J . Phys. Chem. 1990, 94, 7706-7710
7706
Science and Engineering Council of Canada, in the form of an operating grant to Y.K. and a University Undergraduate Student Research Award to W.W.Y.S. T.Y.H.W. thanks the governments of Canada and British Columbia for the Award of Challenge 89 and the Work Study Programme.
relation function, and various kinds of anomalies in the quantities that are proportional to a third and a higher order derivative of free energy.
Acknowledgmenr. This research was supported by the Natural
+
Activity Coefficients for HCi BaCi, Hlgher Order Electrostatic Terms
+ H,O
at Different Temperatures and Effects of
Rabindra N. Roy,* Susan A. Rice, Kathleen M. Vogel, Lakshmi N. Roy, Department of Chemistry, Drury College, Springfield. Missouri 65802
and Frank J. Millero Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, Florida 331 49 (Received: January 30, 1990; In Final Form: April 18, 1990)
+
Activity coefficients for HCI in the system HCI BaC1, + H 2 0 from 278.15 to 3 18.15 K and at ionic strengths between 0.005 and 4.0 mol kg-l have been determined by an emf method using a cell without liquid junction of the type Pt;Hz(g,l atm)JHCl(ml),BaClZ(mz)~AgCI,Ag(A). The thermodynamic properties of asymmetrical mixtures are interpreted by use of Harned’s rule and Pitzer’s virial coefficient approach including higher order electrostatic terms for mixed-electrolyte solutions. and 6$ma/6t, Pitzer’s mixing parameters‘OH& and $Hha,as well as linear representations of temperature derivative&/,6@ have been estimated. A table for the activity coefficients of BaCl2 in the experimental temperature range is reported. The results are compared with the literature data if available. Hydrochloric acid follows Harned’s rule at all experimental ionic strengths and temperatures within experimental error.
Introduction Investigations of the thermodynamic properties of aqueous mixed electrolyte solutions a t high ionic strength are of great interest in fields such as chemical oceanography, geochemical calculations in brines, and industrial processes. In order to facilitate the use of mixed electrolyte, for example, prediction of mineral solubilities in multicomponent ionic solutions a t high concentration, it is highly desirable to know accurate values of the activity and osmotic coefficients of all the components in the mixture. Recent developments of electrolyte solution theories have also generated added interest in this aspect of solution chemistry. While there are alternative approaches’-’ to represent the results of the experimentally determined properties of the electrolyte solutions, ionic interaction models by Pitzer et aL8-” provide the simplest, accurate, and most convenient procedures for calculating the thermodynamic properties of binary mixtures of electrolytes containing a common The most common methods for ( I ) Scatchard, G. J. Am. Chem. SOC.1961,83,2636. (2)Rush, R. M.; Johnson, J. S.J. Phys. Chem. 1968,72,767. (3)Perry, R. L.; Cabezas, H., Jr.; OConnell, J. P. Mol. Phys. 1988,63. 189. (4)Friedman, H.L. Faraday Discuss. Chem. Soc. 1988,85, 1. (5) Blum, L. Primitive Electrolytes in the Mean Spherical Approximation. In Theoreticul Chemistry; Henderson, D., Ed.; Academic Press: New York, 1988;Vol. 5. (6)Millero, F. J.; Schreiber, D. Am. J. Sci. 1982,282, 1508. (7)Rasaiah, J. C. In The Liquid State and Its Electrical Properties; Kunhard, E. E.. Christophoru, L. G.; Lucsscn, L. H., Eds.; NATO AS1 Series 193;Plenum Press: New York, 1989. (8) Ptizer, K. S.J. Phys. Chem. 1973,77, 268. (9)Pitzer, K. S.;Kim. J. J . J. Am. Chem. Soc. 1974,96,5701. (IO) Pitzer, K. S . J. Solution Chem. 1975,4, 229. ( I I ) Pitzer, K. S. In Actiuity Coefficients in Electrolyte Soluriom; Pytkowicz, R. M., Ed.; CRC Press: Boca Raton, FL, 1979;Vol. I, Chapter 3. (12) Roy, R. N.; Roy, L. N.; Farwell, G.D.; Smith, K. A,; Millero, F. J. Submitted for publication in J . Phys. Chem.
