13745
J. Phys. Chem. 1994,98, 13745-13754
Analysis of Gouy Fringe Data and Comparison of Rayleigh and Gouy Optical Diffusion Measurements Using the System Raffinose (0.015 M)-KCl (0.5 M)-H20 at 25 "C Donald G. Miller,* Luigi PaduanoJ Roberto SartorioJ and John G. Albright$ Chemistry and Materials Sciences Department, Lawrence Livermore National Laboratory, Livermore, Califomia 94550 Received: June 6,1994; In Final Form: September 2, 1994@
Thirteen isothermal ternary diffusion experiments have been performed for a single composition of the system raffinose-KCl-H20 at 25 "C. Both Rayleigh and Gouy optical systems were used in all experiments for direct comparison of results from these two configurations. Various data analysis procedures are compared. Good agreement is found between the recommended results from the two optical methods, although diffusion coefficients Dv obtained by the Rayleigh method are generally slightly lower than those obtained by the Gouy method. The sensitivity of the Dv to both experimental error and choices of subsets of the 13 experiments is discussed. Standard errors of coefficients are an inadequate measure of accuracy. This study has led to higher standards for acceptability of individual experiments. This system has the same mean solute concentrations as the four experiments reported by Dunlop, for which a test of the Onsager reciprocal relations (ORR) failed. Our diffusion results are reasonably similar to those of Dunlop. Therefore the reported failure of the ORR test probably results from errors of estimating activity coefficients for this system rather than from diffusion data.
1. Introduction This paper presents an extensive study of precision and accuracy in optical diffusion measurements. To avoid unnecessary repetition, it should be read in conjunction with earlier ~ n e s . l - ~These contain definitions of symbols and details of procedures used with Gouy and Rayleigh interferometric methods of diffusion measurement^.^-* The motives for this research follow. There are published tests of the Onsager reciprocal relations (ORR) for approximately 40 temary compositions; these have been reviewed e l s e ~ h e r e . ~There . ~ ~ are also results for another 25 compositions of NaCl-MgClz-H20 (in preparation at LLNL). All satisfy the ORR within experimental error except for raffinose (0.015 M)-KC1 (0.5 M)-H20 at 25 OC.ll The data for this exception were among the first Gouy data collected on a ternary s y ~ t e m , ~and * . ~the~ ORR test required the estimation of activity coefficients and their derivative^.'^ This ORR disagreement could result from either errors in the measured diffusion coefficients Dv or larger uncertainties in the estimated activity coefficients than those originally assumed. Consequently it seemed desirable to remeasure the Do with a diffusiometer of higher precision than that available to Dunlop.12 Our original purpose in 1986, when these experiments began, was thus to obtain the Do of this system on the high precision Gosting diffu~iometer,'~ then located at Lawrence Livermore National Laboratory.16 The results could be compared with Dunlop's earlier results and used to recheck the ORR. At the same time, we could also compare Dv calculated from Gouy optical data with DQ calculated from Rayleigh optical data6.7 from the same experiment. This was possible because these optical systems can be easily interchanged on this apparatus during a diffusion run. In principle, both methods should give the same values of the D+ TDipartimento di Chimica, Universita Federico II di Napoli, 80134 Naples, Italy. Permanent Address: Department of Chemistry, Texas Christian University, Fort Worth, TX 76129. Abstract published in Advance ACS Abstracts, December 1, 1994.
*
@
0022-3654/94/2098-13745$04.50/0
Although only two diffusion experiments with different ACII AC2 ratios are necessary to determine the four Dv, it is customary and desirable to do at least four to improve statistics and check for a bad experiment. Here Ci is the concentration of solute i in mol dm-3; ACi is the difference of concentration between the bottom and top solutions which are initially separated by a sharp boundary. All the experiments should have the same average concentration for each solute, where
ci= (C, + Ci,)/2
(1)
Dunlop originally chose his four experiments such that two had al close to 0.0 and two had al close to 1.0 in order to use his early special method to calculate D+12 (The refractive index fraction alis defined in eq 4 below.) For this choice, the areas Qo under the s l j versus Azj) graph,17 obtained from the Gouy data, are quite small.13 When more general methods for analyzing Gouy data became available,17J8it became customary to use compositions corresponding to a1 = 0.0, 0.2, 0.8, and 1.0. Upon completion of our four experiments at al = 0.0, 0.2, 0.8, and 1.0, we found that the Qo values at 0.2 and 0.8 were quite large (200 x In such cases, the classical Gouy techniques for analyzing each experiment give rise to problems. In particular, there are problems in obtaining the Gouy quantity C, for each exposure by extrapolation of the Cfj1-3from its low number fringes, and thus problems in determining the derived quantities DA and Qo. How many fringes to use in extrapolation and whether to use linear or quadratic extrapolation w i t h f l ~ ) ~ ' ~ are difficult to decide. Such problems had been observed before but were neither completely resolved nor systematically studied. Since some of us (L.P., R.S.) study high molecular weight solutes whose eo's are often still another order of magnitude higher than for raffinose-KCl-H20, solving this extrapolation problem was even more important. Another problem in Gouy experiments is the determination of the fractional part (fpf) of the total number of fringes J. This is usually done by the Gosting double-slit method.6,7.19.20 However, this procedure can be subject to error if the initial
0 1994 American Chemical Society
Miller et al.
13746 J. Phys. Chem., Vol. 98, No. 51, 1994 boundary is insufficiently sharp. Another fpf method is to linearly extrapolate an appropriate function P, of J versus an appropriate function Q, of fringe positions Y j for fringes near the undeviated slit image (i.e., high number fringe^).^ The extrapolated value is J (including the fpf), and the procedure is iterated until J becomes constant. This PQ procedure is rigorous only for binary systems where all Qj = 0.0 and therefore Qo = 0.0. It does not work as well for ternary systems with large Qo and thus is best used only to get the integer number of fringes and for plate reading diagnostic^.^ As a result of this Cr extrapolation problem, one of us (J.G.A.) suggested a least-squares fit to low number Y j of the theoretical expressions of Fujita and Gosting" for ternary systems. This fit, done for each photographic exposure (Le., for each time), contains C, as one of the parameters. J.G.A. also recognized that the same theoretical expressions involve J indirectly and thus could be fit to the high number fringe positions Y j to get both J and C,, thereby avoiding problems with the PQ technique. At that point, all the authors realized (1) that the best value of C, for each exposure could be obtained by using all the Y j data, provided J were already known accurately and (2) that both C, and J could be obtained (with lesser accuracy) if J were not known accurately. This second case can arise if there is some experimental problem in determining J by the Gosting method. The detailed theory and practice of this method for analyzing Gouy fringe pattern data, exposure by exposure, has been presented elsewhere.z The Rayleigh optical method has fewer of these problems, but the reading of the fringe patterns is more tedious and requires baseline correction^.^^^^ It appeared that these issues might best be studied by a systematic series of measurements on raffinose-KC1-HzO with various values of at,alternately switching between Gouy and Rayleigh optics during each experiment. Consequently, our initial purposes were expanded to provide an examination of the real precision and accuracy of optical diffusion measurements on a high-precision diffusiometer. Thus our 13 experiments include both large and small QOvalues, which correspond to large or small "skewing" of Rayleigh patterns, respectively (see section 3 A). This system also has the advantages that the eigenvalues of the D matrix are significantly different and SA is large; this avoids convergence problems in the data a n a l ~ s i s . ~ After our Gouy data had been analyzed by the above methods, a quite different general method for analyzing three- or morecomponent systems was discovered.' This "Miller method'' uses the same least-squares parameters as does the Rayleigh analysis and uses fringe position data from all experiments directly without needing Cr, DA,or Qo.The present paper contains the f i s t comparison of this new technique with earlier methods for ternary systems. The present work includes (1) a systematic study of this one raffinose-KC1-HzO composition as a function of a', ( 2 ) an intercomparison in detail of results from the various Gouy analysis methods and the Rayleigh method, (3) an examination of the effects of choosing different subsets of input data (i.e., with different values of a'), (4) an examination of sensitivity of Du values to errors in J , etc., and ( 5 ) a discussion of experimental accuracy vs statistical measures of precision. Despite 35 years of ternary diffusion research, this is the first systematic study of any ternary system in which the above issues are examined in We will find that despite the high quality of the Gosting apparatus, the value of D21 is very sensitive to the combination of experiments used and particularly to the values of J and Qo. Our final choice of Dq will still contain errors of the order of (0.02-0.03) x cm2 s-' in 0 2 1 , despite the misleading small
standard error obtained from the propagation of error equations (with or without covariance terms). The two new Gouy analysis are in excellent agreement. The Rayleigh results are in good agreement with the Gouy, within 2 standard errors of the corresponding coefficients. However, such statistics are uncertain for interrelated variables. Recommended values of the four Du are given. On the basis of the original estimates of the activity derivatives and the original estimated errors of those estimate^,'^ the ORR is still not satisfied. We now believe this is due to errors in the estimated activity derivatives. However, the ORR would be satisfied provided these errors are assumed to be larger. Experimental activity data would be very desirable, but unfortunately are very difficult to obtain in solutions so dilute in raffinose. 2. Experimental Techniques
