J . Phys. Chem. 1993, 97, 3885-3899
3885
Isothermal Diffusion Coefficients of NaCI-MgC12-H20 at 25 OC. 5. Solute Concentration Ratio of 1:l and Some Rayleigh Results Donald G. Miller,* John G. Albright,'.+ Roy Mathew,* Catherine M. Lee,# Joseph A. Rard, and
Lee B. Eppsteinll Chemistry and Materials Science Department, University of California, Lawrence Livermore National Laboratory, Livermore, California 94550 Received: December 7. 1992
Isothermal interdiffusion coefficients have been obtained by Gouy interferometry for the ternary system NaClM g C l r H 2 0 a t 25.00 OC and a 1:1 mole ratio of NaCl to MgC12. Data are reported for total molar concentrations of 0.5, 1.0,2.0, 3.0, and 3.72 M. Diffusion coefficients a t mole ratios of 1:l (0.5 and 1.0 M), 1:3 (1.0 and 3.0 M), and 3: 1 (1 .O M) were also obtained by Rayleigh interferometry during the corresponding Gouy experiments by switching the two optical systems back and forth. As expected, good agreement is found between Gouy and Rayleigh results. Diffusion coefficients a t the 1:l ratio are intermediate between the results for 1:3 and 3:l ratios, with Dl2 becoming large with increasing concentration but always less than D I I and D22, and with Dl1 and D22 crossing a t 2.8 M. Densities were measured for all solutions. Rayleigh experimental and data analysis techniques are described.
I. Introduction This is the fifth and last in a series of experimental papers presenting mutual diffusion and density data at 25 OC for the system NaC1-MgCl2-H20. It is part of an international collaboration, organized in 1984, whose purpose is to compare recent statistical transport theories for electrolytemixtures against experimental data. This system was chosen because it contains different valence types, and thus the data provide a more severe test of theories than do existing data for 1: 1 electrolyte mixtures. It is also an important model for seawater and its concentrates. The theoretical quantities of interest are the ionic Onsager coefficientsI , of irreversiblethermodynamics14 and the 'distinct diffusioncoefficients" D,d of statistical mechanical theory.5.6The I, can also be used for semiempirical estimates of the transport properties of mixture~.2*4.~.8 For comparison, experimentally based values of the I , can be calculated from the following data: interdiffusion (mutual diffusion) coefficients, electrical conductances, transference numbers, and activity coefficients.Z4 Similarly the D,: can be calculated from the I , and intradiffusion (self-diffusion) coeff i c i e n t ~D,*. ~.~ Most of the necessary experimental data for the NaC1-MgCl2H20 system at different mole ratios have already been published. These are Leaist's dilute solution interdiffusion(mutual diffusion) data,9 Rard and Miller's thermodynamic activity data,lO Mills et ale's intradiffusion and viscosity data.' Bianchi et ale's dilute and concentrated solution electrical conductance data,12.1jand our earlier mutual diffusion and density data from moderate to high concentrations.I6l7 Other than data reported below, the only remaining data required to calculate experimental values of I , and D,: are the transference numbers. These are being measured by H. Sch6nert and co-workers in Aachen using the Hittorf method. However, that method is by far the most difficult of the required experiments, and resultsarenot yet available. Some additional thermodynamic
' Permanent address: Chemistry Department, Texas Christian University, Fort Worth, TX 761 29. Permanent address: PRC Environmental Management, Inc., 350 North Saint Paul St., Suite 2600, Dallas, TX 75201. 3 Current address: University of California San Francisco, School of Medicine, 513 Parnassus Ave., San Francisco, CA 94143. 1' Permanent address: New Brunswick Scientific Corp., 44 Talmadge Rd., Edison, N J 08818. f
0022-365419312097-3885$04.00/0
data (heat capacities) have also been measured by Bianchi and Tremaine at Memorial University in Newfoundland.18 The Gouy data presented here are primarily for the 1: 1 mole ratio of NaCl to MgCl2 for five sets of mean concentrations whose total concentrations are 0.5, 1.0, 2.0, 3.0, and 3.72 M, where M denotes the molarity in mol dm4. In addition, some Rayleigh resultsatratiosof 1:l (0.5and l.OM), 1:3(l.Oand3.0M),and 3:l (1.0 M) are presented for comparison with the Gouy results at corresponding concentrations, reported here or previously.14J7 Each set of Rayleigh data was collected in the same experiment as its correspondingGouy data by altemating the interferometer's optical configuration between the Rayleigh and Gouy modes. Thus solutions,boundary sharpening, starting time, and diffusion itself were the same for each combined Gouy and Rayleigh experiment. Densities were measured accurately by pycnometry and are reported for the 1:l mole ratio cases; the other densities were reported p r e v i ~ u s l y . These ~ ~ ~ ~data are necessary to convert molal to molar concentrations, to obtain partial molar volumes VI,to convert transport coefficients from the volume-fixed to the solvent-fixed reference frame, and to provide information to calculate the conditions for s t a t i ~ 'and ~ * dynamicZs22 ~~ stability of the diffusion boundaries in the gravitational field. Description of diffusion in an isothermalternary system requires four diffusion coefficients Oil. The flows Ji of solute i are related to the solute concentration gradients by the generalized Fick's law equations
where Ci is the molar concentration of solute i . Here 1 refers to NaCl and 2 to MgC12. We perform 'free-diffusion" experiments, in which two solutions of slightly different concentrations,initially separated by a sharp boundary, are allowed to diffuse in a vertical cell. The total diffusion time is long enough to get good fringe separations but short enough that concentration changes are not observed at theendsof thecell. Becauseconcentrationdifferences across the boundary are relatively small for our experiments, the flows and Dijare effectivelyon the volume-fixed reference frame. 0 1993 American Chemical Society
3886 The Journal of Physical Chemistry, Vol. 97, No. 15, 1993 Although techniques for extracting diffusion coefficients from ternary Gouy data were first published23 in 1956 and given in a general f0rm2~in 1960, no general analysis of ternary Rayleigh data was available until 1970. Following a suggestion of one of us (J.G.A.), twoof us (L.B.E. and D.G.M.) developed theoriginal procedures, plate reading techniques, and computer analysis for this Rayleigh case. Since then, there have been some improvements by D.G.M. and J.A.R., and the analysis methods were extended to four-component systems by D.G.M.2S Only portions of these developments have been published. Consequently, sections 111, IV, and V includea unified description of our Rayleigh experimental and data analysis techniques as developed and refined at LLNL during the last 20 years.
Miller et al.
n
Raylaigh optical system Camera Dlane
11. Couy Procedures and Data Analysis
The apparatus, materials, solution preparation, and general experimental procedures were the same as those reported in our previous publications on this system.I4-l7 All diffusion experiments were performed on the high-precision Gosting diffusiometer26.2' at LLNL with the same high-quality quartz celli4(labeled 4-744) used in the previously reported experiments. Four experiments were performed at each mean composition. As before, all Gouy fringe positions were read on a Gaertner tool makers microscope using a scanning d e v i ~ e ~similar * > ~ ~to the one described in section IV. Definitions of quantities not explicitly given below can be found in the previous paper^.'^-^^ Data from each individual experiment were analyzed by the Albright-Miller procedures30 to obtain DA and Qo. All experiments were analyzed by their F codes, with the exception of AM21-3, AM22-2, and AM22-4, which required 2P codes, and of AM24-1 and AM25-3, which required Y codes. Problems with determining J required the use of J codes for experiments AM22-4 and AM24-1, which were then followed by the use of F or P codes. Finally a slight systematic error in U, positions in AM24-3 suggested the need to add 2 pm to the delta correction. Except asjust indicated, Jvalues were determined by the "Gosting" procedure.3' For each Gouy experiment, DA was obtained from the extrapolation vs 1/t'of the C,values obtained from the AlbrightMiller programs. The quantities Qo and Qi, which were also obtained from these programs for each exposure, are averaged over all exposures to give the experimental QOand Ql as described in ref 30. The volume-fixed Gouy (D,,)" were calculated with the RFG program (denoted by REVQ in paper 417),which is our extended version of a code due to R e ~ z i n .The ~ ~ input data were the four sets of quantities (J, ACl, AC2, DA, Qo) obtained from each experiment: DAand Qo by use of the Albright-Miller programs; J from plate reading; and AC,'s from solution compositions. As diagnostics, calculated values of DA, Qo, and QI were obtained from the RFG program; these can be compared with experimental input values. Although Ql is not used in our calculations (see footnotes 32 and 35 of ref 15), we find that Qo/Ql is a useful measure of the shape of the omega graph and the position of its maximum or minimum.33 111. Rayleigh Procedures
As noted above, in some cases Rayleigh data were obtained on the Gosting diffusiometer during the same run that yielded Gouy data. With practice, theconversion back and forth between Gouy and Rayleigh optics can be done in 1 or 2 min with this apparatus. Rayleigh optical systems for diffusion measurements have been reviewed by L o n g s w ~ r t h ,Dunlop ~ ~ . ~ ~et al.,36Tyrrell and Harris,37 Svensson (Rilbe) and T h o m p ~ o n , and ) ~ Miller and Albright.39 Our system consists of a cell with a diffusion channel, a reference channel, and a mask outside the cell with two vertical double slits, one in front of each channel. The light source and slit are rotated from the horizontal (Gouy) position to give a vertical
Gouy optical system
Figure 1. Schematic diagrams of the Gouy and Rayleigh optical systems of the Gosting diffusiometer. Reprinted with permission from ref 39. Copyright 1991 Blackwell Scientific Publications.
source slit. A cylinder lens is rotated into position; it focuses each level of the cell to a corresponding inverted image position at the camera plane and separates the fringes from a superposed line (Figure 1). This yields Young's double slit patterns for each level of the cell at thecorresponding image positionat the camera. The horizontal shift of these fringes at a position at the camera is proportional to the change in refractive index a t the corresponding position in the diffusion boundary. The net resulting pattern is a Rayleigh fringe pattern (Figure 2). This contrasts with Gouy optics, which (1) has the light source and slit rotated to give a horizontal source slit, (2) needs no reference channel fos the cell, and (3) has no cylinder lens (Figure 1). However, theGouymasking system and taking ofphotographs are much more complicated. The Gosting machine has Rayleigh optics of the Philpot-Cook type,4°-4'which uses the cylinder lens with its axis horizontal. We use this cylinder lens to focus the center of the cell ("i/2" focus position) a t the photographic plane because this focus point is easier to determine accurately, in contrast to the "2/3" focus position of the cell recommended by Svensson (Rilbe).42 The advantage of the "2/3" focus is that the large Wiener skewness aberration is automatically eliminated, and the remaining timedependent aberrations diminish more rapidly. However, we eliminate the Wiener skewness by using separations of Creeth (symmetric) pairs of fringes to analyze data43(see below) and simply wait a little longer for the other aberrations to become negligible. Details of these considerations are available in the supplementary material to ref 44. The calibration constant j3 for a Rayleigh apparatus depends on the magnification factor, MF, which relates a vertical position in the cell to the corresponding inverted vertical position in the Rayleigh pattern at the photographic plate. The M F was obtained by photographing a transparent ruled scale placed where the center of the diffusion channel of the cell would be located. Both the separation of pairs of ruled lines on the photograph and the actual separation of the corresponding pairs of lines on the ruled scale itself are then accurately measured on a comparator. The ratio of the separation on the photograph to the one on the ruled scale yields a value of M F for that pair. These can be averaged. Alternatively, corresponding line positions on the scale and photograph can be least squared against each other, with the slope being MF. For our cell and apparatus, M F was 1.76179. General procedures for making Rayleigh diffusion measurements have been described by L o n g s w ~ r t hCreeth,43 ,~~ Svensson
The Journal of Physical Chemistry, Vol. 97, No. 15, 1993 3087
Isothermal Diffusion Coefficients of NaCl-MgClrH20
111
‘11111 t t t
I1l1
Figure 2. Rayleigh fringe patterns, showing reference fringes above and below with the diffusion pattern in the middle. The narrow right-hand pattern is from the single slit of the Gosting diffusiometer (Philpot-Cook optics). The broader left-hand pattern is from the BeckmanSpinco Model H (Svensson (Ri1be)-Kegeles optics), which has been adapted to use an 11-slit grating as the light source. The quality of both of these Rayleigh fringes is substantially better than the published similar-lookingMachZender, Jamin, or holographic interferometric fringes.
