612
PETERJ. DUNLOP
VOl. 63
DATA FOR DIFFUSION I N A CONCENTRATED SOLUTION OF THE SYSTEM NaCl-KCl-H20 AT 25’. A TEST OF THE ONSAGER RECIPROCAL RELATION FOR THIS COMPOSITION BY PETERJ. DUNLOP’ Contribution from the Department of Chemistry, University of Wisconsin, Madison, Wisconsin Received October I , 1868
The four diffusion coefficients which are necessary to describe the solute flows in a ternary solution have been measured for the system NaCl-KCl-HZ0 with each solute concentration equal to 1.5 molar. These data together with previously measured activity coefficients for this system are used to test the Onsager reciprocal relation for ternary diffusion. The Onsager relation is verified within 8%, and this value is less than the maxlmum expected error which is calculated from estimated individual errors in the various experimental data.
I n a recent paper2 four tests were made of the Onsager reciprocal relationa“ for ternary diffusion in relatively dilute solutions of the system NaC1-KCl-HZO. In no case was the total salt concentration of the system greater than one molar. Previously measured diffusions and thermodynamic’J data were employed in these tests and in each case the Onsager relation was verified within the expected experimental error of about 8%. Millere has also reported several te&s of the Onsager relation, but he used certain estimated, rather than experimental, values for the solute activity coefficients. The diffusion measurements reported in this paper were undertaken to test the Onsager relation for two reasons. Firstly, at high concentrations it was hoped that the cross-term diffusion coefficients’o would be larger than those in the previous measurements.s Secondly, a t high concentrations the derivatives of the logarithms of the activity coefficients contribute a greater proportion of the chemical potential derivatives, which appear in the Onsager relation for a three-ion system (see equations 1 of ref. 2), than they do a t low concentrations. Thus a t the high salt concentrations used in this study these chemical potential derivatives are not as accurately known as they are at lower concentrations. This paper must be read in conjunction with two previous publications2P1l since frequent reference will be made to certain equations and tables in those works. When the letters D or F follow equation and table numbers in this paper, they refer to references 1 and 11, respectively. Experimental The Gouy diffusiometer used to measure the reduced heightrarea ratios, PA,and the differential refractive incre(1) Department of Chemistry. University of Adelaide, 8outh Australia. (2) P. J. Dunlop and L. J. Gosting. THISJOURNAI,, 68, 86 (1959). (3) (a) L. Oneager, Phye. Rev., 87, 405 (1931); (h) 88, 2265 (1981). (4) L. Onsager and R. M. Fuoss, THIBJOURNAL, 86, 2689 (1932). (5) L. Onaaner, Ann. N . Y. Acod. Sci., 46, 241 (1945). (6) I. J. O’Donnell and L. J. Gosting, a paper presented in a Symposium at the 1957 meeting of the Electrochemical Society in Washington, D. C . ; the Symposium papers are to be published as a monograph by John Wiley and Sons, New York. (7) R. A. Robinson, in “Electroohemical Conatanta,” National Bureau of Standards Circular 524, 1953. (8) R. A. Robinson and R. H. Stokes, “Electrolyte Solutions.” Appendix 8.3, Butterworths Scient,ific Publiaations, London, 1955. (9) D. G. Miller. THISJOURNAL, 61, 767 (1958). (10) P. J. Dunlop and L. J. Gosting, J. A m . Chem. Soo., 77, 5238
(1955). (11) H. Fujita and L. J. Gosting, ibid., 78, 1099 (1956).
