J. Phys. Chem. 1993,97, 1464-1469
1464
Gravitational Stability of Taylor Dispersion Profiles. Revised Diffusion Coefficients for Barium Chloride-Potassium Chloride-Water Derek C. Leaist’ and Ling Hao Department of Chemistry, University of Western Ontario, London, Ontario, Canada N6A 587 Received: October 22, 1992
The ternary interdiffusion coefficients of aqueous BaCl2-KCl solutions were measured recently by Gouy interferometry. Convection appeared to develop in the free-diffusion boundaries even when both of the heavy BaC12 and KCl components were more concentrated in the lower end of the solution column. This unusual behavior was attributed to large countercurrent coupled flows of KC1 driven by the diffusing BaC12. Yet the diffusion potential generated by a BaCl2 concentration gradient is expected to drive cocurrent coupled flows of KCl. T o check the Gouy results, diffusion of aqueous BaC12-KCI is remeasured by the Taylor dispersion method. In dispersion experiments, unlike free-diffusion methods, tests show that arbitrary initial concentration gradients can be used without errors from gravitational instabilities. In contrast to the Gouy results, the Taylor results indicate that BaC12 drives cocurrent coupled flows of KCl. Diaphragm-cell diffusion measurements support the Taylor results. In addition, the Gouy data are shown to be inconsistent with Onsager’s reciprocal relation and are therefore erroneous. The revised stability diagram BaClrKCl-water shows that free-diffusion boundaries are stable if BaC12 and KCl are both more concentrated in the lower solution.
Introduction
Common sense suggests that a stable diffusion boundary is formed when a less-densesolutionis layered over a denser solution in a gravitational field. Yet convection can develop in such boundaries despitetheir initial stability. This surprisingbehavior, called double-diffusiveconvection,I-l0can occur when there are at least two independent driving forces, such as a solute concentration gradient and a temperature gradient in a nonisothermal binary solution, or two independent concentration gradients as in isothermal ternary diffusion in a three-component solution. There has been considerable interest in double-diffusive convection’-l0in order to understand the dynamics of processes as diverse as crystallization,the growth of convective fingers when seawater is layered over cooler seawater of lower salinity, and the turnover of stratified liquids in tanks, ponds, and lakes. Recently Gouy interferometry-a sophisticated optical technique-was used to follow double-diffusive convection during isothermal ternary diffusion in aqueous BaC12-KCI solutions.1° Convection was reported in the diffusion boundaries even if BaC12 and KCl (the two heaviest components) were more concentrated in the lower end of the free-diffusion column. This remarkable behavior was attributed to strongly coupled diffusion. At the single composition that was studied, each mole of diffusing BaClz countertransported about 1.6 mol of KCl. The diffusion boundaries were stable only for a narrow range of initial conditions. Double-diffusive convection, though interesting, must be avoided in free-diffusion experiments in order to obtain reliable diffusion data. In some cases time-consuming reconnaissance runs may be needed to map out the initial conditions that can be trusted to give stable diffusion columns. The work reported here was undertaken to see if the practical difficulties connected with double-diffusive convection can be avoided entirely by using the Taylor dispersion method’ I - 1 8 to measure multicomponent diffusion. In a dispersion experiment a small sample of solution is injected into a laminar carrier stream of different composition confined within a long capillary tube. The injected sample spreadsout as it flowsalong the tube, forming a nearly Gaussian distribution. Diffusion coefficients can be calculated from the refractive index profile across the eluted Unwanted convection, including double-diffusive convection, should be negligible because the fluid is confined 0022-3654/93/2097- 1464304.00/0
between the walls of horizontal fine-bore tubing. To test this suggestion, the diffusion coefficients of aqueous BaCl2-KC1 solutions are measured by injecting denser or less-dense solution samples into the carrier stream. The results are examined for possible discrepanciesthat would indicate systematic errors from convection. Why diffusing BaClt should countertransport large amounts of KC1 was not discussed in the Gouy study. Actually, such behavior is unexpected. In water, C1- ions are about three times more mobile than Bat+ ions.Is The diffusion potential generated by diffusing BaC12 speeds up the Ba2+ions and slows down the C1- ions so that the solution stays electrically neutral. The diffusion potential along a BaClt concentration gradient should therefore pull K+ions in the same direction as the diffusingB a c k leading to cocurrent coupled transport of KCl. These considerations gave the present work a second purpose: to verify the puzzling Gouy results. Unhappily, it soon became clear that there were grpss discrepancies between the Gouy and Taylor results. Faced with two sets of conflictingternary diffusion data, is there a way to decide if either set is incorrect? To answer this question, Onsager transport coefficients’4q20.21 arc calculated from the Taylor and Gouy diffusion coefficients. The data that disobey Onsager’s law of reciprocal relation^'^*^^*^^ are judged to be incorrect. An additional check is performed by using a diaphragm cell to independently measure the diffusion coefficients of BaC12KCI-water . Multicomponent diaphragm-cell experiments22-2s have the enormous advantage that the diffusion of each solute can be followed directly and unambiguously by chemical titration. (In Taylor and Gouy experiments diffusion is followed indirectly by measuring changes in the refractive index, a bulk solution property that depends on the concentrations of all the solution components.) For this reason the diaphragm-cell technique may prove to be the method of choice in future studies of manycomponent diffusion.
