7478
J. Phys. Chem. 1992, 96, 7478-7483
(47) Bales, B. L.; Schwartz, R. N.; Kevan. L. Ber. Bunsen-Ges. Phys. Chem. 1974, 78, 194. (48) Hase, H.; Ngo, F. Q.H.; Kevan, L. J . Chrm. Phys. 1975,62,985. (49) Jain, M. K. In Inrroducrion to Biological Membranes; Jain, M., Wiley: New York, 1980; Chapter 4. Wagner, R. C., as.; (50) Bratt, P. J.; McManus, H. J. D.;Kevan, L. J. Phys. Chem., in press. (51) Casal, H. J. J . Phys. Chrm. 1989,93,4328. (52) Tanford, C. The Hydrophobic Effect: Formation of Micelles and
Biological Membranes, 2nd 4.; Wiley: New York, 1980; Chapter 1 1 . (53) BHdt, G.; Gally, H.; Seclig, A.; Seclig, J.; Zaccai, G. Nature ( h n don) 1978, 271, 182. (54) Kalyanasundaram, K. Photochemistry in Microheterogeneous Sysrems; Academic: New York, 1987; Chapter 1 . (55) McManus, H. J. D.;Kang, Y. S.; Kevan, L. J . Phys. Chem. 1992, 96, 2274. (56) McManus, H. J. D.;Kevan, L. J . Phys. Chrm. 1990, 94, 10172.
Measurement of the Mutual Diffusion Coefficients at One Composltlon of the Four-Component System cu-Cyciodextrln-L-Phenylalanine-Monobutylurea-H2O at 25 O C Luigi Paduano, Roberto Sartorio, Vincenzo Vitagliano,* John G. Aibright,+ and Donald G. Miller8 Dipartimento di Chimica dell’Universitd di Napoli. Federico II, Via Mezzocannone 4, 80134 Naples, Italy (Received: March 3, 1992)
The nine mutual diffusion coefficients D, (i, j = 1-3) of the quaternary system a-cyclodextrin (0.02 M)-L-phenylalanine (0.06 M)-monobutylurea (0.06 M)-H20 have been determined at 25 OC. Eight reliable experiments were performed at these mean concentrations by the freediffusion method with a Gouy optical interferometer. A minimum of three experiments is required to obtain the nine diffusion coefficients. However, several more were required to reduce the errors to an acceptable level. A computer program is described that analyzes data from four-component Gouy free-diffusion experiments. Values of those diffusion coefficients for the flow of L-phenylalanine and monobutylurea caused by a concentration gradient of a-cyclodextrin were particularly sensitive to the choice of experiments used in the data analysis. Because of the large uncertainty of these coefficients, any mechanistic interpretation of these data should be viewed with caution. However, the behavior of this system is strongly influenced by the formation of inclusion compounds of L-phenylalanine and monobutylurea with a-cyclodextrin. The diffusion coefficients may reflect this interaction.
Introduction Mutual diffusion coeffcients for four-componentsystems have not previously been measured by an interferometric method. Equations which could have been used for the analysis of data from such experiments have been available since 1966 for the Rayleigh’ or equivalent interferometric In 1969 Kim4 discussed methods for analyzing data for four-component systems obtained by the Gouy method,2v3but did not give a general procedure. However, appropriate general equations for the Gouy interferometricmethod were presented in 1988, along with general methods for extracting the diffusion coefficients directly from Gouy and Rayleigh fringe position data.5 It is important to note that these methods and equations depend on the assumption that the initial concentration differences in the experiments are small enough that the diffusion coefficients can be considered as constants. Multicomponent diffusion studies using the free-diffusion method or any other open channel method require an understanding of density stability in order to avoid experiments with unstable boundaries. Inclusion of data from such experiments will invalidate the calculated diffusion coefficients. Kim6derived equations for static density stability in four-component systems. Recently Vitagliano et al.’ have developed an extended multicomponent theory that gives criteria for both static and dynamic density stability of systems with four (or more) components. With the availability of that theory’ and the general data analysis procedure: it has become possible to extend optical diffusion experiments to systems with four components. The first four-component system to be studied interferometrically was HCI-NaCl-KCl-H20, for which Gouy and Rayleigh
data had been collected by Kim. However, the analysis was never completed, and the photographic plates were lost* before the general analysis method5 became available. As the first four-component system to be fully studied by the Gouy interferometric method, we chose the system cy-cyclodextrin-L-phenylalanine-monobutylurea-H20 at 25 OC. Mean concentrations for the solutes of this study are 0.019 98,0.059 90, and 0.059 85 mol dm-3, respectively. For the remainder of the paper we abbreviate these solutes as CD, PHE, and MBU, respectively. This study is an extension of a previous one on the system9CD-PHE-H20. These systems are interesting because of the formation of inclusion compounds with CD. This interaction between solute molecules has a large effect on the values of some cross-term diffusion coefficients.