menta, Ri, has been previously described,12Js as have also the necessary experimental conditions and methods1°J4 used to obtain the DA, the Ri and the graphs of the fringe deviations, n, versus the reduced fringe numbers f(r). The reader is referred to previous publications1OJ4 for definitions of these experimental quantities. A single quartz Tiselius cell, 9 cm. in height, was used in all experiments. The cell dimension, a , along the optic axis was 2.5061 om. and the optical lever arm, b, of the Gouy diffusiometer was 306.86 cm. All diffusion measurements were made with the 5460.7 A. mercury line isolated from a G.E.A-H4 lamp with a Wratten 77A filter. Materials and Solutions.-The sodium and potassium chlorides used in these experiments had been recrystallized and fused previouely.10J4 All solutions were prepared by weight using doubly-distilled water, saturated with air, as solvent. The weight fraction of each solute, corrected to vacuum, was converted to the corresponding molarity, Ci, by means of densities, p , in g./ml., measured in tri licate in 3 0 4 . P ex pycnometers. The molecular weigits of NaCl and &l were taken to be 58.448 and 74.557, respectively. To facilitate preparation of solutions, an equation was derived for predicting solution densities in the region C1 = CZ = 1.5 by using the empirical relation = NOVO0 N161 N z h (1) where V is the total volume of solution, No, N1, and NZare the number of moles of water, NaCl and KCl, respectively, and is the volume of a mole of water at 25’. The dependences on concentration of +I and d2,the apparent molal volumes for binary systems, are given approximately byla
v
+
+
61 = 16.40 4- 2.153fii (2) qb2 = 26.52 2.327dG (3) For this ternary system C1 and C2 in these expressions were replaced by the total solute concentration ( C1 CZ).“ The resulting expression in the region of CI = CI = 1.5 1s
+
+
p =
1.120786 -k o.O3644(ci
- 1.5) + 0.04216(c~- 1.5)
(4) The density of water was taken to be 0.997075 g./ml. in this calculation. This equation predicts solution densities with an accuracy of about 6 parts in the fifth decimal place (see Table I) and it was used for preparing solutions of the desired concentration for each experiment.
Results A11 the experimental quantities for the determination of the four diffusion coefficients are summarized in Table I. Lines 2-9 give the solution concentrations used in each experiment together (12) L. J. Goating, E. M. Hanaon, G. Kegeles and M. 8. Morris, Rev. Sci. Inah., 20, 209 (1949). (13) P. J. Dunlop and L. J. Goating, J. Am. Chem. Soo., ‘76, 5078 (1 953). (14) D. F. Akeley and L. J. Gosting, ibid., 76, 6685 (1953). (15) H. 9. Harned and R. B. Owen, “The Physical Cheniistry of
Eleotrolytic Solutions,” 3rd Ed., Reinhold Publ. Corp., New York, N. Y., 1958, p. 253. (16) €7. E. Wirth, 1.Am. Chem. Soc., 69, 2549 (1937).
DIFITJSION IN A CONCENTRATED SOLUTION OF NACL-KCL-H~O
April, 1959
613
TABLE I" (D& are given in Table I11 and it is estimated that The double DATAFROM EXPERIMENTS IN WHICH NaCl AND KCl DIF- they are accurate to -I 0.03 X subscripts on the values of the ( D i j ) are ~ retained FUSED SIMULTANEOUSLY IN WATERAT 25' only for calculating predicted values of the DA,the = 1.5 1 = NaCl, 2 = KCl; ZI 1.5,
-
W m and the fringe deviations. Although both these figures are in doubt, they are necessary in these calculations because of certain restrictions which exist between the values of the four diffusion coefficients if they are determined by the method used in this paper. The (Dij)v were 1.60028 1.125974 rounded to the third decimal place for use in 1.125918 testing the Onsager relation. Because of the 1.6000s relatively small concentration increments used in e; 1.49994 each diffusion experiment, it is believed that any 0.04948 0.20068 errors in the four diffusion coefficients due to their 96.40 concentration dependences are negligible when com96.42 pared to the errors of lt0.03 X which were 0.2007 estimated from the expected errors of f 2 X lo-' 1.8641 1.8646 in the experimental deviation graphs. The sub0.4 script V has been given to the values of the Dij 1.9044 in Table I11 because they correspond to a volume-0.1961 Units in Table I: concentrations, CC,moles/l.; densi- fixed frame of reference which becomes identical ties, p , g./ml.; reduced height-area ratios, DA,and re- with the cell frame of reference (a) if the partial duced second moments, D 2 m r cm.