Experimental Section A Gilson Model MP4 metering pump maintained a steady flow of carrier solution through a 3115-cm-longTeflon dispersion tube. The tube’s inner radius, 0.045 1 5 cm, was calculated from the weight of the tube when empty and when filled with water 0 1993 American Chemical Society
Diffusion Coefficients for BaC12-KCl-H20 of known density. Solution samples were injected through a Rheodyne Model 50 injection valve with a 0.020-cm3 sample loop. Retention times were about 6000 s. The dispersion tube was coiled in a 75-cm-diameter helix and held at 25.00 f 0.05 OC in a thermostat. The compositions of the carrier stream and injected solutions usually differed by f0.020 for BaClz or f0.040 mol dm-3for KCl. Smaller initial concentrationdifferencesgave identical results. A Gilson Model 131 deflection-typedifferential refractometer ( 10-mm3flow cells) monitored the broadened distribution of the eluted samples. Refractive index changes as small as 1 X lo-* could be detected. The refractometer's output voltage was displayed on a chart recorder and measured at accurately-timed 15-s intervals with a Hewlett-Packard Model 3478A digital voltmeter. The voltage readings were stored in a microcomputer for analysis. A few additionaldiffusion measurements on BaClrKCI-water were made with a Stokes diaphragm cell.14J9*26 The upper and lower solution compartments of the cell were separated by a 3.0cm diameter fine-porosity sintered Pyrex disk. Each compartment was fitted with a greaseless Teflon stopcock and an iron-cored glass stirrer. External magnets rotated the stirrers at 60 rpm. The lower stirrer floated and the upper stirrer sank so that the diaphragm surfaces were continuously swept free of stagnant solution. Each compartment held 34 cm3 of solution. The initial concentrationdifferences across the diaphragm were f0.400 mol dm-3 for each salt. After a preliminary diffusion period to establish steady-state diffusion through the diaphragm, thecell compartmentswere rinsed and refilled with fresh solutions to start a run. The solutions were retrieved for chemical analysis after a timed interval of 46 h. A 25.00-cm3sample of each BaCl2-KCl solution to be analyzed was first diluted to 1000.0 cm3by using a pipet and a volumetric flask. The concentration of BaC12 in the diluted solution was determined by titrating 25.00-cm3 solution samples with 0.01 50 mol dmw3EDTA at pH 12. Methyl thymol blue indicated the end point.27 To determine the total chloride concentration, additional 25.00-~m-~ samples were titrated against 0.0400 mol dm-3 AgN03 by using dichlorofluorescein to indicate the end point.27 The concentration of KCl was calculated from the measured concentrations of BaC12 and total chloride. For increased precision, the same pipet and volumetric flask were used for each analysis. Each analysis was repeated at least three times. Titrant volumes agreed to within better than 0.05 cm3. The diaphragm cell was calibrated2*by diffusing 0.500 mol dm-3 aqueous KCl into pure water, which gave 0.3654 cm-2 for the cell constant. Solutions were prepared in calibratedvolumetric flasks by using distilled, deionized water and BDH reagent grade BaC12.2H20 and KCl. Stock BaCl2 solutions were analyzed by titration against EDTA and AgNO3 and then diluted as required. The EDTA and AgNO3 analyses agreed within 0.1%.
