Thermodynamics of Electrolytes. 13. Ionic Strength ...

Journal of Solution Chemistry, Vol. 28, No. 4, 1999

Thermodynamics of Electrolytes. 13. Ionic Strength Dependence of Higher-Order Terms; Equations for CaCl2 and MgCl2 Kenneth S. Pitzer,1,4 Peiming Wang,1 Joseph A. Rard,2,* and Simon L. Clegg3 Received December 16, 1998 While the original ion-interaction (Pitzer) equations of 1973 were adequate for many electrolytes to the limit of solubility, additional terms are needed for some systems of large solubility. A simple pattern of ionic-strength dependence is proposed for third, fourth, and higher virial coefficients. It is found to be very effective in representing the complex behavior of CaCl2(aq) as well as that of MgCl2(aq) at 25°C. Equations without ion association are presented for each system that are valid for the full range to 11.0 and 5.9 mol-kg -1 , respectively, as well as simpler equations for limited molality ranges. KEY WORDS: Electrolyte solutions; osmotic coefficient; activity coefficient; thermodynamics; calcium chloride; magnesium chloride.

1. INTRODUCTION The primary advance in papers 1 and 2 of this series(1,2) arose from the recognition and accurate representation of the ionic-strength dependence of

1

Department of Chemistry and Lawrence Berkeley National Laboratory, University of California, Berkeley, California 94720-1460. Current address for Peiming Wang: OLI Systems, Inc., 108 American Road, Morris Plains, New Jersey 07950. 2 Geosciences and Environmental Technologies, Earth and Environmental Sciences Directorate, Lawrence Livermore National Laboratory, University of California, Livermore, California 94550. 3 School of Environmental Sciences, University of East Anglia, Norwich NR4 7TJ, U.K. 4 Deceased December 26, 1997. f We dedicate this paper to the memory of Kenneth S. Pitzer in recognition of his many invaluable contributions to solution chemistry.

265 0095-9782/99/0400-0265$16.00/0 C 1999 Plenum Publishing Corporation

266

Pitzer, Wang, Rard, and Clegg

the second virial coefficient in the low molality range. Basic theory for electrolyte solutions(3) provides for ionic-strength dependence of the various virial coefficients. However, for most systems and with accurate second virials, the third virial coefficient is so small that any ionic-strength dependence is negligible. Thus, the ion-interaction (Pitzer) equations were presented with third virial coefficients independent of ionic strength.(1,2,4,5) They are, of course, dependent on pressure and temperature. This was found satisfactory for mixed as well as pure electrolytes(4) and even extending to solid saturation molalities, in most cases, in the extensive investigations of Harvie and Weare(6) and of Filippov and associates.(7) Subsequently, fourth and higher virial coefficients were added for electrolytes extending to very high molality. Examples are CdCl2, ZnCl2, LiCl, and CaCl2. For the CdCl2, ZnCl2, and LiCl, these treatments were quite satisfactory,(8,9) but for the important case of CaCl2, there are now very accurate data(10) and the 1985 fit of Ananthaswamy and Atkinson (11) with virial coefficients through the sixth order, although good, can now be improved. This paper presents and tests for CaC!2(aq) and MgCI2(aq) a simple formulation of ionic-strength dependent terms extending to any order. Recently, Archer(12) introduced an ionic-strength term at the third virial level for NaBr, which is soluble to 9 mol-kg - 1 at 25°C. He also found that this addition improved the fit to the very accurate data for NaCl(aq).(13) Archer's extension with an ionic-strength dependent third virial term was generalized by Clegg, Rard, and Pitzer(14) to self-associated electrolytes and to electrolyte mixtures. They applied these equations to solutions of H2SO4(aq), which contain equilibrium mixtures of H+, SO2-, and HSO4-. Their model provides a very accurate representation of the available osmotic and activity coefficients, enthalpies of dilution, and heat capacities to 6.1 mol-kg -1 . The degrees of dissociation of the HSO4 ion calculated with this model are in good agreement with experimental values from spectroscopic measurements. Rard and Clegg(10) used similar self-association models for CaCl2(aq), which include CaCl+ ion pairs. There is ample evidence for ion pair formation above about 5 mol-kg-1 in these solutions,(15) but neutron diffraction data(16) show no significant association at 4.5 mol-kg -1 . Rard and Clegg(10) obtained an accurate model for CaCl2(aq) valid to 8.0 mol-kg -1 , and, by including additional virial terms, a second model valid over the full molality range to 11.0 mol-kg -1 . However, the association constant K°(CaCl+) is poorly known at 25°C, in contrast to the more precisely known value of K(HSO4). This uncertainty for K°(CaCl+) gives rise to large uncertainties in the predicted degree of dissociation of CaCl+ and the corresponding ionic activity coefficients for CaCl2(aq) solutions, which then depend strongly on the assumed value of K°(CaCl+) and on whether higherorder virial terms are included in the model. Both models presented by Rard and Clegg yield very accurate values for the stoichiometric quantities c and G±, and thus their equations are

