −1.6339 −1.5175 −1.4036

−1.3381 −1.5800 −1.8921 −2.0974 −2.2102 −2.2032 −2.1787

ln g (p = psat )

m/m°

on molality. Most of the plots at high temperatures for MgCl2 (aq), SrCl2 (aq), and BaCl2 (aq) have been estimated from the parameters of the CaCl2 (aq) model.(1) This paper presents models for the thermodynamic properties of MgCl2 (aq), SrCl2 (aq), and BaCl2 (aq) based on the ion-interaction treatment of Pitzer(2) which is a realistic representation of the available experimental results for these three salts. Based on the previous model for CaCl2 (aq),(1) for which much more extensive results are available, estimates for the thermodynamic properties of the other three systems can be obtained in those regions of temperature and pressure where no experimental values exist. Although we believe these estimates to be reasonable and useful, there are void regions where the model would clearly benefit from reliable experimental results. As a convenience for the reader and an aid in checking calculations, tables 6, 7, and 8 are abbreviated tabulations of the stoichiometric activity coefficients of MgCl2 (aq), SrCl2 (aq), and BaCl2 (aq), respectively, as a function of temperature, pressure, and molality. The authors greatly appreciate Drs H. R. Corti and J. M. Simonson making their experimental enthalpy of dilution results for MgCl2 (aq) available to us prior to publication. We also appreciate discussions with Drs D. A. Palmer and D. J. Wesolowski concerning their studies of the hydrolysis of MgCl2 (aq) at elevated

1356

H. F. Holmes and R. E. Mesmer TABLE 8. Activity coefficients (as ln g) of BaCl2 (aq). (m° = 1 mol·kg−1 ) 273.15

298.15

323.15

0.1 0.2 0.5 1.0 2.0 3.0 4.0

−0.6861 −0.8103 −0.9395 −0.9542 −0.8219 −0.6836 −0.6116

−0.7022 −0.8238 −0.9434 −0.9428 −0.7818 −0.6179 −0.5252

−0.7322 −0.8580 −0.9834 −0.9901 −0.8422 −0.6926 −0.6174

−0.8232 −0.9680 −1.1293 −1.1829 −1.1124 −1.0277 −1.0100

0.1 0.2 0.5 1.0 2.0 3.0 4.0

−0.6746 −0.7949 −0.9184 −0.9274 −0.7824 −0.6257 −0.5291

−0.6906 −0.8088 −0.9237 −0.9192 −0.7493 −0.5712 −0.4580

0.1 0.2 0.5 1.0 2.0 3.0 4.0

−0.6633 −0.7801 −0.8983 −0.9023 −0.7456 −0.5712 −0.4500

−0.6794 −0.7944 −0.9052 −0.8974 −0.7197 −0.5282 −0.3952

T/K:

373.15

423.15

473.15

523.15

−0.9620 −1.1397 −1.3609 −1.4871 −0.5274 −1.5238 −1.5652

−1.1650 −1.3892 −1.6907 −1.9067 −2.0749 −2.1578 −2.2542

−1.4654 −1.7470 −2.1422 −2.4573 −2.7586 −2.9223 −3.0592

ln g −0.7201 −0.8424 −0.9633 −0.9661 −0.8100 −0.6471 −0.5524

(p = 20 MPa) −0.8084 −0.9413 −0.9494 −1.1140 −1.1053 −1.3279 −1.1541 −1.4472 −1.0744 −1.4754 −0.9758 −1.4564 −0.9384 −1.4770

−1.1318 −1.3484 −1.6384 −1.8436 −1.9956 −2.0611 −2.1360

−1.4065 −1.6745 −2.0496 −2.3477 −2.6286 −2.7752 −2.8936

ln g −0.7085 −0.8276 −0.9442 −0.9439 −0.7805 −0.6052 −0.4918

(p = 40 MPa) −0.7945 −0.9221 −0.9318 −1.0901 −1.0826 −1.2968 −1.1269 −1.4089 −1.0382 −1.4244 −0.9264 −1.3895 −0.8701 −1.3891

−1.1008 −1.3103 −1.5889 −1.7827 −1.9164 −1.9622 −2.0135

−1.3477 −1.6025 −1.9575 −2.2373 −2.4941 −2.6194 −2.7145

ln g (p = psat )

m/m°

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(Received 22 March 1996; in final form 31 May 1996)

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