Results Binary BaClrWater and KCI-Water Systems. Before the results for BaCl2-KC1-water aregiven, evidence will be presented for the gravitational stability of binary dispersion profiles for BaCl2-water and KC1-water. Fick's law
J = -DVC (1) describes thediffusion of a solute in an isothermalbinary solution. J is the molar flux of solute in volume-fixed coordinates, D the binary diffusion coefficient, and VC the gradient in the concentration of the solute. To measure D by the dispersion method a narrow band of solution of composition I? + AC is injected at time t = 0 into a carrier stream of composition I?. As the injected solute flows
The Journal of Physical Chemistry, Vol. 97, No. 7, 1993 1465
[ " " ' " " ' " " I
5500
6500
6000 t / s
Figure 1. Binary dispersion peaks (refractometer signal plotted against time) for 0.200 mol dm-' aqueous BaC12 carrier stream: upper peak, AC = 0.020 mol dm-'; lower peak, AC = 4.020mol dm-'. The baselines have been offset for clarity.
along the tube it slowly spreads out, forming an approximately Gaussian distributioni4of variance & ~ / 2 4 D ,where r is the inside radius of the dispersion tube, t~ the sample retention time, and D the binary differential diffusion coefficient at the carrier stream composition Figure 1 shows dispersion profiles measured against a 0.200 mol dm-3 BaC12 carrier stream. The upper peak was obtained by injecting a sample of 0.220 mol dm-3 BaC12: AC = 0.02 mol dm-3 (denser than the carrier stream). The lower peak was obtained by injecting 0.180 mol dm-3 BaC12: AC = -0.02 mol dm-3 (less dense than the carrier stream). The peak shapes shown in Figure 1 appear to be identical whether the injected solution is denser or less dense than the carrier stream. This was confirmed by fitting peaks such as those shown in Figure 1 to the working equation29
e.
V ( t ) = Bo + B,t
+ (tR/t)i/2Vm,,exp[-12D(t
- t,)2/r2t] (2)
for the refractometer output voltage, treating t ~D,, the peak height V,,,,,, and the baseline parametes BOand BI as adjustable least-squaresparameters. Repeated injections of 0.220 mol dm-3 BaCl2 gave 1.150, 1.157, and 1.149 for 105D/cm2s-l, while the 0.180 mol dm-3 solution gave 1.156, 1.152, and 1.151. These results are in excellent agreement with the value 1.151 X 10-5 cm2 s-l for the binary diffusion coefficient of 0.200 mol dm-' aqueous BaC12 suggested by Rayleigh interfer~metry.~~ The binary diffusion coefficient of 0.200 mol dm-3 aqueous KCl was measured by injecting 0.240 or 0.160 mol dm-3 KCl samples into a 0.200 mol dm-3 KCl carrier stream. Repeated determinations gave 1.828, 1.832, 1.828 and 1.833, 1.834, 1.830 for 105D/cm2s-l for the respective injection solutions. The value for the binary diffusion coefficient of 0.200 mol dm-3 aqueous KCl from Rayleigh interferometry30 is 1.838 X cm2 s-I. The binary results suggest that the Taylor profiles were unaffected by gravitational instabilities. Taylor Results for BaC12 (1)-KCI (2)-Water. Diffusion in aqueous BaC12( 1)-KC1(2) solutions is described by the coupled Fick equationsI4
Ji = -Dl,VC, - Dl2VC2
(3)
J2 = -D2lVCl- D22VC2 (4) Ternary diffusion coefficient D,k gives the flux of salt i caused by the gradient in salt k. When a sample of solution of composition + ACi, + AC, is injected into a carrier stream of composition el, c2, the refractometer signal generated by the dispersed sample is the
ci
Leaist and Hao
1466 The Journal of Physical Chemistry, Vol. 97,No. 7, 1993 t
TABLE I: Ternary Diffusion Coefficients of B8Cl2 (1)-KCl (2)-Water at 25 O C from Taylor Dispersion Profiles Cl/mol dm-3 0.230 0.459 0.918 limiting C2/mol dm-3 0.114 0.229 0.458 valueso D11/10-~cm2s-I 1.07 1.08 1.09 1.272 D12/10-5 cm2 s-1 -0.01 0.01 0.02 0.014 10-5cm2 s-1 0.21 0.23 0.26 0.245 D22/ cm2s-I 1.I9 1.72 1.63 1.624 0.32 0.31 &/RI 0.33 0.33b Equations 9-12. * References 33 and 34.