Ionic Strength Dependence

267

satisfactory when CaCl2(aq) is used as an isopiestic reference standard or to calculate the emf of a reversible cell. However, use of distorted values of ionic activities can cause difficulties when the thermodynamic properties of mixed electrolytes containing CaCl2 are being analyzed, since any anomalous predictions of the extent of formation of CaCl+ must be compensated for the values of the ionic activity coefficients of the other electrolytes. For such mixture calculations, there are advantages in having an accurate thermodynamic model in which CaCl2(aq) is treated as a fully dissociated electrolyte. There are alternate forms of expression for an ionic strength dependence of the third virial coefficient—as there were for the second virial term.(l) We have considered alternate forms not only for the third virial but also for the fourth and higher terms. A form is selected that retains the relatively simple expressions of the second (virial) coefficient level for all higher coefficients. At the third virial level, its contribution is very similar to that of Archer's expression. With reoptimization of all parameters, the fit for NaBr and NaCl should remain excellent, but there is no reason to change Archer's equations for these cases. It is, for cases using fourth or higher coefficients, that the advantages of our new form become important. These equations are used here to represent the osmotic coefficients of CaCl2(aq) and MgCl2(aq) at 25°C, and good quality fits were obtained for both systems. These same equations should be applicable to even more highly soluble electrolytes. However, for miscible or nearly miscible systems, e.g. H2SO4(aq) and HNO3(aq), where the molality can approach or become infinite, equations based on the mole fraction scale are more suitable.(5) 2. EQUATIONS FOR PURE ELECTROLYTES We first recall the expression for the second virial coefficient for the osmotic coefficient of a pure electrolyte

with possible additional terms in B(3), a3, etc.; in addition, B(2) is often zero. If we define

Eq. (1) becomes

268


Then the second virial expression for the excess Gibbs energy is

For the natural logarithm of the mean activity coefficient, one has

Note that if a = 0, x = 0, exp(-x) = 1, and limx->0 g(x) = 1; thus, one can write the ionic-strength dependence of each term in the same form. Then the B(0) term may be viewed as having the form B(0) exp(-AoI1/2), but with a0 = 0. For higher virial coefficients, we generalize Eqs. (4) and (5) for any one term in the nth virial for the excess Gibbs energy to

Then appropriate differentiation yields, for the osmotic coefficient

and for the natural logarithm of the mean activity coefficient

Thus, for the fourth virial, if there are three terms, one has for the excess Gibbs energy and with xDO = 0

with

Then the fourth virial for the osmotic coefficient is


269

and for the natural logarithm of the mean activity coefficient is

A complete expression through the sixth virial for the osmotic coefficient is then

with |zMzx|fP the Debye-Huckel term. If there is no ionic-strength dependency for the third and higher virials, this reduces to the expression adopted for CaCl2(aq) by Ananthaswamy and Atkinson.(10) We now consider CMX, DMX, ... to have possible ionic strength dependences with

