J . Phys. Chem. 1989, 93, 4370-4374
4370
as has been observed b e f ~ r e . ' - ~Some , ' ~ modest improvement is obtained up to moderate concentrations with thermodynamic and viscosity factors, as we and Leaist'O have independently observed. However. good approximations for higher concentrations are still elusive. The thermodynamic (Onsager) diffusion coefficient^^^.^^^^^.^' ( L,,)o and (L,,)" can be calculated by using data from this paper and activity derivatives from osmotic coefficient data." Preliminary calculations of this type have been made. The Onsager reciprocal relations are found to be satisfied within reasonable estimates of the experimental errors of the L,. These results, and methods to estimate D,, using I,, or other procedures, will be presented elsewhere so that data for all C , / C ratios can be given together (to be submitted to this journal). Partial Molal Volumes. Partial molal volumes of NaCI, MgCI2, and H 2 0 are plotted in Figure 2. The values at infinite dilution for NaCI. MgCI,, and H 2 0 used in this plot were 16.62. 14.49, (47) Miller. D. G. J. Phys. Chem. 1959, 63, 570
and 18.07 cm3 mol-', respectively. The first two were calculated from the ionic molal volumes a t infinite dilution tabulated by miller^.^^ It is interesting that our graphs of vl versus C1I2are relatively straight. However, care should be taken in the interpretation of this fact because of the changing concentration ratios of the solutes.
Acknowledgment. This work was principally performed under the auspices of the U S . Department of Energy a t the Lawrence Livermore National Laboratory under contract W-7405-ENG-48. J.A.R. and D.G.M. thank the Office of Basic Energy Sciences (Geosciences) for support. D.G.M., L.P., and J.G.A. thank Dr. Christopher Gatrousis for C R R support. J.G.A. also thanks TCU for research fund grant 5-23824. R.M. thanks TCU for supporting him through a research fellowship. This research is based in part on the Ph.D. dissertation of R.M., T C U . Registry No. NaCI, 7647-14-5; MgC12, 7786-30-3. (48) Millero, F. J. Chem. ReL: 1971, 71, 147
Isothermal Diffusion Coefficients for NaCI-MgCI,-H,O at 25 "C. 3. Low MgCI2 Concentrations with a Wide Range of NaCl Concentrations Roy Mathew,+ Luigi Paduano,* John G . Albright,*,$ Donald G . Miller,* and Joseph A. Rard University of California, Lawrence Livermore National Laboratory, Lioermore, California 94550 (Received: October 25, 1988)
Isothermal interdiffusion coefficients have been measured by Gouy interferometry for the system NaCI( 1)-MgCl2(2)-H20 at 25 "C. Diffusion coefficients have been obtained for five sets of total mean concentrations of C, + C2 = 0.536, 1.053, 2.05, 3.05, and 3.80 mol d ~ n - for ~ , which the concentrations C2 of MgC12 are kept small. The main-term diffusion coefficients D,, and D2, vary modestly from their values at infinite dilution as the NaCl concentration is increased. However, the cross-term diffusion coefficient D12,which relates the coupled flow of NaCl to the concentration gradient of MgCI2, increases sharply as the total concentration increases. It becomes even larger than both main-term diffusion coefficients at higher NaCl concentrations. Thus, a gradient of MgCI2 will cause a greater flow of NaCl than will the same gradient of NaCl itself. Also a gradient of MgCI2 will cause NaCl to diffuse faster than MgC12 itself. When combined with diffusion data for other salt ratios, extrapolation of D22to C2 = 0 at constant C = C, + C2 will yield the tracer-diffusion coefficient of Mg2+ in NaCl solutions. Densities were also obtained in this study. These diffusion and density data are part of a systematic study of this system at LLNL.