Theoretical Considerations The theory for the analysis of Gouy interferometric data from free-diffusion experiments on quaternary systems is given by Miller,5and is based in part on results given by Kim? We give only a brief overview of the essential equations here. Flow equations for the diffusion of a four-component system may be written as -J1 = DIIVCI + D12VC2 + DI3VC3 (1)
+ D22VC2 + D23VC3 -J, = DSIVCI + D32VC2 + D33VC3 -J2
= DzlVCl
(2) (3)
‘Visiting Professor at Universita di Napoli, Spring 1991. Permanent a d d m : Depsrtment of Chemistry, Texas Christian University, Forth Worth,
In this study components 1,2, and 3 denote CD, PHE, and MBU, respectively. In our experiments the gradients VCi were in the vertical direction. If volume changes on mixing are small, the measured flows and diffusion coefficients will be for the volume-fixed frame of reference, which is defined by the equation
*Visiting Professor at Universita di Napoli, Spring 1990. Permanent address: Lawrence Livermore National Laboratory, University of California, Livermore, CA 9455 1.
EJ,K. = 0 i=0
TX ... 76129.
3
0022-3654/92/2096-7478S03.00/0Q 1992 American Chemical Society
(4)
The Journal oLF&sical Chemistry, Vol. 96,No. 18, 1992 7479
a-Cyclodextrin-L-Phenylalanine-Monobutylurea-HZO Here component 0 is the solvent. The quantities Ji and F. are the volume-fixed flow and partial molar volume of component i, respectively. Equations for fringe positions5 of a Gouy pattern from an experiment in a four-component system may be written as 3
yi* = Kzrisiexp(-s,2y?) i= 1
Here U,* is equal to U,t112,where is the distance of fringe minimum j from the undeviated slit image in that Gouy fringe pattern at the camera plane, and t is the time elapsed from a perfect step function starting boundary. The constant K is proportional to the total number of fringes, the wavelength of the light, and the distance 6 from the center of the diffusion cell to the camera plane.5 The constants ri depend on the initial concentration differences of the solutes across the starting boundary. The si are the square roots of the reciprocals of the eigenvalues of the diffusion coefficient matrix.I0 Finally, the parameter yj for each of the fringes is defined by the equation
The functionflx) in eq 6 is defined as 1
erf(x) - ( 2 ~ / r ’ / ~exp(-x2) )
(7)
andf(zj) for each of the fringes may be evaluated from f(zj) = Z ( j A / J
(8)
The values of Z(i,J) are approximately (j + 3/4),but the more precise values used in our data analysis were obtained from wave theory approximations for Gouy fringe patterns.Il-’* The sum of the coefficients ri is unity: rl r2+ r3= I (9)
+
Thus, only two rivalues are independent. Our concentration differences are relatively small; therefore, the volume change on mixing is small. Consequently, to a good approximation, the refractive index at any point in the solution will be linearly related to the mean concentrations by the equation 3
n = Ti
+ i=C R i ( C i- Ci) 1
(10)
Here A is the refractive index when all Ci = Ci,Le., at the mean concentrations of an experiment. The Ri are refractive index increments for each of the solutes at the given set of mean concentrations. The difference of refractive index across an initial boundary, An, is equal to A / a , where J is the total number of fringes, X is the wavelength of the light source, and a is the inside width of the diffusion cell along the optic axis. Consequently, J is related to the values of ACi, the initial concentration of solute i in the bottom of the cell minus the initial concentration of solute i in the top of the diffusion cell, according to the equation
+
+
A / a = R I A C I R2AC2 R3AC3
(11)
Experimentally, the values of Rj are obtained by the method of least squares when eq 11 is applied to values of J and ACi from three or more experiments. (In reality only ratios of Ri values are used in the analysis of data, so that A/a is not required.) Once the Ri are determined, eq 11 may be used to calculate smoothed values of J for comparison with the experimental values. To determine the nine diffusion coefficients, three or more experimentsare performed at the same set of mean concentrations but with different initial AC,. For each experiment, a set of refractive index fractions ai are defined by the equation ai =
3
RiACi/j = l RjACj
(12)
Each of the two independent Pi (choose i = 1,2) may be related to the ai by the equationss
(1 3) (14)
The a,, b,, and s, have fixed values for a series of experiments with the same mean concentrations.