0022-3654/90/2094-7706$02.50/0
the determination of the activity coefficients in mixed electrolyte solutions are the isopiestic vapoi-pressure and emf techniques. -In this study, we have applied the emf method which yields the activity coefficient of at least one of the solutes directly. There are several gaps in the recent compilation of Pitzer,” Plummer et a1.,I8 and Kim and FrederickIg for some binary mixtures with a common ion. One such system is HCI BaCI, H20. The main objectives of this paper are (i) to provide precise emf data at 14 different ionic strengths ( I = 0.005-4.0 mol kg-I) in the temperature range 278.15-318.15 K, (ii) to test the applicability of the simple Harned’s rule to hydrochloric acid, (iii) to calculate the activity coefficients of BaCl, in terms of the Pitzer formalism including higher order effect for unsymmetrical mixing, and (iv) to supply values for Pitzer’s mixing parameters 6 and # in the experimental range of temperatures. The emf cell consisting of hydrogen and silver-silver chloride electrodes for the system HCI BaCI2 + H 2 0 was studied by Khoo et aI.,,O but only at 298.15 K and five different ionic strengths ( I = 0.1,0.5, 1.0, 2.0, and 3.0 mol kg-I). Downeszl studied the cell a t 298.15 K for I = 0.1 mol kg-’ whereas Harned and Gary22 first reported the
+
+
+
(13)Roy, R. N.;Gibbons, J. J.; Peiper. J. C.; Pitzer, K. S. J . Phys. Chem. i983,87,2365. (14) Millero, F . J. Thalassia Jugosl. 1982,18, 253. (15) Roy, R. N.; Gibbons, J. J.; Roy, L. N.; Greene, M. A. J. Phys. Chem. i986,90,6242. (16) Amanthaswamy, J.; Atkinson, G.J. Chem. Eng. Data 1985,30,120. (17)Pitzer, K. S. In Activity Coefficients in Electrolyte Solutions; Pytkowicz, R. M., Ed.; CRC Press: Boca Raton, FL; Vol. 111, Chapter 3, in press. (18)Plummer, L. N.; Parkhurst, D. L.; Fleming, G. W.; Dunkle, S. A.
US. Geological Survey, Reston, VA, Water Resources Investigations Report
88-44153,1988. (19)Kim, H.T.;Frederick, W. J. J. Chem. Eng. Data 1988,33,177 and 278. (20)Khoo, K. H.;Chan, C. Y.; Lim, T. K. J. Chem. SOC.,Faraday Tram. I 1978,74,837. (21)Downes, C. J. J . Phys. Chem. 1970,74,2153.
0 1990 American Chemical Society
Activity Coefficients for HCI
+ BaC12 + H 2 0
The Journal of Physical Chemistry, Vol. 94, No. 19, 1990 7707
TABLE I: Emf of Cell A from 5 to 45 'CO YI
5OC
35'C
45OC
Y1
5OC
0.492 01 0.494 07 0.497 61 0.506 25 0.509 97 0.5 I6 86 0.533 65 0.544 44
15'C 25'C I = 0.005 mol kg-l 0.495 63 0.498 93 0.497 89 0.501 97 0.501 65 0.505 55 0.51073 0.51452 0.51472 0.52088 0.521 91 0.521 91 0.539 61 0.547 02 0.55099 0.55909
1.ooo 00
0.500 99 0.504 65 0.508 3 1 0.51563 0.52242 0.528 05 0.548 74 0.565 28
0.502 74 0.507 15 0.51 1 55 0.520 35 0.524 74 0.526 86 0.551 40 0.569 48
1.OOOOO 0.858 06 0.518 74 0.274 79 0.151 91
1.00000
0.900 02 0.800 05 0.599 97 0.499 94 0.400 05 0.200 00 0.099 97
0.469 66 0.473 18 0.477 18 0.485 49 0.490 04 0.496 34 0.51520 0.528 13
I = 0.008 0.474 90 0.478 67 0.48 1 88 0.489 02 0.493 83 0.500 42 0.51970 0.533 68
0.477 17 0.481 94 0.486 74 0.491 40 0.496 94 0.503 56 0.523 79 0.539 25
0.478 36 0.485 43 0.492 45 0.495 39 0.500 11 0.507 91 0.529 75 0.542 25
1.00000 0.90009 0.80008 0.600 09 0.50009 0.400 1 1 0.20000
0.460 IO 0.463 12 0.46642 0.474 8 1 0.48037 0.487 15 0.50085
0.46261 0.465 72 0.469 21 0.478 06 0.48381 0.490 70 0.50868
0.464 49 0.467 63 0.471 02 0.479 57 0.485 25 0.492 22 0.51109
0.465 75 0.469 14 0.47269 0.481 46 0.487 27 0.49443 0.51399
0.466 48 0.469 92 0.47363 0.482 89 0.488 95 0.496 30 0.51594
1.00000 0.800 32 0.600 45 0.500 37 0.40041 0.20021 0.100 10