Introduction Because evolving theories for transport in aqueous strong electrolyte solutionsI4 require testing against experimental data, it is desirable to measure accurate data for representative systems. This paper is part of an international collaboration, initiated in 1984, for an extensive study of the system NaCI(1)-MgCI2( 2 ) - H 2 0 at 25 O C . At present, dilute ternary interdiffusion (mutual-diffusion) data,s thermodynamic activity data,6 and intradiffusion (self-diffusion) and viscosity data7 have been published. Some of the ternary interdiffusion and density data a t moderate to high concentrations have also been published or have been submitted.8-10 Work in progress a t other laboratories includes the measurement of Soret coefficients, transference numbers, and conductances. Ultimately, there will be sufficient data to calculate the generalized ionic transport coefficients of irreversible thermodynamics, the I , coefficients (ionic Onsager coefficient^),'^-'^ as well as their constituent parts, the intradiffusion and distinct-diffusion coefficient^.^
'
Present address: Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455. *Present address: Dipartimento di Chimica, University of Naples, Naples, Italv. #'Permanent address: Department of Chemistry, Texas Christian University. Fort Worth, TX 76129.
0022-3654/89/2093-4370$01.50/0
The system NaCI-MgCI2-H2O is of particular interest because it is a higher valence electrolyte mixture and thus will provide an important and demanding test system for transport theories. It is also of practical importance as a simplified model for sea water and its evaporites. Intradiffusion coefficients are also of interest because they are ( I ) Altenberger, A. R.; Friedman, H. L. J. Chem. Phys. 1983, 78,4162. (2) Thacher, T. S . ; Lin, J.-L.; Mou, C. Y. J . Chem. Phys. 1984,81, 2053. (3) Kremp, D.; Ebeling, W.: Krienke, H.; Sandig, R. J. Stat. Phys. 1983, 33, 99. (4) Zhong, E. C.; Friedman, H. J. Phys. Chem. 1988, 92, 1685. (5) Leaist, D. E. Electrochim. Acta 1988, 33, 795. (6) Rard, J. A.; Miller, D. G. J. Chem. Eng. Data 1987, 32, 85. (7) Mills, R.; Easteal, A. J.; Woolf, L. A. J. Solution Chem. 1987, 16, 835. (8) Albright, J. G.;Mathew, R.; Miller, D. G.; Rard, J A. J. Phys. Chem.
J. Phys. Chem., preceding paper in this issue. (10) Some earlier mutual-diffusion work exist for this system: three dilute compositions by Wendt and Shamim" using the less accurate diaphragm cell, and the sea water composition by Miller et a1.12 using the more accurate Rayleigh optical method. (11) Wendt, R. P.; Shamim, M. J. Phys. Chem. 1970, 74, 2770. (12) Miller, D. G.; Ting, A. W.; Rard, J. A,; Eppstein, L. B. Geochim. Cosmochim. Acta 1986, 50, 2391. (13) Miller, D. G. J. Phys. Chem. 1966, 70, 2639. (14) Miller, D. G. J . Phys. Chem. 1967, 71, 616. ( 1 5 ) Miller, D. G. J . Phys. Chem. 1967, 71. 3588.
8 1989 American Chemical Society
Isothermal Diffusion Coefficients required for calculating velocity correlation coefficientsIG1*and because they are useful for analyzing generalized transport p h e n ~ m e n a . ~In addition, diffusion of radioactive wastes into natural waters or brines involves diffusion of the trace species. Tracer-diffusion coefficients have traditionally been measured with isotopically labeled components by using diaphragm or capillary diffusion cells, with analysis by radioactive counting or by mass spectrometry. N M R techniques have also been used. I n this study, the low MgCI2 concentrations correspond to “near-tracer” concentrations of Mgz+ in NaCl solutions. As indicated in paper 2 of this ~ e r i e sextrapolation ,~ of the DZ2to C2 = 0 at constant C = C , C2 will yield the tracer-diffusion coefficient D2* of Mgz+ in NaCl solutions a t that total concentration. This method is a new application of optical techniques to obtain tracer-diffusion coefficients of the noncommon ions without the use of any isotopically labeled ions. Our optical diffusion methods have the added advantage of yielding all four interdiffusion coefficients, but of course cannot yield the tracer-diffusion coefficients of the common ion or of water. The optical technique also cannot yield the intradiffusion coefficients when both solutes are at finite concentrations; such measurements do require isotope labeling or N M R methods. Our extrapolated D2*,to be reported elsewhere, were used by Mills et aL7 to calibrate their radioactive-tracer measurements using tracer Ca2+ as a substitute for tracer Mg2+ in NaC1-MgC12-Hz0.