Annlysis of Data A computer program, named QTNY, was written to analyze data from four-component Gouy diffusion experiments. The analysis has the following steps. (1) Values of R, are first calculated by applying the method of least squares to fit the parameters of eq 11 to the values of J and ACi. Values of a, are then obtained from eq 12. (2) Fringe position data from each experiment are fmt analyzed as if the experiment were an ideal binary experiment without concentration dependence. This gives the At correction, yi*, and D A described below. For a given fringe pattern, the value of C,, is calculated for each of the outer eight fringes (j = 0-7) by using the equation5*” G j
7
fix)
rl = ala1+ a2a2+ a3a3 r2= blal + b2a2 + b3a3
= Y,/exp(-z,2)
(15)
These values are averaged for this pattern, and the corresponding averages for all the patterns of the experiment are linearly extrapolated versus llt’to get the At correcti~n.’~ By adding this At to the experimental elapsed time t’, the true time t , relative to a hypothetical ideal step function starting boundary, is obtained. Fringe positions 5 from each fringe pattern obtained during the experiment are then multiplied by t1l2for that pattern to get Y,*. Values of ?* for each fringe j are then averaged over all fringe patterns. This yields a set of average U,* values, one from each j , which gives one average, representative fringe pattern for that experiment.s Also,the experimental value of C, (the maximum displacement of light according to ray theory2) is obtained for each fringe pattern of each experiment by linearly extrapolating values of C,j of the outer eight fringes ~ e r s u s f ( z ) ~ toflz) / ~ = 0. From these C, values, apparent values of D A , defined e l s e ~ h e r e ? *are ~ J ~calculated for each fringe pattern. These are linearly extrapolated versus I/t’ to l/t’= 0 to obtain the correct value of D A for that experiment. The value of At obtained from this calculation is not used further, but its comparison with the previous one gives diagnostic information. (3) The values of D A calculated from each experiment in the previous step are related to l’, and s, by the equation4
= rlsl+ r2s2+ r3s3
(16) A preliminary estimate of the three values of s,for the quaternary system is obtained by assuming the cross-term diffusion coefficients Dij (i # j ) are zero. This assumption implies that ri = ai (i = 1-3). These starting values of s, (i = 1-3) can now be obtained by the method of least squares by fitting the parameters of eq 16 to the set of DA and ai data from all experiments. (4) By holding these approximate values of s, fixed, the parameters ai and 6, which best fit eqs 5,6, 13, and 14 are determined by the Taylor series expansion method of nonlinear least squares. The equations (appropriate derivatives) used in the least-squares procedure are those given by Miller.s ( 5 ) The new values of a, and 6,are held constant, and a new set of s, are determined by the Taylor series method of nonlinear least squares, again using Miller’s equation^.^ Steps 4 and 5 are cycled four times. Each value of s, at any stage of the iteration should be the same for all experiments in the set and independent of their initial AC,. ( 6 ) Now, all nine parameters, a,, 6,,and s, (i = 1-3), are directly obtained from the data by the method of least squares. In this last procedure, Marquardt’s methodIs is introduced, which mixes the Taylor series expansion method with the steepest descent method. The value of X, defined by M a r q ~ a r d t , is ’ ~set to 1.3 for the first 200 loops in the analysis. (This X symbol should be distinguished from the h used for the wavelength of light elsewhere in this paper). Then X is set to 0.0001, and the program is run for
7480 The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 TABLE I: Diffusion Coefficients Calculated from Simulated htaavb input three sets of data, value of J diffusn coef correct 0.3200 0.3200 -0.0010 -0.0010 -0).0010 -0).0010 -0.1290 -0.1300 0.6490 0.6500 0.0599 0.0600 -0.07 19 -0.0700 0.0029 0.0010 0.6303 0.6300
six sets
of data, J correct 0.3200 -0.0010 -0.0010 -0.1293 0.6489 0.0599 -0.07 13 0.0032 0.6302
three sets of data,
six sets of data,
J
J
modified 0.3215 -0.001 1 -0.0015 -0.2079 0.6516 0.0789 0.0982 -0.0021 0.5913
modified 0.3188 -0).0005 -0.001 3 -0,0927 0.6292 0.0714 -0.1367 0.0437 0.6066
’All coefficients have units of meters squared per second. bSee text for explanation of the table.
another 2000 loops. Tests show that 2000 loops give good convergence. Our tests also show that there is no advantage to setting X to a lower value, and in some cases a lower X will cause divergence. (7) Once the three values of ai, the three values of bi, and the three values of si are obtained by the method of least squares, equations given by Miller5are used to obtain the nine diffusion coefficients Di,. To test the computation procedure, this program was applied to artificially generated test data that were similar to the results being obtained for this experimentalsystem. The test data were generated for a system with the diffusion coefficients given in Table I. Si artificial sets of data were generated with (aI,a2,a3) values of (l,O,O), (O,l,O), (O,O,l), (0.5,0.5,0), (0.5,0,0.5),and (0,0.5,0.5), and the refractive index coefficients R I ,R2, and R3 were those taken from Table IV. In the first test calculations, values of J were assumed to be 100.0 in all cases. The diffusion coefficients were recalculated with two combinations of these data sets: (1) with the first three artificial sets of data and (2) with all six sets of artificial data. The recalculation for each combination was performed using the outer 98 fringes from each of its data sets. The results are given in columns 2 and 3 of Table I. Overall the fit to the artificial data is good, and is about equally as good with both the three and six sets of combinations. To determine how sensitive the derived D , values are to errors in determining J values, a second set of test calculations involved altering J in four of the six artificial data sets without changing fringe positions. The values of J for the respective data sets became 100.05, 100.00,99.95, 100.00.99.95, and 100.05 fringes. Given in column 4 of Table I are the recalculated diffusion coefficients that were obtained when the first three sets of data were used. In column 5 are the results when all six data sets were used. These results show that the values of Dij are relatively sensitive to errors in determining values of J. It is clear that averaging with six data sets gave a large improvement over results obtained by using only three data sets. In the first case the error in D3]is about 26% of the value of the larger main term coefficient (0.62). In the second case this error is reduced to about 10%. Experimental Section Apparatus. The experiments were performed with the Gouy diffusiometer located at the University of Naples. This instrument has been automated to scan Gouy fringe patterns and record fringe positions during an experiment. A Model “I1 fx” Maclntosh computer is used to control the scanning apparatus and to calculate fringe positions from fringe intensity profiles. The pattern is scanned down to the outer (bottom) end of the Gouy pattern with a photodiode, starting from slightly above the reference fringes (which are slightly above the undeviated slit back up to just image). The pattern is then immediately above the reference fringes.I6 Fringe positions obtained by scanning down are averaged with positions obtained by scanning back up. These averaged positions are recorded along with the mean time of the two scans. The average of the two scans cor-
Paduano et al.