(Rilbe) and Thompson,38 Sundeli)f,46and most recently by Miller and Alb1ight.3~ The procedures are exactly the same as for a Gouy diffusion experiment until after the cell has been filled and has come to temperature equilibrium in the water bath. The remaining procedures are briefly described as follows. Our diffusion cell is the Tiselius type. Its Rayleigh reference channel consists of cell windows that extend out into the water bath, together with the water in between them. Before the boundary is sharpened, the denser solution completely fills the diffusion channel of the cell. With Rayleigh masks and optical configurationin place, vertical, straight, parallel Rayleigh fringes are formed at the camera plane. These are photographed and denoted as base line patterns. Owing to slight optical distortions from thelenses, windows, and cell, these fringes are not perfectly straight and their deviations from straightness can be measured on a base line photograph. These horizontal deviationsare later subtracted from the Rayleigh diffusion pattern photographs as a base line correction.
These base line correction photographs were taken on the same Kodak Metallographic photographic plate as the Gouy delta correction pictures. The Metallographic plates used for our Rayleigh photographs are no longer available, but have been replaced by the equivalent or better substitutes, Kodak TMAXl 00 or Technical Pan. Once the boundary is sharpened, the Rayleigh procedure is quite simple. Masks do not need to be moved and photographic exposure times are constant. The first photograph was taken at an initial time of t 1 (4.8 X 1c8) a.P12/D, where a is the path length in the cell, D is the estimated DA of the system, and all quantities are in SI units. This initial time is sufficient to reduce aberrations to less than 0.02 fringe if the center of the cell is focused on the photographic plate, the light source is on the optic axis39+42 (also see ref 44 supplementary material), and symmetrically paired fringes are used in the data analysis?’ The last photograph was taken about 3.5 times the initial time. This ensures the free-diffusion condition that negligible changes of concentrations occur at the ends of our 10 cm long cell during the experiment. Within the time difference between the first and last photographs, photographs were taken at approximately equal intervals of l/t. Exposure times are much shorter than for Gouy photos because Rayleigh patterns are more intense than Gouy patterns. This makes Rayleigh measurements possible for systems which absorb some of the source light, for which Gouy exposures are too faint to read. The simultaneous Gouy and Rayleigh experiments were performed to compare the quality and agreement of the results from both methods. They were restricted to a few concentrations at 1 M and below (with one exception) for the following reason. Our Tiselius cell in unsuited for experiments at high concentrations because its reference channel is open to the thermostat water. The difference between the refractive index of water and that of diffusing solution can be very large with concentrated solutions. In this circumstance, the slight frequency dispersion of our incoherent light source (Hg green line at 546.1 nm) gives rise to increasingly blurred fringes as solution concentration and corresponding refractive index increase from system to system. We did perform one series of experiments at 3 M (AM-7) with both optical systems, which gave blurred Rayleigh fringe patterns that were barely However, the Gouy fringes could be read easily at all concentrations. We now turn to a description of our plate reading techniques.
IV. Rayleigh Plate Reading Each Rayleigh fringe pattern has three segments (Figure 2). There is the Rayleigh pattern from the diffusion cell itself in the middle segment. There are also straight fringe patterns (reference fringes) in the segments above and below the middle one, which are formed by double slit masks located slightly to the side of the optic axis where the light path bypasses the diffusion cell and goes through the water of the water bath. These reference fringe patterns are used toalign the photographic plateon thecomparator for each pattern, so that all patterns will have the same relative alignment when scanned. Figure 3 shows a schematic of the diffusion part of the pattern. The centers of bright fringe positions on the photographic plates were read on a Grant x-y comparator. Since the developed plates are negatives, a bright fringe on the plate (whose center is read) corresponds to the center of a dark fringe as seen by eye during the actual experiment. Recording of fringe positions is done by digitizing the analog positions of the x and y screws of the comparator stage. The digitized positions were punched on IBM cards, with both x and y coordinates punched together. This now obsolete system is to be replaced by modern encoders with data being sent directly to a PC.
Miller et al.
3888 The Journal of Physical Chemistry, Vol. 97, No. 15, 1993 6 d A I
..... T T
Symmetric Readings
1
xS3-x, x7 3-x3
Figure 3. Schematic diagram of the diffusion segment of a 10.3-fringe Rayleigh pattern, showing the jog in the middle of the pattern to obtain symmetric fringe readings. Reprinted with permission from ref 39. Copyright 1991 Blackwell Scientific Publications.
The Grant comparator has a scanning device which superposes on a cathode ray tube the right-to-left and left-to-right mirror images of a fringe intensity profile. These mirror images of a fringe move in opposite directions as the photographic plate is moved in the x-direction. A fringe position (center of a bright fringe) is the point for which the mirror images coincide with the same maximum. As shown in Figures 2 and 3, the straight outer fringes bend and become steeper toward the center of the pattern. When the fringes have a slope of 4 5 O or more, positions along the x-axis can be reproduced with an precision of 1 pm. However, distortion in photographic images and difficulties in locating centers of fringes yield an inaccuracy of about 1-2 pm. In contrast, as the homogeneous solution region is approached on either the top or bottom of the boundary, the slope of the associated fringes becomes less than 45'. They also become broader as they become more nearly horizontal, and their centers are increasingly harder to locate accurately. The y-position of the center of a nearly horizontal fringe cannot be found by the coincident images device when scanning in the x-direction and, consequently, must be located visually. The uncertainty in this y-position is 5-10 pm. The plate reading procedure is as follows. 1. Alignment of Base Line Pattern. The chosen base line pattern, which consists of a set of nearly straight fringes, is aligned so that the fringes lie as closely as possible along the y-axis of the comparator. This can be done by taking one specific y point at the center of one of the interior fringes near the top of the fringe pattern and another specific y point at the center of the same fringe near the bottom of the fringe pattern, then rotating the comparator stage iteratively until the two y points have the same x reading. The relative x-position of a specific fringe a t a fixed distance (e.g., 1 mm) into the top set of reference fringes is then determined with respect to a specific fringe at a fixed distance into the bottom set of reference fringes. This relative displacement (offset) between the top and bottom reference fringes should not change during the experiment. Consequently, it is used to line up all the succeeding fpf (fractional part of a fringe) and diffusion patterns. This is necessary for each pattern, because the patterns are not at exactly the same level on the photographic plate and do not have the same rotational position, due to small mechanical movements in repositioning the plate holder during the run. One end of the pattern is also chosen as a reference position for distance along the long axis of the pattern, since placement of plates on the comparator table is not uniform. This end-of-pattern reference position is used for all other patterns. 2. Base Line Reading. Once aligned as in part 1 above, and after the end-of-pattern reference position is recorded, the base line pattern is read by recording a series of pairs consisting of x-positions of the centers of two adjacent fringes at a given y-position. Two hundred such fringe pair locations are read at more or less preselected y values. The density of readings is 3 times higher in the center portion of the pattern, where fringes from the diffusion boundary region will be found on diffusion patterns. The distance between the two adjacent fringes is the interfringe distance d for that pair. A systematic change in this
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60 40 50 30 Dlstancdmm Figure 4. Rayleigh diagnostics. Base line (deviations from ideal straight fringe) in micrometers vs distance in millimeters from the end-of-pattern reference position for experiment AM23-4R. Anoptically perfect system would have all points on a horizontal line. Since the scatter in the center of this very good cell is only 2 pm, base line corrections could have been ignored for this particular experiment. The solid line represents the 15thorder polynomial. 20
distance withy would be a measure of vertical distortion in the optical system. However, this is not found for the Gosting apparatus and its cell or for our Beckman-Spinco Model H and its cell. The average interfringe distance is used in subsequent calculations, and is about 270 pm for the Gosting diffusiometer (450 pm for our Model H). The average x value of each adjacent pair is the base line position of a dark fringe (bright to the eye) at its corresponding y. The y-positions are now referred to the end of the pattern by subtracting the end-of-pattern reference position. A 15th order polynomial x(y) is then fit to these 200 positions by least squares, and this polynomial represents the deviations from a "perfect" straight fringe fairly well. The deviations are a measure of horizontal distortion in the whole optical system. The maximum distortion for our Tiselius cell is typically only 0.1 fringe but is usually only a few hundredths of a fringe in the center region of the cell (Figure 4). (For the tantalum-glass cell used in the Model H, it is 0.1 fringe in the center region and about 0.2 maximum). This polynomial makes it possible to subtract deviations due to optical distortion from arbitrary fringe positions in the fpf and diffusion patterns. Other functions could have been used for this purpose, such as a Fourier series. Fitting short segments would also be suitable. 3. Fractional Part of a Fringe (fpf). The plate with the first diffusion exposure is placed on the comparator, again with the long axis of the fringe pattern parallel to the comparator y-axis. Each pattern is aligned parallel to this y-axis by rotating the stage iteratively until the top and bottom reference fringes corresponding to those used in the base line alignment have the same offset found in part 1. The end-of-the-pattern reference position (y-position), corresponding to the one obtained on the base line pattern, is recorded. Then the x-positions of two adjacent vertical fringes are recorded at three y-positions above the diffusion boundary. These y-positions must be sufficiently away from the diffusion boundary that the closest y-position is well within the "straight" fringe portion of the pattern but also away from the optically poorer ends of the cell. The three y-positions may be separated by about 2 mm. Finally the x-positions of two adjacent fringes are recorded at three y-positions below the boundary in a similar manner. Each pair of x-positions is then averaged to get the position of a dark fringe (bright to the eye), and is corrected by subtracting
Isothermal Diffusion Coefficients of NaC1-MgCl2-H20 the base line correction corresponding to its y-position. The corrected top three positions are averaged, and the corrected bottom three are averaged. The fpf is the difference of these averages, divided by the average separation of all six pairs of adjacent x-positions (the interfringe distance 6). This alignment and reading procedure is repeated for the next four patterns. The fpf values from all five patterns are then averaged toget the fpf for the experiment. Generally the standard deviation of this average is 0.003-0.007 fringe. The average value of this fpf in micrometers is also needed to make the "jog" in the horizontal (narrow) direction at the center of the diffusion patterns, as described below (see Figure 3). These fpf values should not depend on when the photographs were taken. Consequently, a systematic trend in the five fpfs with time, if present, indicates a suspect experiment; in such a case, the cell probably has a leak. Obtaining fpf from the first five diffusion patterns has been the practice at LLNL since 1976, although not specifically mentioned in the published papers. However, there are other equally satisfactory ways of getting the fpf. For example, three to five photographs can be taken just after the needle is withdrawn,49which is the practice at TCU, or while the boundary is being sharpened just before the run starts.43-50-52In these latter two cases, three or more positions on each side of the boundary (after base line correction) are separately fit by linear least squares. The fpf is the difference of the intercepts at the center of the boundary, divided by the interfringe distance. Since the fringe positions can be read closer to the center of the boundary, this short extrapolation minimizes the effect of errors in the base line corrections. 4. Alignment of Diffusion Patterns. For diffusion patterns, the photographic plates are put on the comparator so that the long axes of the patterns are parallel to the x-axis of the comparator. Compared to the base line and fpf procedures, this interchanges the roles of x- and y-positions on the comparator in the alignment and reading procedures, as well as in the base line correction polynomial. The alignment requires an iterative rotational adjustment of the comparator stage to get the correct offset between the two sets of reference fringes. As noted above, the Grant comparator is such that the centers of the straight reference fringes on either side of the boundary have to be estimated (with an uncertainty of about 5-10 pm) in the end to end alignment of the reference fringes. However, the distance between the top and bottom (now left and right) sets of reference fringes is about 8 cm. This is large enough that any resulting angle of tilt gives deviations of fringe positions that are too small to matter for the diffusion fringes, which are located near the center of the pattern. 5. Preparationfor Reading the First Diffusion Pattern. Once the first pattern is properly aligned, the end-of-pattern reference position, corresponding to the one located on the base line pattern, is recorded. Although this position (now an x-position) is also uncertain by 5-10 pm, this gives only a small error in the base line corrections, since they vary slowly with cell position. To get the total number of fringes J. this first pattern is now traversed across the boundary at the constant y of an approximate center of one of the bright, horizontal straight fringes (denoted by fringe 0). As the plate is moved across the diffusion boundary, the bright fringes are counted as they move across the cross hairs on this horizontal line. The number of the last fringe counted just before the fringes become horizontal again is the integral number of fringes. This number plus the value of fpf is J. Typically the y-position associated with horizontal fringe 0 on the other side of the boundary is not at the center of a bright horizontal fringe on this side; Le., J will not be an integer. The pattern is now moved (by moving the comparator bed) to an x-position somewhere in this horizontal part of the pattern, provided it is far enough from the boundary for diffusion to have