Mallinckrodt analytical reagent grade potassium chloride was dried at 110 "C. Each type of impurity was reported by the supplier to be 0.005% or less.23 Raffinose-5HzO from U.S. Biochemical was taken directly from the bottle. However, it is slightly hygroscopic so is not exactly stoichiometric. Therefore its water content was determined by heating samples in a vacuum oven for two weeks with periodic weighings, starting at 75 "C and ending at 85 "C. The final mass ratio of dry/ hydrate was 84.90 f 0.02% (the 5.hydrate theoretical is 84.849%), and this was used to obtain the correct quantity of raffinose. All solutions were prepared by mass using doublydistilled water as solvent. To convert weight concentrations to volume concentrations, densities were measured for all solutions at 25.00 f 0.02 "C using 30 cm3 pycnometers. These were calibrated with doubledistilled water, using a value of 0.997 045 g cm-3 for the water density. These densities were well represented by the equationz4
d =2
+ H,(Cl - E,) + Hz(Cz - Cz)
(2)
where d is the density in g ~ m - C ~i ;are the molarities in mol dm-3; E, are the average values of C, for the set of experiments; 1 refers to raffhose, 2 refers to KC1; and 2, HI, and HZ are fitting parameter^.^^ Molar masses used were KCl, 74.551; anhydrous raffinose, 504.444; Hz0, 18.0152, all in g mol-'. The techniques for Gouy experiments have been described previously.zo*26Rayleigh experimental and plate reading techniques were described generally by Creethzl and reviewed by Miller and Alb~ight.~ More details are available for a Beckman Spinco Model H4,z3,27-31 and for the Gosting diffusiometer: All the diffusion experiments were performed with the Gosting diffu~iometer'~ using a Hg arc lamp (546.07 nm). As noted earlier, it has both Gouy and Rayleigh optics and can be switched from one optical arrangement to the other in less than 1 min. Consequently Gouy and Rayleigh photographs can be taken alternately during the same run. The cell, originally used by Kim, has an a distance of 2.5064 cm.26 Its Gouy b distance in its cell-holder, as well as the Rayleigh magnification factor M,were redetermined in 1981, with the values 308.885 cm and 1.7618, respectively. Kodak 11-G plates were used to photograph most Gouy patterns. For all Rayleigh and a few Gouy patterns, Kodak Metallographicplates (no longer available) were used. The decimal part of J (fpf) was determined independently from Gouylg and Rayleighzl runs using the appropriate mask arrangements and plate reading techniques? To obtain the Rayleigh fpf, baseline corrections were used to correct for fringe displacements due to optical imperfection^.^^^' The integer part of J for Gouy experiments was determined using the PQ
Analysis of Gouy Fringe Data extrapolation procedure noted above;3 the Rayleigh integer J was obtained by direct counting. Fringe positions were obtained from the Gouy photographs taken during a diffusion experiment by measuring their vertical displacement from the undeviated slit image.lg These were read to the nearest 1 p m using a comparator consisting of a Gaertner model M200IRS tool makers microscope with a special projection a t t a ~ h m e n t . ~For ~ . ~each ~ run, 10 exposures were taken. In each exposure, fringe positions were read for every fringe from fringe 0 to fringe 9 and then for every other fringe until near the undeviated slit image. However, to improve the extrapolationcalculation for estimating J, every fringe was again read for the last 10-12 readable fringes. Reading becomes impossible for the last 10-15 actual fringes due to increasing blackness and decreasing fringe separation as the undeviated slit image is approached. Rayleigh fringe positions of symmetrical p a i r ~ ~ were 3 ~ l each read to 1p m on a Grant comparator. Every other bright fringe was read to the center of the pattern; the comparator moved upwards by the fpf in micrometers, and then every other corresponding symmetrical fringe was read. The fringe pairs retained for data analysis were the same for every fringe pattern of an experiment and were chosen using the Albright and Miller criterion for skewed fringe pattems.4~~8~~ All AClIAC2 ratios and thus alvalues were tested for the possible onset of convective instabilities by using the Hi and ternary diffusion coefficients D,.34,35 The calculated stability region is between -0.25 (fingering) 5 a1 5 1.027 (overstability). The region suitable for Gouy e x p e r i m e n t ~is~ -0.12 ~,~~ 5 al 5 1.027 and also depends on the refractive index increments R, defined in eq 3 below. Since our a1 values lie between 0.000 and 1.018, all compositions are stable and suitable for Gouy measurements, although experiment 9 is close to the onset of overstability. Because the AC, are relatively small, the diffusion coefficients are those of the volume-fixed reference frame.
3. Data Analysis Methods A. Rayleigh. The ternary Rayleigh diffusion coefficients Dv were calculated from the combined data of an appropriate set of experiments. For each experiment, these data consist of fringe positions from each exposure together with the J and AC, associated with that experiment. The fringe positions were paired symmetrically as recommended by Creeth4,7,21,3* to eliminate optical aberrations and concentration dependence of the Dg and refractive index n. The nonlinear least-squares procedure used was developed by Miller, Eppstein, and Albright in 1972. It is described in detail by Miller et al.? briefly described by Miller et aL30 and Rard and Miller,31 and more briefly noted by o t h e r s . l ~ ~ ~ - ~ ~ Baseline corrections and elimination of outlying fringe pairs are briefly described by Miller et aL30 In particular, baseline positions were fitted with a 15th-order polynomial, and this deviation from straightness was subtracted from all fringe position^.^,^^,^^ The starting time correction At was obtained from the slope of the plot of the pseudobinary D vs reciprocal experimental time. These issues are described in more detail in refs 4 and 7. For diagnostic purposes, each experiment was f i s t treated as if it were a binary system. The skewing of Rayleigh patterns is defined as the deviation from the behavior of an ideal binary system (Le., with constant 0).It can be illustrated by a plot of Djl/* vs (z,*)~, where Dj and zj* are defined in ref 4. All exposures are plotted together to check for consistency. For an ideal binary system, the plot of Dj'" vs (zj*)* is a horizontal straight line. In general such plots are curved in
J. Phys. Chem., Vol. 98,No. 51, 1994 13747 ternary systems; they become straight but not horizontal as (z]*>~ approaches 0. If the D, are not concentration dependent, the intercept of the plot is DA. This extrapolated DA can be compared with the DA calculated from the R, and D,, as well as with the DA obtained from Gouy measurements. When Qo of a Gouy measurement is large, so is the skewing of the Rayleigh fringes. Large skewing is a numerical advantage in the AM and Miller Gouy analyses and in the Rayleigh analysis but a disadvantage in the classical Gouy analysis because of the Cr extrapolation problem. Typical examples of baseline fit and skewing plot are seen in Figures 4 and 5, respectively, of ref 4. The precision of J , using the average fractional part of a fringe obtained from the first five photographs, is generally f0.0030.007 fringe. The accuracy, however, appears to be only about 0.01-0.02 fringe and is due in part to uncertainty in ACl values. This has a significant effect on the final results, as will be seen in section 6. B. Gouy. There are several ways to extract ternary Gouy D, from fringe position data. Two of these involve calculating Cr for each exposure of an experiment in different ways but obtain DA, the S2 graph, and QOfor the whole experiment in the same way. The classical way to obtain Cr for an exposure is to extrapolate C, (defined in refs 1-3) from the low number fringes (typically the first 8) toflz) = 0. This involves the extrapolation problems noted in the Introducti~n.~-~ A second, better way is to use the Albright-Miller2 programs (henceforth denoted by AM) to obtain Cr from all the fringe position data of the exposure. The ternary Gouy diffusion coefficients D, are then calculated from the DA, Qo, J , and AC, of each experiment for an appropriate set of experiments. The D, calculation procedure is due to Fujita and G o ~ t i n g , ' and ~ , ~our ~ program (denoted by RFG) is our extension of a version of it due to R e ~ z i n . ~ ~ Definitions Of DA, n,and Qo are given by Fujita and Gosting,17 who denote QO by Q43 A third, different procedure (Miller method) is to directly fit the appropriate least-squares parameters to all the fringe position data from all the exposures of all the experiments.' This new method uses "reduced" fringe positions ?* = 5t1l2,where t is the corrected time of the exposure and is the distance of fringe j from the undeviated slit position. For each experiment a time correction At has to be added to the experimental clock time t' to get the corrected time t. In all three Gouy methods, it is obtainedu by plotting the apparent DA' (calculated from Cr and t' of the exposure) vs Ut'. The intercept of this line is the extrapolated DA for the experiment, and its slope is DAAt. In the AM programs, D A is also calculated from Cr for each pattern by using its corrected time, and these DA are averaged over all the patterns of that experiment. The extrapolated and average values of DAusually agree well. Similarly Qo for each pattern, calculated from the least-square parameters, is also averaged over all the patterns to give an average Qo for that experiment. The AM technique works best for systems with large Qo. However the programs have been adapted to handle experiments with low Qo by using preliminary D, results from all the experiments to reduce the number of least-squares parameters. As noted in the Introduction, still other programs can use J as an additional least-squares parameter when the experimental J is unobtainable or uncertain due to experimental problems. The AM programs thus provide extensive diagnostics.2 Although diagnostics for the Miller method are not yet fully worked out, they will be analogous to those for the AM
Miller et al.