4
0 h
v 1
6000
5500
6500
I
t / s
Figure 2. Ternary dispersion peaks for 0.918 mol dm-3 BaC12 (1) + 0.458 mol dm-3 KCI (2) carrier stream: peak I, ACI = 0.020, AC2 = 0.OOO;peak 11, ACI = O.OOO, AC2 = 0.040; peak 111, ACI = 0.000, AC2 = -0.040; peak IV, ACI = -0.020, AC2 = 0.000.
7
1
1.51
sum of two overlapping Gaussians15
V ( t )=
0
0
Dl 1
O
0
0
%I
O
0
0
Ql2
-0.5 " 0.0
DI and D2 are the eigenvalues of the 2 by 2 matrix D of Dik coefficients. The normalized preexponential weighting factors are given byI8 (a
+ ba,)D;I2
w,= ( a + ba,)Df'2+ (1 - u - bal)D:'2 w2= 1 - w,
(6)
(7) where a and b areconstants for a given carrier streamcomposition and a1 stands for the fraction of the initial refractive index contributed by BaCl2 (salt 1)
'
1
'
"
0.5 11"
/
'
1
1 .o
'
'
'
"
1.5
"
$
1 i
"
2.0
(mol dm")'''
Figure 3. Ternary diffusion coefficients of BaC12 (l)-KCI (2)-water at 25 OC and salt ratio Cl:C2 = 0.918:0.458 plotted against the square root of the ionic strength: (3c1 + C2)Il2. (0)Taylor results; ( 0 )limiting values (eqs 9-13). R I was evaluated by taking the ratio of peak areas per mole of excess BaC12 or KCl in the injected solution samples. Since the change in the refractive index per mole of BaC12 is about three times larger than that for KCl (see Table I), the diffusion of BaC12, and hence DII and 0 1 2 , could be determined more precisely than 91 and 0 2 2 for KCl. To learn more about diffusion in aqueous BaC12-KCI solutions, dispersion measurements were also made at concentrations onehalf and one-quarter as large as those used in the Gouy study, keeping the molar BaC12:KCl ratio constant. The results are summarized in Table I. Table I includes accurate limiting values for the Dik coefficients calculated from the Nernst equati0ns3~
= RlACl/[RlACI + R2AC21 (8) RI = dn/dC, and R2 = dn/dC2 give the change in refractive index per unit change in the concentration of BaC12 (1) and KCI (2). Figure 2 shows ternary dispersion profiles measured against a carrier stream containing 0.91 8 mol dm-j BaC12 + 0.458 mol dm-3 KCI (the composition used in the Gouy study). The two Doll = DoBa tOBa(DoCI- DoBa) (9) uppermost profiles were obtained by injecting denser solution samples with 0.02 mol dm-3 excess BaC12 (ACI = 0.02, AC2 = 0 ' 1 2 = foBa(DoCI- D 0 , ) / 2 (10) 0.00 mol dm-j) or 0.04 mol dm-3 excess KCI (ACI = 0.00, AC2 = 2t0~(D0c1 DoBa) DO21 (1 1) = 0.04 mol dm-3) relative to the carrier stream. The two lower profiles were obtained by injecting the corresponding less-dense D O 2 2 = D o , + t°K(DoCI - DOK) (12) solution samples (ACI = -0.02, AC2 = 0.00 mol dm-3 or AC1 = together with the limiting ionic diffusion coefficientsIg DOB~= 0.00, AC2 = -0.04 mol dm-3). Once again the profiles for the 0.847 X DOK= 1.957 X and Doc) = 2.033 X cm2 denser or less-dense injection solutions appear to be identical. S-I calculated from limiting ionic conductivities. t, is the limiting Ternary diffusion coefficients can be evaluated from the transport number of ion j: moments of dispersion profiles.31 Significantly better accuracy and precision can, however, be achieved by direct least-squares t o j = c ~ ; D ~ ~ / ( +c cKz;DoK ~ ~ z + ~ C~C ~ DZ ~ & D~~ C ~(13) ~) fitting of eqs 5-7 to measured dispersion peaks.I6-l8 The ternary diffusion coefficients reported here were calculated from the leastIn Figure 3 the diffusion coefficients of BaCl2-KCI-water squares values of a, b, DI, and D2 according to the expressions calculated from the dispersion profiles are plotted against the developed in ref 18. Within the experimental precision the square root of the ionic strength. The measured coefficients different injection solutions gave identical results: lO5DlI = 1.09 appear to be qualitatively consistent with the limiting values. A (*0.01), lO5DI2= 0.02 (f0.01). 