TABLE U h i t i e s of Solutions at 25 O C CI * CZ, expt, s o h
1, top 1, bottom 2, top 2, bottom 3, top 3, bottom 4, bottom 5, top 5, bottom 6, top 6, bottom 7, top 8, top 8, bottom
mol dm-) 0.01741 0.022 57 0.01831 0.021 67 0.017 39 0.022 57 0.019 77 0.019 66 0.020 34 0.017 72 0.022 25 0.014 59 0.01978 0.02023
mol dm-’ 0.048 78 0.071 11 0.052 79 0.067 08 0.060 32 0.059 70 0.070 60 0.058 29 0.06 1 69 0.049 93 0.069 94 0.059 97 0.079 43 0.040 12
c3,
mol d d
d. R
0.06036 0.059 65 0.042 35 0.077 34 0.035 19 0.084 54 0.084 00 0.020 22 0.098 97 0.055 37 0.064 58 0.059 98 0.059 33 0.060 69
1.00593 1.008 80 1.00633 1.008 39 1.006 28 1.008 46 1.007 90 1.006 96 1.00783 1.00606 1.008 64 1.005 38 1.008 06 1.00662
responds to an exposure on a photographic plate. This scanning procedure is repeated 15-20 times during an experiment to obtain a set of fringe patterns for the experiment. The set of reference fringes is generated by passing light from a region of homogeneous solution through a double slit and then bending the double slit pattern with a prism to the appropriate position above the undeviated slit image. This prism is located just outside the bath window on the camera side. The relative positions of the undeviated slit image and these reference fringes are obtained (1) by scanning the “double slit” fringe pattern from a double slit placed at the center of the cell, before the initial boundary is formed and while there is homogeneous solution throughout the center of the cell, and then (2) by scanning the reference fringes. The center of the pattern from step 1 corresponds to the relative position of the undeviated slit image.ll Once the pattern has been scanned, the fringe position data are transferred to a VAX computer for later analysis and for transfer of data between our laboratories by electronic mail. The Gouy diffwiometer is a tw&.nsapparatus in which parallel light is focused through the diffusion cell. The light source of the diffusiometer is a Unifas PHA SE 0.8-mW neon-helium laser operating at X = 632.8 nm. It is focused through a spatial filter which defines the actual location of the light source. In this parallel light arrangement, the b distance of the apparatus is the optical distance from the principal plane of the lens on the detector side of the apparatus to the plane in which the detector scans the patterns. It has the value 193.39 cm. S o l ~ t i All ~ ~solutions . were prepared by mass. Sigma analytical reagent grade CD, Sigma reagent grade PHE, and Fluka reagent grade MBU were used without further purification. The water content of air-equilibrated CD was determined by drying at 120 OC for several days under vacuum. The initial sample of CD was found to be 9.50% water, in good agreement with the value calculated from the formula CD-6H20. The molar masses used for CD, PHE, MBU, and water were 972.9, 165.2, 116.16, and 18.016 g mol-’, respectively. Demities. The densities of all solutions were measured to obtain values of molar concentrationsand of density increments Hi for all the solutes. Densities were measured with a Mettler Paar DMA 602 density meter with an accuracy of &O.O0001 g cm-3. Water and air were used to calibrate the meter. The assumed density of water was 0.997 045 g ~ m - ~The . density values for the top and bottom solutions are given in Table 11. Values were not obtained for the top solution of experiment 4 and bottom solution of experiment 7. An equation of form” d = d + HI(CI - el) + Hz(C2 - C2) + H3(C3 - C3) (17) was fitted by the method of least squares to the densities. The and H3 were, respectively, 1.0074 g cmm3, values of d, H1,H2, 0.3689 kg molb1,0.043 36 kg mol-I, and 0.005 95 kg mol-’. Number of Fringes J . The total number of fringes is determined as follows. The fractional part of a fringe was determined for each fringe pattern by the Gosting method,]*except for experiment 6. This
The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 7481
a-Cyclodextrin-L-Phenylalanine-Monobut ylurea-H20 TABLE [u: Experlwatd Purwters'
values for each experiment 1
5.16 22.33 -0.7 1 0.4726 0.5355 -0,0081 60.13 60.16 0.4248 0.4249 82.69 78.98
2 3.36 14.29 34.99 0.2937 0.3270 0.3793 63.12 63.04 0.5191 0.5227 102.21 98.30
3 5.19 -0.62 49.35 0.4655 -0.0146 0.549 1 61.39 61.43 0.4592 0.4646 115.93 122.75
4 -0.27 22.14 48.95 -0.0233 0.4998 0.5235 63.63 63.90 0.7307 0.7221 -13.91 -13.29
5 0.68 3.40 78.77 0.0599 0.0785 0.8616 62.70 62.49 0.7161 0.7078 16.65 26.71
6 4.53 20.01 9.21 0.4152 0.4801 0.1047 59.70 60.12 0.4517 0.4522 85.76 89.93
7 10.83 0.04 0.01 0.9989 0.0010 o.Ooo1 59.91 59.73 0.2954 0.2925 -36.21 -40.03
8 -0.45 39.33 -1.33 -0.0465
1.0635 -0.0170 53.68 53.35 0.669 1 0.6676 -23.76 -16.73
'Units: ACi, mol dm-3; DA, m2 s-l. MBU
TABLE Iv: Dah Fitting Parameters i= 1 Ri, dm3 mol-' 0.13946 si, cm-I s1/2 563.30 ai 1.12040 bi 0.16451
A
~~
/ 7 CD
expt 1 expt 2 expt 3 expt 4 expt 5 expt 6 expt 7 expt 8
\ L
L1
.