0.38660 0.393 74 0.402 26 0.407 42 0.41438 0.43283 0.450 99
I = 0.05 mol kg-I 0.38663 0.38608 0.393 99 0.393 63 0.402 81 0.402 76 0.409 00 0.408 86 0.41528 0.41567 0.434 38 0.438 20 0.453 14 0.454 83
0.38506 0.392 51 0.402 12 0.407 99 0.41554 0.438 75 0.455 90
0.38345 0.39074 0.400 88 0.406 82 0.41488 0.438 82 0.456 45
1.00000 0.63 1 30 0.483 16 0.41392 0.14687
0.355 14 0.369 68 0.377 41 0.38209 0.41001
I = 0.1 mol kg-l 0.35420 0.35258 0.369 22 0.368 17 0.377 45 0.37667 0.38237 0.381 76 0.41097 0.411 38
0.35044 0.366 53 0.375 32 0.38058 0.411 21
0.34770 0.364 28 0.373 36 0.37880 0.41041
1.00000
0.31321 0.33567 0.343 89 0.36795
I = 0.25 mol kg-l 0.31092 0.30792 0.334 19 0.33207 0.342 70 0.34087 0.36763 0.36671
0.30434 0.32927 0.338 39 0.35787
0.30021 0.32595 0.335 36 0.35545
0.267 76 0.278 11 0.28026 0.294 60 0.31608
0.262 67 0.271 78 0.27553 0.292 79 0.29460
0.899 98 0.799 96 0.599 95 0.50005 0.400 05 0.20000 0.099 92 1.NO00
mol kg-I 0.475 30 0.478 67 0.484 3 1 0.491 58 0.496 99 0.503 78 0.523 74 0.530 19
I = 0.01 mol kg-l
0.491 04 0.369 83 0.151 90
I = 0.5 mol kg-' 1 .OOOOO
0.89748 0.71992 0.484 93 0.14690
0.279 76 0.28355 0.291 00 0.303 48 0.326 IO
0.276 45 0.28035 0.28800 0.297 74 0.32572
0.272 32 0.27643 0.28443 0.296 49 0.32480
35 'C
45OC
0.242 62 0.248 47 0.26602 0.285 77 0.301 67
15OC 25OC I = 1.0 mol kg-' 0.238 26 0.233 23 0.244 35 0.239 54 0.262 64 0.258 42 0.282 73 0.279 30 0.29972 0.29691
0.227 59 0.23407 0.253 52 0.275 08 0.29327
0.221 58 0.228 28 0.248 32 0.27060 0.28931
0.83407 0.486 95 0.36072 0.152 19
0.21849 0.225 98 0.245 74 0.255 52 0.281 02
I = 1.5 mol kg-l 0.213 15 0.20732 0.22093 0.215 43 0.241 45 0.236 62 0.251 65 0.247 17 0.277 70 0.274 18
0.201 45 0.209 63 0.23 1 24 0.24204 0.269 74
0.19463 0.20302 0.225 33 0.23633 0.265 06
1.00000 0.820 IO 0.68940 0.488 66 0.313 87 0.161 98
0.19774 0.20679 0.21417 0.227 68 0.24402 0.26289
I = 2.0 mol kg-I 0.19228 0.18636 0.201 52 0.19566 0.20909 0.20351 0.22287 0.217 54 0.238 39 0.233 46 0.25893 0.25481
0.18008 0.18955 0.19756 0.21 1 87 0.228 35 0.25003
0.17320 0.18305 0.191 12 0.205 74 0.22249 0.24483
1.00000
0.17995 0.18849 0.20022 0.211 84 0.22295 0.248 94
I = 2.5 mol kg-' 0.17463 0.16840 0.18277 0.17676 0.19487 0.189 IO 0.20688 0.201 35 0.21820 0.213 13 0.244 72 0.240 70
0.161 69 0.17030 0.18298 0.19546 0.20746 0.235 92
0.15474 0.16338 0.176 19 0.18904 0.201 37 0.230 56
0.16459 0.171 92 0.18635 0.18751 0.19549 0.213 15 0.21492 0.244 66 0.250 36
I = 3.0 mol kg-l 0.158 15 0.1 52 02 0.16564 0.161 96 0.18045 0.17582 0.181 94 0.177 20 0.19053 0.18597 0.209 22 0.203 57 0.21095 0.205 02 0.241 10 0.235 59 0.246 75 0.243 13
0.14532 0.153 12 0.16845 0.170 78 0.179 80 0.198 45 0.199 21 0.230 71 0.236 12
0.137 81 0.145 85 0.161 43 0.16380 0.17296 0.19204 0.19284 0.225 16 0.230 74
0.89642 0.70508 0.59455 0.51271 0.31620
0.14959 0.15596 0.16881 0.177 12 0.18374 0.20298
I = 3.5 mol kg-I 0.14329 0.13690 0.14984 0.14334 0.16272 0.15653 0.171 17 0.16523 0.17789 0.17201 0.19753 0.191 97
0.12981 0.13633 0.14982 0.15842 0.16532 0.18577
0.12257 0.12900 0.14279 0.15144 0.15853 0.17661
1.00000 0.901 17 0.70750 0.48845 0.34977 0.096 22
0.13545 0.14208 0.15660 0.17442 0.18733 0.230 28
I = 4.0 mol kg-l 0.12886 0.12255 0.13555 0.12922 0.14959 0.14336 0.16803 0.16200 0.18236 0.17647 0.225 42 0.22063
0.11481 0.12192 0.13664 0.15622 0.171 24 0.215 99
0.107 13 0.11416 0.12907 0.14902 0.16452 0.21 1 36
0.85109 0.65505 0.491 20 0.36207 0. I58 30 1.ooo 00
0.869 6 1 0.642 66 0.61459 0.503 84 0.323 20 0.31154 0.122 37 0.094 20 1.00000
0y* = 1 -y1.