13748 J. Phys. Chem., Vol. 98, No. 51, 1994
TABLE 1: Densities of Raffinose (0.015 M)-KCl (0.5 M)-HzO Mixtures at 25 "Ca It lb 2t 2b 3t 3b 4t ml 0.015 156 0.015 236 0.003 056 0.027 428 0.012 734 0.017 673 0.005 474 mz 0.417 365 0.605 056 0.509 062 0.512 874 0.435 713 0.586 641 0.490 737 CI 0.014 868 0.014 867 0.003 001 0.026 735 0.012 494 0.017 241 0.005 375 Cz 0.409 429 0.590 392 0.499 911 0.499 914 0.427 517 0.572 308 0.481 820 1.020 806 1.025486 d 1.019 009 1.027 278 1.019 365 1.026 931 1.020 461 ml m2 C1 Cz
d
ml m2 C1
Cz
d r?
6t 0.008 957 0.464 338 0.008 792 0.455 770 1.019 961
6b 0.021 475 0.557 891 0.020 943 0.544 071 1.026 354
lot 0.013 944 0.426 542 0.013 681 0.418 483 1.019 205
10b 0.016 454 0.595 859 0.0160 53 0.581 345 1.027079
7t 0.006 840
0.480 343 0.006 715 0.471 603 1.020 268 llt
0,011 523 0.444 892 0.011 308 0.436 576 1.019 560
7b 0.023 609 0.541 760 0.023 020 0.528 240 1.026037 llb 0.018 891 0.577 433 0.018 427 0.563 251 1.026 725
4b 0.024 987 0.531 332 0.024 361 0.518 018 1.025 850
5t 0.015 093 0.417 838 0.014 806 0.409 893 1.019011
5b 0.015 279 0.603 760 0.014 900 0.589 137 1.023 117
9t 0.002 841 0.510 694 0.002 790 0.501 526 1.020 844
9b 0.027 645 0.511 232 0.026 946 0.498 307 1.025461
6Bt 0.008 958 0.464 340 0.008 793 0.455 766 1.019 949
6Bb 0.021 475 0.557 894 0.020 943 0.544 073 1.026 353
8t 0.002 840 0.507 366 0.002 789 0.498 306 1.020 699
8b 0.027 647 0.514 604 0.026 946 0.501 547 1.025 611
0.002 841 0.507 367 0.002 789 0.498 306 1.020 697
8Bb 0.027 647 0.514 600 0.026 945 0.501 541 1.025 605
12t
12b 0.026 206 0.522 104 0.025 546 0.508 960 1.025 656
1Bt 0.015 156 0.417 365 0.014 868 0.409 428 1.019006
1Bb 0.015 237 0.605 068 0.014 868 0.590 398 1.027 269
0.004 265
0.499 901 0.004 188 0.490 877 1.020 656
8Bt
Units: mi,mol (kg HzO)-';Ci, mol dn~-~; d, g ~ m - ~ ,
TABLE 2: Primary Diffusion Data for Raffinose (0.015 M)-KCl(O.5 M)-HzO at 25 "Cd 1B
2
3
4
5
6B
7
8
8B
9
10
11
12
0.014 868 0.499 913 0.000 OOO 0.180 970
0.014 868 0.499 913 0.023 734 0.000 003
0.014 868 0.499 913 0.004 747 0.144 792
0.014 868 0.499 919 0.018 986 0.036 198
0.014 858 0.499 915 0.000 103 0.179 244
0.014 868 0.499 920 0.012 150 0.088 307
0.014 868 0.499 922 0.016 305 0.056 637
0.014 868 0.499 927 0.024 157 0.003 241
0.014 867 0.499 924 0,024 156 0.003 235
0.014 868 0.499 917 0.024 156 -0.003 219
0.014 867 0.499 914 0.002 372 0.162 862
0.014 868 0.499 914 0.007 119 0.126 675
0.014 867 0.499 919 0.021 358 0.018 083
80.133 80.091 14.30 0.00000 -0.0044
80.059 80.083 16.25 0.99998 0.9806
80.068 80.097 14.03 0.19997 0.1926
80.055 80.082 17.68 0.79995 0.7836
79.643 79.675 18.81 0.00436 -0.oo01
80.122 80.075 12.77 0.51197 0.4999
Rayleigh 80.087 80.081 19.21 0.68700 0.6723
82.939 82.944 33.41 0.98271 0.9636
82.906 82.938 31.86 0.98274 0.9636
80.136 80.081 21.32 1.01779 0.9982
80.061 80.081 11.38 0.09994 0.0940
80.101 80.083 12.85 0.29995 0.2911
80.072 80.071 16.90 0.90005 0.8822
80.092 (nC" 80.069 (DAY 1.8388 (DA)P 1.8398 Qo -9.86 (Qo)~' -8.99 Q1 -3.78 (QI)~' -3.44 At 12.34 ala 0.00000 r,* -0.0050 E P ( 1)
80.038 80.062 0.4412 0.4415 17.28 18.16 9.20 9.67 13.36 0.999 98 0.9791 F2M(1)
80.031 80.075 1.2604 1.2602 232.40 232.60 96.07 96.09 10.29 0.199 97 0.1918 FW)
80.024 80.060 0.5480 0.5475 169.15 168.49 85.00 84.71 10.77 0.799 96 0.7823 F3(2)
79.630 79.653 1.8228 1.8232 0.58 -1.16 0.22 -0.45 15.85 0.004 36 -0.0007 F2P(3)
80.095 80.053 0.7831 0.7827 295.87 295.63 135.88 135.91 8.78 0.511 96 0.4988 F3(1)
Gouy 80.077 80.060 0.6252 0.6254 232.34 233.22 112.86 113.31 15.28 0.687 00 0.671 1 F3(5)
82.887 82.922 0.4487 0.4498 31.46 32.64 16.65 17.30 28.05 0.982 71 0.9621 F3(3)
82.900 82.916 0.4495 0.4494 32.27 32.64 17.18 17.30 25.21 0.982 74 0.9622 F3(4)
80.121 80.060 0.4337 0.4336 4.58 2.96 2.45 1.58 21.24 1.017 79 0.9967 F2M(4)
80.062 80.059 1.5093 1.5099 138.45 139.00 54.97 55.34 8.37 0.099 94 0.0934 F3(l)e
80.080 80.061 1.0682 1.0686 284.41 284.55 121.71 121.80 9.53 0.299 95 0.2902 F3(1)
80.064 80.050 0.4902 0.4903 99.17 98.30 51.36 50.88 13.17 0.900 05 0.8808 FW)
Cz AC1 ACz J
Ar ai' rlb
J
From all data taken together (13 runs). Experimental Gouy values of DA, Qo,and Ql are obtained from the Albright-Miller programs. Subscript from F4J, then F3 used. Units: Ci and Act, mol dm-3; At, s; DA, ~. c denotes calculated value. * Value calculated from si and ( D A ) ~ J~obtained cm2 s-l.
J = R,AC,
programs for individual runs and analogous to Rayleigh diagnostics for a set.
a, = R,AC,lJ
4. Experimental Results In Table 1, density values d i n g cm-3 for all top and bottom solutions are reported with the experimental molalities mi in mol (kg HzO)-' and the computed molarities Ci in mol dm-3. The least-squares parameters reported in Table 9 are based on all density data, including those for solutions It, lb, 6t, and 6b whose corresponding diffusion runs were rejected due to experimental problems. Table 2 compares both Gouy and Rayleigh data for all 13 runs. Rayleigh rows 1-5 contain the experimental and calculated values of the total number of fringes J , as well the At (time correction), al,and rl. These provide a test of plate reading and timing errors. The Gouy rows also contain the experimental and calculated values of J, DA,Qo,and Ql, as well as At. The experimental D A and QOare the extrapolated and average values respectively obtained from the Ah4 programs.2 The calculated values are obtained from the equations
+ R2AC2
II&
Qd&
= = Eo
+ sAa,
+ E,a, - E2aI2
QII& = [aEo (bEo &,)a, (-a
+
+
+
(3) (4)
(5) (6)
+ b E , ) a I 2- ba13]E2(7)
using the quantities R1,R2; IA, SA; EO,El, E2; and SI,s2; obtained by least-squaringthe experimental q~antities~."*~~ for all 13 runs. These least-squares coefficients are reported in Table 9. The quantities Ei are defined by Fujita and Gosting," and a and b are defined by Revzin$2 as is Q1.42*43In the last four rows are reported the values of At; al;rl; and the type of AM program with the rank (in parentheses) of internal consistency of the single runs in the order: 1 good, 2 fairly good, 3 fair, 4 fair to poor, and 5 poor.