0 PHE
Figure 1. ai diagram. al= 1 in the CD corner; a2 = 1 in the PHE comer; and a3 = 1 in the MBU comer. Points outside the triangle indicate some negative ab
method consists of measuring the shift of fringe positions in a double slit fringe pattern when the bottom solution replaces the top solution in front of the bottom slit. In both cases there is top solution in front of the top slit. This shift divided by the interfringe distance of the double slit pattern is the fractional part. The integral number of fringes is obtained by the PQ extrap 01ation.~~ Addition of the fractional part of a fringe to the integer number gives the total number of fringes. Due to an experimental problem, the value of J for experiment 6 could not be obtained in this way. cohsequently, it was obtained by applying a special J code program for fouramponent systems, and is similar to those described by Albright and Miller13 for three-component systems. This code, designated by QTNYJ6, was written to obtain J, sl,s2, s3,rl,and r2simultaneously from the fringe position data of one experiment. Equation 11 was fit to the experimental J and the ACi values for the set of experiments to determine the values of Ri.The a distance is 2.50 cm. These calculated values of Ri are listed in Table IV and have an uncertainty of about f0.3%. Calculated values of J were obtained from these values of Ri by using eq 11. In Table I11 are listed the values of AC,, experimentalJ, calculated J, and their differences. The standard deviation of the fit is f0.3 fringes. Part of the 0.3 variation in J arises from uncertainties in solution concentrations. We believe that the tabulated value of J for each experiment is correct within f0.05 of the actual number of fringes for that experiment. a,Values. Values of ai calculated from eq 12 for each component of each experiment are included in Table 111. These are also shown in Figure 1, where they are plotted on a triangular graph (note a1+ a2+ a3 = 1). By analogy to analysis of data from ternary diffusion experiments, the most critical experiments to include in an analysis of data from quaternary experiments are those with sets of a,values near the vertices of the triangular graph. Data from experiments with sets of ai placed well away from vertices of the triangular graph should also be included in the analysis of quaternary systems. This was indicated by the results
i s 2 0.036513 398.60 0.011 59 0.12923
i=3 0.017299 392.46 0.021 18 -5.950 88
r,'
r2'
r3a
0.535 53 0.340 84 0.533 00 -0.009 21 0.086 33 0.472 93 1.119 17 -0.040 1 1
0.19495 -2.166 87 -3.192 70 -3.054 59 -5.106 98 -0.492 73 0.163 77 0.231 19
0.269 52 2.826 03 3.659 70 4.063 80 6.020 65 1.01980 -0.282 94 0.808 92
"Defined in eq 9 and computed from eqs 13 and 14. of analysis of the artificial data sets given above. DA Values. Extrapolated values of DA obtained from each experiment (as described above) and values calculated from eq 16 are included in Table 111. It is seen that the average of [DA(calCd) - D~(eXtrapd)]is -0.0014 x 10-9 m2s-'. The standard m2 s-'. deviation of the differences is 0.0051 X QoValues. The quantity, Qo, denoted by Q in refs 1 and 20, is a valuable diagnostic tool. Although not used in our calculations to obtain the De it is a measure of the deviations of fringe positions from what they would be for a concentration-independent binary system with diffusion mficient DA'. The value of Qo is obtained by integration of il versusflz) fromflz) = 0 tofiz) = 1, where il is defined in ref 20. A modified Simpson's rule for integration with unequally spaced values offlz,) was used to obtain Qo directly from each experiment. Kim' has shown that Qo/DA1/2is related to the values of ri and s, (denoted by in the original paper) by the following equation, valid if the Dij are constant:
where g(S,Sj)
=
(Si
+ Sj)/(2(2'/2))
- SiSj/(S?