The Journal of Physical Chemistry, Vol. 97, NO.15, 1993 3889 had no influence but away from the ends of the pattern where base line corrections are usually larger. A horizontal bright fringe in the middle of the diffraction pattern is chosen as fringe 0. Its center is located visually using the comparator cross hairs, and both x and y positions are recorded. The uncertaintly is also about 5-10 pm in y. This y-position of fringe 0, after base line correction, is needed for the remaining base-line corrections. However, an error in this position will cancel to first order in the final base-line corrections (see section V, 3d below). The center of each fringe subsequently crossed a t the constant y-position of fringe 0 is the position of an "integer" fringe. 6. Reading the First Diffusion Pattern. The optimum analysis of Rayleigh patterns requires symmetric pairing of fringes.43 Symmetric reading with non-integer J requires the following considerations. Location of the center of a bright diffusion fringe with a comparator is relatively easy, but location of the position of a noninteger fringe is not reproducible. Therefore, suppose one starts on they-position of the center of the bright fringe 0. Then to find the centers of bright fringes symmetrically placed on the other side of the boundary center requires a jog upward in y at the boundary center of exactly fpf in micrometers. With this procedure, when the cross hairs of the comparator are finally in the center of a bright horizontal fringe far on the other side of the boundary, that fringe number will be J. Once the 0th fringe reference position is recorded, the plate is moved in the x direction keeping constant they coordinate of fringe 0, with the x- and y-positions of every fringe recorded, until just before J / 2 (the boundary center) is reached. At this point, the pattern (plate) is moved up in the y direction by a distance exactly equal to the fpf in micrometers. This "jog" will make the center of a bright fringe counted as j actually be the position of fringe (j + fpf). Then reading of x- and y-positions on the other side of the boundary begins at the center of the symmetric fringe corresponding to the last fringe read. The readings of every fringe a t this new y-position continue until the integer of fringe (J- 1) is reached. Continued plate movement into the horizontal part of the pattern should result in the cross hairs being in the center of a bright horizontal fringe, i.e., of fringe J (see Figure 3). When fringes are read in this way, the first fringe crossed is fringe number 1 and the last fringe crossed is fringe number J - 1. For every fringe numberj that has been read, there will be a symmetrical fringe J - j that has also been read on the other side of the boundary. This symmetrical fringe pair is called "Creeth pair j".43 Separations of Creeth pairs are used in our data reduction procedure to calculate diffusion coefficients. For example, Creeth pairs in a 100-fringe system are fringe 1 and 99, 2 and 98, ..., 49 and 51. Creeth pairing removes the largest aberration, the Wiener skewness4*(which remains due to the center-of-cell f o c u ~ ) ; ~ ~ ~ ~ ~ it eliminates to first order (by analogy to binary system~)~3.~3 the effects of concentration dependence of diffusion coefficients and refractive index; and it avoids the necessity of knowing the location of the center of the b o ~ n d a r y . Consequently, ~~,~~ except for the need to make base line corrections, the end-of-pattern reference would be unnecessary. It should be noted that every other fringe may be read instead of every fringe. This is our usual practice for J larger than 40 since the procedure is less tedious and time-consuming, although the principles remain the same. 7. Reading of SubsequentDiffwion Patterns. Each subsequent pattern must be aligned in turn as in part 4 above, as noted earlier. The end of pattern and 0th fringe reference positions are again determined as in part 5 . Finally the fringes are read symmetrically, using the fpf jog in the center of the pattern as in part 6. Typically 9 or 10 patterns are read per experiment. With the Rayleigh optics of the Gosting Diffusiometer and its
3890 The Journal of Physical Chemistry, Vol. 97, No. 15, 1993 single vertical source slit, there are only five distinct Rayleigh fringes visible in the narrow direction of the diffraction pattern. Consequently, if the fpf shift is more than 0.5 fringe relative to they-position of fringe 0, then different intensities can result for the fringes being read symmetrically. Although this effect could slightly influence the apparent location of fringe centers, we have not noticed any systematic errors from this effect. This intensity variation could be avoided by using a light source grating to increase the number of fringes,55as we have done on our Model H.44.56 This is not possible for the Gosting machine because of the need to switch between Gouy and Rayleigh optics, both of which use the same light source slit. 8. Comments on Plate Reading. Plate reading can be partially or fully automated as described be lo^.^^-^^ For conformity of description in the following, we assume that the fringes curve downward in the viewer as the boundary is traversed; that y increases on the comparator as the cross hairs appear to move downward (the pattern actually moves upward); and that x increases as the cross hairs appear to move from right to left (the pattern actually moves from left to right). There are several techniques of obtaining symmetric fringe readings from diffusion patterns besides the one described above. a. Albright T e c h n i q ~ e . ~ ~After - ~ * determining the fpf and locating the fringe 0 position, move the pattern up by fpf/2 in micrometers. Then read the pattern straight through without any further jog, since on one side of the boundary the center of a fringe corresponds to (j fpf/2) and on the other side to ( J - (j fpf/2)). Thereforesymmetric pairs areobtained by moving in an equal number of fringes from both outer “edges” of the boundary. A Creeth pair is now identified as pair (j fpf/2). This procedure is the basis of an automated ~latereader.5~ Details are found in the supplementary material of ref 58. b. SundelM Technique.46 First record the end-of-pattern reference position. Then go to the horizontal fringes region on the right side of the pattern image and pick a horizontal fringe just above the middle of the diffraction pattern. Call it fringe 0; locate its center at position yo as before; and record the x- and y-positions. Follow the y-positions of that fringe downward as x is increased, recording both x and y at some suitable interval of x values, such as 15-20 pm. When the y-position of the center of the fringe differs from yo by approximately the interfringe distanced, move the pattern vertically upward by d to the center of the next fringe immediately above. The effective y-position of this new fringe is ( Y =y + d). Follow this fringe downward until it also differs from yo by approximately d. Again move up vertically by d, to a new y-position effectively (Y= y + 2 4 . This procedure is continued n times until the horizontal part of the pattern is reached on the other (left) side of the boundary. After making base line corrections for all Y-positions, the center of the left horizontal fringe is (Yh = yh + nd); (YI,- yo)/d is J; and Qh -yo)/d is fpf. This piecewise reading of fringes is equivalent to following a single fringe through the whole pattern, had the diffraction pattern not been limited in width. The x locationsof the base line corrected Y-positions generally do not correspond to the locations of integer fringes. However, if the x-positions are read closely enough together, the position corresponding to any desired Y value can be obtained by interpolation. Since ( Y - yo)/d corresponds to fringe j at the corresponding x-position, the x-positions of any fringe j , integer or not, and its symmetriccreeth pair fringe ( J - j ) can be obtained. Consequently Creeth pairs can be obtained at other than integer fringe positions. This technique or the one below are essential for a system with a small number of fringe^.^^,^' For example, the above technique has been used for as few as one to ten fringes.46.6’ In contrast, it is not possible with the Gouy method to get precision data successfully from such a small number of fringes. If J is less than the number of fringes in the horizontal part
+
+
+
Miller et al. of the diffraction pattern, then a single fringe can be followed completely through the pattern. This makes theabove technique easier because the pattern does not have to be moved by multiples of d during the reading. If an optical system has a light source gratings5 with a sufficient number of lines (e.g. loo), then even systems with a large number of fringes can have a single fringe read in this way. c. Yphantis T e c h n i q ~ e . ~If~fringe - ~ ~ intensities are obtained by photodiode scanning of the photographic plate, then the following procedure, due to Yphantis and co-workers,s9.60can be used. Consider a linear photodiode array (PDA) which is perpendicular to the 0th (horizontal) fringe (see Figure 3) and which covers all the fringes in the diffraction pattern. At any given x-position, intensities are obtained for the set of fixed y-positions corresponding to the fixed pixel positions. These intensities follow the form of a sine squared function. In general, they do not directly locate the maximum fringe intensity (center of a bright fringe). However, thecenters in they direction (intensity maxima) can be obtained by curve fitting using Fourier62 or Walsh transform^^^.^^ (after base line correction) at this x-position. The distance between the maxima is the interfringe distanced. More importantly, these transforms yield an amplitude and a phase. The phase is proportional to the relative vertical displacement of the fringes, Le., relative fringe shifts. Scanning in the x direction begins by choosing an x-position of “fringe 0” on the right-hand side of the pattern where the fringes are horizontal. The phase at this position locates the effective y-position of fringe 0. As x increases and the fringes turn down, the phase of the fitted function shifts. This shift divided by 2~ is effectively the j (not necessarily an integer) associated with its x-position. As a visual fringe is passed in the horizontal line, the calculated phase appears to have the same value as for fringe 0. This corresponds precisely to passing an integer fringe and is counted as such inj. This process is continued until x moves into the horizontal fringe portion on the left side of the boundary. The cumulative value of j at this point is J . This plate reading technique has been used so far only for ultracentrifuge p l a t e ~ . ~ ~ fWe ~ O are currently considering its application to diffusion plates, since the positions of both integer and noninteger Creeth pairs can be obtained by interpolation of the fringe shifts. This allows the determination of the x which corresponds to any value o f j (not necessarily an integer), as well as the x of its symmetric pair ( J - j ) . Again this is useful if the number of fringes is small, since more pairs can be generated for improved statistics. 9. Comments on Real Time Data Acquisition. The Yphantis technique can also be used for real time data acquisition, provided the x-direction scan along the actual fringe pattern can be done within a few s e c o n d ~ . s ~ ~ ~ ~ If a two-dimensional PDA is used, all the x,y intensity data are collected for a series of x values all at once. However, the analysis of fringe shifts and interpolation to get the x values for j and (J - j ) is exactly the same as in part 8c above. Highprecision TV cameras are important examples of two-dimensional arrays, which have been used by Yphantis and co-workers59.60.6* for recording real time Rayleigh patterns from ultracentrifuges. Real time Rayleigh diffusion data will be best collected with a large two-dimensional array. At present these arrays are very expensive. Another real time alternative is a long linear array positioned parallel to the long axis of the pattern, Le., in the x direction, which can be moved by a stepping motor perpendicular to the pattern (in the y direction) using a precision micrometer screw. If the perpendicular movement is fast enough, diffusion will not significantlyalter the positions of intensitymaxima. If the motion is too slow, so that diffusion does alter the y-positions of intensity maxima of later relative to earlier ones, the PDA can be moved
The Journal of Physical Chemistry, Vol. 97, No. 15, 1993 3891
Isothermal Diffusion Coefficients of NaCl-MgC12-H20 forward and promptly back. The intensities at a given y are then averaged for each pixel (x-position) of the array and reported at the average time of the scan. A long linear array, moved perpendicular to the pattern, can also be used to automate a plate reader. An important advantage of this method is that the sensitivity of each pixel need not be calibrated.