The Journal of Physical Chemistry, Vol. 93, No. 10, 1989 4311
3
+
Experimental Section The apparatus, plate reading, and experimental procedures for Gouy interferometry were essentially the same as described bef ~ r e . ~All! ~measurements were done a t 25.00 f 0.02 OC, with the temperature constant to f0.005 OC during an experiment. Density measurements were done pycnometrically, and the water calibration density at 25.00 OC was 0.997045 g The integer part of the total number of Gouy fringes J was determined either from a Rayleigh photograph or by an extrapolation of the highnumbered fringes to the undeviated slit image.I9 The fractional part of J was obtained by the usual procedure.20 Molecular masses of NaCI, MgCI2, and H 2 0 were 58.443,95.211, and 18.015 g mol-’, respectively.
Analysis of Diffusion Data Programs Used f o r Analysis of Fringe Patterns. A set of nonlinear least-squares programs have been written at this laboratory for the analysis of Gouy data from isothermal ternary diffusion experiments. These are fully described elsewhere2’ and will be identified here by the labels used there. We used the F3 program when Qo> 25 X lo4 and J is known and the F2P program when Qo 25 X lo4 and J is known. Here, Qois the area under the R graph.22 Note that for this series of experiments, the conditions were such that the use of the F2M program is not justified. Generally, the 6 correctionsz4are very similar for all experiments of a set a t the same mean concentrations. However, there were
(16) Hertz, H. G.; Harris, K. R.; Mills, R.; Woolf, L. A. Ber. Bunsen-Ges. Phys. Chem. 1977, 81, 664. (17) Woolf, L. A.; Harris, K. R. J . Chem. SOC.,Faraday Trans. 1 1978, 74, 933. (1 8) Miller, D. G. J . Phys. Chem. 1981, 85, 1 1 37. ( 1 9) Miller, D. G.; Sartorio, R.; Paduano, L., manuscript in preparation. (20) Gosting, L. J.; Morris, M. S. J . Am. Chem. SOC.1949, 71, 1998. (21) Albright, J. G.; Miller, D. G.J . Phys. Chem. 1989, 93, 2169. (22) The quantity Qois -fJn dcf(z)). The usual notation has been Q, but Qo is introduced to distinguish it from Q,,which is identified as the first moment of the area under the omega graph, i.e., -folQflz) dmz)). The first moment is used in a least-squares analysis procedure developed by Revzin2’ to calculate Di . That procedure, in turn, is based on an unpublished derivation by Fujita as described by R e v ~ i n . * ~ (23) Revzin, A. Ph.D. Thesis, University of Wisconsin, Madison, WI. 1969. We thank Dr. Revzin for sending us his version of the RFG code. (24) Gosting, L. J. J. Am. Chem. SOC.1950, 72, 4418.
Figure 1. Partial molal volumes of NaC1, MgCI2, and H 2 0 versus the square root of volumetric ionic strength I, where I = (C, 3C2) mol dm-’ at ‘near-tracer” concentrations of MgC12.
+
two cases, reported in Table I, where the measured 6 seemed significantly out of line with the others of that series (4 or more pm). The source of this difference was not clear but could involve dirt or grease on the diffusion cell window or slipping of a photographic plate in the plate holder between reference and cell exposures. Using the average 6 correction from the remainder of the experiments of the set greatly improved consistency of this previously discrepant experiment. Our programs yield a C:5926 and Qofor each fringe pattern, which are obtained from the least-squares parameters for that pattern. An apparent DA‘ can then be calculated from C,, but it is based on the experimental clock time t’. Because the initial boundary is never infinitely sharp, t’is not the “true” time t but is related to the true time by t = t ’ + At. The At correction can be obtained by plotting DA’ versus 1/ t ’ for all fringe patterns of a single experiment. The intercept is the correct DA for that experiment, and the slope is DAAt.27 Accuracy of Fits. One diagnostic from programs F3 and F2P is the standard deviation of measured minus calculated fringe positions for each fringe pattern. In this series of experiments, this deviation was typically less than f 3 pm for early fringe patterns and less than f 2 pm for later patterns. Another diagnostic is the scatter of the values of DA, which are calculated from the C, and corrected time for each fringe pattern. These were typically within i0.1% or better of their mean. This mean is ordinarily very close to the extrapolated value. However, we use the extrapolated value in subsequent calculations because experience shows it gives slightly more consistent results. Calculation of Di,. The Dij are obtained from DA, Qo,J , and ACi by using the RFG code. This code is our extended version of a program originally developed by and is based on the Fujita and Gosting procedure^.^^*^^ The ACi are the concentration differences between the upper and lower solutions.
