J . Phys. Chem. 1989, 93, 2176-2180
2176
Isothermal DMuslon Coefficients for NaCI-MgCI,-H,O Concentration Ratio of 3:l
at 25 ‘C.
1. Solute
John G. Albright,*lt Roy Mathew,’ Donald G . Miller,* and Joseph A. Rard University of California, Lawrence Livermore National Laboratory, Livermore, California 94550 (Received: May 24, 1988; I n Final Form: September 7, 1988)
Isothermal interdiffusion coefficients have been obtained by Gouy interferometry for the ternary system NaC1-MgC12-H20 at 25.00 OC and at a 3:l molarity ratio of NaCl to MgC1,. Data are reported for total molar concentrations of 0.5, 1.0, 2.0, 3.0, and 3.79 M. One cross-term diffusion coefficient, DI2,becomes greater than all the other diffusion coefficients above 3.2 M. This shows that at higher concentrations a flow of NaCl resulting from a concentration gradient of MgC12 will be greater than the flow resulting from the same concentration gradient of NaCl itself. Also, a concentration gradient of MgC12will cause a greater flow of NaCl than of MgC12itself. Densities were measured for all the solutions; they were used to calculate partial-molal volumes of all components in the solutions and to interconvert reference frames for diffusion.
Introduction There have been few systematic studies of any multicomponent transport property. In particular there is no mixed-electrolyte system for which all the necessary transport data over a wide range of concentration are available to calculate either the generalized ionic transport coefficients lik (Onsager coefficients) of linear or their equivalents such as velocity irreversible thermodynami~sl-~ correlation coefficient^.^" Such coefficients are directly related to statistical mechanical electrolyte and experimental values are needed to test the theoretical models. In 1984, an international collaboration was organized (now involving 10 laboratories), whose objective is to collect the appropriate data for a suitable ternary aqueous mixture. A higher valence mixture provides a more severe test of theories, and the system chosen was NaC1-MgCl2-H20 at 25 O C . This system is also important as a model of seawater and its evaporites. The data being collected are mutual- and tracer-diffusion coefficients, transference numbers, conductances, Soret coefficients, densities, viscosities, and activity coefficients, all at three mole ratios (3:1, 1:1, and 1:3) from dilute solution to near saturation (3.6-3.8 mol dm-3). Our own mutual-diffusion work will also include “near tracer” compositions of both NaCl and MgCI2 at the total molar concentrations of 0.5, 1.0, 2.0, 3.0, and 3.79 M. So far, activity coefficient,I0 tracer-diffusion coefficients and viscosities,” and mutual-diffusion coefficients in dilute solutionsI2 are available. Reported here are the first set of our mutualdiffusion coefficients from 0.5 to 3.8 M, at a mole ratio of NaCl to MgC12 of 3:l. There is good agreement at the overlap concentration of 0.5 M between our results from free-diffusion Gouy interferometry and Leaist’s resultsf2from the restricted-diffusion conductometric method. The only previously reported diffusion data for the NaC1MgCI2-H20 system are the data of Wendt and ShamimI3 and of Miller et al.14 Wendt and Shamim used the diaphragm-cell method at three sets of mean concentrations. None of their three sets corresponded to a 3:l concentration ratio of NaCl to MgC12, so no direct comparison can be made. Moreover, this method is less precise than the optical methods used here. Miller et al.I4 used Rayleigh interferometry for the seawater composition (0.489 M NaCl, 0.051 M MgC12), corresponding to a 9.57:l ratio. Four diffusion coefficients are necessary to describe isothermal mutual diffusion in a three-component system. According to the linear laws for this case, the flows, Ji,can be related to the solute concentration gradients by the equations -J1 = Dll
ac, ac2 - + D12 ax
ax
(1)
*Address correspondence to either author. Permanent address: Chemistry Department, Texas Christian University, Fort Worth, TX 76129. *Present address: Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455.