and D is the determinant of the diffusion coefficient matrix. The si are functions only of the Dij, are related to the eigenvalues Xi of the Di, matrix, and are given by
V. Rayleigh Data Analysis The following is a general procedure to analyze the data from a set of experiments which has the same mean concentrations but (Dl 1 + 4')+ [(Oil + D2,l2 - 4 DI1l2 different ACI/AC2 ratios. We first obtain diagnostics on each A, = 2 experiment by treating its fringe position data as if they were from a binary system. Then the data for all the experiments are si = 1/Xi'/2 (12) combined, including the ACI and AC2data for each experiment, and these combined data are least squared to get the Do. We From eq 3 it is seen thatfli) =AJ- j) and x, = -x(J..~).These begin our discussion with the theoreticalbasis of t h e a n a l y ~ i s . ~ ~ . ~are ~ ~the ~ two x-positions of a Creeth pair, and their difference is 1. Tbeory. Rayleigh fringes reflect the refractive index twice each one. Consequently&) and Xj distribution in the cell, and their positions shift as the boundary j' - '(J-j) is traversed. This results from the change in refractive index sij = with vertical position in the diffusion channel, compared to the 2 refractive index of water or a solution of uniform composition in are sufficient to characterize Creeth pairs. Only Creeth pairs the reference channel. whose j values lie between 0 and 512 are required, since they Since concentration differences in the diffusion channel are contain all the fringe position information. As mentioned earlier, small in our experiments, a truncated Taylor series expansion of the position of the center of the pattern is not needed for Creeth the refractive index around the mean concentration is adequate pair calculations. Note that bothfli) and X j are negative for all to relate refractive index to concentration, and yields for a ternary j < J / 2 , so signs can be reversed for programming purposes. system It is convenient to define a "reduced fringe position" y, as n = H R l ( C ,R2(C2(2) Yj = where n is the refractive index at position x, Ciis the concentration of solute i at position x, Ci is the average concentration of solute This y, is independent of the corrected time (Le., should be the i, A is the refractive index at the mean concentrations, and Ri is same for all exposures) for a given j. For a given j , if there is the refractive index increment of solute i. The bar denotes mean a large deviation of y, in one pattern from the average y, of all concentrations of that experiment. the patterns, it signifies an error in reading a fringe position. For the usual case where the Dij matrix has distinct eigenvalues, Consequently, that y, can be discarded and the remaining y, substitution of the appropriate solution of Fick's law for freereaveraged. Since we have only 9 or 10 exposures, we actually diffusion boundary conditions23into eq 2 yields use all individualvalues of y, after any such outliers are discarded. The average y, is the representative position for that j , The set of average y, for all j thus forms a representative Rayleigh pattern. Use of this representative pattern will be essential with automated data acquisition, which could yield the equivalent of 20 to 1000 exposures. For the least-squares analysis of data, eq 3 can be rewritten wherefCi) is the reduced fringe number and x is the distance as from the center of the boundary and is positive downward. The difference in refractive index across the boundary, An, is related xi)= ( a ba,) erf(s,y,) (1 - a - ba,) erf(s9,) (15) to J by The four least-squares parameters are a, b, SI,and s2. An = ( A l a ) = R,AC, R 2 X 2 (4) Equation 15 refers to a given fringe in a given exposure in a where X is the wavelength of the light. The r i coefficients are given experiment, for which the experimental quantities are (j, independent of the boundary conditions, and their expressions x,, x(~-,),t ) , taking into account eqs 13 and 14. The quantities J are given below. Since the Ri appear as ratios in subsequent and a,are experimental quantities associated with each experequations, X/a in eq 4 need not be determined. iment. Finally the quantities a, b, s,, and s2 and the D,, calculated For the distinct eigenvalues case from them are derived quantities associated with the combined set of experiments. Although I',, sI,and s2 can be obtained by rl r2= 1 (5) least squares from a single experiment, the a and b (and R I and R2 as well) cannot be obtained without data from two or more and rl can be written as25-63.66 experiments with different a1 values, Le., from two or more r, = a + ba, (6) different ACl/AC2 ratios. where The four diffusion coefficients can be determined from a, b, sI,and s2using the equation^^^^^^^^^"^
c,)+
+
c2)
+
+
+
+
ai = ( R , A 2 2 2 A C 2 a and b are the following functions of both the Dij and Ri
(7)
(a
4 ,=
(
+ b)(1 - a)s; - a ( l - a - b)sI2 b~,~s,~
Miller et al.
3892 The Journal of Physicul Chemistry, Vol. 97, No. 15, 1993
( a + b)(l - u ) s I 2 - u ( l - a - b ) s ,
D22 =
b~,*s,~ We note that if the eigenvalues of the diffusion coefficient matrix are exactly equal or nearly equal, then a different analysis will be required, as summarized in section 3f below. 2. Binary System Analysis. After each experiment is performed, it is first analyzed as if it were a binary system. This provides useful diagnostics. Equations for a binary system are simpler, since there is only a single error function term. Thusfl) for a binary can be written
1.064 l-O'ti.".
The quantity DJis the "effective" diffusion coefficient associated with a fringe pair (j, J - j).39370Using eq 20 and the definition of yj, DJ can be written
where zj* = e r f i n v m ) ] (22) and where XI is the actual fringe position on the plate. The @ is related to the square of the magnification factor MF.71 Our binary system diagnostics include calculations of DJ for each Creeth pair associated with fringe number j o f each exposure. After base line and Ar corrections are made, these D, should be the same for all exposures for a given j . If binary diffusion were ideal (Le., linear dependence for n and no concentration dependence of D), then all individual (and thus average DJ values would be the same for every value ofj. For binary systems with a polynomial dependence of D and n on C, these DJ values will be nearly the same if AC is ~ma11.~3%~O For ternary systems, except at certain special A C I / A Cratios, ~ these average DJ are not the same, but are functions ofj; Le., the fringe pattern is skewed from the "ideal case". It is precisely this skewing which provides the information to extract the D,J of a multicomponent system from fringe position data. It can be shown from a Taylor series expansion of the Creeth functions which describe a polynomial concentration dependence for D and n of a binary system,43that d ( D J vs (z,*)~should be the proper curve to graph. That curve should become a straight line as the center of the pattern is approached (Le., wherej = J / 2 and z,* = 0). Moreover, the points for e a c h j from all patterns should superpose on each other. Experimental problems are indicated by a failure to superpose for each j within reasonable error limits or by a shift of DJ with exposure number for all j. Consequently, this plot serves as a diagnostic for both binary and ternary systems. For a given j, the scatter around the average of individual DJ from the different exposures is a measure of precision of both the experiment and plate reading. This scatter is typically larger at low values o f j (low number Creeth pairs and high values of z,*) because the fringes are broader and tilted more horizontally, making their centers harder to locate. It is also larger a t j values nearer the center of the pattern (high number Creeth pairs and low values of zJ*) because the fringes are closer to each other. Here the increased scatter in Dj results from the constant error of 1-2 km in each position, which is a larger fraction of the distance between the centers of the inner Creeth pairs than it is of the outer ones. Consequently it is desirable to reject the lowest number fringe pairs as well as the highest number fringe pairs. It has been
1
0
1
1
0.2
1
0.4
1
l
0.6
~
l
1
0.8
1
1.0
(ZiI2
1
1.2
1
~
1.4
~
1.6
1
1
1.8
l
l
1
2.0
Figure 5. Rayleigh diagnostics. Plot of (D,)'12vs ( z , * ) ~for experiment AM23-4R. The DJvalues for each exposure have been calculated from the corrected time t ' + AI. Units of Dl are m2/s. Plotted points at each value of (zJ*) are difficult to distinguish, but the symbol for each point is the number associated with its exposure. The scatter indicates the relative fringe position errors. Note the larger scatter at large values of z,* (low number fringe pairs) and at low zJ* (high number fringe pairs). The horizontal line is the square root of the pseudobinary D (see Figure 6).
empirically found70.72that fringe pairs should be retained only between 0.84 > - - ) > 0.28. In a 100-fringe system, these correspond toCreeth pairs 8 (fringes 8 and 92) through 36 (fringes 36 and 64), respectively. The reasons for these cutoffs can be illustrated with the aid of the Rayleigh pattern schematic shown in Figure 3 and a typical experimental plot of d(D,) vs ( z ] * ) ~in Figure 5 . For ideal binary systems, the cutoff for high number Creeth pairs can be a minimum separation between the pair of fringe positions, such as 2 mm.49 This is successful for the extrapolation to get D and At (described below), because all the Dl are the same for a given pattern within measurement errors. Consequently, they can be averaged over differing numbers of fringes to get an apparent D for that pattern. However, for the skewed fringe patterns of dilute electrolyte or multicomponent systems, a consistent calculation requires the same set of fringe pairs to be averaged in every pattern, Le., with fixed-) cutoffs. 3. Ternary System Analysis. a. At and Pseudobinary D. A starting time correction is required because the initial boundary cannot be made infinitely sharp; its finite width corresponds to the effective time it would have taken for an infinitely sharp boundary to diffuse to that ~ i d t h . ~There ~ - ~can ~ also be small delays in starting the clock which records the experimental time t 'after closing stopcocks, etc. These delays can be accounted for by plotting the apparent diffusion coefficient D'vs l/r'. The D' is obtained for each pattern by averaging Dl calculated using eq 21 with its appropriate r'. It can be s h o ~ n ' ~that - ~ ~to good approximation
D'= D
+ D($)
Linear least squares yields the pseudobinary diffusion coefficient D (intercept) and DAr (slope). The true time t of an exposure is thus
t = t'+ At (24) Figure 6 shows a typical plot of eq 23 along with its least-squares line. b. Ri. The quantities R I and R2 are obtained by a linear least squares of J, A C I , and ACz from all the experiments by using
I
I
I
Isothermal Diffusion Coefficients of NaCI-MgC12-H20
Creeth pairing, the x locations of the true centers of fringesj and (J - j) must be found. The true center of diffusion fringe j (Le., x,) can be obtained from (a) the apparent center of fringe j on the horizontal line corresponding to fringe 0, (b) the vertical y shift Aw at that apparent x, due to the base line correction, and (c) the slope of the fringe curve dw/dx at that apparent x,. This slope can found from the derivative of eq 15, since the base line shift in micrometers can be transformed into an equivalent fringe shift by
1.162 1.160 1.158
b
-
1.156 -
X 0)
$ 1.154 -
1.148 1.146
I-/ -
0
The Journal of Physical Chemistry, Vol. 97, No. 15, 1993 3893
I 0.008
I
I 0.016
I
I 0.024 lit’
I
I 0.032
I
I I 0.040 0.048
Figure 6. Rayleigh diagnostics. Plot of the apparent pseudobinary diffusion coefficient D‘vs 1 lr’for experiment AM23-4R. The intercept of the line is the pseudobinary D and its slope is DAt. The deviations of the points from this least-squares line do not exceed about 0.05%. Units for D and t are m2/s and min, respectively.