For the mean activity coefficient, the corresponding equations are

270


At the third virial level, the only difference from the Archer term for the osmotic coefficient is the use of the first power of I instead of I1/2 in xCL. However, for the Gibbs energy and the activity coefficient, the expressions corresponding to Eqs. (5) and (17b) are more complex for the Archer term and progressively much more complex for the fourth or higher virials. At each level, the ionic-strength-independent term, B(0), C(0), D(0), etc., is obtained with a = 0, x = 0. It is only at the second virial level that there is useful guidance from basic theory for the A's. The other nonzero a values are chosen for optimum fit to experimental data. While purely empirical, the higher order terms with nonzero a are very useful in fitting experimental data since their contribution is relatively localized in a limited molality range. 3. EQUATIONS FOR MIXED ELECTROLYTES While no new, detailed, treatments for particular mixed system are presented here, a few comments are appropriate. Above the third virial level, a complete expression with all possible mixing terms becomes very complex. It seems best to follow the method introduced by Filippov etal.(8) for mixtures, including CdCl2, and used by Anstiss and Pitzer(9) for ZnCl2-NaCl-H2O. For these 2-1 electrolytes, the fourth and fifth virial terms of the M 2+ , Clelectrolyte should be dominated by (M 2+ )(Cl - ) 3 and (M 2+ )(Cl - ) 4 interactions (or complexes). It is then assumed that the excess Gibbs energy from the fourth and fifth terms is

Comparison with Eqs. (7), (8), and (10) then yields, for the D and E terms

For the contribution of the (M 2+ )(Cl - ) 3 and (M 2+ )(Cl - ) 4 components to the higher virial terms of the osmotic and activity coefficients, one finds

Then, for mixtures, terms of high order such as O(Na, Cd, 3Cl) are added if


271

needed and the resulting equations successfully represent mixed electrolyte properties.(8,9) At the third virial level, the situation for 1-1 electrolytes is straightforward. The equations in general use(4-6) remain satisfactory if the CMX, CMX, and CMX quantities become ionic-strength dependent. In particular,

For cases involving multiply charged ions, the established definition(4,5) of CMX is

It is the factor |zxzx|1/2 that is inconsistent. It arose from rather complex considerations of the mixing terms PMNX, P MXY and their definitions for ions of differing charge.(1,4,5) Clegg et al.(14) treat sulfuric acid as a mixed electrolyte with ionicstrength-dependent third virial terms. Their Appendix I includes appropriate general equations for Archer-type terms. With amendments, these equations would serve for the present ionic-strength function. We prefer to delay a full treatment of mixed systems until one or more examples are considered, including fourth virial and possibly higher terms. At this time, we suggest only that an ionic strength-dependent CMX obtained from Eq. (23) or from Eqs. (15b), (17b), and (24) be divided by |zMzx|1/2 for use in the familiar equations for mixed electrolytes.(1,4,5) 4. EXPERIMENTAL DATA AND RESULTS FOR CaCI2 Rard and Clegg(10) presented, for CaCl2(aq) at 25°C, a very extensive and carefully reviewed data base. Included were new isopiestic measurements in the range 4-10 mol-kg -1 referenced to H2SO4(aq). That data base was used for the present calculations. Since a full description has been published,(10) only a few comments are needed here. In general, the change from the 1977 review of Rard et al.(17) or from the older review of Robinson and Stokes(18) is very small. Rard and Clegg(10) give full consideration to various electrochemical cell measurements, but find only a few to merit inclusion for the general equation. It is only in the very dilute region, below 0.2 molkg - 1 , that their effect is important. Since their inclusion would complicate the present calculations, the cell data were omitted. In their place the P values at or below 0.2 mol-kg -1 , from the final table of recommended values of Rard and Clegg, were added.