Experimental Results Concentrations. Concentrations of the solutions in mol dm-3 (M) for all experiments are listed in Table I. The number of significant figures for concentration values are representative of the precision of the solution preparations; however, their relative accuracy is probably only i O . I % . The ACi are also listed in Table I. (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) Longsworth, L. G. J . Am. Chem. SOC.1947, 69, 2510. (28) Revzin’s program can use values of ACi, J , DA, Qo,and/or Q,to calculate the four diffusion coefficients based on the iterative least-squares methods given by Fujita and G o ~ t i n g . ~We ~ .have ~ ~ found that Q,or combined Qoand Q,do not lead to improvement in the consistency of Di calculations. Consequently, we use only Qo.This Q,is a measure of the shape of the n graph.
4372
The Journal of Physical Chemistry, Vol. 93, No. 10, 1989
Mathew et al.
TABLE I: Data from Diffusion Measurements“
experiment 2
I Cl
c2 ACl A c2
J(expt1) J(ca Icd) ffl
CI c 2
ACl Ac 2
J(expt1) J(calcd) cy I
CI
c2 JCl Ac 2
J(expt1) J(calcd) ff1
Cl c 2
ACl ’Lc 2
J(expt I ) J(calcd) ffl
CI
c2 ACl Ac 2
J(expt1) J(calcd) BI
experiment 3
1
4
2
3
4
1.0291 1.0300 87.44 87.26 41.09 40.14 1.01707 1.022 93
1.4900 1.4889 -10.99 -1 0.47 -4.59 -4.42 1.01641 1.023 77
1.3537 1.3545 3 I .89 31.33 14.1 I 13.51 1.01651 1.023 49
1.5000 1.4984 -9.06 -9.62 -3.77 -4.04 1.035 42 1.046 59
1.3725’ 1.3742’ 30.26b 30.93’ 13.16b 1 3.24b 1.035 70’ 1.046 24’
1.0630 1.0630 94.61 94.59 43.58 42.99 1.036 62 1.045 3 1
Series AM-12; Total Concentration = 2.05 M 2.000 12 1 09DA(exptl) 1.5294‘ 0.050 00 109D,(calcd) 1.5290‘ 104Qo(exptl) -5.83c 0.051 08 0.072 17 IO4Qo(calcd) -6.48‘ 1O4QI(exptl) -2.39c 92.782 92.810 1O4QI(calcd) -2.6V 4top) 1.074 OY 0.2245 d(bottom) 1.083 6OC
1.0406 1.0403 123.43 123.85 56.46 55.88 1.075 59 1.08206
1.4 I96 1.4199 32.27 33.26 14.06 13.97 1.074 35 1.08341
1.1268 1.1271 I 12.96 112.19 50.81 49.67 1.075 28 1.082 39
Series AM-13; Total Concentration = 3.05 M 3.00041 lO9D,(exptl) 1.5651 0.050 0 1 lO9D,(calcd) 1.5643 0.053 88 1O4Q0(exptl) -5.23 0.072 17 1 04Qo(calcd) -5.27 104Q,(exptl) -2. I O 90.800 90.578 1O4Q1(calcd) -2.14 0.2320 d(top) 1.11079 d( bottom) 1.120 58
1.0981 1.0981 147.71 147.68 66.18 64.77 1.11253 1.11887
1.4643 1.4652 34.85 34.89 14.22 14.36 1 . 1 1 1 15 1.12022
1.1853 1.1852 127.93 127.96 56.02 55.15 1.11226 1.11922
Series AM-14; Total Concentration = 3.80 M 3.750 35 lOgD,(exptl) 1.5868 0.050 00 109D,(calcd) 1.5864 0.055 94 1O4Q0(exptl) -4.53 0.072 17 1O4Q0(calcd) -4.76 89.210 1O4Q1(exptl) -1.80 89.137 104Q,(calcd) -1.91 d(top) 1.13775 0.2375 d(bottom) 1 . I47 66
1.1238 1.1239 164.23 164.08 71.52 70.75 1.13960 1.14583
1.4895 1.4901 36.77 36.99 15.13 15.00 1.138 13 1.14723
1.2133 1.2130 139.32 139.47 59.69 59.12 I .I39 23 1. I46 21
0.500 21 0.035 83 0.000 49 0.071 66 76.837 76.862 0.0028