105D21= 0.26 (f0.02), and precise extrapolation cannot be made because the concentration 105D22= 1.62 (f0.02) cm2s-l. Notice that the ratio D ~ I / D = ~ ~ dependence of the Dik coefficients is not known. Nevertheless, 0.24 suggests that each mole of diffusing BaCl2 cotransports the dispersion results and the limiting coefficients suggest that 0.24 mol of KCI. diffusing BaC12 cotransports KCl over the entire composition RZIRI, the ratio of refractive index increments per mole of range that was studied. BaC12( 1) and KC1(2), is required to calculate the ternary diffusion Gouy Results for BaC12 (0.918 mol dm-3)-KCI (0.458 mol coefficients from the refractive-index profiles. In this study R2/ dm-3)-Water. The previously reported Gouy resultsI0 10sDlI =
+
Diffusion Coefficients for BaC12-KCI-H20
The Journal of Physical Chemistry, Vol. 97, No. 7, 1993 1467
TABLE 11: Initial and Finn1 Concentration Differences from Diapluap-Cell Runs’ at the Mean Cell Composition 0.918 mol dm- BaCI2 (1) 0.458 mol dm-’ KCI (2)
+
run B
run A Ac(O)/mol dm-3 AC?(O)/mol dm-3 A&)/mol dm-3 Ac(r)/mol dm-3
0.4001 0.4001 0.2052 0.1 171
AC?(O)/mol dm-3
A&O)/mol dm-3 A&t)/mol dm-3 A&t)/moI dm-3
0.4002 -0.4000 0.2088 -0.1795
“ r = 165 600 s.
0.982 (*0.009), 105D12= 0.054 (*0.002), 105D21= -1.539 (*0.040), and IO5& = 1.62 (f0.014)cm2 s-I for 0.918 mol dm-3 BaC12 + 0.458 mol dm-3 KCl are in very poor agreement with the present Taylor results. The Gouy value for D2] has the opposite sign and is six times larger than the Taylor value. The refractive-index ratio R2/RI = 2.36 used in the analysis of the Gouy interference patterns suggests that the refractive index change per mole of added KCl is more than twice as large as that for BaC12. This clashes with the present results and with measurements on BaClrwater” and K C l - ~ a t e r which , ~ ~ suggest R ~ / R I 0.3. Diaphragm-CellResultsforBaC12 (0.918m0Idm-~)-KCI(0.458 mol dm-3)-Water. One way to resolve the contradictory Gouy and Taylor results is to measure the diffusion coefficients by an independent third technique. Diaphragm-cell experiments were therefore run at the mean cell composition 0.918 mol dm-3 BaCl2 0.458 mol dm-3 KCl. The difference in the concentration of each salt across the diaphragm, ACi(t) = Ci(f)iOwer - Ci(t)upper, was followed by chemical titration. In run A both BaC12 and KCl were initially 0.400 mol dm-3 more concentrated in the lower solution compartment: (0) = +0.400;A e ( 0 ) = +0.400mol dm-3. In this run both salts diffused upward. After diffusion for a timed interval of 165 600 s (46 h), the concentration difference for KCI had decayed to 29% of its initial value. In run B, BaC12 was initially more concentrated in the lower compartment, while KCI was more concentrated in the upper compartment: AG(0) = +0.400; AC(0) = -0,400mol dm-3. In this case the two salts diffused countercurrently, and the concentration difference for KCI decayed to 45% of its initial value after 46 h. KCI clearly diffuses more rapidly when BaC12 is simultaneouslydiffusing in the same direction. In other words, BaC12 cotransports KCl. In both runs the lower solution was always denser than the solution in the upper cell compartment. The initial and final concentration differences for the diaphragm-cell experiments are summarized in Table 11. Ternary diffusion coefficients were calculated from this information by Kosanovich and Cullinan’s elegant matrix procedure:22
-