precise data at total ionic strengths I = 1 .O, 2.0, and 3.0 mol kg-I but only a t 298.15 K. Hence, we were prompted to extend and supplement the emf measurements for a wide range of ionic strengths and temperatures (278.15, 288.15, 308.15, and 318.15 K and ionic strengths I = 0.005, 0.008, 0.01,0.05, 0.25, 1.5, 2.5, 3.5, and 4.0 mol kg-I).
The cells were of the all-glass type described by Bates.23-24The silver-silver chloride electrodes were of the thermal electrolytic type2s and the hydrogen electrodes were platinized according to the recommendation of Bates.2s The standard potential of the silver-silver chloride electrode for cell of the type A was deter-
Experimental Section
mined by measuring the emf in 0.01 mol kg-' hydrochloric acid. = 0.222 58 V, These measurements at 298.15 K gave EoAk-AgCI which is in excellent agreement with earlier results.26 Emf measurements were made with a Keithley (Model 191) digital
Hydrochloric acid solutions were prepared from twice-distilled constant-boiling acid that had been standardized by gravimetric analysis assayed as silver chloride with an accuracy of better than *0.01%. Stock solution of barium chloride (ACS certified reagent grade), about 5 mol kg', was prepared. Triplicate analyses as silver chloride agreed within *0.02%. All weighings of the experimental cell solutions were corrected to mass. (22) Harned, H. S.;Gary, R. J . Am. Chem. SOC.1954, 76, 5924.
Pt;H2(g)lHCl(m1),BaCl2(mz)lAgC1,Ag
(A)
(23) Bates, R. G. NBS Tech. Note (US.)1965, No. 271, 18. (24) Gary, R.;Bates, R. G.; Robinson, R. A. J . Phys. Chem. 1964, 68, 1186. (25) Bates, R. G. In Determination of pH, 2nd ed.; Wiley: New York, 1973; p 331. (26) Roy, R. N . ; Gibbons, J . J.; Trower, J. K.; Lee, G . A. J . Solution Chem. 1980, 9, 535.
7708 The Journal of Physical Chemistry, Vol. 94, No. 19, 1990 TABLE II: Parameters of the Harned Equation, (2): HCI-BaCI, Mixtures at 298.15 K 11 mol kg-' 0.005 0.008
0.01 0.05 0.1 0.25 0.5
I .o
I .5 2.0
2.5 3 .O 3.5 4.0
-log yIo 0.03175 0.03752 0.04413 0.08066 0.09871 0.1I9 17 0.11930 0.09002 0.04713 -0.00622 -0.06027 -0.11894 -0.18080 -0.24354
a12"
-0.0203 f 0.0010 -0.0084 f 0.0017 -0.0028 f 0.0007 0.0047 f 0.0008 0.0091 f 0.0002 0.0182 f 0.0002 0.0352 f 0.0002 0.0667f 0.0004 0.0991 f 0.0003 0.1339f 0.0013 0.1656f 0.0004 0.1986 f 0.0011 0.2338 f 0.0012 0.2686 0.0001
*
Roy et al. TABLE 111: Ion Interaction Parameters at 298.15 K
HCI
BaCI2
0.1778' 0.2628" (0.2907)d (0.2164)' (0.2579)f 8"' 0.2945" 1.4963" (1.2499)d (1.8459)( (1.1624)f C* 0.0008" -0.0198" (-0.03046)d (0.0000). (-0.00276)f 104(6i3(0)/ -3.O81Oc -1.7956 8'0'
1O'(rmsd)
0.6 0.9 0.4 0.4 0.0
6n 6n
0.1
IO4(&$')/
1.4197' 117.162*
IO'(&@/
0.6213c
-9.1896
6T)
0.2 0.3 0.2 0.9 0.3 0.9 0.7
+.
a From ref I 1 used here for the calculation of 0 and bFrom this work. CFrom ref 36. dFrom ref 19. 'From ref 18. /From this work; results are given only for comparison.
0.1
= -eA in ref 20.