0022-3654189 12093-2 176%01S O ,I O I
,
In this article, subscripts 1 and 2 denote, respectively, NaCl and MgC12. Here, D l l and DZ2are main-term diffusion coefficients that relate flows of solutes to their own concentration gradients. D I 2and D21are cross-term diffusion coefficients that relate the coupled flow of one solute to the concentration gradient of the other. The Cts are molar concentrations, and x is the vertical distance. Because our experimental concentration differences are small, to a good approximation it may be assumed there is no volume change on mixing in this system. Consequently, the diffusion coefficients are relative to the uolume-fixed reference frame. Accurate densities were measured for the solutions used to perform the diffusion experiments. These were used to calculate molar concentrations, determine partial-molal volumes which can be used to interconvert frames of reference, and provide the necessary information to check the s t a t i ~ ’ and ~ J ~dynamicls stability of the experiments. Experimental Procedures Diffusion Measurements. The diffusion measurements were done by Gouy interferometry with the precision “Gosting” diffusiometer” now located at the Lawrence Livermore National Laboratory.18 All measurements were made at 25.00 OC. General ~
~~~
(1) Miller, D. G. J . Phys. Chem. 1966, 70, 2639. (2) Miller, D. G. J . Phys. Chem. 1967, 71, 616. (3) Miller, D. G. J . Phys. Chem. 1967, 71, 3588. (4) Woolf, L. A.; Harris, K. R. J . Chem. SOC.,Faraday Trans. 1 1978, 74, 933. (5) Hertz, H. G.; Harris, K. R.; Mills, R.; Woolf, L. A. Ber. Bunsen-Ges. Phys. Chem. 1977, 81, 664. (6) Miller, D. G. J . Phys. Chem. 1981, 85, 1137. (7) Altenberger, A. R.; Friedman, H. L. J. Chem. Phys. 1983,78,4162. (8) Thacher, T. S.; Lin, J. L.: Mou, C. Y . J . Chem. Phys. 1984,81,2053. (9) Kremp, D.; Ebeling, W.; Krienke, H.; Sandig, R. J . Stat. Phys. 1983, 33 99
(10) Rard, J. A,; Miller, D. G. J . Chem. Eng. Data 1987, 32, 85. (11) Mills, R.; Easteal, A. J.; Woolf, L. A. J . Solution Chem. 1987, 16, 0,c
03.’.
(12) Leaist, D. G.Electrochim. Acta 1988, 33, 795. (13) Wendt, R. P.; Shamim, M. J . Phys. Chem. 1970, 74, 2770. (14) Miller, D. G.; Ting, A. W.; Rard, J. A.; Eppstein, L. B. Geochim. Cosmochim. Acta 1986, 50, 2397. (15) Miller, D. G.;Vitagliano, V. J . Phys. Chem. 1986, 90, 1706. (16) Wendt, R. P. J . Phys. Chem. 1962.66, 1740. (17) Gosting, L. J.; Kim, H.; Loewenstein, M. A,; Reinfelds, G.; Revzin, A. Reu. Sci. Instrum. 1973, 44, 1602. (18) This instrument was originally built at the Institute for Enzyme Research in Madison, WI, by Dr. Gosting and co-workers. After Dr. Gosting’s untimely death, Prof. Kegeles moved the instrument to the University of Connecticut. Before Prof. Kegeles retired, he assisted D.G.M. in having the instrument moved to its new location at LLNL.