0.499 98 0.035 00 0.037 04 0.057 61 78.160 78.138 0.21 1 5
Series AM-IO; Total Concentration = 0.536 M 0.499 99 0.50004 109DA(exptl) 0.9447 0.03641 0.03500 lO9D,(CalCd) 0.9440 0.18536 0.14824 104Qo(exptl) 84.75 0,00000 0.01440 104Qo(calcd) 84.98 82.673 1O4Q1(exptl) 40.49 8 1.578 82.709 81.539 1O4QI(calcd) 39.95 1 .oooo d(t0P) 1.01733 0.8112 d( bottom) 1.022 79
I ,000 0 I 0.053 80
0.000 05 0.107 60 I 1 1.864 1 1 1.859 0.0002
0.999 65 0.053 78 0.286 94 -0.000 03 123.907 123.799 1.0003
Series AM-1 I ; Total Concentration = 1.053 M 1.00008 1 09DA(exptl) 0.9767 0.053 01 109DA(calcd) 0.9765 0.053 OOb 0.230 1 I’ 0.057 52 1O4Q,(exptl) 97.22 0.021 69’ 0.08673 I 04Qo(calcd) 97. I3 121.705’ 114.991 1 04Ql(exptl) 45.8 1 121.847’ 1 14.962 1O4QI(calcd) 45.07 0.8150’ 0.2159 d(t0P) 1.037 02 d(bottom) 1.045 03
2.000 12‘ 0.050 OOc 0.255 07 0.000 OOC I 04.077c 1 04.046c 1 .ooooc
2.000 09 0.050 01 0.00007 0.090 2 1 90.026 89.997 0.0003
2.000 22 0.050 0 1 0.204 32 0.018 05 101.321 101.353 0.8223
3.000 23 0.050 00 0.269 07 0.00001 105.015 104.945 0.9999
3.000 27 0.050 00 0.000 28 0.090 22 86.917 87.066 0.0013
3.00024 0.050 00 0.21 5 22 0.01 8 04 101.181 101.324 0.8284
3.750 36 0.050 00 0.278 13 0.000 00 105.168 105.261
3.750 37 0.05000 0.000 34 0.090 22 85.017 85.095 0.00 15
3.749 34 0.049 99 0.22043 0.01801 100.487 100.388 0.8310
1 .oooo
1.OOO 03’
+
‘Units: C, and X i , mol dm-); D,, m2 s-l; d, g cm-3. The refractive index fraction a I is given by a I = R I A C l / ( R I A C l R2AC2). ’The 6 for this r u n was the average of the values of the 6’s for runs I , 2, and 4. ‘The 6 for this r u n was average of the 6’s for runs 2, 3, and 4.
Densities. The densities measured for all experimental solutions are listed in Table I. The parameters of eq 129 were fit by
d = d(C,,C,)
+ H’(C, - C’) + H2(C2 - C2)
(I)
least-squares methods to each set of densities from those solutions that had the same mean concentrations Cl and C2 of NaCI and MgCI2,respectively. Values of d, HI,and H 2 are listed for each series of experiments in Table 11. The coefficients H I and H , together with measured diffusion coefficients were then used to test for static and dynamic stability30,31 of the experimental diffusion boundaries. In all cases the boundaries were calculated to be stable. These H I ,H,, and d were also used to calculate the partial-molal volumes vi for each component, by using an equation from Dunlop and These Vivalues are listed in Table 11. They are also shown in Figure 1, where they have been plotted against the (29) Dunlop, P. J.; Gosting, L. J . J . Phys. Chem. 1959, 63, 86. (30) Miller, D. G.: Vitagliano, V. J. Phys. Chem. 1986. 90, 1706. (31) Vitagliano, P. L.; Della Volpe, C.; Vitagliano, V. J . Solution Chem. 1984. 13, 549. Misprints are corrected in J . Solution Chem. 1986, 15, 8 1 1 , and in ref 30.