OaI2
voltmeter. The cells were immersed in a thermostatic water bath, controlled to fO.O1 K. The cells were measured at intervals of 10 K from 298.15 to 278.15 K, returned to 298.15 K, and then measured at intervals of 10 K from 298.15 to 318.15 K with a final measurement at 298.15 K. The three measurements at 298. I5 K never differed on the average by more than 0.07 mV. Details of the experimental aspects of the emf measurements including the preparation of cell solutions and the purification of the hydrogen gas have been described ~ r l i e r . ~ ' -Cell ~ potentials corrected to a hydrogen fugacity of 1 atm (or 101.325 kPa) are listed in Table 1. The correction to a constant hydrogen fugacity is significant for cells containing solutions of high ionic strength. Hence, allowance was made for the change in vapor pressure of the solutions with the variations in the barium chloride concentration. The procedure for the required correction by iterative computer calculation has been discussed in previous
where y I o and yZ0are the activity coefficients of HCI and BaC1, in their pure solutions at the same total ionic strength as the mixture and a12and a Z Iare the Harned interaction coefficients for the acid and the salt, respectively. Harned's rule is obeyed for the HCI-BaCI, system from I = 0.005 to I = 4.0 mol kg-I. The nonlinear terms of the complete Harned eq 2 were not needed to fit the data within the experimental error. Table I1 lists the values of the Harned coefficients at 298.15 K. Our result of a I 2= 0.1986 from Table I1 at 298.15 K for I = 3.0 mol kg-I is in excellent agreement with the value of 0.1984 obtained by Khoo and his co-workers.20 Pitzer's Formalism. The ion interaction treatment developed for mixed electrolytes by Pitzer and Kim9 has been shown to adequately represent the activity coefficients as functions of temperature and solution composition. We have found the Pitzer treatment to satisfactorily describe our experimental results in earlier papers.I2J3Js The expression for the activity coefficients of hydrochloric acid in a solution containing barium chloride is given by
Theory and Equations Studies of cell A give the activity coefficient of hydrochloric acid ( T i ) in a mixed solution of molality ml with respect to HCI and m2 with respect to BaCI,. The value of y , is derived by means of the Nernst equation:
1'12
2 b
'f = -A4[ -+ - In (1 1
+ bI'/2
+ b11/2)
E =
(1) where Eo is the standard emf; F, R, and T have their usual significance. The values of E corrected to 1 atm pressure of hydrogen, for 14 different values of the total ionic strengths and various vaues of y I are collected in Table I. The term y l is the ionic strength fraction of hydrochloric acid in the HC1/BaC12 mixture; that is, y I = m l / ( m l 3m2) and y 2 = 3m2/(ml + 3m2). The total ionic strength I is given by I = m1 + 3m2, Po = 101.325 kPa, mo = 1 .O mol kg-I, and the physical constants R and F a r e taken from Cohen and Taylor.3' Harned's Equations. The form of Harned's rule can be conveniently expressed in the experimental activity coefficient of HCI in the mixture:
+
1% 71 = log TI0 - a12Y2 - P I z y 2 2
(2)
and for BaCI2 ( 7 , ) 1%
72
= 1% YZ0 - a2IYI - P21Y12
(3)
(27)Robinson, R. A.; Roy, R. N.; Bates, R. G. J. Solution Chem. 1974, 3, 837. (28) Roy, R. N.;Gibbons, J. J.; Bliss, D. P.; Baker, B.; Casebolt, R. G. J . Solution Chem. 1980,9, 12. (29) Bates, R. G. NBS Tech. Note (US.)1965, No. 271, 28. (30)Simonson, J. M.;Roy, R. N.; Gibbons, J. J. J. Chem. Eng. Data 1987, 32, 41. (31) Cohen, R. R.; Taylor, B. N. J. Phys. Chem. ReJ Dola 1973,2, 363.