272

Since the basic objective was equations that are not only accurate, but also convenient, for use for either pure CaCl2 or for mixed solutions, several equations of increasing complexity are presented for increasing ranges of molality. We consider first the complex equation, but still without ion association, for the full range from 0 to 11 mol-kg -1 (i.e., 3.5 mol-kg -1 above saturation). It extends to the fifth virial coefficient with nine terms in all. The parameters are given in Table I, along with those for the simpler equations. The Debye-Huckel parameter AP = 0.391476 kg l / 2 -mol - l / 2 is based on the dielectric constant equation of Archer and Wang.(l9) There are five ionic strength-dependent terms with a values selected after various trial calculations. The B, C, D, and E values are obtained by the least-squares method. For mixtures the molality of CaCl2 seldom exceeds 7 mol-kg -1 and the ionic strength is less than 21 mol-kg -1 . In addition, the interpretation of the fifth virial is ambiguous with respect to CaCl4 and Ca2Cl3 interactions. Hence, an equation limited to the fourth virial is desirable; we present one valid to 7 mol-kg -1 with seven terms. The fourth virial is certainly dominated by CaCl3 interactions and can be treated as a w(Ca, 3Cl) term in the Filippov system described above. This equation, valid to 7 mol-kg - 1 , also suffices for Table I.

Parameter Values for CaCl(aq) in Eqs. (14-17) with 3, 5, 7, and 9 IonInteraction Parameters

Parameter

B(0)

B(1) (2)

B

ABI C(0)

C(2)(1)

C

ACI AC2 D(())

D(1)

D(2) ADI AD2 (0)

E

mmax

Units kg-mol-1 kg-mol -1 kg-mol -1 kg1/2-mol1/2 kg 2 -mol -2 kg2- mol - 2 kg 2 -mol -2 kg-mol -1 kg-mol-1 kg-mol-3 kg3-mol-3 kg 3 -mol -3 kg3/2-mol-3/2 kg3/2-mol-3/2 kg4-mol-4 mol-kg -1

Sf

Number of points a

This study. Pitzer and Mayorga (Ref. 2). c Standard deviation of the fit.

b

np = 9a

np = 7a

0.569081 0.464899 1.55840 1.59385 0 0 3.0 2.5 -0.0556802 0.00459539 0.215755 -0.00941176 -0.317714 -0.0729924 0.15 0.15 0.25 0.25 0.00282839 -0.000745381 0.0170624 -0.00241580 -0.0244910 0.030 0.030 0.035 -0.0000509632 11.0 7.0 0.0033 0.0026 430 355

np = 5a

np = 3b

0.446867 0.31588 1.57819 1.6140 0 0 2.45 2.0 -0.00663855 -0.00017103 -0.0151049 -0.0421769 0.15 0.25

5.2 0.0022 293

2.5


273

the interpretation of many isopiestic measurements where CaCl2 is the reference electrolyte; those for MgCl2, as discussed below, are an example. For solutions of still lower ionic-strength, it is convenient to have parameters that can be used in the programs in common use that extend only to the third virial coefficient. Parameters are given in Table I for a treatment with C(1) and C(2) terms for the range to 5.2 mol-kg -1 . We also considered the appropriate treatment for a still more limited range and with only B(0), B(1), and C(0) terms. The parameters originally given in 1973(2) for the range to 2.5 mol-kg-1 yield such good results and are now so widely used that we recommend their retention without change. These parameters are included in Table I. Each of these functions is compared with the recommended values of Rard and Clegg(10) in Fig. 1 and in Tables II and III. The maximum deviations

Fig. 1. Comparison of (a) osmotic coefficients and (b) activity coefficients of CaCI2(aq) from Rard and Clegg (Ref. 10) with those calculated using equations from this study (Eqs. 14 and 16; np = 5, 7, 9) and from Pitzer and Mayorga (Ref. 2; np = 3). np = 9; np = 7;

np = 5;

np = 3.