square root of volumetric ionic strength. In Figure 1, the values of partial molal volumes at infinite dilution for NaCl and MgCI,, 16.62 and 14.49 cm3 mol-‘, respectively, were calculated from the ionic molal volumes at infinite dilution reported by miller^.^^ The infinite dilution value used for water was 18.07 cm3 mol-]. Total Fringe Numbers J . The J for all experiments are listed in Table I. For each set of four experiments, the value of J from each experiment was used as part of the input to the R F G code. This code then calculates the refractive index increments R , and R2 for the set by least-squares fitting of Ri with the equation
given the J , AC,, and AC2 for that set. Here a is the diffusion cell path length (2.5064 cm) and X is the wavelength of the light source (546.1 nm). It is seen in Table I that least-squares values of J calculated by the R F G code agree with measured ones within h0.22. The larger J errors may be partly due to uncertainties in the measured AC, values, especially a t higher concentrations where it becomes (32) Millero, F. J . Chem. Rec. 1971, 7 1 , 147
The Journal of Physical Chemistry, Vol. 93, No. 10, 1989 4373
Isothermal Diffusion Coefficients TABLE 11: Experimental Results’
series AM-IO
+ c2
0.53561 0.50005 C2 0.035 56 ml(CI,C2) 0.50642 m2(Cl,C2) 0.036 01 10’RI 9.723 103R2 23.290 1.02004 a H, 39.710 H26 16.057 18.781 VI 19.202 v2 vo 18.061 10-9ut 1.4371 10-9u0.6861 109(Dll), 1.4429 109(D12)v 0.7978 109(D21), 0.0136 109(D22), 0.7104 109(Dll)0 1.4567 109(D12)0 0.8123 lO9(D2JO 0.0146 109(D22)0 0.71 14
C’
AM-11
AM-12
AM-13
1.05335 0.99995 0.053 40 1.02300 0.054 63 9.401 22.647 1.04099 38.866 74.358 19.613 20.892 18.048 1.4846 0.6800 1.4542 0.9666 0.0132 0.6900 1.4836 1.0007 0.0148 0.6918
2.05014 3.05029 2.000 14 3.00029 0.05000 0.05000 2.08960 3.20682 0.052 24 0.053 45 8.886 8.504 21.732 20.986 1.07884 1.11570 37.657 36.351 71.870 69.892 20.787 22.023 23.343 25.240 18.017 17.959 1.6241 1.8085 0.6614 0.6445 1.4968 1.5365 1.3596 1.7400 0.0098 0.0085 0.6309 0.5681 1.5623 1.6460 1.4494 1.9094 0.01 14 0.0103 0.6332 0.5709
AM-I4 3.80010 3.750 1 1 0.05000 4.081 62 0.054 42 8.247 20.532 1.14271 35.635 69.034 22.681 26.030 17.915 1.9582 0.6346 1.5612 1.9685 0.0077 0.5252 1.7074 2.2079 0.0096 0.5284
b
r
”’ 1.5
.5
Dl1
i
1 021 .’
a Units: C,, mol dm”; mi, mol (kg of H20)-’; Ri, dm3 mol-’; vi, cm3 mol-’; ut, u-, m-2 s; Di,, m2 s-I; H i , g mol-’; a, g cm-’. bThe values of H i in this table must be divided by 1000 when inserted into eq 1 to give densities in g
more difficult to prepare solutions with accurate ACi values. DA Values. The extrapolated value of DAfor each experiment of a set was obtained as described in the Analysis of Diffusion Data. They are listed in Table I together with least-squares values calculated with the RFG code. The agreement between measured and calculated DAis 0.1% or better. This agreement shows the. overall precision of the “Costing” diffusiometer and the accuracy of solution preparation. However, it is not quite as good as for our previous two s e r i e ~ . ~ * ~ Qo Values. The experimental value of Qofor each experiment of a set is listed in Table I and was obtained by averaging the Qo values from each fringe pattern. The Qofor each pattern had been obtained from the least-squares parameters for that pattern, by using eq 4 and 13 of ref 26. T h e scatter of Qofrom all the patterns within a single experiment is another important diagnostic. These individual Qowere usually in agreement within f0.5 X lo4 of their average. Least-squares values of Qofrom the R F G code are also listed in Table I. The agreement between experimental and least-squares values of Qo was within fl.O X This agreement, while satisfactory, is not quite as good as those in papers 1 and 2 of this serie~.~.~ The values of the ternary diffusion coefficients Dii are very sensitive to the values of Qo.Therefore, in all cases, we examined graphs of measured R versusf(z), to be sure that the quality of the R graph appeared satisfactory for that experiment. Ordinarily, when values of R from all fringe patterns are plotted versusflz), the values lie in a band with a scatter of 10 X or less. This band should extend smoothly fromf(z) = 0 to f(z) = 1 and extrapolate to 0 at bothflz) = 0 andflz) = 1. If the band is much wider than 10 X lo4, or is not smooth, or does not appear to extrapolate to 0 atf(z) = 1, then a problem is indicated for that e ~ p e r i m e n t . ~Such ~ experiments were considered unreliable and discarded. Di, Values. The four DUthat describe ternary diffusion at each composition are given in Table 11. They are given for both the experimental volume-fixed (Di,), and derived solvent-fixed (Dij)o ~
~~~
~~
(33) If the n band has a range much greater than IO X IO4 or spreads out systematically at higher values of Ar), then a serious problem is indicated-such as a leak in the diffusion cell. A single discrepant pattern usually implies an error in reading fringe positions. If the graph fails to extrapolate smoothly to zero as A I ) I , then an error in J is possible.