A, = 0.3770
+ 4.684 X
+
104(T/K - 273.15) 3.74 X lO"(T/K - 273.15), (10)
Herein, f ' is the Debye-Huckel function for the activity coefficient with parameter A,. The parameter 6 has the standard value 1.2 kg1IZmol-1/2. The molality of ion i is given by mi and the ionic strength by I; both have dimensions of mol kg-I. The second and third virial coefficients for a pure electrolyte ij are Bij and Cij and have dimensions of kg mol-' m o P , respectively. a has the value 2.0 kg1l2mol-1/2. E : . is the ionic strength derivative of B, with dimensions of kg2 mol-i. Here m, is the molality of cation c with charge z, and correspondingly for anion a. Sums over c or a cover all cations or anions. The detailed discussion of the physical significance of the pure electrolyte parameters as well as those for the mixed electrolyte parameters is given in our previous paper.I3
Activity Coefficients for HCI
+ BaClz + HzO
The Journal of Physical Chemistry, Vol. 94, No. 19, 1990 7709
TABLE I V Mixing Parameters from Isothermal Fits to Data of Table I 5 'C
'BH.a/(kg mol-') $H,h,cl/(kg' mol-') uat/mv
h , r / ( k g mol-') #H,Ba.Cl/(kg2
-0.049 f 0.013 0.032 f 0.007 0.56
Without and -0.056 f 0.013 0.037 f 0.007 0.53
25 'C
45
O C
oc
Included 0.069 f 0.008 0.002 f 0.005 0.44
0.076 f 0.009 -0.002 f 0.005 0.41
0.093 f 0.017 -0.013 f 0.012 0.67
-0.045 f 0.009 0.026 f 0.006 0.43
-0.042 f 0.018 0.025 f 0.013 0.69
Included -0.044 f 0.008 0.026 f 0.005 0.46
0.0
P .27
35
,
oat/m"
'2
0.055 f 0.012 0.011 f 0.007 0.55
15 OC With E6 and 0.060 f 0.013 0.012 f 0.006 0.52
1
5
%,Ba
-
L
Lo
0 0
.-I
.-. -0.8
L
I
I
I
I
0.8
1.6
2.4
3.2
( m,c m 2 ) I 2
moll k 9
The quantities 8, e', and J. are properties characteristic of the mixture. According to the work of Pitzer'O they are defined by
O'H,Ba
= 'OH,Ba + E'H.Ba
(1 1)
= '@'H,Ba + E8'H,Ba
(12)
where s@'H,ea is set equal to zero by analogy with YH,&, but E@H' e,a is significant. The terms %H,ea and are due to contributions arising from the higher order electrostatic terms that are associated with the mixing of the ions of the same sign but different charge. The following equations were derived by Pitzer based on cluster-integral theory32with the omission of short-range forces for these terms: E8H.Ba
=
( Z B ~ I ~ H / ~[J(xH,Ba) I)
- f/2J(xH,H)
- !!2J(xBa.Ba)l (13)
E8'H.Ba
= -[EeH,Ba/fl
[xH,BaJ'(xH,Ba)
+ (zBazH/8p) x - !!ZxNi.NiJ'(xNi.Ni)]
- !hXH,HJ'(XH,H)
15
25
35
45
T/OC
Figure 2. Temperature dependence of the mixing parameters '6H.h and $ H , ~ , C I . The fitted lines are represented by eqs 17 and 18.
Figure 1. Fits of eq 21 for HCI + BaCI' + H 2 0 at 298.15 K. The upper plot shows the concentration dependence of (A In YHa)/mh vs (mH + mcI)/2. Data for (mH + mc1)/2 < 0.2 mol kg-I are not included because of the large uncertainties and negative deviations which become very , ~ neglected. The straight lines indicate the large if E6H,), and E f ) ' ~ are least-squares fit to the emf data. The lower plot refers to the values of the residuals (Eobd - E,Id) for the emf data at 298.15 K, for leastsquares calculations with and without higher order electrostatic terms, solid circles and open circles, respectively. It is clear that the definite dish-shaped curvature for the fit without E6H,$? and is better represented at low ionic strength with the inclusion of these terms.
OH.Ba
5
1
( 4,
where XH,ea= 6ZHZBa A,I1l2 and ZHand ZBaare the valencies of the ions. The univariant functions J(X) and J'(X) are given (32) Peiper, J. C.; Pitzer, K. S. J . Chem. Thermodyn. 1982, 14, 613.
by the equations of Pitzer'O or can be evaluated numerically. A variety of numerical integration technique^'^*'^ may be used to compute these integrals, but we have chosen a 200-point Gauss-Legendre method as has been done in previous paper~.~2'~J~ Table 111 lists the values for all HCI and BaC12 pure-electrolyte parameters at 298.15 K, along with their temperature derivatives. The virial coefficients needed in eqs 4 and 5 are assumed to be known over the range of experimental temperatures and are given by PL?,Cl(T)
=
88?,c1(298.15K) + ( T - 298.15 K)(~PfP,!ci/~r)zmi, K (15) and similar expressions for the remaining pure-electrolyte parameters. The temperature dependence of E8H,Ba and E8'H,Ba can ~ ~ mixing be derived mainly from the values of A, a ~ f ( T ) . The parameters 8H,Ba and J.H,Ba,CI (without higher order electrostic effect) and % H , B ~and J.H,Ba,CI (with electrostatic effects) can be readily evaluated as functions of temperatures from the temperature dependence of and and the precise emf data by using eqs 1 and 4.
Results and Discussion Linear least-squares procedures were used to fit eqs 1 and 4 to the equally weighted experimental emf data. The values of %H,Ba, J.H,B~,cI, and the standard deviations of fit determined at each temperature (isothermal fit) are given in Table IV. The values of Pitzer's mixing parameters (8, J.) are obtained by using the pure-electrolyte parameters listed by Pitzer." It is interesting (33) Harvie, C. E.;Moller, N.; Weare, J. H. Geochim. Cosmochim. Acta 1984, 48, 723. (34) Pitzer, K. S.; Roy, R. N.; Silvester, L. F. J . Am. Chem. SOC.1977, 99, 4930.