Analysis of Gouy Fringe Data
J. Phys. Chem., Vol. 98, No. 51, 1994 13749
B. Du from Subsets. Although the classical “linear 8” procedure is part of our diagnostics (along with sets of fringes other than 8), the Ah4 programs are more consistent and have smaller SEC. Therefore we use only AM results in the FujitaGosting calculations and denote them by %G. All cases use the extrapolated DA and average Qo. Miller calculations are denoted by & and Rayleigh calculations by R. As noted in the Introduction, the 13 runs (case 1) include a wide variety of al values. We have also examined a large 1 number of subsets of these 13 and have tabulated seven in Table 4 along with case 1. Of interest are the following: The la lb “standard choice” (case 5 ) has the a1 values (0.0,0.2, 0.8, 1.0) Figure 1. (a) Experimental values of I/& vs al from Gouy and usually includes some large values of Qo. This choice experiments and AM programs.2 (b) Experimental values of Qd partially emphasizes the two ends of the a1 range. Dunlop’s & vs alfrom Gouy experiments and AM programs,2where Qo is choice (case 6) has two runs close to a1 = 0.0 and two close to the Qo(average). a1 = 1.0, for which all four QO are small. Two cases of 3 symmetric al (cases 7 and 8) were considered to see if fewer runs than the customary four would be suitable. In addition, other subsets were based on the internal diagnostics of the Gouy and Rayleigh calculations. Six Gouy runs were ranked 1 (see Table 2) based on the high internal consistency of the individual Gouy experiments and give the “internally good Gouy” set (case 4) . The “best Gouy” (case 2) were obtained by deleting from all 13 the two runs whose calculated values of Qo were most discrepant, redoing the RFG calculation on the remaining 11, deleting the next two runs with the most discrepant Qo, and again redoing the RFG calculation 2a 2b (experiments 4,5, then 9,12 Finally, the Rayleigh Figure 2. (a) Experimental values of Qo (solid circles) and Ql (open diagnostics for all 13 runs showed that four had large fi) circles) vs al from Gouy experiments and AM programs.* (b) residuals (lB, 3, 5 , 10). Therefore the remaining 9 were rerun Experimental values of QdQl vs a1 from Gouy experiments and AM programs.2 with the Rayleigh program to get the “best Rayleigh” (case 3). The deleted runs for the “best Gouy”and “best Rayleigh” have Figure l a shows a plot of 1/&, which by eq 5 is linear in only run 5 in common. It is interesting that this run is the only al. The least-squares line is an excellent fit. Figure l b shows one whose is slightly discrepant from the overall average. Q d A , which by eq 6 is quadratic in al. This curve is A visual comparison of D,j variations from subset to subset reasonably good, with the maximum deviation in Qo of 1.7 x is shown in Figure 3. The variations in 0 1 2 are too small to Figure 2a shows QOand Ql plotted together vs al; Ql is see and so have been omitted. always smaller than Qo. Figure 2b shows QdQl vs al. Values of this ratio below 2.0 give poor values of J from the PQ 1. Comparisons of Diferent Gouy Calculations. Comparisons for %G and & show that the Dll are all very close and agree within 1 CSD for a given subset, except for case 8 which 5. Calculated Results and Comparisons is within 2 CSD. The same is true for Dl2 and 0 2 2 , although the errors for 0 2 2 are generally larger. The worst case for 0 2 2 A. D” from All 13 Experiments. Previously, RFG values is case 5, the “standard choice,” which is still within 1 CSD. of the Dv from a number of AM program variants were The agreement within 1 CSD is also true for D21 for all 8 cases, compared using the Gouy data for all 13 runs.2 Table 3 contains but the errors are much larger, from 2 to 6 times the errors of a similar comparison of the Dv and for the two best ones 0 2 2 . It is very gratifying that these two very different types of (DA(extrapolated), Qo(ave), and DA(ave), Qo(ave)) and the Gouy calculations yield close results within each subset. In classical “linear 8” method, plus the new Miller yj* Gouy method‘ and the Rayleigh nonlinear least-squares m e t h ~ d . ~ , ~ addition, if Qo and DA obtained from the & calculation are used as input for the %G program, the corresponding D, agree Table 3 shows that the newer Gouy methods are all in within computer round-off errors, Le., at most 2 in the fourth excellent agreement with each other, within one “combined figure of D21 and 0 2 2 . Consequently the two algorithms are standard deviation” (CSD). (In this paper, “within 1 CSD’ essentially equivalent. A more detailed comparison awaits the means within the sum of the two standard deviations of the further exploration of the diagnostics. However, the classical “linear 8” method cases considered.) However, for each type of Gouy calculation, the agreement has larger standard errors of the coefficients (SEC) than the of corresponding D, between subsets is much less encouraging. other Gouy calculations and is in poorer agreement even if it is The different combinations of experiments give rise to differwithin 1 CSD. These results are of course all based on the ences of 2-3 CSDs for Dll and 0 1 2 in the worst cases (the same fringe position data. The Rayleigh results, based on &G is more consistent). The variations are quite large for different fringe position data, are also in excellent agreement 021, with a maximum range of 0.026 for &G and 0.032 for except for 0 2 1 . However, the disagreement between the &. (These deviations are given in units of lop5 cm2 s-l to Rayleigh and best Gouy values of D21 is also just at the 1 CSD save space.) This reflects the sensitivity of D21 to adding or level. Consequently, based on all 13 runs, the agreement deleting an experiment. This sensitivity will be explored below. between Gouy and Rayleigh methods is very good. UnfortuMore encouraging is the reasonably close agreement of all nately, the error in D21 is relatively large. D,, for the “best” cases (cases 2-4), as well as the standard The next section contains the analysis of subsets of the 13 choice (except for the & 0 2 2 ) . However, the agreement experiments. The various agreements are not always as good, between these and the cases 1 (all 13) and 6 (Dunlop) is not as and variations in D21 are still larger.
Miller et al.
13750 J. Phys. Chem., Vol. 98, No. 51, 1994
TABLE 3: D” from Various Data Analysis Techniques for AU 13 C a s e case classical linear 8 (RFG) DA(av), Qdav) (RFG) DA(ext), Qdav) (RFG) Miller q* = I;tln Rayleigh a
Units: Dij,
DII
D12
0.4314 0.4323 0.4322 0.4321 0.4321
0.0007 0.0009 0.0009 0.0010 0.0008
D21 0.2260 0.2198 0.2200 0.2210 0.2037
O.OOO4
0.0002 0.0002 0.0001 0.0001
OD,,
OD,,
OD22
0.0002 0.0001 0.0001 0.0000 0.0000
0.0096 0.0041 0.0040 0.0045 0.0024
0.0046 0.0022 0.0021 0.0008 0.0003
cm2 s-l.