0 1989 American Chemical Society
Isothermal Diffusion Coefficients for NaCI-MgCI2-H20
The Journal of Physical Chemistry, Vol. 93, No. 5, 1989 2177
TABLE I: Data for Diffusion Measurements' expt 1
expt 2
expt 3
0.37503 0.12503 0.229 83 0.000 00 102.192 102.203 1.oooo
0.375 02 0.12503 0.000 00 0.091 04 96.887 96.912
0.0000
0.374 99 0.12502 0.18377 0.018 19 101.092 101.086 0.8085
0.75008 0.25003 0.235 88 0.000 00 100.868 100.8 15 1.oooo
0.750 10 0.250 02 0.000 05 0.092 61 95.620 95.616 0.0002
0.75008 0.250 02 0.188 70 0.018 53 99.709 99.778 0.8083
1.500 28 0.500 19 0.247 52 0.000 02 99.565 99.510 0.9998
1.500 25 0.500 19 0.00005 0.095 44 93.872 93.818 0.0002
1.50019 0.500 18 0.19808 0.01909 98.329 98.384 0.8092
2.251 12 0.752 11 0.258 76 0.000 02 98.892 98.878 0.9999
2.251 13 0.752 12 0.000 11 0.098 30 92.543 92.529 0.0005
2.251 04 0.75209 0.207 01 0.01968 97.590 97.605 0.8103
2.84499 0.948 5 1 0.267 10 0.000 06 98.539 98.510 0.9994
2.845 07 0.948 54 0.000 11 0.10009 91.678 91.564 0.0004
2.845 06 0.948 53 0.213 61 0.02005 97.070 97.071 0.8109
expt 4
expt 1
expt 2
expt 3
exDt 4
1.5158 1.5161 -32.36 -32.56 -13.98 -13.83 1.01734 1.026 44
0.9752 0.9749 59.27 58.86 28.57 27.77 1.01844 1.025 34
1.3827 1.3821 7.50 7.59 3.22 3.29 1.01756 1.026 22
1.060 1 1.0607 60.39 60.90 28.27 28.14 1.018 22 1.025 56
1.5325 1.5324 -42.27 -42.80 -18.13 -18.04 1.04139 1.050 48
1.0173 1.0171 66.90 66.58 31.56 30.90 1.042 5 1 1.049 37
1.4070 1.4070 -2.52 -1.99 -1.07 -0.85 1.041 61 1.050 26
1.1003 1.1006 62.14 62.44 28.69 28.42 1.042 28 1.04961
DA X lo9 (expt) DA X lo9 (calcd) Qo X lo4 (expt) Qo X lo4 (calcd) Q, X lo4 (expt) Q, X lo4 (calcd) d (top) d (bottom)
1.5675 1.5676 -57.38 -58.00 -24.07 -23.98 1.087 96 1.096 86
1.0905 1.0904 83.02 82.40 37.69 37.00 1.088 92 1.095 69
1.4553 1.4551 -15.81 -15.27 -6.58 -6.42 1.087 98 1.096 62
1.1703 1.1705 67.65 68.33 30.23 30.17 1.088 70 1.095 95
Series AM9; Total Concn = 3.0 M 2.251 09 DA X lo9 (expt) 0.752 11 DA X lo9 (calcd) 0.051 83 Q, X lo4 (expt) 0.078 64 Qo X lo4 (calcd) 93.779 Q, X lo4 (expt) 93.792 Q, X lo4 (calcd) 0.21 11 d (top) d (bottom)
1.5893 1.5887 -69.62 -69.22 -28.65 -27.94 1.132 63 1.141 91
1.1437 1.1434 98.23 98.65 43.39 42.79 1.13391 1.14065
1.4857 1.4862 -23.61 -23.95 -9.92 -9.80 1.13288 1.141 65
1.2198 1.220 1 76.9 1 76.42 33.47 32.66 1.13365 1.140 90
1.5727 1.5741 -79.21 -79.03 -31.87 -31.34 1.166 19 1.17630
1.1490 1.1499 106.90 107.26 45.82 45.46 1.168 18 1.174 87
1.4790 1.4777 -3 1.46 -3 1.54 -12.55 -12.67 1.167 10 1.17595
1.2252 1.2242 80.58 80.13 34.62 33.50 1.167 87 1.175 14
Series AM3; Total Concn = 0.5 M 0.374 98 DA x io9 (expt) 0.12503 0.045 96 0.072 83 97.997 97.967 0.2087
DA X lo9 (calcd) Qo X lo4 (expt) Qo X lo4 (calcd) Ql X lo4 (expt) Ql X lo4 (calcd) d (top) d (bottom)
Series AM21; Total Concn = 1.0 M 0.750 08 DA X lo9 (expt) 0.250 02 DA X lo9 (calcd) 0.047 25 Qo X lo4 (expt) 0.074 11 Qo X lo4 (calcd) 96.710 QI X lo4 (expt) 96.698 Ql X lo4 (calcd) 0.2088 d (top) d (bottom)
Series AM4; Total Concn = 2.0 M 1.500 32
0.500 20 0.049 59 0.076 34 94.909 94.963 0.2098
Series AM6: Total Concn = 3.79 M
"Units:
C,and AC,, mol dm-);
DA, m2
2.845 04 0.948 54 0.053 60 0.080 11 92.871 93.013 0.2122
DA X lo9 (expt) DA X lo9 (calcd) Qo X lo4 (expt) Qo X lo4 (calcd) Ql X lo4 (expt) QI X lo4 (calcd) d (top) d (bottom)
, d, g cm-3.