+ s/2)1/2
(19)
Values of Qo obtained by integration of the original data for each experiment and values calculated from eqs 16, 18, and 19 are included in Table 111. Their differences are also listed. The standard deviation of the differences of Qo(calcd) - Q,(directly meas) is 5.63 X 10-4. The average of the calculated values minus the m e a s d values is 2.16 X 10-4. The relatively large differmces may indicate concentration dependence. Values of a,, bt, s,,.ad ri. Values of a, and bi obtained in the calculation by least-squares calculations are included in Table IV. The values of r, obtained for each experiment are also included in Table IV. Since these are intermediate numbers used in the calculations, more digits are tabulated than are significant. Values of D,,. The nine diffusion coefficients for this system are given in Table V.
7482 The Journal of Physical Chemistry, Vol. 96, No. 18. 1992
0 1 1 021 4
1
0.316 (kO.OO1) 01, -0.001 (kO.001) Dl3 = -0.001 (kO.OOOl) 4 . 1 3 2 (M.030) 0~ 0.646 (f0.013) 0 2 3 = 0.064 (k0.003) -0.076 (M.061) 0 3 2 = 0.001 (&0.021) D33 0.632 (a0.003)
‘All diffusion coefficients have the units meters squared pcr sccond, and the table entries are lo9&,. c, = 0.01998 mol dm” for u-cyclodextrin (solute component 1); c2= 0.05990 mol dm-’ for L-phenylalanine(solute component 2); = 0.05985 mol dm” for butylurea (solute component 3).
c3
As a method of estimating the errors in each coefficient Dlj, comparison values of Dlj were calculated by using the following combinations of data sets in the fit: (1) all but experiment 1 included, (2)all but experiment 2 included, (3) all but 6 included, (4)all but 2 and 6 included, and (5) all but 1 and 2 included. In all cases the critical experiments near the vertices of the a triangle (Figure 1) were included, along with at least one experiment centrally located along each edge of the alpha triangle. The standard deviation for each Dljcalculated from these comparison values is included in Table V. It is clear that values of Dzl and 0 3 1 are very sensitive to the choice of experiments used for the calculations. Consequently, the uncertainties of these coefficients are large. For example, if only the comer experiments 5, 7,and 8 are included in the fit, a D31 value of 1.4X l p m2 s-l is found, in contrast to the smaller, negative value found when more experiments are included. Thus, it was critical that as many as eight experiments were performed in this case. The errors given in Table V may be very optimistic. At this point it is important to consider that errors in D21and D3I could be as high as 20-30% of the value of the larger main term diffusion coefficients Dz2 and D33. The sensitivity of D21and D3l to experimental error is expected because the value of R I is 3.82 times greater than R2 and 8.06 times greater than R3. The consequence of this may be seen by examining eqs 15 and 16 given in ref 5. For the same reason, values of D I 2and D I 3tend to be rather more precise. The study of systems where the values of R, are closer would give a set of values of D, with more uniform precision. This system by its very nature has diffusion coefficients that are very dependent on concentration. Consequently, this may be a significant source of error in the extraction of D,, since the extraction is based on the assumption of constant Dl,. It has been shown,21for example, that concentration dependence in binary systems will lead to nonzero values of Qo;without concentration dependence, Qois zero. Modification of Qoby concentration dependence is equivalent to modification of each 5’. Consequently, some error will be introduced in the direct analysis of U,* data from multicomponent systems as well. An example can be found with the earlier type of analysis of three-component system^'^^^^^^^ in which Qowas directly used in the calculation of the four diffusion coefficients. Any alteration of U,* due to concentration dependence will alter Qo,and thus affect calculated values of D,j which were calculated using Qo.
Discussion This study shows the feasibility of measuring diffusion coefficients in quaternary liquid systems by the Gouy method. Most of the values of D, seem reasonable. The value for the main term diffusion coefficient D l l of CD, 0.316 X m2 s-l, is consistent with the values of (0.31-0.32) X found in the ternary study of the system CDm2 PHE-HzO. There the concentration of CD was 0.2 mol dmF3, and the concentration of PHE ranged9 from 0.025 to 0.10 mol dm”. The main term diffusion coefficient DZ2of PHE, 0.646 X m2 s-I, is consistent with the values of (0.62-0.63)X m2 s-l found in the same ternary study.9 It is somewhat lower m2 s-l found by Paduano et al.23in than the value 0.685 X the binary system PHE-HzO at -0.06 mol dm-3. It is also m2 s-l found by somewhat lower than the value of 0.70 X Longsworthz4for the binary diffusion coefficient of PHE at 0.021 mol dm-3 in water. This difference can be explained by the resistance to diffusion caused by the association with the CD. There are no direct data for MBU to compare with the value of D33given in Table V. Here again association with CD will have
Paduano et al. lowered its value from what it would be in a binary system with water. The small values of DI2and Dl3 are expected since there would be little coupled flow of the CD molecules due to gradients of the much smaller PHE and MBU molecules. In terms of the species model, which as noted below will be the subject of another paper, the fact that D12and D I 3are small indicates that the diffusion rate of complex4 and uncomplexed CD is about the same. The somewhat larger values of D z l ,023, and D3l reflect the formation of the inclusion compounds of both PHE and MBU with CD. The molar association constants with CD are about 10 for PHE9sZ5and 125 for MBU.z6 The smaller value for D3* may reflect the fact that MBU has a much greater association constant for the association with CD. Consequently, a gradient of PHE will not significantly displace the MBU and thus not cause a flow of MBU. There are some problems with this interpretation, however. For example, one would expect D3I to be considerably larger than Dzl because of the much greater association between MBU and CD than between PHE and CD; this is not the case. Of some concern is the possible effect of concentration dependence on the Gouy fringe position data. The calculated values of DZ1,0 2 3 , D 3 ] ,and 0 3 2 may be very sensitive to this effect. Consequently, there is a need to develop equations that correct for it. Theoretical analysis of concentration dependence in four-component systems will be very complex, but we hope to carry out such an analysis with the help of symbolic algebra programs such as MAPLE or MACSYMA. Another approach would be to do a computer simulation of diffusion in four-component systems, as was done in two-component system^^'^^^ in order to analyze effects of concentration dependence. A direct experimental approach would be to decrease the values of AC,, which also reduces the value of J. However, on the basis of the experience from a number of studies of other systems, values of J of about 60 are about the minimum that can be used for good experimental precision. If a diffusion cell with double the path length were used, values of ACi could be reduced by half. In this case,the same number of fringes J would be retained, and so would the corresponding precision. However, by analogy with the situation in binary systems,21~27*28 the effect of concentration dependence should be reduced by a factor of about 4.