274


Table II. Comparison of the Osmotic Coefficients of CaCl2(aq) from Table 11 of Rard and Clegg with Those Calculated Using Eq. (14) and Ion-Interaction Parameters in Table I

ma

Rard and Clegg

np = 9b

np = 7b

np = 5b

np = 3c

0.001 0.002 0.005 0.01 0.02 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.20 1.40 1.60 1.80 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75

0.9623 0.9493 0.9275 0.9078 0.8871 0.8628 0.8527 0.8588 0.8748 0.8944 0.9163 0.9401 0.9655 0.9924 1.0205 1.0495 1.1100 1.1733 1.2393 1.3079 1.3793 1.4725 1.5699 1.6705 1.7731 1.8759 1.9778 2.0788 2.1796 2.2805 2.3808 2.4792 2.5743 2.6643 2.7479 2.8241 2.8920 2.9514 3.0022 3.0447 3.0796 3.1074 3.1291 3.1453

0.9623 0.9494 0.9275 0.9079 0.8871 0.8627 0.8527 0.8589 0.8753 0.8957 0.9181 0.9419 0.9668 0.9928 1.0200 1.0482 1.1078 1.1715 1.2387 1.3089 1.3816 1.4754 1.5717 1.6702 1.7705 1.8723 1.9751 2.0785 2.1818 2.2841 2.3845 2.4820 2.5753 2.6635 2.7457 2.8209 2.8887 2.9486 3.0005 3.0443 3.0804 3.1092 3.1314 3.1476

0.9622 0.9492 0.9271 0.9074 0.8866 0.8627 0.8533 0.8593 0.8751 0.8950 0.9172 0.9411 0.9664 0.9929 1.0204 1.0489 1.1087 1.1721 1.2387 1.3084 1.3807 1.4744 1.5712 1.6703 1.7713 1.8736 1.9765 2.0795 2.1820 2.2834 2.3828 2.4795 2.5727 2.6615 2.7450 2.8220 2.8918 2.9531 3.0049 3.0463 3.0762

0.9622 0.9492 0.9272 0.9074 0.8867 0.8629 0.8535 0.8594 0.8750 0.8948 0.9170 0.9410 0.9664 0.9929 1.0205 1.0490 1.1088 1.1721 1.2386 1.3082 1.3805 1.4743 1.5712 1.6706 1.7718 1.8740 1.9767 2.0793 2.1812 2.2819 2.3811 2.4784 2.5735 2.6661

0.9621 0.9490 0.9269 0.9072 0.8866 0.8636 0.8553 0.8613 0.8758 0.8941 0.9150 0.9382 0.9633 0.9900 1.0180 1.0474 1.1092 1.1745 1.2426 1.3129 1.3850 1.4772 1.5711


275 Table II. Continued

a

m

Rard and Clegg

np = 9b

8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00 10.25 10.50 10.75 11.00

3.1570 3.1649 3.1697 3.1721 3.1727 3.1720 3.1704 3.1683 3.1659 3.1635 3.1610 3.1586 3.1563

3.1588 3.1658 3.1694 3.1707 3.1702 3.1688 3.1671 3.1654 3.1641 3.1633 3.1629 3.1628 3.1625

np = 7b

np = 5b

np = 3c

a

Units, mol-kg-1. This study: np is the number of ion-interaction parameters used in the fit. c Pitzer and Mayorga (Ref. 2). b

remain below the range of deviations of the more accurate measurements (±0.3% of P) as shown in the figures of Rard and Clegg.(10) A comparison of the present np = 5 fit for m < 5.2 mol-kg-1 to the corresponding Archer equation(12,13) fit to a comparable molality range indicates that the present model gives a better representation of the osmotic coefficients of CaCl2(aq) by up to 0.004 in P. However, the present model has one more adjustable parameter. 5. EXPERIMENTAL DATA AND RESULTS FOR MgCl2 Goldberg and Nuttall (20) in 1978 and Rard and Miller (21) in 1981 reviewed the available data for MgCl2(aq) at 25°C. Rard and Miller also reported important new isopiestic measurements at higher molalities. Above 2.5 molkg~', the new measurements give significantly higher osmotic coefficients than the earlier data that were primarily from Stokes.(22) The difference, which exceeds 0.01 above 3.0 mol-kg -1 , is surprising and probably arose from contamination with alkali chlorides in some of the earlier studies.(21) The Rard and Miller values were adopted for the isopiestic ratios. These and other P values from isopiestic measurements were recalculated for the more recent equations for the reference solutes: Archer(13) for NaCl(aq), Clegg et al.(14) for H2SO4(aq), and our new np = 7 equation for CaCl2(aq). Below 2.5 mol-kg-1 there are several sets of isopiestic measurements(22-29) The Rard-Miller(21) and Goldberg-Nuttall(20)evaluations of osmotic coeffi-