TABLE 4: Intercomparison of D” from Different SubseW case R 1 All 13 expts (1B-2-3-4-5-6B-7-8-8B-9-10-11-12) 2 “best Gouy” (1B-2-3-6B-7-8-8B-10-11) 3 “best Rayleigh” (2-4-6B-7-8-8B-9-11-12) 4 “intemally good Gouy” (1B-2-3-6B-10-11) 5 standard choice (a1= 0.0,0.2, 0.8, 1.0) (1B-2-3-4) 6 Dunlop choice (1B-2-5-9) 7 symmetric three (a1= 0.0,0.5, 1.0) (1B-2-6B) 8 alternate three (al= 0.1, 0.5,0.9) (6B-10-12)
a
OD,,
022
1.8230 1.8213 1.8206 1.8193 1.8198
Units: D,, and OD,,,
DII
0.4321 GRFG 0.4322 GM 0.4321 R 0.4317 GkpG 0.4324 GM 0.4324 R 0.4320 G W G 0.4324 GM 0.4322 R 0.4315 G W G 0.4324 0.4324 GM R 0.4324 G W G 0.4323 GM 0.4328 R 0.4313 G W G 0.4319 GM 0.4318 R 0.4318 GRFG 0.4324 GM 0.4324 R 0.4305 GkFo 0.4320 GM 0.4314
D12 0.0008 O.OOO9 0.0010 0.0009 0.0010 0.0010 0.0007 0.0009 0.0010 0.0009 0.0010 0.0010 0.0009 0.0010 0.0012 0.0008 0.0009 0.0009 0.0008 0.0010 0.0010 0.0011 0.0009 0.0009
D21 0.2037 0.2200 0.2210 0.2081 0.2099 0.2125 0.2070 0.2152 0.2191 0.2149 0.2121 0.2116 0.1933 0.2169 0.2091 0.2266 0.2246 0.2262 0.2068 0.2117 0.2099 0.2417 0.2359 0.2407
D22 1.8198 1.8206 1.8193 1.8176 1.8197 1.8186 1.8210 1.8207 1.8224 1.8177 1.8198 1.8187 1.8198 1.8195 1.8157 1.8208 1.8208 1.8205 1.8208 1.8195 1.8187 1.8131 1.8188 1.8189
OD,,
0.0001 0.0002 o.Ooo1 0.0002 0.0002 0.0001 0.0001 0.0002 0.0001 0.0003 0.0002 0.0001 0.0003 o.Oo04
0.0003 o.OOO4
0.0005 0.0001 0.0002 0.0003 0.0002 0.0002 0.0001 0.0002
OD,,
002,
OD22
0 . m 0.0024 0.0003
0.0001 0.0040 0 . m 0.0045 0 . m 0.0031 0 . m 0.0020 0 . m 0.0042 0.0001 0.0024 0.0001 0.0037 0.0001 0.0093 O.oo00 0.0051 o.oo00 0.0030 o.oo00 0.0037 O.oo00 0.0053 0.0001 0.0069 0.0001 0.0089 0 . m 0.0071 0.0002 0.0118 0.0001 0.0058 0.0000 0.0036 0.0001 0.0057 0.0001 0.0055 0.0000 0.0035 0.0001 0.0029 0.0001 0.0056
0.0021 0.0008 O.Oo04
0.0014 0.0008 0.0008 0.0033 0.0019 0.0005 0.0011 0.0006 0.0005 0.0032 0.0015 0.0003 0.0045 0.0009 0.0004 0.0028 0.0009 O.OOO4
0.0011 0.0010
cm2 s-l
good, and the two symmetric 3 cases (cases 7, 8) have quite different values of Dll and DZI. 2. Comparison of Rayleigh and Gouy. There are significant differences between the Rayleigh and Gouy results within subsets. Rayleigh D , are usually slightly lower than Gouy ones. The differences are usually between 1 and 2 CSDs in comparisons of R vs GRFGor R vs 6. However, the absolute errors are particularly large for 0 2 1 . The two cases of our initial interest, cases 1 and 5, have the largest absolute differences (0.024 for case 5). This probably arises because the lower quality Rayleigh runs are included in these cases (2 out of the 4 runs in case 5). Between subsets, the maximum variation in D21 is larger for R (0.048) compared to 6 (0.042) and &G (0.036). Some differences between Gouy and Rayleigh results can arise in part from plate reading problems and in part because each Rayleigh fringe j and its corresponding Gouy fringe j originate from different parts of the diffusion boundary. It is encouraging that the agreement is somewhat better (within 2 CSDs) for Rayleigh and Gouy results when their corresponding “best” runs are chosen, i.e., in cases 2, 3, and 4. These do agree within 1 CSD for D21 and within 2 CSDs for the others whose absolute errors are smaller. This will be discussed further when our final recommended values of D, are presented. C. Values of J and DA. Table 5 contains a comparison between Rayleigh and Gouy J and DA results, denoted as JG and DAGfor Gouy and JR and DARfor Rayleigh. Included are expenmental JR and JG, (JR - JG), J P Q ~and , J from the AM programs2 F3J (denoted by JEPand JEM)and F4J (denoted by J D ) , whenever these programs converge. JG indicates the experimental J obtained by Gosting’s method,Ig except for experiment 10 (J obtained from F4J). Note the relatively
constant difference of about 0.02 between the experimental JR and J G . ~The J from the AM and PQ programs differ from each other and the JG. As will be seen below, these differences have a significant impact on the D+ Table 5 also shows the values of DA from Gouy and Rayleigh experiments. The experimental Rayleigh values come from an extrapolation of (D,)ln vs (z,*)~;the calculated values come from least-squares values of the Dv. The experimental Gouy DA come from the AM programs; the calculated values from the RFG programs. The quantity ~ D I D A G = (DAR- DAG)JDAG(based on calculated values of DA) shows that the DAR are also systematically lower than the DAGby a quite constant 0.2%. Although there are systematic differences between both J and DA from Gouy and Rayleigh, the Do are close-except for D21 of course. The authors do not have an explanation for these differences, which in principle should not exist when the data are taken alternately on a high-precision diffusiometer with highquality cells, followed by careful plate reading. Nonetheless, whatever their cause, there is an internal compensation which fortunately yields similar numerical values of the Do from both types of optical measurements. 6. Discussion
A. Sensitivity of Calculated D p In all ternary systems studied so far by optical means, certain Do are quite sensitive to which experiments are included in the set to be analyzed and to experimental errors. In general, one cross-term Dij, i # j , has the largest variation, typically the one for which R,/Ri is larger than 1.0. An extensive examination has been done of the Gouy and Rayleigh results from the various combinations of the 13 experimentspresented in Table 4, as well as many others. These
J. Phys. Chem., Vol. 98, No. 51, 1994 13751
Analysis of Gouy Fringe Data
Table 6. In brief, the errors in D , are relatively small for errors in a AC, of fO.OOO1 mol dm-3 (a few tenths of a percent to 3% for D z ~ )and , not so large for errors in DA of f0.0005. For errors in J of k 0.05 taken alone, the effect on D21 is about 1%, but this is misleading as we will see. The effect of Qo is much larger. For errors in Qo of f1 x the error in D,, is only 0.2%, but the error in Dzl is 7% or about 0.016. These variations in D , are shown to the same scale in the “left-hand” Errors column of drawings in Figure 4 for AQo = f l x of 1.0 x shown by QO are not uncommon. For example, in the “all 13 runs” case, there are two (runs 5 and 9) for which this difference is nearly 2 x For Gouy data, the major problem comes from the hierarchy of errors. As noted above, the quantities DA and QOdepend on J . Therefore, we recalculated with the AM programs all Gouy data for the 27 combinations, using an error in J of only 0.02. The corresponding errors in DA, Qo, and J give rise to errors in 0.004 in Dll (0.1%), 0.003 in 0 2 2 (0.18%), but 0.016 in Dz1 or 6.8%. For an error of 0.05 in J , the maximum deviation in D z ~ was 0.04 (16.8%). In addition, the calculated D, had large SEC. If the maximum values of the error bars were taken, the error reached about 23% in 0 2 1 . These variations in D , are shown to the same scale in the “right-hand” column of drawings in Figure 4 for AJ = f0.02. Note that errors in J from different AM methods of calculation are usually 0.01-0.05, as is seen in Table 5. Moreover, the root-mean-square differences of J - JCdcfor all 13 Gouy runs was 0.0149 (maximum 0.060) and for all 13 Rayleigh runs was 0.0155 (maximum 0.054). The average errors based on this analysis were about half the maximum values noted above. Consequently just the expected errors in J alone give rise to errors in D21 of about 0.02. Another useful “average” diagnostic is to subtract 1.O x from each value of Qo and recalculate the D,. In the RKC system, this yields an error in D21 of 0.013. Similar calculations for the impact of J alone are easier with the Miller method because there are no intermediate calculations of DA and Qo. A few such calculations indicate the same impact of AJ = 0.02 on the D , as did the RFG calculations. Consequently the Miller method makes it possible to do the 50 or so sets of random variations in fringe positions and times necessary to get a reasonable measure of errors in D,.49 The major conclusion from these, and many other comparisons not shown, is that the precision of diffusion measurements as indicated by least-squares statistics is very misleading. A rough rule of thumb is that the accuracy of the D, is 4xSEC obtained from least-squares statistics and the propagation of error equations. B. Choice of Best Values of a1 for a Set of Experiments. Values of alshould be chosen to maximize the precision of the results for a set of experiments, typically four. Clearly the “orthogonal” a1 cases of 0.0 and 1.O should be included if their associated diffusion boundaries are dynamically stable. In terms of the Fujita-Gosting analysis, Gosting’s laboratory used experiments with intermediate values of al but not too far away from a1 of 0.0 and 1.0. This would get the best straight line for l/& vs al and have intermediate values of Qo; this vs a1 curve. An would help better determine the Qd& extension of t h s approach would be to look for two large values of Qo, usually in the middle of the a1 range and two small ones at the ends of the al range. As noted above, the classical linear extrapolation methods to get C, do not work as well for large Qo, whereas the AM methods work best at large Qo. Dunlop’s original special method1*to get D,, when translated into the Fujita-Gosting analysis, depended on all four Qo being close to zero. This occurs very near a1 of 0.0 and 1.0, provided one or both cross-term D , are small.