Species Model In principle, it should also be possible to model this system by assuming the presence of five solute species: free CD, free PHE, free MBU, PHE-CD complex, and MBU-CD complex. By assuming no cross-term diffusion coefficients between species, main-term diffusion coefficients of species and equilibrium constants in the species representation should be directly related to diffusion coefficients in the stoichiometric component representation. Such a model has been used for three-component systems?-29 Analysis of four-component systems based on this type of model will be the subject of a future paper. IIlStabilitieS One experiment not listed in the tables had a clearly visibile instability in the boundary. The values of ACi were O.OOO1,-0.003, and 0.09274 mol dm-3 for solutes 1,2,and 3. These corresponded to values of ai equal to 0.0093, -0.0729,and 1.0636,respectively. Instability calculations showed that this experiment should have been stable, but the values of aidid put the experiment very close to a predicted region of gravitational instability at the center of the boundary.
Conclusions This study demonstrates that diffusion coefficients can be measured in four-component systems by the Gouy method. This system has three features which would add to experimental or numerical error. These are Ri that are fairly different from each other, as noted earlier, two eigenvalues that are as can be seen by the near equality of sz and s3 in Table IV, and the
Additions and Corrections possible concentration dependence, as discussed above. As a result, it is too early to set general expectations of experimental error for the study of other systems. The results do suggest the need to examine the effect of concentration dependence on the analysis of Gouy data from fourcomponent experiments by this method.
Acknowledgment. We thank Dr.Donato Ciccarelli for his help during the experiments. J.G.A. and D.G.M. thank the University of Naples for supporting them during recent visits to Naples, Italy. We are indebted to Dr. Joseph A. Rard for a critical reading of the paper and helpful comments. The research was supported by the Italian MURST and the Italian CNR. Registry No. a-Cyclodextrin, lO(316-20-3; L-phenylalanine, 63-91-2; monobutylurea, 592-31-4.
References and Notes (1)Kim, H.J . Phys. Chem. 1966.70, 562-575. (2) Tyrrell, H. J. V.; Harris, K.R. Dif/usion in Liquids; Butterworth: Stoneham, MA, 1984. (3) Miller, D. G.; Albright, J. G. Optical Methods. In Measurement of the Transport Properties of Fluids: Experimental Thermodynamics; Wakeham, W. A., Nagashima, A., Sengers, J. V., Eds.;Blackwell Scientific Publications: Oxford, 1991;Vol. 111, Section 9.1.6,pp 272-294 (see also references pp 3 16-3 19). (4)Kim. H.J. Phvs. Chem. 1969. 73. 1716-1722. (5j Miller, D. G. j . Phys. Chem. 1988,92,4222-4226. (6) Kim, H. J. Phys. Chem. 1970,74,4577-4584. (7)Vitagliano, P. L.;Ambrosone, L.; Vitagliano, V. J . Phys. Chem. 1992, 96, 1431-1437. (8)Kim, H. Private communication, 1981
The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 7483 (9) Paduano, L.; Sartorio, R.;Vitagliano, V.; Albright, J. G.; Miller, D. G.; Mitchell, J. J . Phys. Chem. 1990, 94,6885-6888. (10) Miller, D. G.; Vitagliano, V.; Sartorio, R. J. Phys. Chem. 1986,90, 1509-1 519. (1 1) Gosting, L.J.; Morris, M. S . J . Am. Chem. Soc. 1949,71,1998-2006. (12)Gosting, L.J.; Onsager, L.J. Am. Chem. Soc. 1952,746066-6074. (13) Albright, J. G.; Miller, D. G. J . Phys. Chem. 1989,93,2169-2175. (14) Longsworth, L.G. J. Am. Chem. Soc. 1947,69,2510-2516. (15) Marquardt, D. W. J. Soc. Ind. Appl. Math. 1963, 11, 431-441. (16) The fringe pattern extends down from the undeviated fringe position in the usual case that the bottom solution has a higher refractive index than the top solution, but it will extend up if the refractive index of the bottom solution is less