-
On the basis of our experience and careful analysis of these and other results, we estimate that the main-term diffusion coefficients D,, and D2, are accurate to 1 .O% of their values or better. We estimate that the cross-term diffusion coefficient D,, for the flow of MgCl, due to a gradient of NaCl is accurate to 1 .O% of the value of DZ2.The magnitude of D,, is small in this series of experiments because the concentration of MgCl, is small. For NaC1-MgC12-H20, the coefficient D I 2 (for the flow of NaCl due to a gradient of MgC12) has generally been the most sensitive to variations in input values used to calculate the D,,, in particular to Qo. In the current series of experiments, the value of DI2increases as the NaCl concentration increases and becomes greater than both main-term diffusion coefficients. We believe its accuracy to be better than 2%of the magnitude of D l l . These estimated errors are approximately 4 times the standard error of the coefficients obtained from propagation of error equations. We have also found for other systems, Le., aqueous NaC1-SrC1235,36 and ZnC12-KC1,37 that one of the cross terms always has the largest uncertainty. Values of the volume-fixed D, coefficients are plotted versus total solute concentration in Figure 2. Values a t zero total concentration are not included because the ratio of the concentration of NaCl to MgCl, is not constant for this series.
Discussion The modest concentration dependences of D,,, DZ2,and D2, in Figure 2 indicate no particularly unusual features. It is not surprising that D2, is very small, since it must vanish under true trace conditions. However, the large increase of D I 2with increase of total concentration is very striking. We see that at the highest concentrations, D I 2is larger than both main-term coefficients D I and D22, Thus, a t higher concentrations of this “trace” MgC1, system, any given gradient of MgC1, moves more NaCl than the same gradient of NaCl would. Also a given gradient of MgC12 (34) Woolf, L. A.; Miller, D. G.; Gosting, L. J . J . Am. Chem. SOC.1962, 84, 317.
(35) Rard, J. A.; Miller, D. G. J . Phys. Chem. 1987, 91, 4614. (36) Rard, J . A,; Miller, D. G. J . Phys. Chem. 1988, 92, 6133. (37) Miller, D. G.; Ting, A. W.; Rard, J. A. J . Electrochem. SOC.1988, 135, 896.