eocale
I
0.tao-l 1.830
0
RnrW9h
e
WI*I
Figure 3. Comparison of 105Dufor the various subsets, including the error bars obtained from the statistical standard error of the coefficients and propagation of error equations. Open circles for Rayleigh calcula-
tions; solid circles for Miller calculations; solid triangles for RFG (Revzin) calculations. show only small changes in Dll, 0 1 2 , and 0 2 2 but relatively large ones in 0 2 1 . This sensitivity of D21 to the set of experiments was initially quite surprising to us. As seen above, the variations from one Gouy set to another are usually several times more than the 1 standard error of the coefficient (SEC). This SEC is determined from the propagation of error equations:’ using the covariance matrix for the least-squares parameters. Besides errors from making up solutions, evaporation of solutions in the reservoirs before beginning the run, reading plates, the lack of baseline corrections for Gouy experiments, etc., the statistical problems may well arise in the following way. Standard least-squares methods, linear or nonlinear, assume that the x vectors are without error and only the y vector has errors. However, almost all experimental situations involve errors in all the variables. Although the error-in-variables( E N ) problem has been studied to some there is a further complication. In most experimental situations, the errors form a hierarchy. In the Gouy example, besides the error in J itself, an error in J also affects the values of DA and Qo, which in turn enter into the RFG calculation of the Dib For Rayleigh, the value of J affects the way the plates are read and therefore the final results. The case of EIV with an error hierarchy will require development of different and more complicated statistical analysis procedures. As a substitute, we have done an extensive sensitivity analysis of the FG type of Gouy calculations by examining the effects of errors in the five input variables AC1, ACz, J, DA, and Qo, each taken separately. We have chosen runs 6B, 10, and 12 (case 8), which cover a large enough range of al and which involve the 27 combinations of -, and 0 errors given in
+,
13752 J. Phys. Chem., Vol. 98,No. 51, 1994
Miller et al.
TABLE 5: Comparison of J and DA for Raffiose (0.015 MkKCl(O.5 M)-H20 at 25 OCU Gouy Rayleigh exp JR JG JPQ JEP JEM JD J R - JG J G - J E DA @A)C DA PA), 1B 80.133 80.092 80.14 80.104 0.041 -0.012 1.8388 1.8398 1.8382 1.8369 2 80.059 80.038 79.87 80.016 0.021 0.022 0.4412 0.4415 0.4405 0.4407 3 80.068 80.031 80.04 79.992 79.997 0.037 0.039 1.2604 1.2602 1.2600 1.2582 4 80.055 80.024 79.988 80.060 0.031 0.036 0.5480 0.5475 0.5458 0.5466 5 79.643 79.630 79.63 79.597 0.013 0.033 1.8228 1.8232 1.8206 1.8203 6B 80.122 80.095 80.00 80.170 80.127 80.061 0.027 -0.032 0.7831 0.7827 0.7815 0.7814 7 80.087 80.077 80.04 80.318 80.094 80.215 0.010 -0.017 0.6252 0.6252 0.6243 0.6241 0.052 -0.023 0.4487 0.4494 0.4480 0.4486 8 82.939 82.887 82.63 82.910 8B 82.909 82.900 82.64 82.851 0.009 0.049 0.4495 0.4494 0.4481 0.4486 9 80.136 80.121 80.03 80.021 0.015 0.100 0.4337 0.4336 0.4330 0.4328 0.004 1.5086 1.5093 1.5068 1.5069 10 80.061 80.062 80.07 80.058 80.062 -0.001 0.OOO 1.0685 1.0682 1.0609 1.0665 11 80.101 80.080 80.08 80.080 80.111 80.072 0.021 12 80.072 80.064 79.32 80.017 80.181 0.008 0.047 0.4906 0.4902 0.4886 0.4894 JEPand JEM result from F3J programs,2which are used for rl near 0.0 and 1.0, respectively (where Qo are small). is its JD from the F4J program;2 all other JG are from the Gosting method.6~7J9~20 AD = DAR - DAG.Units of DA, TABLE 6: Error Combinations for Sensitivity Analysis“ Combination 1 2 3 4 5 6 7 8 9 10 11 0 - 0 0 0 0 + + RKC6 0 RKC6 RKClO O O O + - O O + O R K C l O + RKC 12 O O O O O + - O + R K C 1 2 + O
+
12 13 14 15 16 17 18
-
0
+
+ + 0 0 0 - + -
0 - - 0
+,
0 RKC6 +RKClO
00 -9.9 17.3 232.4 169.2 0.6 295.9 232.3 31.5 32.3 4.6 138.4 284.4 99.2
QdQi 2.61 1.88 2.42 1.99 2.60 2.18 2.06 1.89 1.88 1.87 2.59 2.34 1.93 The JG for experiment 10 cm2 s-l. 0ID.a 0.0016 0.0018 0.0016 0.0016 0.0016 0.0017 0.0018 0.0018 0.0018 0.0018 0.0016 0.0016 0.0016
19 20 21 22 23 24 25 26 27 0 -
+
+
-RKCl2++
-
-
- + + + - ++ +- + - + - - - +
The 0, and - mean no change, an increase, and a decrease, respectively, by the amount of the chosen value of the error, in the appropriate quantity for the given experiment AO,
=l
.-
.....
0.440
0.420
0.420 Comblniilon 0.400 0
s
10
1s
0.040
0.000 .0.020
2s
0
s
15
20
2s
AO,
s
S
0
10
15
20
25
20
I6
10
Each Do from the average of its four correspondingDo from and 6 using case 2 (“best Gouy”)and case 4 (“intemallygood Gouy”). Estimated errors are the realistic errors discussed in the text. From case 3 (“best Rayleigh). Estimated errors are the realistic errors discussed in the text. (I
0.020
o.ooo]
4
-0.020
......-..- ......
0
10
S
10
1S
20
I 25
10
=l
0.200
10
1S
il
-0.040
10
0.210
0
0.400
10
1......................*...
-0.040
0.200
20
AO,
101 2 0.020
I
n . . . -
C.