276


Table III. Comparison of the Activity Coefficients of CaCl2(aq) from Table 11 of Rard and Clegg with Those Calculated Using Eq. (16) and Ion-Interaction Parameters in Table I

ma

Rard and Clegg

np = 9b

np = 7b

np = 5b

np = 3c

0.001 0.002 0.005 0.01 0.02 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.20 1.40 1.60 1.80 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75

0.8886 0.8508 0.7871 0.7290 0.6651 0.5786 0.5187 0.4716 0.4538 0.4476 0.4479 0.4527 0.4609 0.4721 0.4859 0.5021 0.541 1 0.5891 0.6467 0.7153 0.7965 0.9191 1.0702 1.2554 1.4811 1.7534 2.0795 2.4695 2.9377 3.5010 4.1762 4.9786 5.9208 7.0112 8.2526 9.6418 11.169 12.820 14.575 16.416 18.322 20.276 22.263 24.270

0.8886 0.8509 0.7872 0.7292 0.6652 0.5786 0.5187 0.4716 0.4541 0.4484 0.4490 0.4540 0.4621 0.4730 0.4864 0.5021 0.5405 0.5884 0.6467 0.7164 0.7989 0.9227 1 .0736 1 .2568 1.4791 1.7487 2.0753 2.4703 2.9463 3.5170 4.1965 4.9985 5.9351 7.0155 8.2448 9.6228 11.144 12.796 14.562 16.422 18.352 20.329 22.334 24.350

0.8884 0.8505 0.7865 0.7282 0.6640 0.5777 0.5183 0.4714 0.4536 0.4476 0.4481 0.4530 0.4613 0.4724 0.4859 0.5017 0.5403 0.5881 0.6460 0.7152 0.7973 0.9207 1.0715 1.2552 1.4784 1.7488 2.0759 2.4702 2.9440 3.5106 4.1845 4.9803 5.9120 6.9914 8.2265 9.6190 11.162 12.837 14.612 16.439 18.257

0.8884 0.8505 0.7865 0.7283 0.6642 0.5779 0.5186 0.4717 0.4538 0.4477 0.4482 0.4532 0.4615 0.4726 0.4861 0.5020 0.5406 0.5884 0.6463 0.7154 0.7974 0.9209 1.0720 1.2561 1.4797 1 .7505 2.0774 2.4708 2.9427 3.5068 4.1787 4.9758 5.9176 7.0257

0.8882 0.8503 0.7860 0.7277 0.6636 0.5781 0.5197 0.4735 0.4552 0.4485 0.4483 0.4526 0.4606 0.4715 0.4852 0.5013 0.5408 0.5900 0.6493 0.7197 0.8024 0.9256 1.0744

277

Ionic Strength Dependence Table III. Continued

ma

Rard and Clegg

np = 9b

8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00 10.25 10.50 10.75 11.00

26.291 28.321 30.360 32.411 34.477 36.566 38.685 40.841 43.042 45.292 47.598 49.960 52.377

26.366 28.378 30.386 32.399 34.427 36.485 38.589 40.754 42.993 45.315 47.722 50.207 52.752

a,b,c

np = 7b

np = 5b

np = 3c

See footnotes, Table II.