procedures and photography sequences are those developed by Gosting and coworkers (e.g., see ref 19-21). The light source was a mercury vapor lamp with a Wratten 77A filter (A = 546.1 nm). A quartz diffusion cell of good optical quality was used for all the experimentsz2 Its inside dimension, the a distance, was 2.5064 cm. The distance from the center of the diffusion cell to the photographic plate, the b distance, was 3.088852 m. All reference-fringe photographs were taken on Kcdak Metallographic 4 X 5 in. glass plates, and Gouy pattern photographs were taken on Kodak 11-G 4 X 5 in. backed glass plates. Fringe positions were measured with a Gaertner Tool Makers Microscope. This unit is fitted with a scanning device that makes it possible to measure fringe minima to within f l pm.23,24 (19) Gosting, L. J.; Morris, M. S. J . Am. Chem. SOC.1949, 72, 1998. (20) Gosting, L. J. J. Am. Chem. SOC.1950, 72, 4418. (21) Woolf, L.A.; Miller, D. G.; Gosting, L. J. J. Am. Chem. SOC.1962, 84, 317. (22) This cell had originally been used by H. Kim at the Institute for Enzyme Research, University of Wisconsin, for a series of ternary experiments. We tested the quality of the cell by photographing Rayleigh patterns from the cell in which there was a solution with a high refractive index in the diffusion channel and bath water in the reference path. These base-line patterns were straight to better than 0.05 fringe, showing that the windows were very flat. (23) Albright, J. G. Ph.D. Thesis, University of Wisconsin, Madison, 1962.
The initial top and bottom concentrations were chosen so that the total number of Gouy fringes would be about 90-100 fringes. Usually 10 photographs of Gouy fringe patterns were taken at approximately even intervals of l/t: where t'was the clock time from the start of an experiment. The first photograph was usually taken when the Gouy pattern was between 3.5 and 4.0 cm in length. The l/t'interval between the last two photographs was half the interval between the other photographs. This was done so that the last pattern would still be at least 1.5 cm in length. When reading the plates, each of the positions of the outer nine fringes (fringes 0-8) were measured, and from there on the position of every other fringe was measured. Fringe positions were recorded as close to the (overexposed) undeviated slit image as possible, while still being able to locate accurately the positions of the minima.
Experiments were performed at each of five total concentrations, Le., at C2) = 0.5, 1.0, 2.0, 3.0, and 3.79 total molarity. Four reliable experiments were performed at each set of mean conand c2. These were at ratios of a I = RIAC1/ centrations, (RIAC1 R2AC2)that were typically 0.0,0.2,0.8, and 1.0. These a , are included in Table I. The ACi values are the differences of the solute concentrations across the initial free-diffusion (i.e.,
(cl+ +
cl
(24) Wendt, R. P. Ph.D. Thesis, University of Wisconsin, Madison, 1960.