TABLE 7: Recommended Dy for Raffinose (0.015 M)-KCI (0.5 M)-H20 at 25 “C i05D1, 105D12 1O5Dzl 1O5DZ2 Gouy“ 0.4324 0.0010 0.2115 1.8194 estimated errors fO.OO1 f0.0002 f0.02-0.03 f0.004 Rayleighb 0.4320 0.0007 0.2076 1.8210 estimated errors fO.OO1 f0.0002 f0.03 f0.004
10
0
5
10
1s
20
2s
a0
1.000 1.710 0
I
10
1s
20
2s
a0
Figure 4. Sensitivity analysis for Dv based on the 27 possible cases of the presence (+ or -) or absence (0) of errors in experiments 6B, 10, and 12. The various combinations of errors are listed in Table 6. The “left-hand” column of drawings indicates the Do and their associated standard error of the coefficients (as error bars) for each combination. These are based on a AQo alone of 1 x The “righthand” column of drawings indicates the & and their associated standard error of the coefficients for each combination. These are based on a AJ of 0.02, taking into account the corresponding changes in DAand QOfrom the different value of J . All drawings are to the same scale. Note the large effect of small errors in Qo and J on 021. Our motive for examining the various subsets was the question: what are the most suitable combinations? For example, should alvalues which yield the largest Qo serve
better, since the Dv are so sensitive to Qo? Unfortunately, for most of the small sets, the differences among them and with their corresponding Rayleigh sets were significant, especially for D21. Consequently we are not able to suggest any special optimum choice of a1 values. However, from a numerical analysis viewpoint, the largest range of alinside the limits of hydrodynamic stability is important (alvalues of 0.0 or 1.0 sometimes yield unstable boundaries for other systems). An analysis of this issue is in progress using simulated fringe positions. These are obtained from eqs 27, 28, and Bl-B4 of ref 1 for Gouy and eqs 4, 10, and B 1-B4 of ref 1 for Rayleigh, using assumed Dv and Ri and imposing random errors. The calculated vs input Di, for various al combinations should indicate whether optimum or near optimum choices exist. C. Recommended Values of D p Despite our extensive discussion of the large errors in D21 and the variations of Dv among the different sets, it is possible to recommend “best” Gouy and Rayleigh values of Do for this raffinose (0.015 M)KCl (0.5 M)-H20 system. These are given in Table 7. The Gouy values are the respective averages of &G and & from subsets 2 (“best Gouy”) and 4 (“internally good Gouy”). The Rayleigh values are from case 3 (“best Rayleigh”). The errors are the more realistic ones based on the examination of the subsets, rather than the purely statistical errors. Consequently the recommended Gouy and Rayleigh Dv are in fact in good agreement. Table 8 includes Dunlop’s results13for this same composition. Our recalculation with his values of input datal3 gave his results exactly, after conversion of concentrations from his gram basis
Analysis of Gouy Fringe Data
J. Phys. Chem., Vol. 98, No. 51, 1994 13753
TABLE 8: Comparison of Dunlop and Our Analog lO’D11 l@Diz 1@Dzi 105Dzz 0.0008 0.2416 1.8179 Dunlop’s Dip 0.4300 f0.0006 f0.0002 f0.0126 f0.0050 SEC our Dunlop analogb 0.4319 O.OOO9 0.2246 1.8208 f0.0005 f0.0002 f0.0118 f0.0045 SEC Dunlop’s resultsI3as recalculated with the RFG program. The M program could not be used because his fringe positions are not available. The errors are the least-squares SEC. * Our “Dunlop case” (case 6). For comparison with Dunlop, the SEC are also given. TABLE 9: Overall System Dataa = 0.014 868 Ri = 3373.27 c 2 = 0.499 918 Rz = 442.44 co E 5j.309 IA = 233.14 mi(Ci,C2) = 0.015 197 SA = 242.79 mz(c&) = 0.510 964 Eo = -0.209 61 2 = 1.023 150 E1 = 41.995 HI= 197.136 E2 = 40.921 Hz = 45.730 SI = 481.08 ?I = 308.122 sz = 234.36 vz = 28.897 di = 0.4321 = 18.063 1 2 = 1.8207
CI
vo
Gouy Di Do, = 0.4345
Db2= 0.0018 DI, 10.2825 D,,- 1.8464 Rayleigh Di, Do, = 0.4341 Db2= 0.0015 Dbl = 0.2785 D,! = 1.8479
?i is average of the from $1 the data of all experiments except expt 5 , which is slightly-off. d, Hi, and Vi are obtained from all experiments=usingC( = ci in eq 2. his leas_t-sq_uaresvalue of ;ijs identical to d, the average of the experimental d. (d = 1.023 180 if C1 and C2 are taken as the round values 0.015 and 0.500, respectively). Divide the Hi in this table by lo00 to get din g cm-3 from eq 2 when Ci are in mol dm-3. The remainder of the least-squares parameters are obtained from the 13 retained Gouy experiments, except the Di, which are based on the recommended Dv values in Table 7. R1and R2 are referred to J but can be converted to n values by multiplying by ala where a = 2.5064 cm and 1 = 5461 x cm. The 11 and 1 2 are the eigepalues of the diffusion coeficient-rn$ix D,and Ai = lh?. Units: Ci, mol dm-3; mi, mol (kg HzO)-’;d, d, g ~ m - H~i,;g mol-’; Vi, cm3mol-’; Ri,dm3mol-’; IA, SA, s1IZ cm-’; ai, lo-’ cm2s-l; si,s112 cm-1; D;,10-5 cmz s-1.
to our mole basis. Dunlop did not estimate his errors, other than to note that QOwas accurate to better than 2 x lop4based on his Qo residuals. The realistic errors of his data are about the same as those for our best Gouy values. However, our RFG recalculation does give the SEC; these are only a few figures in the fourth decimal higher than those for our case 6 (Dunlop) shown in Table 4. We could not do a Miller calculation because Dunlop’s fringe positions were not available to us. Our own “Dunlop case” is also shown in Table 8. There are some slight differences between our results and his, but these are well within the expected errors. The large difference in 9 1 (0.017) is within 1 CSD, and all values in Tables 7 and 8 are in agreement within realistic errors. This agreement is testimony to Dunlop’s careful experimental technique, since his were among the fiist ternary Gouy experiments and were done on a good but less precise diffusiometer than ours. Table 9 contains the solvent-fixed diffusion coefficients Duo based on the recommended volume-fixed Do in Taye 7. Table 9 also contains the overall system data including: Ci,the average gf the average ei; the molalities based on these the average Ci; the density parameters 2 and Hi obtained by least-squgres from eq 2 for the overall composition (the same value 2 is obtajned by averaging the measured 2 from all experiments); the Vi calculated from the Hi;24the least-squares parameters Ri, ZA, SA, EO,El, and E2 from the RFG analysis; and the parameters si and the eigenvalues of the D matrix Ai = Us?. See the footnote to Table 9 for the experiments used. D. Comments on Precision and Accuracy. The AlbrightMiller programs, developed as a result of problems recognized during the original experimental work, have improved diag-
nostics and thus raised the standards for the acceptability of individual experiments. Consequently a major result of this research has been the recognition that some runs which would have been acceptable before 1986 are less acceptable now. Since the large variations of D21 for this raffinose-KClH 2 0 system have counterparts in other systems, some Do have a larger internal sensitivity to experimental and calculational errors than has been previously recognized. The large impact on Di, of small errors in J indicates that considerably more care may be necessary both in making up solutions and in determining the fractional part of a fringe. As mentioned in section 5.B.2, differences between Gouy and Rayleigh results can arise from errors introduced in reading photographic plates and the subsequent data reduction and because the two optical systems give information about concentration distributions with different relative precisions as a function of positions in the diffusion boundary. Although photographs do provide a permanent record, there are errors in reading grainy plates. These errors can be avoided by real-time capture of fringe positions using photodiode arrays and by automating these procedures. Such automated scanning of Gouy patterns at the University of Naples has substantially improved their precision. Similar automation of the more precise Gosting diffusiometer, now at Texas Christian University, is in progress. Not only are the fringe positions located better, more of the close together inner Rayleigh fringes (center of boundary) and inner Gouy fringes (edges of boundary) can be resolved. Moreover, the equivalent of many more exposures can be taken, thereby improving measurement statistics. These newer techniques, as well as automated plate reading, involve the same issues discussed above. For the present, the results of this research support the rule of thumb mentioned above: the accuracy of Do is approximately 4 x SEC obtained from standard least-squares statistics and the propagation of error equations. Ultimately a consideration of real statistical errors will require error-in-variable analyses, involving error hierarchies. Although Gouy and Rayleigh interferometry are the most precise and accurate of liquid diffusion measurements, the above comments suggest that the accuracy of ternary solution diffusion measurements is less than most workers have previously believed. It is also possible that we are seeking a greater precision and accuracy than can be achieved in view of experimental, numerical, and statistical limitations. E. Onsager Reciprocal Relations. To test the ORR, we have taken from Table 7 the Do and their estimated errors for the better Gouy and Rayleigh sets, using 0.02 (Gouy) and 0.03 (Rayleigh) for 8021. The calculations require the activity coefficient derivatives yij = a In yi IaCj, where yi is the activity coefficient on the molarity basis. As described in ref 14, all four derivatives can be obtained from values of yii = a In yi I ami and y12, together with the vi, where yi is the activity coefficient on the molality basis. The errors dy, are obtained analogously from the errors in yii and y12. Since our Ci and mi are slightly different from Dunlop’s, the yii quantities were recalculated but are nearly unchanged. We retain the originalI4 estimated value of y12= 0.25 and estimated errors of yii. These estimated values and their estimated errors were entirely based on ana10gy.l~ The ORR will be satisfied if the (rhs - h s ) of eq 16 of ref 14 is less than the probable error based on the values and errors of the Do and yo. The overall probable error is taken as the square root of the sums of the squares of the probable errors of all individual terms. The results are in Table 10. Analysis of the ORR calculation shows that y12 and its error have the most significant effects on the ORR error estimate.
13754 J. Phys. Chem., Vol. 98, No. 51, 1994
Miller et al.
TABLE 10: Onsager Reciprocal Relation Calculations“ yll = 0.5537 ylz = 0.25 ~ 2= 1 0.3889 y22 = -0.1921 6yii = 0.103 6yiz = 0.25 By21 = 0.123 6y22 = 0.022
GOUYDij ~ D Z=I0.02
rhs - lhs probable error
= =
0.428 0.474b
Rayleigh Dij 6021 = 0.03‘
rhs - lhs probable error
=
0.423 0.481
=
These calculations are based on the concentrations and vi in Table 9 and on the activity assumptions y l 1 = 0.2,6yll = 0.1; y~ = -0.2136, dyzz = 0.021 (10% of ~ z z ) ;and y12 = 0.25, 6 y 1=~ 0.25. The Do and 6Dij are taken from Table 7, with the specific values of 6 9 1 given above. If 6y12 = 0.1 (its 1959 estimate), then the probable error for this Gouy case is 0.220. Note that all other yij and 6yij change when Y I Z or 6y12 are changed. If 8021 = 0.02 for this Rayleigh case, the probable error is 0.474 and is not much different from the table value of 0.481.