eq 4. The calculated and experimental values of J are usually in good agreement for the Gosting diffusiometer, with Nusually averaging 0.02-0.05. The errors for J are sometimes larger at high concentrations, owing to difficulties in making up solutions. c. Nonlinear Least-SquaresTechnique. A general technique of nonlinear least squares (NLLS) follows. However, the need for base line corrections requires calculations in stages, as described in part 3e below. Equation 15 is nonlinear, both in its least square (LS) parameters (a, b, sl,s2)and its experimental variables y,). Because the error function and its derivatives are well-behaved, we can expand eq 15 in a Taylor series. If we keep only firstorder terms and denote the LS parameters temporarily by gi, then
w),
where&) is obtained from the value of j and J for the given pattern and experiment. The analytical expressions forf(i)o and [dm)/agi]0 obtained from eq 15 are evaluated from the current values of the g, and the experimental value of the time-corrected y, from the Creeth pair associated with fringe j (eqs 13 and 14). A linear LS of eq 25 over all the retained fringe pairs of every pattern of all the experiments provides Agi, the corrections to gi. The quantities (gi + Agi) are then used as the new current values of gi,and this process is iterated until all Agi/givalues are smaller than O.oooO1. This NLLS procedure usually converges in three to seven iterations. Suitable initial values of a, b, sI,and s2 are (0, 1, ( I / d D l ) , and (1 /dDz), respectively. These starting values result from assuming the cross terms D12 and 41 equal 0 and that DII DZZ. Here D1 and DZare the pseudobinary D whose calculation was described in section V2 for the two experiments with a l = 1 (AC2 = 0) and a l= 0 (AC,= O), respectively, or as close as possible to these a l values. More powerful NLLS techniques could be used, such as that of L~venberg-Marquardt,~~ but have not been necessary so far. d. Base Line Correction. As noted earlier, optical distortions result in deviations from absolute straightness of the fringe pattern from a cell filled with a homogeneous solution. Consequently for a diffusing system, the apparent x-position of a fringe center on the photographic plate is in general not exactly where the real center would be in the absence of such distortions. To get exact
A j = Aw/d (26) where d is the interfringe separation in micrometers. However, the derivative of eq 15 depends on already knowing the values of D, or their equivalents (a, 6, S I ,S Z ) . Therefore iterations are necessary. A somewhat elaborate geometric and Taylor series analysis of this base line correction shows that for a single position x,, the correction for the starting reference position xo of fringe 0 and thecorrection at x,are both necessary. However, if Creeth pairing is used, the correction at xo cancels to first order, and only the corrections at x, and x(J-,) are needed. This is another advantage of Creeth pairing. However, it helps to have good quality optics and cells, because if base line correctionsare very large, the secondorder term involving the xo correction could be required. e. Calculation Procedure. The procedure begins with a calculation of the pseudobinary D and the associated Ar for each experiment. The Ar is then used to obtain the time-corrected y, value for that experiment. All experiments are treated this way. The next step is to calculate both the R, and a,from the J , ACl, and AC2 of all the experiments. Finally the 15th order base line correction polynomial for each experiment is calculated from its base line measurements. The diffusion coefficients are now calculated by the NLLS technique in steps. In the first step, data from the two experiments as close as possible to a1 = 1 and a1 = 0 are used without base linecorrections to get preliminary values of the NLLS parameters, thereby minimizing computer time. This calculationprovides better initial values for the next step. The second step uses the data for all experiments. It is also done without base linecorrectionsand yields a more representative set of parameters. The third step, also without base line correction, starts with the data set which has deleted from it all&) points which deviate by more than 2 standard deviations (2a) from the calculatedm) of the previous step. The parameters from this “cleaned” data set are the input for the derivative o f n ) needed for the base line correction in the next step. The fourth step restores all the data deleted in the previous step. Now corrected center positions for every Creeth pair are calculated using the base line correction polynomial evaluated at both x-positions and the fo’) derivative obtained from the parameters of the third step. The corrected y, are obtained from these corrected x-positions. Then the NLLS technique yields a new set of parameters, based on base line corrected data. This process could be iterated within this step to get better input values for the base line correction, but trials have shown that this is unnecessary. The fifth step uses the parameters from step 4 in a new calculation of the base line correction derivative, but uses the data set =cleaned” in the same way as in step 3. Again, trials have shown that iteration to get a better base line correction is unnecessary. The points deleted at this step are usually but not always the same as those deleted in step 3. This last NLLS step yields our final results for the LS parameters. At every step, our program provides diagnostics which include values of the LS parameters for every iteration, and after convergence, values of the fi) residuals for every Creeth pair, plus values of the D , and their standard errors. If convergence is not reached after nine iterations, an additional diagnostic is
3894
Miller et a].
The Journal of Physical Chemistry, Vol. 97, No. 15, 1993
printed and the procedure goes on to the next step. It seldom requires going to step 2 or 3 before convergence is reached, even in difficult cases. After step 5 , the results include the final LS parameters, s, and A,, R,, calculated and experimental J , a,,r l ,DA,23-24 and SA (discussed below), and the final values of D , and their standard errors. f. ConvergenceProblems. Generally least-squares convergence is rapid with the usual initial conditions mentioned above. Sometimes it is incomplete in step 1 of the NLLS procedure but is complete in step 2. After that, convergence takes place in the remaining steps in three or four iterations. However, there are two situations which have severe convergence problems. The first situation is the case of nearly equal or exactly equal eigenvalues. In these cases, the respective values of both a and b become very large or infinity. The nearly equal case can be treated in two ways.77 An exact way is to do least-squares fits using the parameters SI and As = sI-s2. This has one large parameter and one small, which avoids round off errors from the difference of two nearly equal large quantities to give a small one. The second (less exact) way is to expand eq 15 in a Taylor series in terms of SI and As, and then least square with this new functional form. The exactly equal eigenvalue case involves an expression for &) withdifferent functions (an error function and an exponential) and a different expression for rl.65.78 However,&) is a function of the same experimental variables C f o ) , ~ , ) and , it can be least squared in a fashion65 similar to eq 15. which results The second situation involves the quantity from rewriting the expression23
+
i/DA1I2 = r,S, r2s2
set to sIand s2. Equation 28 yields I A and SA,and finally a and b are obtained from eq 29 and 30, respectively. Sets a, b, and c have the advantageof needing only the minimum number of least-squares parameters. However, calculations with set c were frequently numerically unstable and thus did not converge. Set b works well but cannot easily be directly extended to four or more components, because there are no simple closed forms for the si.79 Set d has as many extra rl parameters as there are experiments exceeding two. All the rl values can be linearly least squared with eq 6 to get a and 6. This has the advantage that any large errors in a1 can be detected. However, this larger number of fitting parameters leads to greater uncertainty in the “best values” of SI and s2 and, ultimately, in a and b. Set e dependson an accurate extrapolation toget DA. However, this extrapolation depends primarily on the fringes close together near thecenter ofthe pattern. These have the largest measurement errors, so that DA from Rayleigh experiments are less certain. If Gouy measurements aredone at thesame time, thesecould provide a more accurate DAfor each experiment. However, this approach, suggested by Revzin81 for Gouy analysis, cannot be extended to four or more components. There are also other ways to write eq 6. Thus
r, = a + ba, = r l a I+ r2a2=
where the ri are symmetric and obviously related to Xi. In this paragraph, symbols Xi refer to
(27)
in the form
X,a,+ X2a2- x2 (31) A, - A2
1)
x2 = D22 + D12(
+
1/DA1l2 = IA SAal It can be seen from eqs 27, 5 , and 6 that SA = b(s, - s2)
+
1, = a(s, - s2) s2
so that SA is also a function of the Di, and If SA = 0 because sI = SI, then the equal eigenvalue solution is valid. However if SA= 0 because b = 0, then the expression forf0’) is the same for all experiments. Thus each experiment looks like all the others and like the same binary system. In this case, it is not possible by any means to extract the D , from fringe data by any refractive index method at this value of A. The only conceivable alternative would be to try another wavelength A for the diffusiometer light source in the hope the R, would change enough to yield a nonzero b and thus a nonzero SA. Experience with real and simulated data have shown that if cm-I, then reasonably rapid convergence is greater than 25 and reasonable standard errors of the LS parameters and of the resulting D, areobtained. However, as FA( decreases below about 25, convergence becomes increasingly more difficult and standard errors of the D, increase rapidly. g. Alternative Variable Sets. Equation 15 can be rewritten in terms of several other variable sets, giving rise to other leastsquares procedures. Among potential variable sets considered in detail some time ago by LBE and DGM were (a) the set used above (a, b, sI, sz), (b) the D,, themselves, (c) (SI,s2, IA, SA),(d) (SI,s2) and as many rl variables as there are experiments, and (e) (sI,s2) and the DA for all experiments, where DA for each experiment is obtained from either extrapolating d ( D , ) vs ( z , * ) ~to zJ* = 0 or l / d ( D J ) vs ri12&)/yJ to yJ = 0. DAis an experimental quantity, and thus canreplacer1 bymeansofeq 27. ThisreducestheNLLSvariable
(33)