TABLE III: Ternary Diffusion Coefficients of BaC12 (C, = 0.918 mol dm-+KCI (C2 = 0.458 mol du~-~)-Water at 25 OC Taylor diaphragm cell“ GouylO D11/10-5cm2s-I 1.09 f 0.01 1.09 0.982 f 0.009 D12/10-~cm2s-I 0.02 f 0.01 0.02 0.054 f 0.002 D21/10-5cm2SKI 0.26 f 0.02 0.29 -1.539 f 0.040 D22/ cm2s-I 1.63 f 0.02 1.64 1.882 f 0.014 Integral values. between diaphragm-cell data and Taylor or Gouy data. Nevertheless, the diaphragm-cell diffusion coefficients are very close to the Taylor values, within 0.03 X 10-5 cm2 s-1. Discussion Onsager Reciprocal Relation. The Taylor and diaphragm-cell results support each other and suggest that the Gouy diffusion data for BaCl2-KC1-water are incorrect. Skeptics might need more evidence3ssince the Taylor and diaphragm-cell techniques obviously lack the sophistication of Gouy interferometry. To settle the issue an appeal will be made to a higher authority: Onsager’s law of reciprocal relations.21,22If a set of diffusion coefficients leads to Onsager transport coefficients that clearly disobey the reciprocal relations, those data are certainly incorrect. The Onsager coefficients for isothermal ternary diffusion in the solvent-fixed (0) frame are defined as follows:
+
AC
QIand
Q2
are the eigenvalues of matrix Q defined as follows
Jl(0)
= - ~ l l ( o ) v c L I- 4 2 ( 0 ) V k
(16)
JZ(0)
= -4l(o)vP, - L22(O)VP2
(17)
Vpi is the gradient in the chemical potential of solute i. There is one reciprocal relation: Ll2(0)= LZl(0). The first step in the calculation of the Lik(O)coefficients is to transform the measured differential diffusion coefficients from volume-fixed to solvent-fixed(0) coordinates.21 2 Dik(0)
( i = 1,2) (18)
= Dik + ( c i / c O V O ) ~ V @ k j k= 1
vk and ck are the partial molar volumes and the molar concentrations of the components. For BaC12 and KCl the partial molar volumes are calculated from2I - Hk
Vk = P - HICI - H2C2 Mk
(k = 1,2)
is the density of the solution, M I and M2 are the molecular weights of the salts, and Hk = (ap/aCk)T,p.,,,. The concentration COand the partial molar volume Voof water, the solvent, are evaluated from the identities
p
MOCO+ M,C,+ M2C2= p
(20)
COVO+ ClV1+ C,V2 = 1
(21)
The Onsager coefficientsand diffusion coefficientsare related as follows D(o)= L ( o pwhere pik = (api/aCk)T,p,i+k. Robinson and Bower36 have reported isopiestic data for BaC12-KC1-water at 25 OC. Their isopiestic data together with Pitzer’s fitting equations (including higher-order and 8 t e r m ~ ) can ~ ~ be J ~used to evaluate the molality-scalechemicalpotential derivatives(&/ amk)T,p,mk+k. The C(ik derivatives are then calculated from2I
+
The columns of matrix P are the corresponding eigenvectors. /3 is the cell constant and f is the duration of runs A and B, 165 600 s. In Table I11 the diaphragm-cell diffusion Coefficients are compared with the Taylor and Gouy values. Since Taylor and Gouy experiments employ small concentration differences, differential diffusion coefficientsare determined directly. In the diaphragm-cell experiments, larger initial concentration differences were used, so the measured diffusion coefficients are integral values representing averages of the differential coefficients over the range of composition between the upper and lower solutions. For this reason it would be unreasonableto expect exact agreement
(19)
2 pij
= x(api/amj)T,p,mi+jBjk j= I
Leaist and Hao
1468 The Journal of Physical Chemistry, Vol. 97, No. 7, 1993 TABLE I V Onsager Coefficients of BaC12 (CI = 0.918 mol dm-3)-KC1 (CT= 0.458 mol dm-%Water at 25 ‘C Taylor Gouy RTLll(o)/lO+’mol cm-I s-I 3.67h 0.04 3.02 f 0.03 RTL12(0)/10+’mol cm-’ s-’ -1.10 f 0.05 -0.67 f 0.02 -1.08 t 0.07 -7.36 f 0.13 RTL21(0,/10-9mol cm-’ s-I 10.02 f 0.08 6.64 f 0.09 RTL22(0)/1O+’ mol cm-l s-I ~~
\
TABLE V Supplementary Data Used To Calculate the Onsager Coefficients of BaCl2 (CI= 0.918 mol dm”)-KCI (Cz = 0.458 mol dm-”-Water at 25 ‘C
SOLUTIONS
TAYLOR
0.05
0.00
c
I
I
\
?--A
ua I 0“
v
-0.05
2.990 0.813 0.813 2.376 3.467 1.030 1.011 2.672 1.16 0.08 0.30 1.66
18.016 208.246 74.55 1.1440 150.1 35.2 18.20 58.73 39.74 1.Ooo
0.498 0.9186
1a-005/, ,