2178
The Journal of Physical Chemistry, Vol. 93, No. 5, 1989
step-function) boundary of the experiment. The coefficients, Ri, are the derivatives of refractive index with respect to the concentrations, Ci. According to t h e ~ r yat, least ~ ~ ~two ~ experiments ~ at the same mean concentration but two different ACl/AC2 ratios are required in order to have sufficient data to calculate the four diffusion coefficients. However, at least four reliable experiments were performed in order (1) to give additional statistical diagnostic information, (2) to increase the accuracy of the calculated Dik, (3) to be sure the boundaries are statically and dynamically stable,ls and (4) to verify that all experiments are consistent. A few experiments that appeared to be of poor quality were repeated. Calculations. A set of new computer programs were written and used for the analysis of fringe-position data from Gouy fringe patterns. These programs are fully described in a companion a r t i ~ l e . ~ Program ' F3 was used for the analysis of data if lQol 1 25 X loy4,and program F2P was used if lQol< 25 X lo4 (see note 28). The quantity Qo was introduced by Fujita and Gosting,26who denoted it by Q (See note 29). These programs provide a value of DA and a value of Qo for each individual experiment. The sets of DA and Qo for each set of experiments are subsequently used to get the Dik. The standard deviation of the difference between measured fringe positions, Yj, and positions calculated with the regression parameters were usually better than f 4 pm for the early photographs and often better than f 2 pm for the later photographs. I n some cases, the average of these standard deviations over all the patterns was better than f 2 pm. Density Measurements. Two methods were used to measure the density of the solutions. A Sodev vibrating densimeter was used to measure densities for experiments in series AM3, AM4, and half of those in series AM6. The meter was fitted with a platinum-iridium vibrating tube because the standard stainless steel tube was not inert to salts with high chloride concentrations. The meter was calibrated with water and an electrolyte solution whose density had previously been determined by pycnometry. The density measurements with this instrument appeared to have a precision of f0.00005 g cm-3 at higher concentrations and a better precision at lower concentrations. Densities for all experiments at 1.0 and 3.0 total molarity and half of those a t 3.8 total molality were measured with a single-stem pycnometer. These measurements appeared to have a precision of better than fO.OOOO1 g ~ m - In ~ .general, we believe the pycnometer results are more reliable. Whenever possible, partial-molal volumes were computed from pycnometer data, since these volumes are very sensitive to an inaccurate measurement. Preparation ofSolutions. All solutions were prepared gravimetrically to 0.1-0.3-mg level of accuracy, and all weights were corrected to masses. A preliminary estimate of the density to calculate the weights S obtained from a modified corresponding to the desired C ~was form of Young's rule:30
d = Y,d1° + Y2d2O + BYIY2I
(3)
where I is the ionic strength, and ionic strength fractions are given by (25) Fujita, H.; Gosting, L. J. J. Am. Chem. Soc. 1956, 78, 1099. (26) Fujita, H.; Gosting, L. J. J . Phys. Chem. 1960, 64, 1256. (27) See: Albright, J. G.;Miller, D. G.J. Phys. Chem., accompanying paper in this issue. (28) F3 and F2P denote computer programs that are described in the companion article. It should be noted that it was never appropriate to use program F2M for this set of experiments because no values of r- were small. (See ref 27.) (29) The quantity Qo is SbQd(rcz)). The notation Qo was introduced to distinguish it from , which is identified as the first moment of the area under the omega graph, EQj(z) d(rcz)). The first moment is used in a least-squares analysis procedure developed by Revzin (ref 34) to calculate Din. That procedure, in turn, is based on an unpublished derivation by Fujita. Although Q,was not used in the Din calculations, it is a measure of the shape of the omega graph. (30) It has been shown by one of us (D.G.M.) that eq 3, without the B term, is equivalent to Young's rule for apparent-molar volumes in terms of molarities (to be published).
Albright et al. (4) (5)