7710 The Journal of Physical Chemistry, Vol. 94, No. 19, 1990 TABLE V Activity Coefficients for [HCl(m,) Strength Fraction Y of BnCI,
+ BaCl,(m,)(aq)]
T/K 278.15
288.15
298.15
308. I5
1 .oo 1S O
318.15
2.00 2.50 3.00 3.50 4.00 0.005 0.008 0.010 0.10 0.25 0.50 I .oo ISO
2.00 2.50 3.00 3.50 4.00
at Ionic
vestigators. The unknown mixing parameters may also be estimated by a graphical procedure. For the activity coefficient of HCI in this mixture one has
( A In Y H c i ) / m B a = '0H.Ba + YAW + %)$H,Ba,CI (16) where A In yHClis the difference between the experimental value of In yHClobtained from eq 1 and that calculated from eq 4 with ~ H , = B ~0 and $ H , B ~ , C I = 0. The quantity on the left was plotted against the coefficient of $ on the right to obtain a linear plot with intercept 0 and slope (see Figure I ) . This type of presentation of results leads to an assignment of estimated error since the uncertainty in (A In Y H C ) ) / m B a is greatly enhanced at lower molalities of Ba2+. The construction of such a plot has been recommended by Pitzer.lo Figure 1 shows this plot for the data at 298.15 K. In order to judge the quality of the measurements and of the fit, the deviations between the measured and calculated values of E are also shown in the lower part of Figure 1. The deviations rarely exceed 0.6 mV. With the omission of higher order terms (open circles), the systematic departures are quite apparent particularly at dilute solutions. Figure 2 shows plots of the temperature dependence of the results of the mixing parameters. The error bars correspond to the standard deviations of the parameters as determined by the fitting procedure. These mixing parameters are expressed by the equations
YhBaCI2
YLHCl
[/(mol of kg-I) 0.005 0.008 0.010 0.050 0.10 0.25 0.50 I .oo 1S O 2.00 2.50 3.00 3.50 4.00 0.005 0.008 0.010 0.050 0.10 0.25 0.50 1 .oo 1S O 2.00 2.50 3 .oo 3.50 4.00 0.005 0.008 0.010 0.050 0.10 0.25 0.50 I .oo IS O 2.00 2.50 3.00 3.50 4.00 0.005 0.008 0.010 0.050 0.10 0.25 0.50
Roy et al.
y=
y =
y =
y =
y=
y =
0.0 0.930 0.920 0.903 0.836 0.804 0.771 0.775 0.841 0.927 1.074 1.237 1.428 1.675 1.958 0.930 0.920 0.905 0.832 0.801 0.765 0.766 0.826 0.914 1.044 1.189 1.389 1.598 1.869 0.930 0.9 17 0.903 0.831 0.797 0.761 0.759 0.813 0.897 1.014 1.149 1.315 1.516 1.752 0.926 0.924 0.905 0.833 0.794 0.756 0.753 0.802 0.875 0.983 1.109 1.260 1.446 1.664 0.928 0.925 0.904 0.790 0.751 0.745 0.788 0.859 0.952 1.068 1.21 I 1.372 1.590
0.5 0.976 0.928 0.909 0.830 0.796 0.756 0.746 0.780 0.829 0.906 1.005 1.127 1.242 1.389 0.969 0.928 0.908 0.828 0.79 1 0.750 0.736 0.765 0.815 0.889 0.979 1.083 1.205 1.345 0.951 0.926 0.906 0.826 0.788 0.744 0.730 0.753 0.800 0.869 0.950 1.046 1.158 1.286 0.990 0.932 0.934 0.827 0.785 0.741 0.724 0.744 0.787 0.849 0.922 1.012 1.120 1.228 0.982 0.932 0.936 0.782 0.736 0.717 0.731 0.774 0.829 0.895 0.981 1.078 1.171
1.0 1.025 0.937 0.914 0.824 0.787 0.741 0.719 0.723 0.742 0.763 0.817 0.889 0.921 0.986 1.008 0.936 0.91 I 0.825 0.781 0.735 0.707 0.708 0.728 0.757 0.807 0.845 0.908 0.968 0.974 0.935 0.909 0.822 0.780 0.729 0.701 0.697 0.7 14 0.745 0.785 0.833 0.885 0.944 1.057 0.940 0.965 0.820 0.777 0.726 0.696 0.689 0.709 0.734 0.767 0.812 0.868 0.906 1.039 0.938 0.969 0.774 0.721 0.691 0.678 0.698 0.722 0.751 0.794 0.847 0.862
0.0 0.865 0.837 0.822 0.692 0.63 1 0.562 0.532 0.544 0.593 0.665 0.761 0.883 0.988 1.097 0.864 0.835 0.820 0.689 0.628 0.560 0.531 0.542 0.589 0.649 0.750 0.873 0.98 1 1.093 0.862 0.833 0.818 0.687 0.626 0.558 0.537 0.542 0.587 0.65 1 0.735 0.834 0.970 1.062 0.860 0.830 0.684 0.680 0.622 0.555 0.535 0.539 0.582 0.642 0.731 0.829 0.945 1.049 0.859 0.828 0.812 0.619 0.552 0.529 0.537 0.578 0.634 0.704 0.785 0.874 0.960