cients from these data are essentially the same and need not be discussed in detail here. Only the values given large weights in these reviews were included in the present calculations. A few more recent measurements of Kuschel and Seidel(30) were included. For the very dilute range, the values from the freezing point measurements of Gibbard and Gossmann(31) can be converted using thermal data to yield accurate osmotic coefficients at 25°C; these were included up to 1.02 mol-kg -1 . In addition, in the extremely dilute range below 0.01 mol-kg-1, activity coefficient values from diffusion data of Harned(32) were included. For an accurate representation over the full range to 5.9 mol-kg -1 , an equation extending to the fourth virial with six terms was required. Since there is no ambiguity of interpretation of the fourth virial as an MgCl3 term, the equation is fully satisfactory for use in mixed electrolytes. Nevertheless, it is convenient to have a simpler equation if it is valid over a substantial range. Thus, we also present an equation with only B(0), B(1), C(0), C(1) terms that is valid to 4.5 mol-kg -1 . This is essentially the same pattern as the Archer equations(12,13) for NaBr(aq) and NaCl(aq), the only difference being in the simpler ionic strength dependence of the C(1) term in the present equation. The parameters for these equations are in Table IV. It is interesting to compare the equation first presented in 1973(2) and evaluate its validity for further use. As expected, the MgCl2 np = 3 equation deviates in the range above 2.0 mol-kg-1 where the Rard-Miller measurements(21) differ significantly from the earlier values. Below 2.0 mol-kg -1 , the agreement shown in Fig. 2 is acceptable for many purposes. However, the new np = 4 equation

278


Table IV. Parameter Values for MgCl2(aq) in Eqs. (14-17) with 3, 4, and 6 IonInteraction Parameters Parameter

B(0) B(1) (2)

B

ABI

C(0)

C(1) C(2)

ACI AC2

D(0)

mmax of

Number of weighted points

Units kg-mol -1 kg-mol -1 kg-mol -1 kg1/2-mol-1/2 kg 2 -mol -2 kg 2 -mol -2 kg 2 -mol -2 kg-mol -1 kg-mol -1 kg3-mol-3 mol-kg -1

np = 6a

np = 4a

np = 3b

0.563113 1 .95628 0 3.0 -0.0279311 0.0468272 -0.129805 0.15 0.25 0.00131506 5.9 0.0028 176

0.512290 1 .97367 0 2.8 -0.00676278 -0.0665035

0.35235 1.6815 0 2.0 0.0025959

0.23 4.5 0.0030 150

2

a

This study. Pitzer and Mayorga (Ref. 2). c Standard deviation of the fit. b

Fig. 2. Differences of the experimental osmotic coefficients (symbols) of MgCI2(aq) from the model equation, Eq. (14), and parameters given in Table IV, fit to 5.9 mol-kg -1 (np = 6): + Ref. 21; • Ref. 27; A Ref. 29; X Ref. 28; * Ref. 22; * Ref. 24; - Ref. 25; - Ref. 26; a Ref. 30; 0 Ref. 31; o Ref. 23. Differences are also shown for the model fit to 4.5 mol-kg-1 (np = 4, dotted curve) and the model by Pitzer and Mayorga (Ref. 2; np = 3, dashed curve) from those fit to 5.9 mol-kg-1 (np = 6).


279

is no more complex than the Archer equations(12,13) that are coming into general use; hence, the new np = 4 equation should be used for new research. Tables V and VI give osmotic and activity coefficient values for these equations; Fig. 2 shows comparisons with the array of measurements for the appropriate range of molality. 6. DISCUSSION If there is significant ion association at very low concentration, it is unambiguous and can be evaluated from thermodynamic, conductance, or structural measurements. Aqueous sulfuric acid is an example. In addition, for cases where there is partial association at low concentration with redissociation at higher molality, as in MgSO4(aq) and other 2-2 electrolytes, the method with B(2) = -K/2 is satisfactory(5,33) At higher concentrations, the population of positive and negative ion near neighbors is substantial on a Debye-Huckel basis and is subject to large variation for differences in shortrange forces. Thus, any assignment to ion association is ambiguous and alternate descriptions may be both simpler and preferable for various reasons. The method described in this paper is shown to give an accurate description of the CaCl2(aq) system at 25°C where any significant ion association occurs only at high molality. This system has important advantages over a representation that includes an assumed ion-association constant. One is convenience of calculation, even for the pure electrolyte. The calculation of the fractional association at each point and the corresponding ionic strength is avoided. For mixed electrolytes, it is more important to avoid the unnecessary complexity introduced by the assumption of additional species of ambiguous molality. Thus, we strongly recommend the present system when higherorder terms are needed, in addition to the long-established formulation (1,2,4,5) with B(0), B(1), B(2), and C terms. ACKNOWLEDGMENTS The work at Berkeley was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Division of Chemical Sciences of the U. S. Department of Energy under Contract No. DE-AC03-76SF00098. The contribution of J. A. R. was performed under the auspices of the Office of Basic Energy Sciences (Geosciences) of the U. S. Department of Energy by the Lawrence Livermore National Laboratory under contract No. W-7405ENG-48. That of S.L.C. was supported by an Advanced Research Fellowship (GT5/93/AAPS/2) from the Natural Environment Research Council of the U.K.