4374
J . Phys. Chem. 1989. 93. 4374-4382
moves more NaCl than it moves itself. These results were quite and derivatives of activity data;6 the O R R were found to be uncxpected. satisfied with reasonable estimates of experimental errors. These This surprisingly large D , , underscores the fact that a cross-term results, plus estimates of the D,, based on ionic Onsager coefficients cocfficient need not be small and can exceed one or both main-term 1,,,13,'4 will be reported elsewhere. cocflicients. contrary to commonly held opinions. Clearly, models It is interesting that the partial molal volumes of these systems of diffusion processes that ignore cross terms can often be subappear to be linear in the square root of ionic strength (see Figure stantiall) inaccurate, as discussed in ref 12. Moreover, the large 1 ) . However, it is noted that the ratio of concentrations of the D I 2at high concentrations simply cannot be predicted from the solutes is not constant in this series of experiments, so care should infinitc dilution value calculated from the Nernst-Hartley be taken in interpreting this linearity. c q u ; i i i o n ~ . ~ ~Inclusion . ~ ~ . ~ ~ of thermodynamic gradients and Acknowledgment. This work was primarily performed under vibcoaity factors in Nernst-Hartley estimates improves matters the auspices of the U S . Department of Energy a t the Lawrence somewhat up to about 1 mol dm-3, as both we (unpublished) and Livermore National Laboratory under Contract W-7405-ENG-48. Leaist' have found. However, other types of ionic interaction have J.A.R. and D.G.M. thank the Office of Basic Energy Sciences a major influence on the values of D , even in moderately con(Geosciences) for support. D.G.M., L.P., and J.G.A. thank Dr. centrated electrolyte solutions, and good approximations a t high Christopher Gatrousis for C R R support. J.G.A. also thanks TCU concentrations have yet to be found. These issues will be discussed for research fund Grant No. 5-23824. R.M. thanks T C U for elsewhere. after we present the D, extrapolated to C, = 0 and supporting him through a research fellowship. The research c2= 0. published here is based in part on the Ph.D. dissertation of R.M., W e have made preliminary calculations of the Onsager reciprocal relations ( O R R ) for ternary d i f f ~ s i o nusing ~ ~ ,our ~ ~D, ~ ~ ~ TCU. Registry No. NaCI, 7647- 14-5; MgCI,, 7786-30-3 (38) Gosting, L. J. Adcances in Protein Chemistry: Academic Press: New York, 1956; Vol. XI. pp 429-554. (39) Wendt. R . P. J. Phys. Chem. 1965, 69, 1227.
(40) Miller, D. G. J . Phys. Chem. 1959. 63, 570
Structure, Dynamics, and Molecular Association of LIAsF, and of LiCIO, in Methyl Acetate at 25 OC Mark Salomon,* Michelle Uchiyama, Army Power Sources Laboratory, Fort Monmouth, New Jersey 07703-5000
Meizhen Xu, and Sergio Petrucci* Weber Research Institute and Department of Chemistry, Polytechnic University, Farmingdale, New York I I735 (Received: August I I , 1988; I n Final Form: November 14, 1988)
The nature of ion solvation and complex formation of LiAsF, and LiCIOI in methyl acetate (MA) at 25 OC has been studied over the concentration range of to 0.5 mol dm-). The properties of these electrolyte solutions were studied by infrared and microwave relaxation spectroscopy and by audio frequency electrolytic conductance. The infrared spectra of LiAsF6 solutions over the concentration range 0.1-0.5 mol dm-3 suggest the existence of contact ion pairs, and solvent separated ion pairs, and possibly dimers. Microwave complex permittivities for LiAsF, m d L i C Q solutions over this same concentration range confirm the existence of both contact and solvent separated ion pairs and indirectly confirm the existence of contact to 0.03 mol dm-3) reveal apolar dimers. The audio frequency electrolytic conductivity data obtained in dilute solutions ( a minimum in the molar conductances of both salts at around 0.01 mol dm-3. Simultaneously we have found that the solution permittivity and viscosity both increase as concentration increases. These results are interpreted in terms of alternative models either involving triple ions or neglecting triple ions. The analyses suggest that the effect of increasing solution permittivity and viscosity as concentration increases is not an important factor leading to the appearance of conductivity minima. Instead, the important factor may be attributed to either ion-dipole and/or dipole-dipole interactions that result in nonelectrolyte activity coefficients significantly less than unity.
Introduction The search for a new electrolyte-solvent system that could give the best suitable combination for the construction of secondary lithium batteries is a problem of pragmatic importance in the area of portable power sources. On the theoretical side, the knowledge of the structure and dynamics of the species in solution is a prerequisite for a nonempirical approach to the selection of electrolyte solutions for use in lithium batteries. It was with these ideas in mind that we investigated the properties of LiAsF, and LiC104 in methyl acetate. The methods selected for these studies were infrared spectroscopy, and audio frequency electrolytic conductance. Since various species (ions, ion pairs, and higher agglomerates) may predominate over specific concentration ranges, the advantage of combining the above three 0022-3654/89/2093-4374$01.50/0
experimental techniques is that we were able to investigate electrolyte solutions over the extremely wide concentration range of to 0.5 mol dm-3. For example, in dilute solutions (