The values di" are the densities of the binary s o l ~ t i o n s at ~ ~the -~~ same ionic strength of the actual ternary density measurement. The term with the coefficient B gives a small correction to Young's rule and can be used if one density exists from which B can be calculated. The NaCl was Baker analytical reagent grade which had been carefully dried at 80 to 120 O C . Stock solutions of MgCI2 were prepared by dissolving Mallinckrodt AR MgCI2.6H20in purified water, and all solutions for the experiments were then prepared from these stocks. Three stock solutions were used in this series of experiments. Each one was carefully analyzed in two ways. In one procedure, the samples of MgCI2 were converted to MgSO,, dried at 500 "C, and weighed. In the other procedure, samples of the MgCI2 were mass titrated with standardized AgN03 solution by Fajans' method. The two methods of calibration of the stock solutions agreed with each other to within f0.1%, and the mean value was used. Double-distilled water was used in the preparation of all the solutions. The molecular weights used for NaCI, MgCI2, and H 2 0 were 58.443,95.211, and 18.015 g mol-], respectively. Experimental Results
Diffusion Measurements. Given in Table I are both measured (expt) and calculated (calcd) values of J , DA, Qo, and QI. The calculated values were obtained by applying the equations of Fujita and G o ~ t i n g . This ~ ~ , was ~ ~ done once Ri, IA, SA,Eo, E , , and E2 were obtained in the process of calculating the four diffusion coefficients by the Fujita-Gosting method.21,2s*26 We use a program that is an extended version of one originally described by R e ~ z i nfor~ these ~ calculations, denoted here by RFG. Note that there are two degrees of freedom for the fit of each of J , DA, and Qo. If only two diffusion experiments had been performed, the measured and calculated values would be the same for each of these parameters. As can be seen from Table I, the measured and calculated values of J among four experiments generally agree within 0.10 fringes, and agreement to 0.05 fringes or less is typical. Discrepancies in J can be attributed to two basic causes. One is the error associated with reading photographs of the fractional part of a fringe.20 The other is the error resulting from uncertainty in the preparation of the solutions for the experiments, and possibly some uncertainty due to evaporation of the solutions in the diffusion cell reservoirs prior to the start of the experiment. In relation to the latter source of error, the agreement shows that almost all RiACi values are accurate to a part per thousand or better relative to the sum R I A C , + R2AC2. The largest discrepancy in J occurs at the highest concentrations, series AM6, for which precise preparation of the solutions was the most difficult. The agreement between measured and calculated values of DA is remarkably good, which is an indication of the high precision of the Gosting diffusiometer. The agreement here between measured and calculated values of Qo is always better than f 1 X IO-, and approaches the internal agreement of Qo calculated from the fringe patterns within a single experiment. Usually the error in Qo is the most critical to the accuracy of the Dik values. The agreement between experimental and calculated Qo values is also good relative to agreement reported in previous ternary experiments on this d i f f u ~ i o m e t e r . ~ ~ - ~ ~ (31) Rard, J. A.; Miller, D. G.J . Solution Chem. 1979, 8, 701. (32) Miller, D. G . ;Rard, J. A.; Eppstein, L. B.; Albright, J. G. J . Phys. Chem. 1984,88, 5739. (33) The densities listed in ref 31 were based on a density of 0.997072 mL-' for water at 25 "C. They were converted to a base of 0.997 045 g cmfor these calculations. (34) Revzin, A. Ph.D. Thesis, University of Wisconsin, Madison, 1969. (35) Kim, H.; Reinfelds, G. J . Solution Chem. 1973, 2 , 477. (36) Kim, H. J . Solution Chem. 1974, 3, 149. (37) Kim, H. J . Solution Chem. 1974, 3, 271.
9
Isothermal Diffusion Coefficients for NaCI-MgC12-H20
The Journal of Physical Chemistry, Vol. 93, No. 5, 1989 2179
TABLE 11: Experimental Results" series
Cl + c2 Cl C2
ml(cl,c2) m2(CI,C2) R 1 X lo' R 2 X IO3
a
HI H2
6
v2 VO
u+ X u- X lo4 (Dll), X lo9 ( 4 2 ) " X lo9 X lo9 (D22), X lo9 (Dll)0 X lo9 ( 4 2 ) o X lo9 (DZ1),,X lo9
(oZ2),, x 109
AM3
AM21
AM4
AM9
AM6
0.50003 0.375 00 0.12503 0.379 53 0.12654 9.6898 23.191 1.021 89 39.406 74.762 19.080 20.495 18.056 1.3124 0.7094 1.367 0.577 0.045 0.805 1.377 0.587 0.048 0.808
1.000 10 0.750 08 0.25002 0.766 72 0.255 57 9.31 14 22.491 1.04594 38.558 74.057 19.915 21.185 18.042 1.3569 0.7141 1.328 0.692 0.062 0.810 1.349 0.7 16 0.069 0.818
2.00045 1.SO0 26 0.500 19 1.567 67 0.522 66 8.7550 21.417 1.092 30 36.766 70.957 2 1.642 24.214 17.985 1.SO92 0.7083 1.270 0.936 0.092 0.804 1.318 0.999 0.108 0.825
3.003 21 2.251 10 0.752 11 2.409 90 0.805 17 8.3233 20.499 1.13727 35.860 68.536 22.470 26.543 17.926 1.7643 0.6998 1.219 1.181 0.1 16 0.777 1.293 1.295 0.141 0.8 15
3.793 57 2.845 04 0.948 53 3.10957 1.036 72 8.0256 19.932 1.171 52 35.080 67.253 23.179 27.738 17.874 2.0183 0.7086 1.156 1.284 0.131 0.751 1.246 1.437 0.161 0.802
P-=
.- I
=/'
c1
c2
Figure 1, Mutual-diffusion coefficients versus total concentration: 0. this research; X, Leaist's data. dY
'
E
c1/c2=3
..