These ri and Xi forms can be extended to four or more component systems25 In addition, the Xi and Ai variables are especially convenient for the algebraic analysis of the equal eigenvalues case.65 Overall, however, the parameter set (a, b, sl,s2)was found by two of us (L.B.E. and D.G.M.) to be particularly useful for the ternary distinct eigenvalue case. The “ab” code based on them has been successfully used at LLNL for 20 years. VI. Experimental Results Tables 1-111 contain the data for individual experiments, Tables IV and V contain the derived data for the mean compositions, and Tables VI and VI1 contain comparisonsof Gouy and Rayleigh results. Table I includes experiments for which both Gouy and Rayleigh data are new. These are the 1:l systems AM-24 and AM-23 at 0.5 and 1 M, respectively. Table I includes ti,the average concentration of top and bottom solutions, ACl, d,oprdbttom; Rayleigh quantities aI, Ar, and experimental and calculated J; the Gouy quantities al,At, plus experimental and calculated J , DA, Qo, and QI. The were determined from the R,, which were obtained in turn from least-squares fits of eq 4 to the sets of ( J , ACi) from all Gouy experiments. Closely similar values were obtained by using the same procedure with all Rayleigh experiments. For Gouy data, Ar was obtained from the LS of c,vs 1/ t ’;for Rayleigh data, At was obtained from the LS of pseudobinary D’vs I / t < Table I1 contains the analogous Gouy data for the higher concentration 1:l mixtures AM-8, AM-25, and AM-26, where only Gouy experiments could be performed. Table 111 contains the Rayleigh data for experiments whose Gouy data have already been reported, namely, AM-7,I4 AM-2l,I7 and AM-22.17 How-
Isothermal Diffusion Coefficients of NaCI-MgC12-H20
The Journal of Physical Chemistry, Vol. 97, No. 15, 1993 3895
TABLE I: Data from Diffusion Measurements, 1:l Mole Ratio Cases with Both Rayleigh and Couy Data. series AM-23, total concentration = 1 .O M
series AM-24, total concentration = 0.5 M CI
c2 ACI Ac2
Rayleigh J(expt) J(ca1c) a1 At
Gouy J(expt) J(calc) At
IO9 DA(expt) IO9 DA(calc) IO4 Qdexpt) IO4 Qo(calc) IO4 Qdexpt) IO4 Ql(calc) d(top) d( bottom)
expt 1
expt 2
expt 3
expt 4
expt 1
expt 2
expt 3
expt 4
0.25000 0.25000 0.18221 -0.00001 80.416 80.409 1.0002 14.77 8O.35Ob 80.362 1.0002 13.93 1.5463 1.5473 -57.44 -56.80 -24.98 -24.25 1.0228 1 1.02998
0.25002 0.25002
0.25002 0.25001 0.14577 0.01480 79.997 79.996 0.8043 12.34 79.976 79.961 0.8042 11.01 1.4144 1.4139 -17.18 -17.94 -7.5 1 -7.82 1.02297 1.02984
0.25001 0.25001 0.03645 0.05921 78.669 78.710 0.2044 21.99 78.71 1 . 78.713 0.2043 18.38 1.0996 1.0977 34.04 34.01 16.01 15.76 1.02344 1.02934
0.50005 0.49996 0.23185 o.oo001 97.924 97.962 0.9999 16.68 97.907 97.943 0.9999 10.41 1.5652 1.5645 -72.78 -72.80 -31.13 -30.78 1.OS029 1.05916
0.50002 0.49995 -0.ooo03 0.09539 97.222 97.288 -0.0001 19.35 97.224 97.229 -0.0001 13.24 1.OS64 1.0561 39.07 39.08 18.46 18.13 1.05124 1.05822
0.5Oool 0.49996 0.18545 0.0 1909 97.844 97.816 0.8009 16.08 97.830 97.800 0.8009 13.25 1.4370 1.4376 -3 1.57 -3 1.54 -13.74 -13.59 1.05048 1 .OS897
0.50004 0.49997 0.04641 0.07632 97.534 97.459 0.2017 17.97 97.509 97.448 0.2012 15.61 1.1359 1.1362 33.61 33.59 15.63 15.31 1.05106 1.05842
-0.00000 0.07400 78.294 78.261
-0.OOOO 15.36 78.275 78.876
-0).oooo 13.12 1.0126 1.0140 32.83 32.97 15.84 15.58 1.02362 1.02919
*Units: C, and AC, in mol dm-3; 1 in s; DA in m2 SKI; d in g ~ m - ~J .from 4 5 code.30
TABLE 11: Data from Diffusion Measurements, 1:l Mole Ratio Cases with Gouy Results Only. series AM-25, total concentration = 2.0 M series AM-8, total concentration = 3.0 M series AM-26, total concentration = 3.72 M CI
c2 ACI Ac2 Gouy J(expt) J(calc) a1 At
109D~(expt) 109D~(calc) IO4 Qo(expt) IO4 Qo(calc) 104Ql(expt) 104Ql(calc) d(top) d(bottom)
expt 1
expt 2
expt 3
expt4
expt 1
expt2
expt 3
expt4
expt 1
expt 2
expt 3
expt4
1.00024 1. W 2 3 0.23973 0.00012 94.627 94.555 0.9988 15.83 1.5820 1.5822 -101.85 -101.48 -42.90 -41.86 1.10481 1.11360
1.00024 1.00024 -O).OOOOI 0.10060 96.718 96.721 -0.0004 14.16 1.1248 1.1249 48.35 48.44 21.99 21.59 1.10570 1.11270
1.00021 1.00021 0.19178 0.02017 94.848 94.945 0.7958 14.38 1.4692 1.4690 -55.41 -55.86 -23.60 -23.42 1.10498 1.11340
1.00023 1.OOO24 0.04798 0.08056 96.396 96.368 0.1962 15.52 1.1976 1.1975 32.34 32.32 14.61 14.20 1.10551 1.11291
1.50091 150438 0.24768 0.00017 92.684 92.577 0.9983 26.90 1.5496 1.5506 -1 17.94 -117.18 -48.34 -47.02 1.15729 1.16609
1.50093 1.50442 0.00006 0.10553 96.966 96.979 0.0002 12.98 1.1424 1.1429 58.57 59.65 25.58 25.57 1.15815 1.16526
1.50096 1.50443 0.198 15 0.02123 93.299 93.446 0.7912 12.84 1.4507 1.4498 -67.25 -67.85 -27.86 -27.61 1.15748 1.16592
1.50092 1 SO439 0.04962 0.08450 96.210 96.157 0.1926 11.92 1.2084 1.2079 37.77 36.54 16.56 15.47 1.15797 1.16543
1.86082 1.86081 0.20232 0.00003 72.918 72.911 0.9996 20.96 1.4697 1.4694 -129.77 -129.92 -52.07 -51.26 1.19463 1.20165
1.86081 1.86080 -0.00006 0.08637 77.111 76.921 -0,0003 18.18 1.1035 1.1042 59.55 59.77 25.68 25.04 1.19527 1.20098
1.86077 1.86076 0.161 78 0.01730 73.747 73.692 0.7909 23.14 1.3789 1.3795 -79.18 -78.90 -32.07 -31.53 1.19474 1.20152
1.86077 1.86078 0.04048 0.06916 75.943 76.194 0.1914 19.88 1.1639 1.1628 33.59 33.25 14.48 13.77 1.19512 1.20110
Units: same as Table 1.
TABLE III: Data from Diffusion Measurements, Rayleigh Cases Where Couy Data Were Previously Reported. series AM-21, total concentration = 1.0 M expt 1 expt 2 expt 3 expt4 0.75008 0.25003 Act 0.23588 AC: O.ooOo0 Rayleigh J(expt) 100.878 J(ca1c) 100.843 a1 1.0001 AI 6.36 Gouy J(expt) 100.868 J(calc) 100.815 a1 1.OOOO AI 3.74 CI
C:
a
0.75010 0.25002 0.00005 0.09261 95.643 95.646 0.0002 15.31 95.620 95.616 0.0002 10.25
0.75008 0.25002 0.18870 0.01853 99.760 99.807 0.8083 14.40 99.709 99.778 0.8083 10.50
0.75008 0.25002 0.04725 0.0741 1 96.744 96.728 0.2089 18.36 99.710 96.698 0.2089 13.59
series AM-22, total concentration = 1.0 M expt 1 expt 2 expt 3 expt 4 0.25004 0.75007 0.22789 0.00005 95.227 95.178 0.9995 18.51 95.199 95.157 0.9995 15.24
0.25004 0.75005 0.00000 0.09800 98.993 98.945
-0.ooOO 29.55 98.956 98.915
O.OOO0 24.10
0.25000 0.75005 0.18230 0.01963 95.868 95.917 0.7934 18.95 95.851 95.894 0.7934 16.29
0.24999 0.75005 0.04562 0.07842 98.176 98.223 0.1939 17.98 98.154h 98.195 0.1939 13.44
series AM-7, total concentration = 3.0 M expt 1 expt 2 expt 3 expt4 0.75039 2.25638 0.23598 0.00013 86.413 86.352 0.9987 23.70 86.358 86.317 0.9987 19.11
0.75037 2.25635 0.00002 0.1 1079 99.405 99.344 0.0001 19.64 99.321 99.286 0.0001 10.69
0.75034 2.25623 0.18888 0.02252 89.150 89.21 1 0.7737 20.17 89.128 89.171 0.7737 13.68
0.75035 2.25634 0.04723 0.08867 96.700 96.761 0.1784 16.44 96.676 96.708 0.1784 7.65
Units: same as Table 1. J from 3J code.30
ever, the Gouy J(expt), J(calc), and a lare also included for comparison, as well as their previously unpublished Gouy Az. Agreement between calculated(least squares) and experimental J is quite good. For Gouy experiments, differences are mostly less than 0.05 fringe, with the larger maximum values at the higher concentrations: 0.1 at C = 2,0.15 at C = 3, and 0.25 at C = 3.7 M. The largest error in J for Rayleigh experiments at all concentrations is 0.07, with most around 0.05. Differences of calculatedvsexperimental Gouy DAare mostly less than 0.0696, withthelargestbeing0.1796forAM-24at C=O.5 M. Calculated
vs experimental QOare also in good agreement; none differ by more than 0.8 X 10-4, and most are less than 0.4 X 10-4. Since An of the solution is the same for both optical systems, both should yield identical J values. However, the Rayleigh J is generally slightly larger than the Gouy J from the same experiment, with only two exceptions. The differences average about 0.025 at C = 1 and 0.045 at C = 3 M. A similar effect has been observed with the systems raffinose-KCl-H20 and NaCI-KCl-H20, which were also measured in this same ce11.82 A possible explanation is that the Gouy fpf is obtained from two
3896
Miller et al.
The Journal of Physical Chemistry, Vol. 97, No. 15, 1993
TABLE VI: Commrison of Couy and Rayleinb DA'
TABLE IV Experimental Results for 1:l Mole Ratio, Gouy Measurements'
c
TI T2 e0
mI(c1, c2) m2(CI, C2) 1O3R1 1 0 3 ~ ~
a
HI
H2
PI t 2
R 10-9Ut
I 0-9u. SA
IO9(D11), IO9(Dl2), 109(D21), 109(D22)v 109(Dll)o
IO9(D12)o IO9(D2~)o lO9(D22)o
Units:
AM-24
AM-23
AM-25
AM-8
AM-26
0.50002 0.25001 0.25001 54.8417 0.25305 0.25305 9.6108 23.048 1.02640 39.408 75.306 19.078 19.950 18.056 1.1815 0.7460 -59.81 1.275 0.341 0.082 0.912 1.281 0.347 0.089 0.918
0.99999 0.50003 0.49996 54.2822 0.51 1328 0.51 1258 9.2024 22.222 1.05473 38.263 73.127 20.200 22.106 18.033 1.2359 0.7564 -54.90 1.203 0.422 0.1 1 1 0.928 1.217 0.437 0.125 0.943
2.00046 1.00023 1.00023 53.0392 1.04680 1.04680 8.5832 20.949 1.10920 36.576 69.829 21.807 25.312 17.965 1.4500 0.7628 46.81 1.050 0.562 0.167 0.950 1.079 0.600 0.196 0.988
3.00534 1.50093 1.50441 51.6643 1.61261 1.61635 8.1290 20.018 1.16170 35.450 67.349 22.829 27.663 17.887 1.8205 0.7697 41.92 0.930 0.710 0.198 0.919 0.973 0.777 0.242 0.986
3.72158 1.86079 1.86079 50.6356 2.03987 2.03987 7.8487 19.409 1.19813 34.750 66.16 23.450 28.750 17.831 2.1760 0.8123 40.07 0.807 0.691 0.213 0.884 0.858 0.777 0.265 0.970
ti,mol d m 3 ,m,, mol(kg of H20)-'; R,,dm3mol-'; a, g
HI, g mol-'; VI,cm3 mol-]; ut, u-, m2s; (D&, (D,)o,m2 s-l. See text for discussion of errors in (D& Divide the H,of this table by lo3 to get d in g c m 3 from eq 34 when C, are in mol d w 3 .
TABLE V
T TI
e2
TO
1O3R1 IO'R2 10-9a+ 10-9u. SA
109(D1~), 109(D12), 109(D21), 109(D22), 1O9(Dli)o IO9(Dl2)0 109(D21)~ IO9(D22)0
Experimental Results, Rayleigh Measurementsa AM-24'
AM-23'
AM-21'
AM-22d
AM-7d
0.50002 0.25001 0.25001 54.8417 9.6164 23.043 1.1723 0.7460 -60.23 1.278 0.324 0.082 0.916 1.285 0.330 0.089 0.922
0.99999 0.50003 0.49996 54.2822 9.2041 22.224 1.2346 0.7596 -55.08 1.197 0.412 0.112 0.930 1.211 0.426 0.126 0.944
1.00010 0.75008 0.25002 54.3040 9.3148 22.496 1.3545 0.7138 -58.32 1.329 0.688 0.062 0.810 1.350 0.712 0.069 0.818
1.00008 0.25002 0.75006 54.2573 9.0946 21.998 1.1361 0.8140 -52.43 1.084 0.191 0.154 1.024 1.091 0.198 0.173 1.045
3.00669 0.75036 2.25633 51.4443 7.9618 19.534 1.8846 0.8666 -40.57 0.682 0.285 0.251 1.002 0.701 0.314 0.308 1.088
Units: _seeTable IV. For m,, a, H,, and p,,see Table !V. For m,, a, H I , and V,, see Table I1 of ref 14. For m,, 2, H,,and VI,see Table a
I1 of ref 17.
slits on both sides of but very close to the optic axis,39,g3whereas the Rayleigh fpf is obtained from positions further out on both sides of the boundary. Thus optical imperfections such as prismatic or lens effects from tilted or bowed cell windows could be different at the two locations. This explanation may be supported by the larger differences between J values at higher concentrations, although the number of examples is too small to be certain. Similarly, it would be expected that the At from Gouy and Rayleigh data from a given experiment would be the same, since the boundary and the starting time are the same. However, Tables I and I11 show that Rayleigh Af are universally 1-8 s larger than the corresponding Gouy At. A part of this difference is due to the difference in how Rayleigh and Gouy times are recorded. Since Gouy exposure times run from 10 to 40 s, tb is taken as the average time of the exposure, Le., shutter opening time plus l / 2 the exposure time. However, Rayleigh exposure times are constant at 3 s, so it is convenient to use the shutter opening
Gouv AM-21 C= I 3:l ratio AM-22 C= 1 1:3 ratio
9
AM-23 C= 1 1:l ratio AM-24 C=0.5 1:l ratio
of
1
2 3 4 1
2 3 4 1
2 3 4 1 2 3 4
1.5325 1.0173 1.4070 1.1003 1.5800 1.0827 1.4532 1.1592 1.5652 1.0564 1.4370 1.1359 1.5463 1.0126 1.4144 1.0996
Ravleiah 1.5324 1.0171 1.4070 1.1006 1.5803 1.0829 1.4529 1.1590 1.5645 1.0561 I .4376 1.1362 1.5473 1.0140 1.4139 1.0977
1.5324 1.0153 1.4071 1.0994 1.5763 1.0828 1.4544 1.1578 1.5638 1.os43 1A371 1.135, 1.5480 1.0116
1.4154 1.0985
1.53301.0161 I .407 1 1.0998 1.5811 1.0828 1.4535 1.1591 1.5635 1.0543 1.4364 1.1345 1.5493 1.0123 1.4148 1.0965
DA(expt)for Rayleigh comes from the estimated intercept of the plot vs (r*,)2. DA = (intercept)2.