280

Pitzer, Wang, Rard, and Clegg Table V. Calculated Osmotic Coefficients of MgCl2(aq) Using Eq. (14) and IonInteraction Parameters in Table IV

ma

np = 6b

np = 4b

np = 3c

0.0001 0.0005 0.00 1 0.005 0.01 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 5.90

0.9870 0.9726 0.9627 0.9292 0.9107 0.8703 0.8629 0.8724 0.8929 0.9182 0.9463 0.9765 1.0082 1.0413 1.0757 1.1113 1.1863 1.2661 1.3505 1.4391 1.5313 1.6265 1.7242 1.8237 1.9246 2.0265 2.1290 2.2320 2.3352 2.4388 2.5427 2.6734 2.8055 2.9396 3.0765 3.2171 3.3624 3.5135 3.6074

0.9870 0.9725 0.9627 0.9291 0.9107 0.8709 0.8637 0.8727 0.8923 0.9171 0.9451 0.9754 1.0074 1.0410 1.0758 1.1118 1.1874 1.2672 1.3512 1.4390 1.5304 1.6250 1.7222 1.8218 1.9232 2.0260 2.1296 2.2337 2.3378 2.4417 2.5449 2.6725 2.7982

0.9870 0.9723 0.9622 0.9275 0.9083 0.8682 0.8633 0.8753 0.8953 0.9192 0.9459 0.9749 1.0059 1.0388 1.0733 1.1093 1.1850 1.2651 1.3489 1.4358 1.5253

a,b,c

See footnotes, Table II.

281

Ionic Strength Dependence Table VI.

Calculated Activity Coefficients of MgCl2 (aq) Using Eq. (16) and IonInteraction Parameters in Table IV

ma

np = 6b

np = 4b

np = 3c

0.0001 0.0005 0.001 0.005 0.01 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 5.90

0.9613 0.9184 0.8894 0.7900 0.7337 0.5897 0.5332 0.4905 0.4771 0.4761 0.4823 0.4935 0.5087 0.5276 0.5498 0.5753 0.6370 0.7142 0.8097 0.9266 1.0693 1.2427 1.4530 1.7076 2.0152 2.3866 2.8346 3.3749 4.0268 4.8138 5.7653 7.2429 9.1292 11.551 14.682 18.763 24.136 31.284 36.705

0.9613 0.9184 0.8893 0.7899 0.7337 0.5902 0.5342 0.4913 0.4775 0.4761 0.4821 0.4932 0.5086 0.5276 0.5500 0.5759 0.6380 0.7156 0.8109 0.9274 1.0693 1.2417 1.4511 1.7051 2.0130 2.3858 2.8369 3.3817 4.0390 4.8306 5.7820 7.2420 9.0679

0.9612 0.9179 0.8885 0.7870 0.7294 0.5837 0.5288 0.4883 0.4753 0.4739 0.4793 0.4898 0.5045 0.5228 0.5448 0.5701 0.6315 0.7082 0.8022 0.9163 1.0541

a,b,c

See footnotes. Table II.

282


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