"Units: C,,mol dm-); mi, mol (kg of H,O)-'; R,, dm3 mol-': d, g ~ m - HI, ~ : g mol-'; V,, cm3 mol-'; u+, u-, m-2 s; (DIJv, (Di& m2 s-I. See text for discussion of errors in (03,. To use eq 6 directly with d and C, in these units, divide the tabulated Hi by 1000.
Each Ri value listed in Table I1 is an approximation of the derivative of refractive index with respect to the molar concentration of component i. They are calculated from the values of J , ACI, and AC2 along with the a distance and the wavelength of the light source.4o The quantities, Hi (=dd/aCi),listed in Table I1 were calculated by fitting density data from each series of experiments to the equation4I d = d HI(CI - C1) + H2(C2 - Cz) (6)
+
cl
c2
Here, d is the mean density and and are the mean concentrations at each overall composition. These d , Hi, and Ci can also be used to obtain the partial-molal volumes 6 by using the equations of Dunlop and G o ~ t i n g .All ~ ~density data in Table I were least-squared for systems AM3, AM9, and AM21. However, the density of AM4-1 (top) was discrepant and thus not used. For series AM6, only the densities of experiments 3 and 4 were done by pycnometry; therefore only these data were least-squared. For each mean composition, densities at neighboring concentrations can be calculated with eq 6. The quantities Hi together with values of Dik were substituted into equations that show whether the diffusion boundaries were completely ~ t a b 1 e . l Note ~ that having the more dense solution below the initial free-diffusion boundary does not ensure stability within the diffusion boundary itself. However, unless the instability is unusually severe, the problem may not be apparent from either the Rayleigh or Gouy fringe patterns obtained from the standard optical arrangement.42 Wendt16 first gave equations that describe requirements for static stability, and M ~ D o u g a l gave l ~ ~ equations (38) Kim, H.; Deonier, R. C.; Reinfelds, G. J. Solution Chem. 1974, 3, 445. (39) Jordan, M.;Kim, H. J . Solution Chem. 1982, 1 1 , 347. (40) The refractive index difference between top and bottom solution An is given by R , A C I + R2AC2. Since J = uAn/X, it is also possible to write J = R I * A C I+ R2*ACZ.Since the ratios of R , / R , and R,*/Rj* are the same, and only the ratios are used in calculating diffusion coefficients, it is only a matter of convenience which choice is made. Both descriptions are found in the literature. (41) Dunlop, P. J.: Gosting, L. J. J . Phys. Chem. 1959, 63, 86. (42) The standard optical arrangement averages the refractive index acrcss each horizontul plane in the cell. However, if a camera lens and horizontal cylinder lens are added beyond the regular camera position (a type of Philpot-Svensson optical system), the relayed Gouy fringes do show distortions in an unstable diffusion boundary. See ref 15.
Figure 2. Partial-molal volumes of NaC1, MgCI2, and H 2 0 versus the square root of total concentration.
for dynamic stability. McDougall's equations have been extended to the general free-diffusion case and given in our notation by Miller and Vitagliano15and Vitagliano et al." All the experiments listed in Table I were found to be well within both criteria for stability. Given in Table I1 are the volume-fixed diffusion coefficients, (Djk)",for this study. They are shown in Figure 1 as a function of total concentration. Also shown in Figure 1 are data obtained at low concentration by Leaist." Our data and his appear to be in good agreement at the point of overlap (0.5 total molarity) and run smoothly into each other. We believe our results for D l l and D22are accurate to within 1% of their own values. We believe D21is accurate to within 1% of the value of D l l . Because D , 2 is the most sensitive to Qo,it appears to be the least accurately known. Its uncertainty may be 2-3% of its own value. These estimates are based on the examination of a large number of other experiments as well. A rough rule of error estimation for a four-experiment series is that the probable uncertainty in any D , is four times its standard error as calculated by the propagation of error equations. The partial-molal volumes of the two solutes and solvent, Vl, and respectively, are given in Table I1 and shown in Figure 2. They have been used to calculate the diffusion coefficients for the solvent-fixed (H20) reference frame, (Dik)o,which are also listed in Table 11.