0.5 0.865 0.836 0.821 0.687 0.623 0.544 0.500 0.483 0.498 0.532 0.578 0.644 0.704 0.744 0.863 0.834 0.8 19 0.684 0.619 0.541 0.495 0.479 0.492 0.516 0.562 0.610 0.668 0.719 0.861 0.832 0.8 17 0.682 0.618 0.541 0.490 0.480 0.491 0.517 0.553 0.605 0.644 0.688 0.859 0.829 0.699 0.675 0.61 5 0.539 0.510 0.477 0.486 0.508 0.516 0.564 0.597 0.637 0.858 0.827 0.8 I2 0.61 1 0.536 0.477 0.474 0.48 I 0.499 0.525 0.559 0.594 0.636
1.0 0.865 0.835 0.820 0.682 0.614 0.527 0.469 0.423 0.405 0.399 0.398 0.405 0.413 0.424 0.862 0.833 0.8 I8 0.679 0.6 I2 0.525 0.459 0.420 0.404 0.398 0.398 0.403 0.412 0.420 0.861 0.831 0.8 16 0.677 0.6 I O 0.523 0.444 0.418 0.403 0.397 0.397 0.401 0.41 1 0.418 0.858 0.828 0.831 0.670 0.609 0.522 0.440 0.414 0.397 0.391 0.392 0.395 0.402 0.412 0.857 0.826 0.81 1 0.664 0.519 0.425 0.4 10 0.383 0.375 0.376 0.379 0.388 0.398
"Trace activity coefficients of HCI and BaCI, are given in columns 4 and 5, respectively.
+
'eH,Ba
= 0.0708
$ H , B ~ , C I=
+ 0.00091(T - 298.15)
(17)
0.0018 - 0.00063(T - 298.15)
(18) The standard deviations of the fit for eqs 17 and 18 are 0.0036 and 0.0033, respectively. The temperature coefficients of these parameters are then 6SBH,Ba/6T= 0.00091 f 0.0001 (19) 6$H,Ba,cl/6T
= -0.00063 f 0.0001
(20)
All of the earlier emf for the present system are limited only to 298.15 K and only at five different ionic strengths. ~ 298.15 K obtained by Khoo The values of 'OH,& and + H , B a , ~ at et al.35 are 0.076 and 0.000, respectively. Table IV shows our results of %H,Ba = 0.0692 and $H,h,~l = 0.0019. This comparison is satisfactory within limits of experimental uncertainties. The small differences in the reported values of the mixing parameters are largely accounted for by (a) additional emf measurements at nine different total ionic strengths, (b) slight differences in the values of E o A , - A g ~ due to differences in the reported E values for pure HCI solutions, and (c) the effect of including emf data at lower ionic strengths that leads to weight 0 more negatively and $ more positively. The activity coefficients of HCI and BaCl, given in Table V were calculated by using eqs 4 and 5, based on the mixing coefficient parameters (Table IV) with electrostatic effects. The trace activity coefficients of HCI and BaCI, at the experimental temperatures are presented in columns 4 and 5, respectively. The mean molal activity coefficient of pure BaCI2ilast column in Table V) at 298.1 5 K for I = 3.0 mol kg-l is 0.401, which can be compared with the literature value3' of 0.401. This type of agreement gives confidence to the experimental emf data as well as Pitzer's treatment.
Acknowledgment. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for the partial support of this work and to the National Science Foundation through Grant No. CBT-8805882. F.J.M. wishes to acknowledge support from the oceanographic section of N S F for ROA to R.N.R. Registry No. HCI, 7647-01-0; BaCI,, 10361-37-2.
to note that the values of @ O ) , /3('), and 0 (last column in Table 111) calculated solely from our emf data are in good agreement with those obtained by other workers. The slight differences in values of the virial coefficients are attributed to differences in the range of experimental ionic strengths studied by different in-
(35) Khw, K. H.; Lim,T. K.; Chan, C. Y. J . Solution Chem. 1978, 7,291. (36) Silvester, L. F.; Pitzer, K. S. J . Solution Chem. 1978, 7, 327. (37) Robinson, R. A,; Stokes, R. H. In Elecrrolyre Solutions, 2nd ed.; Butterworths: London, 1959; p 498.