41'2
time as t'R. Consequently, using the mean time of a Rayleigh exposure would add 1.5 s to Af. However, this does not explain differences larger than that, some of which are larger than the combined standard error of the two Af values, since the standard error of a Af is usually less than 0.5 s, and the maximum errors are about 1 s. Table IV contains the derived Gouy results, including the mean values of the total concentration C = CI + C2 and individual average concentrations Cifor the set of experiments in each series. The Ri for each series were obtained as described above, using X = 546.07 nm and cell thickness a = 2.5064 cm. The Hi and a were obtained by fitting the equationg4
d = a + H I (C, - Cl) + H,(C, - C,)
(34) to all the densities in Tables 1-111, using the overall average concentrations ci of Table V and the specific C1 and C, of the top and bottom solutions of each experiment (obtained from (ti f ACi/2)). Here ddenotes density (in section IV only, it referred to the interfringe distance). The partial molar volumes 6 are obtained from the expressions in ref 84g5and are used to calculate (Dij)0 from (Dij),.g43g6 The u+ and u- were defined by Fujita and G ~ s t i n g ,are , ~ equal t o ~ ~ ~ a nrespectively,andare ds~~, thereciprocalsoftheeigenvalues of the diffusion coefficient matrixag7As noted above, SAis an indicator of the ease of convergence. All our SAare large enough for good convergence. The experimental volume-fixed (Di,)" and the calculated solvent-fixed (Dij)0 for the Gouy series are also given in Table IV. The (D,), were obtained from the RFG program. The Hi and the calculated Dij were used to test for the static and dynamic stabilities of all diffusion boundaries.20-22 All boundaries were stable, although there were a couple of cases close to a fingering instability at the boundary center. Such an instability was actually found in two analogous NaClSrC12H20 cases at a l = 1 . f ~ ~ Table V contains the analogous Rayleigh results wherever these are specific to Rayleigh experiments or do not duplicate results already given for the Gouy experiments in Table IV or in previous papers in this series.lG17 The (Di,), were obtained as described in section V from the combined sets of J, ACI, AC2, and the fringe positions of the Creeth pairs (after base line corrections) from all the experiments at a given composition. The accuracies of the Gouy Dij are believed to be as estimated in our previous papers;lG17 namely, about 1% in Dll and D22, 2-3% of D II for D12, and 1% of 0 2 2 for 0 2 1 . The same estimates apply to the Rayleigh Dij. For both cases, a rough rule of thumb is that the inaccuracies are about 4 times the standard error of
Isothermal Diffusion Coefficients of NaC1-MgCl2-H20
TABLE VII:
The Journal of Physical Chemistry, Vol. 97, No. 15, 1993 3897
Compfrison of Couy vs Rayleipb Results' ~
AM21 CI
C! AI(,
AIR AZO
A~R
DI1 6 DIIR 8=G-R
DI2 6 DI~R 8=G-R 0 2 1 ~
DZIR 8=G-R
D226 D22~
d=G-R
RIL RIR R~c R ~ R SAG
SAR
0.75 0.25 0.7370 0.7383 1.4003 1.4010 1.3275 1.329 I -0.0016 0.692 I 0.6881 0.0040 0.0621 0.061 8 0.0003 0.8098 0.8102 -0,0004 427.419 427.541 1032.235 1032.563 -58.1 1 -58.32
AM22
(I
0.0022 0.0008 0.0037 0.0020 0.0007 0.0003 0.0011 0.0007
0.25 0.75 0.8848 0.8802 1.2294 1.2285 1.0890 1.0844 0.0046 0.1884 0.1914 -0.0030 0.1521 0.1537 -0.0016 1.0252 1.0243 0.0009 417.334 417.434 1009.353 1009.662 -52.36 -52.43
~
AM23
U
0.0006 0.0014 0.001 3 0.0041 0.0002 0.0005 0.0004 0.0014
0.5 0.5 0.809 1 0.8100 1.3220 1.3165 1.203 1 1.1969 0.0062 0.4225 0.4116 0.0109 0.1 109 0.1 124 -0.0015 0.9281 0.9297 -0.00 16 422.38 1 422.459 1019.948 1020.041 -54.90 -55.08
l7
0.0003 0.0011 00007 0.0028
0.0002 0.0004 0.0004 0.0009
AM24 0.25 0.25 0.8464 0.8530 1.3405 1.3405 1.2748 12780 -0.0032 0.3408 0.3240 0.0168 0.0825 0.0821 0.0004 0.9121 0.9155 -0.0034 441 . I 25 441.384 1057.862 1057.656 -59.81 -60.23
7 l
0.0026 0.0017 0.0046 0.0048 0.0008 0.0006 0.0019 0.00 16
AM7 0.7504 2.2563 0.5247 0.5306 1.1569 1.1540 0.6897 0.6825 0.0072 0.3 107 0.2850 0.0257 0.248 1 0.2513 -0.0032 0.9919 1.002 1 -0.0102 365.290 365.438 896.074 896.596 -40.50 -40.57
(I
0.00 13 0.0019
0.0036 0.0054 0.0005 0.0007
0.001 2 0.0019
Units: all D,, A,, 8, and (I are in m2/s; R, in dm-)/mol; SA in s1i2/cm. R, of Tables IV and V have been multiplied by a/A.
0
1
2
3
4
C, +C,, mol Figure 7. Experimental interdiffusion (mutual diffusion) coefficients of NaCI(I)-MgC12(2)-H20 versus total concentration for the 1:l mole ratio: circles,our Gouy data; pluses, our Rayleigh data; crosses, Leaist's data;9 squares, Nernst-Hartley equations.
the Dij coefficients, which are obtained from the propagation of error equations using the variance4ovariance matrix of the leastsquares parameters (e.g., 0 , 6,sI,and s2for Rayleigh).
VII. Discussion Figure 7 shows the Dij of 1:l mixtures vs total concentration C. The data are quite smooth, and Rayleigh and Gouy points
are in excellent agreement. Leaist's data9 are somewhat more scattered but are in good agreement with ours. Figure 7 also shows that D12becomes large with increasing C, but not larger than either or both of the Dii; that occurred in the 3:1 mixtures14 or trace MgCl2 cases.I6 However, D II for NaC1, which is larger than D22 of MgC12 at infinite dilution, becomes smaller than 0 2 2 a t 2.8 M and above. Because D12and D21 are both fairly large, NaCl is always the faster diffuser when there are equal gradients of both salts, unlike the case for 1:3 mixturesI7 or trace NaCl cases.I5 The Dij a t infinite dilution are obtained from Nernst-Hartley type equations, generalized to multicomponent systems in terms of infinite dilution ionic c o n d ~ c t a n c e s . ~ The J ~ ~Na+ ~ . ~ and ~ C1conductances are taken from Robinson and Stokes,go and Mg2+ from Miller et and are 50.10, 76.35, and 53.32 cm2/(Q equiv), respectively. The resulting values for the 1:l mole ratio
are D l l = 1.425 X D I 2= 0.344 X DZl= 0.097 X and DZ2= 1.075 X m2/s. Experimental and calculated values of DA are given in Table VI. The experimental value of the Rayleigh (DA)Rfor each experiment is obtained by extrapolation of d(D,ave)vs (z,*)2 to (z,*) = 0. The calculated Gouy and Rayleigh DA values are obtained from eq 27. The calculated (DA)Gand (DA)Rusually differ by less than 0.1%, with a maximum of 0.17%. This provides a measure of the similarity of results from the two methods. A detailed comparison of the Rayleigh and Gouy D, and other derived quantities are given in Table VII. There are some small differences between the two methods, as expected. However, the differences are within the combined errors of the estimates or rule of thumb given above, in particular for the D,. This is very gratifying. In addition, as will be shown in a subsequent paper, the Onsager reciprocal relations are satisfied by both the Gouy and Rayleigh Dl, within experimental errors for all experiments. The Nernst-Hartley equations are often used to estimate D,, values. However, they predict constant D, as a function of total concentration for a given mole ratio, i.e., horizontal lineson Figure 7. This isclearly incorrect here. Anextension is touse theNemstHartley analogues of the diffusion Onsager coefficients LJj2Jand data for the chemical potential derivativesdp,/dC, to obtain (DJ0, and then (D,),.239.88.89This technique provides improvement only up to total concentrations of 1-2 M, and is poor at higher concentrations, as we (unpublished) and Leaist9have found. This and other types of estimation procedures will be discussed in a subsequent paper. Certain comments deserve repetition. First, the cross terms are large. Consequently, the common assumption that they are zero is grossly incorrect at the 1:l mole ratio and is even worse at larger Na:Mg mole ratios. Some geochemical examples of the importance of accounting for cross term diffusion were discussed elsewhere.64 Second, it is not yet possible to estimate D,, values at high concentrations from data a t low concentrations. All D, from this and previous papersI4-l7 can be used to obtain extrapolated -trace" D,, for comparison with radioactive tracer results obtained by Mills et a1.11 These will be reported in a later paper. Figure 8 shows partial molar volumes of all components vs the square root of C for the 1:l mixtures. As was observed in our previous papers,I4-I7 values for each component fall on nearly straight lines, and those for theelectrolytes increase with increasing
3898
The Journal of Physical Chemistry, Vol. 97, No. 15, 1993
28k 26
-E --d
241
c"cz=l
77
0
0.5
1
1.5
2
(c, + c ~mO~112dm-312 ) ~ ~ ~ Figure 8. Partial molar volumes of NaCI, MgCI2, and H 2 0 vs square root of total concentration.