v
,
,
I
,
-“:, :00
,,
STABLE C UPPER SOLUTIONS - 0 05
,
0 00
(c, - c,) /
,
,
,
,
,
/
’
-0 10
-0 10
,
0 05
0 10
mol dm-’
Figure4. Stabilityconditionsfor BaC12 (211 = 0.918moldm-’)-KCI (c2 = 0.458 mol dm-’)-water at 25 OC from Gouy interferometry.I0 Line G: limiting conditions for convection at the borders of the diffusion boundary (overstability). Line F limiting conditions for convection at tliecenterof thediffusion boundary (finger formation). Line D: limiting conditions for density inversionsdue to diffusion. Line N: compositions of constant density. Boundaries in Gouy runs 5-7 were believed to be
unstable. Table IV gives the L i k ( 0 ) coefficients for BaClz-KC1-water calculated from both the Taylor and Gouy diffusion data. The uncertainties in the reported L i k ( 0 ) values were calculated from the propagated uncertainties in the diffusion coefficients. Errors in the chemical potential derivatives were ignored, so the reported precision of the &(O) values given in Table IV is optimistic. TheTaylorresultsagreewithL~z(~) = LZI(O) within thesuggested experimental errors. The values of L12(0)and L21(0)derived from the Gouy diffusion data deviate from the reciprocal relation by a wide margin, well outside the suggested experimental error. For this reason the Gouy diffusion data are believed to be incorrect. Table V gives the supplementary data used to calculate the Onsager coefficients from the diffusion data. Revised Stability Conditions for BaCl2 (l)-KCI (2)-Water. Finally, the stability of free-diffusion boundaries for BaC12-KClwater will be reconsidered in light of the revised diffusion coefficients. Figure 4 shows the previously reported10 stability fields for 0.918 mol dm-3 BaC12 + 0.458 mol dm-3 KCl. The boundaries were believed to be stable if the concentrations of the upper and lower solutions lay between lines F and G, the respective limits for instability at the center and edges of the diffusion
STABLE UPPER SOLUTIONS
-0.10
-0.05 (C,
0.00
- F,) /
0.05
0.10
mol dm-’
Figure 5. Revised stability conditions for BaCI2 (e1 = 0.918 mol dm-’)-KCI (e2= 0.458mol dm-3)-water at 25 OC from Taylor dispersion data. All Gouy runs are stable.
(r=
TABLE VI: Slo of Stability Lines for B a t 3 2 (Cl = 0.918 mol d~n-~)-KCl 0.458 mol dm-3)-Water Free-Diffusion Bouadories from Taylor and Couy Data slow Tavlor Gouv line D -5.29 -7.23 central static instability line F -8.50 -25.62 formation of fingers at center lineG -0.47 1.93 edge static instability IineN -4.26 -4.26 neutral density line boundary. LineN gives the compositionsof neutral density while line D gives the limits of instability for density inversions due to diffusion. Gouy run 7 was reported to be unstable, even though BaC12 and KCI were both more concentrated in the lower half of the boundaries. (The equations for calculating the slopes of lines F, G, N, and D are given in refs 8-10.) Figure 5 shows the corrected stability diagram drawn with the revised diffusion coefficients from the Taylor dispersion measurements. Line G now has a negative slope, which opens up a much larger stability field. In fact the diffusion boundaries are stable if BaClz and KCI are both more concentrated in the bottom of the diffusion column. The calculated slopes of the stability lines are summarized in Table VI. It is not clear then why convection developed during run 7 of the Gouy study. It would be interesting to make additional Gouy measurements to clarify this point.
Acknowledgment. We thank the National Sciences and Engineering Research Council for the financial support of this research. References and Notes (1) Turner, J. S. Annu. Rev. Fluid Mech. 1985, 17, 11, (2) Turner, J. S. Bouyoncy Effecrs in Fluids; Cambridge University
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