than that of the top solution. In the latter case the direction of scans will be reversed. (17)Dunlop, P. J.; Gosting, L. J. J . Phys. Chem. 1959,63,86-93. (18)Gosting, L. J. J . Am. Chem. SOC.1950,72,4418-4422. (19) Miller, D. G.; Paduano, L.; Sartorio, R. J. Solurion Chem., in press. (20)Fujita, H.; Gosting, L. J. J. Phys. Chem. 1960,64, 1256-1263. (21)Albright, J. G.; Miller, D. G. J . Phys. Chem. 1980,84, 1400-1413. (22) Fujita, H.; Gosting, L.J. J . Am. Chem. SOC.1956,78, 1099-1106. (23)Paduano, L.;Sartorio, R.; Vitagliano, V.; Costantino J . Mol. Lfq. 1990,47,193-202. (24)Longsworth, L. G. J . Am. Chem. SOC.1953,75, 5705-5709. (25) Barone, G.; Castronuovo,G.; Del Vecchio, P.; Elia, V.; Muscetta, M. J. Chem. Soc., Faraday Trans. 1 1986,82,2089. (26) Barone, G.; Castronuovo, G.; Elia, V.; Muscetta, M. J . Solution Chem. 1986,15, 129-140. (27)Albright, J. G.; Miller, D. G. J . Phys. Chem. 1975,79,2061-2068. (28)Gosting, L.J.; Fujita, H. J . Am. Chem. SOC.1957,79, 1359-1366. (29)Paduano, L.;Sartorio, R.; Vitagliano, V.; Costantino, L. Be?. Bunsen-Ges. Phys. Chem. 1990,94, 741-745. (30)Miller, D. G. Equal eigenvalues in multicomponent diffusion. Fundamentals and Applications of Ternary Dvfusion, Proceedings of the International Symposium, 29th Annual Conference of Metallurgists of CIM, Hamilton, Ontario, Canada, August 27-28, 1990 Purdy, G. R.Ed.; Pergamon Press: New York, 1990;pp 29-40.
ADDITIONS AND CORRECTIONS
1991, Volume 95
C.E.Wslrafen* and Y.C. Cbu: Shear Viscosity, Heat Capacity, and Fluctuations of Liquid Water, All at Constant Molal Volume. Page 89 16. An incorrect conclusion was drawn in the second paragraph [the sentence starting with (fiC, (18.015 78)], righthand column. The crossing of (%$ and (‘‘)Cy,which is the subject of this paragraph, is not related to the change in the sign of 8, above and below 277.15 K, as stated. The divergence shown at temperatures below the crossing point results from extrapolating the function (fiCy= - A / P + B/T - C to temperatures well below 273.15 K, but this extrapolation produces values which are too large for highly supercooled water. [We are now attempting to address this type of problem in the constant 1-atm pressure case , that by using a fit with the above functionality in ( Wexcept we add the terms D / ( T - TH)*,E / ( T - TH), and F,where TH = 228 K and T > 235 K. Unfortunately, data are not available to allow us to do this with (‘‘C,] Of course., the differences shown in Figure 9 between (Wyand (“)Cyat temperatures above the crossing point are real and accurate. It can be shown that
(ac,/av),
= T(a2p/aTz)y = T ~ ~ ~ P / ~ , I=/ ~ T ) ~ ( T / K ~ ) ( a b / a T )+ p (~PT/KT’)(&~/W -T ( B 2 T / K T 3 ) ( a ~ , / a P )(,1) (@/aT),, which occurs in the first and largest term of eq 1, is positive when 0 < 0 below 277.15 K, and this derivative is also positive when @ > 0 above 277.15 K. At 273.15 K, (dCy/av), = 91 bar deg-I, and at 283.15 K, (aCy/dv), = + 84 bar deg-I. (p)Cyisidentically equal to (fiC,near 277.15 K. However, when it is recognized that the volume at constant 1-atm pressure increases progressively above 18.01578 cm’/mol as T decreases below 277.15 K, it becomes evident from the positive sign of (dCy av), that ( Wmust y be smaller than ( p ) C ybelow 277.15 K. ( Cyalso falls below (‘‘)Cyabove 277.15 K, as seen from Figure
+
rl
9.
A similar conclusion results from Figure 8. (fiC,decreaseS as V decreases for all three molal volumes involved. The reader may readily determine that (aC,/av), = + 90 bar deg-l at 273.15 K. [Multiply Cyby 10 to convert units shown to cm3 bar deg-l mol-I. Then determine ACy/AV at 273.15 K (two bottom curves).] Page 8909. At the end of the eighth line from the bottom of the abstract, 17.5636 cm3/mol should read 17.756 36 cm3/mol. Page 891 1 . Right-hand column, seventh line down, last paragraph, mute should read moot.