r2, ro,
Discussion An interesting and quite surprising result is that the cross-term diffusion coefficient, D I 2 ,becomes larger than the other three diffusion coefficients a t high total concentration. This means that the flow of NaCl resulting from a concentration gradient of only MgC12 will be greater than the flow of MgC12 itself; it is also greater than the flow of NaCl resulting from the same concentration gradient of NaCl itself. (43) McDougall, T. J. J. Fluid Mech. 1983, 126, 379. (44) Vitagliano, P. L.; Della Volpe, C.; Vitagliano, V. J . Solution Chem. 1984, 13, 549; misprints corrected in J . Solution Chem. 1986, 15, 8 1 1 and ref 15.
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The Nernst-Hartley equations generalized to multicomponent s y ~ t e m s ~are . ~based ~.~~ on infinite dilution ionic conductances and are often used to estimate Dik values. However, these equations predict constant Dik at a constant mole ratio, i.e., horizontal lines on Figure 1. This is clearly not the case, particularly for D 1 2 . Consequently, the Nernst-Hartley equations are quite unsuitable as an approximation at high concentrations. Other estimation procedures will be the subject of another paper. Partial-molal volumes plotted against the square root of the total concentrations are shown in Figure 2. It is interesting that they give nearly straight lines to quite high concentrations. Note that the cross each other. The values of the molal volumes at infinite dilution for NaCl and MgC12, 16.62 and 14.49 cm3 mol-', respectively, were calculated from the ion molal volumes for infinite dilution tabulated by miller^.^' The value 18.07 cm3 mol-' was used for pure water. The results for the 3:l NaCI-MgCI2-H20 system show that it is not possible to anticipate values of diffusion coefficients at high concentration, given only the diffusion coefficients at low concentration. These results also show that the cross-term diffusion coefficients can become very large in ternary electrolyte solutions at high concentrations. The diffusion Onsager coefficients, (L& and (Lij),,have been calculated from the data of this paper and the activity coefficient
vi
Albright et al. derivatives4* based on osmotic coefficient data for this system.I0 The Onsager reciprocal relations are found to be satisfied with reasonable estimates of the errors of the experimental quantities. These results, and the use of diffusion Lik and ionic l j k for estimating diffusion coefficients, will be presented elsewhere. The authors note that this is the first of several papers from LLNL that will present isothermal diffusion results for this system at different mole ratios. Other types of transport data will be forthcoming from other laboratories around the world. It is hoped that the compilation and eventual interpretation of all these data will give new insights into transport properties of dilute to concentrated aqueous multicomponent electrolyte solutions. Acknowledgment. This work was primarily performed under the auspices of the US. Department of Energy at Lawrence Livermore National Laboratory under Contract No. W-7405ENG-48. J.A.R. and D.G.M. thank the Office of Basic Energy Sciences (Geosciences) for support. D.G.M. and J.G.A. thank Dr. Christopher Gatrousis for CRR support. J.G.A. also thanks TCU for Research Fund Grant No. 5-23824. R.M. thanks TCU for supporting him through a research fellowship. The research published here is based in part on the Ph.D. dissertation of R.M., TCU. We thank Prof. Derek Leaist for permission to use his data points in Figure 1. Registry No. NaCI, 7647-14-5; MgCI,, 7786-30-3.
(45) Gosting, L. J. Advances in Profein Chemisfry;Academic Press: New York, 1956; Vol. 1 1 , pp 429-554. (46) Wendt, R. P. J . Phys. Chem. 1965, 69, 1227. (47) Millero, F. J. Chem. Rev. 1971, 71, 147.
(48) The appropriate equations for calculating the derivatives of the activity coefficients from a Scatchard neutral electrolyte representation of the osmotic coefficients have been derived by one of us (D.G.M.;to be published).