C. The values at infinite dilution were calculated from data tabulated by Millerog1and are 16.62 for NaCl and 14.49 for MgC12, in cmj/mol. The value 18.07 was taken for HzO. Acknowledgment. This paper is dedicated to Dr. J. M. Creeth, whose Rayleigh study of binary systems43is theepitomeofcareful research and clarity. His work inspired our original multicomponent Rayleigh studies. This work was primarily performed under the auspices of the U.S.Department of Energy under Contract W-7405-ENG-48. D.G.M., J.A.R.,andC.M.L. thank the Office of Basic Energy Sciences (Geosciences) for support. J.G.A. thanks TCU for Research Fund Grant No. 5-23824. R.M. thanks TCU for supporting him with a research fellowship. Portions of this work form part of the 1987 Ph.D. dissertation of R.M. L.B.E. thanks the Office of Saline Water of the US. Department of Interior for support during preparation of the original programs for analyzing ternary Rayleigh data and while establishing our techniques for plate reading and base line corrections. The authors thank A. W. Ting for some preliminary experiments on these compositions done on our Beckman-Spinco Model H under NSF sponsorship. We also thank K.S.Spiegler, D. L. Graf, and D. E. Anderson for encouragement over the years. References and Notes ( I ) Miller, D. G. J . Phys. Chem. 1966,70,2639. (2) Miller, D. G. J . Phys. Chem. 1961,71,616. (3) Miller, D. G. J . Phys. Chem. 1967,71,3588. (4) Miller, D. G. Faraday Discuss. Chem. Soc. 1978,No.64,295;also see discussion, pp 346-350. ( 5 ) Zhong, E. C.; Friedman, H. L. J . Phys. Chem. 1988,92, 1685. (6) Friedman, H.L.; Raineri, F. 0.; Wood, M. D. Chem. Scr. 1989,29A, 49. (7) Kim, H.;Reinfelds, G.; Costing. L. J. J . Phys. Chem. 1973,77,934. (8) Felmy, A. R.; Weare, J. H. Geochim. Cosmochim. Acta 1991,55, 113. (9) Leaist, D. G. Electrochim. Acta 1988,33,795. (IO) Rard, J. A.; Miller, D. G. J . Chem. Eng. Data 1987,32, 85. ( I I ) Mills, R.; Easteal, A. J.; Woolf, L. A. J . Solution Chem. 1987,16, 835. (12) Bianchi, H.;Corti, H. R.; FernBndez-Prini, R. J . Solution Chem. 1989,18,485. (13)Bianchi. H.;Corti, H. R.; FernBndez-Prini, R. J . Solution Chem. 1992,21. 1107. (14)Albright. J. G.; Mathew, R.; Miller, D. G.; Rard, J. A. J. Phys. Chem. 1989,93,2176. ( I 5 ) Paduano, L.; Mathew, R.;Albright, J. G.; Miller, D. G.; Rard, J. A. J . Phys. Chem. 1989,93, 4366. (16) Mathew, R.; Paduano, L.; Albright, J. G.; Miller, D. G.; Rard, J. A. J . Phys. Chem. 1989,93.4370. (17) Mathew, R.; Albright, J. G.; Miller, D. G.; Rard. J. A. J . Phys. Chem. 1990, 94,6875. (18) Bianchi, H.;Tremaine, P. R., in preparation. (19) Wendt, R. P. J. Phys. Chem. 1962,66, 1740. (20) Miller, D. G.; Vitagliano, V. J. Phys. Chem. 1986,90, 1706. (21)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 20.
Miller et al. (22) Vitagliano. P. L.; Ambrosone, L.; Vitagliano, V. J. Phys. Chem. 1992, 96, 1431. (23) Fujita, H.; Costing, L. J. J . Am. Chem. Soc. 1956,78, 1099. (24) Fujita, H.; Costing, L. J. J . Phys. Chem. 1960,64, 1256. (25) Miller, D. G. J . Phys. Chem. 1988,92,4222. (26)Gosting, L. J.; Kim, H.;Loewenstein, M. A.; Reinfelds, G.; Revzin, A. Rev. Sci. Instrum. 1973,44, 1602. (27) Thisoutstanding apparatus wasconstructed by Professor L. J. Gosting at the University of Wisconsin during 1960-1970. After his untimely death, it was transferred to Professor G. Kegeles a t the University of Connecticut, and in 1981 to D. G. Miller at LLNL. Since 1991,it has been under thecare of J. G. Albright a t Texas Christian University. (28) Wendt, R. P. Ph.D. Thesis, Universityof Wisconsin, Madison, 1960. (29)Albright, J.G.Ph.D.Thesis.UniversityofWisconsin, Madison, 1962. (30)Albright, J. G.; Miller, D. G. J . Phys. Chem. 1989,93, 2169. (31)Costing, L. J.; Morris, M.S. 1.Am. Chem.Soc. 1949,71,1998.Also see ref 83. (32) Revzin, A. Ph.D. Thesis, University of Wisconsin, Madison, 1969. (33) Miller, D. G.; Sartorio. R.; Paduano, L. J . Solution Chem. 1992,2/, 459. (34) Longsworth, L. G. In Electrophoresis: Theory, Methods, and Applications; Bier, M., Ed.; Academic Press: New York, 1959;Chapter 4, pp 137-177. (35) Longsworth, L. G. In Physical Techniques in Biological Research, 2nd ed.; Vol. 11, Part A, Academic Press: New York, 1968;Chapter 3, pp 85-120. (36) 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. I , Chapter IV. (37)Tyrrell, H.J. V.; Harris, K. R. Diffusion in Liquids; Butterworths: London, 1984. (38)Svensson (Rilbe). H.;Thompson, T. E. A Laboratory Manual of Analytical Methods of Protein Chemistry; Alexander, P.. Black, R., Eds.; Pergamon Press: New York, 1961;Vol. 3, Chapter 3, pp 57-1 18. (39)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; Blackwell Scientific Publications: Oxford, UK, 1991;Vol. 111, Section 9.I .6,pp272-294 (references pp 316319). (40) Philpot, J. S.L.; Cook, G. H. Research 1948,I, 234. (41) The Philpot~mkopticalsystem contrasts with the Svensson (Rilbe)Kegeles form:5 which is used in our other diffusiometer (Beckman-Spinco Model H). That instrument has the cylinder lens in the vertical position with a better aperture, but requires an additional camera lens. In this form, that camera lens focuses the center of the cell. The appearance of the fringe patterns is the same for both configurations. (42) Svensson (Rilbe). H. Opt. Acta 1956,3, 164. (43) Creeth, J. M. J. Am. Chem. Soc. 1955,77,6428. (44) Miller, D. G.; Rard. J. A.; Eppstein. L. B.;Albright, J. G. J . Phys. chem. 1984,88, 5739. See supplementary material. (45) Svensson (Rilbe), H. Acta Chem. Scad. 1950,4,399. (46)Sundelbf, L.-0. Ark. Kemi 1965,25 ( I ) , I . (47) This result is based on Svensson (Rilbe)'s thorough analysis of aberrations in Rayleigh optical systems.J2 (48) The fringe blurring problem with Tiselius cells could be avoided by a different cell design. in which the reference arm is closed off from the water bath and is filled with the denser solution. We have successfully used such a cell for high concentrations in our Beckman-Spinco diffusiometer, but unfortunately that cell does not fit into theGosting diffusiometer cell holders. The blurring problem can also be avoided by using laser light sources. However, the resulting patterns are of somewhat poorer quality.4u (49) Albright, J. G.; Miller, D. G. J . Phys. Chem. 1972, 76, 1853. (50) Longsworth, L. G. Rev. Sei. Instrum. 1950,21, 524. (51) Longsworth, L. G. J . Am. Chem. Soc. 1952,74,4155. (52) Longsworth, L. G. J. Am. Chem. Soc. 1953. 75,5705. (53) Costing, L. J.; Fujita, H. J . Am. Chem. Soc. 1957,79, 1359. (54)There is an alternate pairing, Longsworth pairing,",'' '1 which uses constant Aj. For example, in a 100-fringe system, fringe 1 is paired with fringe 51,2with 52,...,49 with 99. This pairing avoids locationoftheboundary center and has similar values of the fringe separations. However, the disadvantages are that the Wiener Skewness and concentration dependencies are not removed. (55) Svensson (Rilbe), H.Acta Chem. Scand. 1951.5, 1301. (56) Rard, J. A.; Miller, D. G. J . Solution Chem. 1979,8, 701. (57)Sherrill, 6 . C.; Albright, J. G. J . Solution Chem. 1979,8, 217. (58)Albright, J. G.; Mathew, R.; Miller, D. G. J. Phys. Chem. 1987,91, 210.
(59)Yphantis, D. A. Private communication, 1981. (60) Laue, T. M. Ph.D. Thesis, University of Connecticut, Storrs, 1981. (61) Longsworth, L. G. J. Colloid Interface Sei. 1966,22, 3. (62) Laue, T. M.; Yphantis, D. A.; et al. In preparation. (63) Miller, D. G. J. Phys. Chem. 1981, 85, 1137. (64) Miller, D. G.; Ting, A. W.; Rard, J. A.; Eppstein, L. B. Geochim. Cosmochim. Acta 1986,50, 2397. (65) Miller, D. G. Equal eigenvalues in multicomponent diffusion: the extraction of diffusion coefficients from experimental data in ternary systems. In Fundamentals and Applications of Ternary Dijjusion; Proccedings of the International Symposium, Purdy, G. R.. ed.; 29th Annual Conference of Metallurgists of CIM, Hamilton, Ontario, August 27-28, 1990 Pergamon Press: New York, 1990 pp 29-40.
Isothermal Diffusion Coefficients of NaCI-MgCI2-H20 (66) Albright, J. G.; Sherrill, B. C. J . Solution Chem. 1979, 8. 201. (67) Miller, D. G.; Ting, A. W.; Rard, J. A. J . Electrochem. SOC.1988, 135. 896. (68) Rard, J. A.; Miller, D. G . J . Phys. Chem. 1988, 92, 6133. (69) Rard, J. A.; Miller, D. G . J . Solution Chem. 1990, 19, 129. (70) Albright, J. G . ; Miller, D. G. J . Phys. Chem. 1975, 79, 2061. (71) The value of B is obtained from the expression B = 1 /[4 X 60(s/min) X IO'>(mm'/m') X MF'] = 4.166667 X IO q/MF? which yields diffusion
coefficients in m2/s. (72) Miller, D. G.; Rard, J. A.; Eppstein, L. B.; Robinson, R. A. J.Solution ..
Chem. 1980, 9, 467. (73) Lonasworth, L. G . J. Am. Chem. SOC.1947, 69, 2510 (74) Fujiia, H. J . Phys. SOC.Jpn. 1956, 11, 1018. (75) Ambrosone, L.; Vitagliano, V.;Della Volpe, C.;Sartorio, R. J.Solution Chem. 1991, 20, 271. (76) Marquardt, D. W. J. SOC.Ind. Appl. Math. 1963, 1 1 , 431. (77) Miller, D. G.; Albright, J. G . In preparation. (78) Toor, H. L. AlChE J . 1964, 10, 460. (79) However, a technique suggested by Leaistxocould be used for four
or more components. The values of s,are determined numerically from the characteristic equation of the diffusion coefficient matrix. That matrix is
The Journal of Physical Chemistry, Vol. 97, No. 15, 1993 3899 based on the current values of the D!,. The s,are put into the error function terms as "known" constants, and the next NLLS iteration involves only the coefficients of the error functions. This increases the number of iterations but is a small price to pay for the results. (80) Leaist, D. G. Private communication, 1991. (81) Revzin, A. J . Phys. Chem. 1972, 76, 3419. (82) Miller, D. G.; Paduano, L.; Sartorio, R. In preparation. (83) Costing, L. J. J . Am. Chem. SOC.1950, 72, 4418. (84) Dunlop, P. J.; Costing, L. J. J . Phys. Chem. 1959, 63, 86. (85) With the Dunlop-Costing equations,x4 these are sonstants characteristic of the data set used, and are thus independent of the C, chosen. (86) Woolf, L. A,; Miller, D. G . ;Costing, L. J. J . Am. Chem. SOC.1962, 84, 317. (87) Miller, D. G.; Vitagliano, V.; Sartorio, R. J . Phys. Chem. 1986, 90, 1509. (88) Wendt, R. P. J . Phys. Chem. 1965,69, 1227. (89) Leaist, D. G.; Lyons, P. A. Ausr. J . Chem. 1980, 33, 1869. (90) Robinson, R. A.; Stokes, R. H. Elecrrolyte Solutions, 2nd ed.; rev., Butterworths: London, 1970; pp 294-299. (91) Millero, F. J. Chem. Reu. 1971, 71, 147.