Therefore their values largely determine whether the ORR are satisfied or not. In 1959, we assumed that an error of 0.1 in y12 was reasonable. However, in the absence of any guidance from experiment, it may well be that 6y12 was substantially underestimated. If 6y12 is taken as 0.25, Table 10 shows that the ORR is satisfied in terms of eq 16 of ref 14. (The minimum value of that error to have the ORR just satisfied is 0.23.) This raffinose-KC1 composition is the only failure of the ORR out of the 60 or so compositions of various ternary systems. Consequently the authors believe that the original estimates of activity derivatives are in substantial error, rather than the Du values or the ORR itself. This case can only be resolved by experimental measurements of the activity coefficients. Unfortunately this is very difficult in the raffinoseKCI system because the raffinose is so dilute. Isopiestic measurements are unlikely to be sufficiently precise without a large number of experiments and extensive data analysis; emf experiments may be preferable.
Acknowledgment. This work was done by all the authors both at LLNL and at the UniversitA di Napoli. The work at LLNL was done under the auspices of the U.S. Department of Energy, Office of Basic Energy Sciences (Geosciences), at Lawrence Livermore National Laboratory under Contract No. W-7405-ENG-48. The work at Napoli was done under the auspices of the Italian CNR. The authors thank Dr.Joseph A. Rard and Prof. Vincenzo Vitagliano for advice and help. References and Notes (1) Miller, D. G.J. Phys. Chem. 1988, 92, 4222. (2) Albright, J. G.; Miller, D. G. J. Phys. Chem. 1989, 93, 2169. (3) Miller, D. G.; Sartorio, R.; Paduano, L. J. Solution Chem. 1992, 21, 459. (4) Miller, D. G.; Albright, J. G.; Mathew, R.; Lee, C. M.; Rard, J. A.; Eppstein, L. B. J. Phys. Chem. 1993, 97, 3885. (5) Dunlop, P. J.; Steel, B. J.; Lane, J. E. In Physical Methods of Chemistry; Weissberger, A., Rossiter, B. W., Eds.;John Wiley: New York, 1972; Vol. 1, Chapter IV. (6) Tyrrell, H. J. V.; Harris, K. R. Difusion in Liquids; Butterworth: London, 1984. (7) Miller, D. G.; Albright, J. G. “Optical Methods” In Wakeham, W. A.; Nagashima, A.; Sengers, J. V., Eds., Measurement of the Transport Properties of Fluids: Experimental Thermodynamics; Vol. III, Blackwell Scientific Publications: Oxford, UK, 1991; Section 9.1.6, pp 272-294 (references pp 316-319). (8) dun lo^. P. J.: Harris. K. R.: Young. D. J. In Phvsical Methods of Chemistry; Roisiter, B. W., Baetzold, R. C.:Eds.; John Wiley: New Yo&, 1992; Vol. VI, Chapter 3. (9) Miller, D. G. In Transport Phenomena in Fluids; Hanley, H., Ed.; Marcel Dekker: New York, 1969; Chapter 11. (10) Miller, D. G. In Foundation of Continuum Thermodynamics; Delgado Domingos, J. J., Nina, M. N. R., Whitelaw, J. H., Eds.; Macmillan: London, 1974; Chapter 10 (see also E. A. Mason, Chapter 11 of this volume). (11) Miller, D. G. J. Phys. Chem. 1965, 69, 3374.
(12) Dunlop, P. J. J. Phys. Chem. 1957, 61, 994. (13) Dunlop, P. J. J. Phys. Chem. 1964, 68, 3062. A recalculation of the data from ref 12 by the method of ref 17. (14) Miller, D. G. J. Phys. Chem. 1959,63, 570; corrections, 1959,63, 2089. (15) Gosting, L. J.; Kim, H.; Loewenstein, M. A.; Reinfelds, G.; Revzin, A. Rev. Sci. Instrum. 1973, 44, 1602. (16) This diffusiometer is now located at Texas Christian University, Fort Worth, TX. A brief history of this instrument is given in footnote 27 of ref 4. (17) Fujita, H.; Gosting, L. J. J. Phys. Chem. 1960, 64, 1256. (18) Fujita, H.; Gosting, L. J. J. Am. Chem. SOC. 1956, 78, 1099. (19) Gosting, L. J. J. Am. Chem. SOC. 1950, 72, 4418. (20) Woolf, L. A,; Miller, D. G.; Gosting, L. J. J. Am. Chem. SOC. 1962, 84, 317. (21) Creeth, J. M. J. Am. Chem. SOC. 1955, 77, 6428. (22) We have also done a related empirical study of concentration dependence of the Dij for the system NaCl-KCl-H20 at one set of mean concentrations, since as yet there is no analytical theory analogous to the binary studies.21.28,50.51 That study is based on fixing the ratios of ACi for each set of four experiments, but increasing the absolute values of ACi to get 30, 60, 90, and 120 fringes. It will compare results from different diffisiometers and from both Gouy and Rayleigh measurements, and is being prepared for publication. (23) Rard, J. A.; Miller, D. G. J. Solution Chem. 1979, 8, 701. (24) Dunlop, P. J.; Gosting, L. J. J. Phys. Chem. 1959, 63, 86. (25) The paramete_rs H1 and HZ depend only on the input values of-d, whereas the value of d also depends on the assigned values of C1 and CZ. (26) Albright, J. G.; Mathew, R.; Miller, D. G.; Rard, J. A. J. Phys. Chem. 1989, 93, 2176. (27) Albright, J. G.; Miller, D. G. J. Phys. Chem. 1972, 76, 1853. (28) Albright, J. G.; Miller, D. G. J. Phys. Chem. 1975, 79, 2061. (29) Rard, J. A.; Miller, D. G. J. Chem. Eng. Data 1980, 25, 211. (30) Miller, D. G.; Ting, A. W.; Rard, J. A.; Eppstein, L. B. Geochim. Cosmochim. Acta 1986, 50, 2397. (31) Rard, J. A.; Miller, D. G. J. Phys. Chem. 1987, 91, 4614. (32) Wendt, R. P. Ph.D. Thesis, University of Wisconsin, Madison, 1960. (33) Albright, J. G. Ph.D. Thesis, University of Wisconsin, Madison, 1962. (34) Miller, D. G.; Vitagliano, V. J. Phys. Chem. 1986, 90, 1706. (35) Vitagliano, P. L.; Della Volpe, C.; Vitagliano, V. J. Solution Chem. 1984, 13, 549; misprints corrected in J. Solution Chem. 1986, 15, 81 1 and ref 34. (36) Vitagliano, V.; Sartorio, R.; Spaduzzi, D.; Laurentino, R. J. Solution Chem. 1977, 6, 671. (37) Vitagliano, V.; Borriello, G.; Della Volpe, C.; Ortona, 0.J. Solution Chem. 1986, 15, 811. (38) Creeth, J. M.; Gosting, L. J. J. Phys. Chem. 1958, 62, 58. (39) Miller, D. G . J. Solution Chem. 1981, 10, 831. (40) Albright, J. G.; Shemll, B. C. J. Solution Chem. 1979, 8, 201. (41) Miller, D. G.; Ting, A. W.; Rard, J. A. J. Electrochem. SOC. 1988, 135, 896. (42) Revzin, A. Ph.D. Thesis, University of Wisconsin, Madison, 1969. (43) The quantity Qo is the integral of the zeroth moment, is., /SZ, djfzj). The integral of the first moment, Jflzj) Szj djfzj), is Ql.Fujita showed that Ql could be used with DA to obtain the Dij, and the results of this unpublished work are given in Revzin’s thesis$2 Revzin’s program allows the use of Qo, Ql. or any weighted combination of Qo and Ql to obtain the D p We have found by experience that QOalone yields more consistent results than the other choices. (44) Longsworth, L. G. J. Am. Chem. SOC. 1947, 69, 2510. (45) This nine-experiment set had quite good consistency, with all QO residuals less than 1 x On the other hand, deleting the four worst cases from the first pass gives a different second pair and a slightly different and poorer set of Di, than the chosen method. (46) In principle, the Rayleigh fpf should be more precise than the Gouy fpf because the interfringe separation is 270 vs 160 pm, respectively. However, a sample Miller Gouy method calculation for case 8 using JR instead of JG showed that the discrepancy between Gouy and Rayleigh Dij became worse. (47) Wentworth, W. E. J. Chem. Educ. 1965.42, 96. (48) Macdonald, J. R.; Thompson, W. J. Am. J. Phys. 1992, 60, 66. (49) Anderson, K. K. Private communication, January, 1988. (50) Gosting, L. J.; Fujita, H. J. Am. Chem. SOC.1957, 79, 1359. (51) Albright, J. G.; Miller, D. G. J. Phys. Chem. 1980, 84, 1400. JP9413612