*/sec. Calculated using molal volumes of al; components are independent equation 4. of concentration or (b) in the limit that the ACr with the measured densities and those predicted are all zero. Using the four ( D i j ) in ~ Table 111, the values of by means of equation 4. The subscripts A and B the ai in Table I and in addition Table I V of on these data denote the upper and lower initial solutions, respectively, used in each diffusion ex- reference 14, values for the DA, the Wrn and the periment. Lines 10-13 list the mean solute con- fringe deviations were calculated for each experiand the concentration increments, ment by means of equations 30F, 31F, 50F, 51F, centrations, AC,, across the initial diffusion boundaries. Con- 55F, 60F-63F and 72F. Table I1 lists the calcentration increments are considered positive if culated and experimental values of DA and Dzm a solute's concentration in a given lower solution is together with values for the intermediate quantity greater than its value in the corresponding upper I?+ (see equation 50F). The agreement is seen to solution. The total number of Gouy fringes, be not unsatisfactory. Figure 1 gives the fringe J , obtained in each experiment is given in line 14. deviations calculated by means of equation 72F Using these values of J and the corresponding (dashed lines); they are to be compared with the values of the ACr, values for the differential re- average experimental points denoted by crosses. fractive increments, Rr, were calculated.6 These Inspection of Table I1 and Fig. 1 shows that the values are reported a t the bottom of Table I and four values of (D& will reproduce, within the were used, together with the AC,,to calculate the error of measurement, not only the height-area values of Joelcin line 15 and also the solute fractions ratios and the redhed second moments, but also on the basis of refractive index (see equations 45F- the deviations of the refractive index gradient 47F), ai, in line 16. Lines 17-19 give the measured curves from Gaussian shape. height-area ratios; the values of the reduced TABLE I1 second moments, Dzm, which were calculated10 COMPARJSON OF CALCULATED WITH EXPERIMENTAL VALUES from the DAand the fringe deviation graphs (see OF DAAND D i m Fig. 1); and the values of the fringe deviations a t T 25'; 1 = NaC1,2 = KC1 f ( S ) = 0.73854. The values of IA, SA,Izm and a)A x 10' W m x 106 Szmin Table I were obtained from the quantities r+ Expt. Calcd. Expt. Calcd. a1 l/& and %m by the method of least squares. -0.0004 -0.0624 1.9142 1.9147 1.9039 1.9039 These latter quantities are linear functions of .2007 -0081 1.8641 1.8640 1.8646 1.8652 al (see equations 55F and 60F); their average .8010 .2185 1.7259 1.7240 1.7520 1.7499 deviations from straight lines were 0.03 and 0,1370, 1.oooo .2882 1.6797 1.6811 1.7046 1.7117 respectively. The Test of the Onsager Reciprocal Relation Evaluation of the Four Difhsion Coefficients A complete description of the methods used to The methods used to evaluate the four diffusion evaluate the diffusion coefficients with respect to coefficients which appear in the flow equations for the solvent frame of reference, (D&, the chemical diffusion in ternary solutionslo have been described potential derivatives and the expected error in a in detail in two previous papers.sJ1 Approximate given test of the Onsager relation already has been values for these coefficients were first obtained from given2; only a short summary will be given here. the DAand the P2m (see method iii of Table IIF) and DBusion Coefficients for the Solvent Reference then final values were computed by means of Frame.-Diffusion coefficients for the solvent frame method iv of Table IIF. These values of the of reference were obtained from the values for the Exp. no. (CdA
1 2 3 1.40071 1.50007 1.37536 1.47496 ((?,)A 1.49996 1.37497 1.115584 1.116180 (Psxp)A 1.116292 1.115518 1.116111 (kalc)Ab 1.116242 1.59976 1.62516 1.49996 (CJB 1.52534 1.49997 1.0250g 7 (CdB 1.125390 1.126122 1.125552 8 (PSXP)B 1.126059 1.125489 9
