J . Phys. Chem. 1990, 94, 6885-6888
6885
Diffusion Coefficients in Systems with Inclusion Compounds. 1. a-Cyclodextrin-L-Phenylalanine-Water at 25 C Luigi Paduano, Roberto Sartorio, Vincenzo Vitagliano,* Dipartimento di Chimica dell’llniversitd di Napoli, Via Mezzocannone 4, 801 34 Naples, Italy
John G.Albright,t Donald G.Miller, and John Mitchellt Chemistry and Materials Sciences Department, Lawrence Livermore National Laboratory, Livermore, California 94550 (Received: July 27, 1989: In Final Form: March 30, 1990)
Diffusion coefficients in the ternary system a-cyclodextrin (at one concentration)-L-phenylalanine (at four concentrations)-water have been measured by using the Gouy interferometric technique. The effect of the inclusion equilibrium on the cross-term diffusion coefficients was observed. The measured diffusion coefficients in the ternary systems were used to calculate values of the binding constants. These values are in good agreement with the value obtained from calorimetric studies.
Introduction Cyclodextrin (CD) is a common name of an homologous series of cyclic polysaccharides composed of six (a-CD), seven (8-CD), or eight (y-CD) glucose monomers linked by 1,4-glycosidicbonds.’ The truncated conical structure of CDs allows them to form inclusion complexes with a number of small molecules in solution.2 The stability of these “host-guest” complexes depends upon the dimensions of the C D cavity and the nature of intermolecular
force^.^ Interest in this aspect of C D chemistry has increased considerably in recent years because of its similarity to the enzymesubstrate interactions. Thus, a number of papers have been devoted to the determination of inclusion equilibrium constants for a variety of molecule^.^" Little attention, however, has been directed to the study of transport properties of CDs or the effect of CDs on the transport properties of other solutes. Three report binary diffusion data for CD-H20, and only onelo is concerned with a ternary system [(a-CD)-polyacrylic acid-H20]. This apparent lack of interest is surprising because of the importance of transport phenomena that are complicated by chemical equilibria. In this paper we present diffusion data for the system (aCD)-(L-phenylalanine)-water at four sets of mean concentrations, 0.02 M of a-CD with 0.025, 0.050, 0.075, and 0.100 M, respectively, of phenylalanine. From these data and some assumptions, a value of the inclusion constant is calculated and compared with a calorimetrically determined value.
Theoretical Background In an n-component system, the flows of solutes i ( i = 1, 2 , ..., n - 1) through the solvent (component 0) are described by a set of N - 1 phenomenological equations as follows:
where J i is the flow of solute i , Ci its concentration, Dii its main-term diffusion coefficients, and Dij ( i # j ) its cross-term diffusion coefficients. At very low solute concentrations, the cross-term diffusion coefficients will approach zero for nonelectrolytes: lim D, = 0 ( i # j ) (2) cp-0
where ck is the total solute concentration. At high solute concentrations, the magnitude of cross-term diffusion coefficients may equal or exceed the magnitude of the main-term diffusion coefficients.”
’
Permanent address: Department of Chemistry, Texas Christian University, Fort Worth, TX 76129.
Large and positive Dij values have been observed in systems where the solutes tend to “salt-out”.” Large negative cross-term diffusion coefficients can occur in systems with large attractive interactions between solute^.^^-'^ While the size and sign of diffusion coefficients for systems with high solute concentrations are understood only in qualitative terms at present, a more detailed interpretation of data is possible for dilute solutions in which significant fractions of solute monomer species have combined to form associated species. In this case, an interpretation in terms of rapid equilibria and their associated equilibrium constants can be explored. In the system studied here, there are monomer species C D and Phe and a complex species CD-Phe, where henceforth CD denotes a-CD. We identify these as solute species A = 1, B = 2 , and C = 3, respectively, which are in equilibrium according to the equation A+B=C
(3)
Flow equations for these solute species may be written
-J,* = D l l * grad C1* + D12*grad C2*+ Dl3* grad C3* (4a) -J2* = DZl*grad CI*
+ D22* grad C2* + D23* grad C3* (4b)
-J3* = D31* grad Cl* + D32* grad C2*+ D3,* grad C3* (4c) ( I ) Szejtli, J. J. Starch 1978, 30, 427. (2) Szejtli, J. Proceeding of the First International Symposium on Cyclodextrins, Budapest Sept 1981; D. Reidel: Dordrecht, 1982. (3) Huber,0.;Szejtli, J. Proceeding of the Fourth International Symposium on Cyclodextrins; Munich, April, 1988 Kluwer Academic Press: Dordrecht, 1989. (4) Lewis, E. A.; Hansen, L. D. J. Chem. SOC.,Perkin Tram. 2 1973,2081. ( 5 ) Barone, G.;Castronuovo, G.; Del Vecchio, P.; Elia, V.; Muscetta, M. J. Chem. Soc., Faraday Trans. 1 1986, 82, 2089. (6) Paduano, L.; Sartorio, R.; Vitagliano, V.; Castronuovo, G. Thermochim. Acta, in press. (7) Miyajima, K.; Sawada, M.; Nakagaki, M. Bull. Chem. SOC.Jpn, 1986, 56, 3556. (8) Paduano. L.; Sartorio, R.; Vitagliano, V.; Costantino, L. J. Solution Chem., in press. (9) Uedaira, H.; Uedaira, H. J. Phys. Chem. 1970, 74, 221 1. (10) Uemura, T.; Moro, T.; Komiyana, J.; Iijima, T. J . Am. Chem. SOC. 1979, 12, 731. (1 1) Vitagliano, V.; Sartorio, R.; Spaduzzi, D. J. Solution Chem. 1974, 7, 605. (12) Vitagliano, V.; Sartorio, R. J . Phys. Chem. 1970, 74, 2949. (13) Albright, J. G. Ph.D. Thesis, University of Wisconsin, Madison, 1963 (contains a typographical error corrected in ref 14). (14) Kim, H.J. Solution Chem. 1974, 3, 271.
0022-3654/90/2094-6885$02.50/0 0 1990 American Chemical Society
Paduano et al.
6886 The Journal of Physical Chemistry. Vol. 94, No. 1 7 , I990 Here the cross-term diffusion coefficients Dji* indicate the interactions of solute species caused only by the diffusion process. They are independent of the chemical reaction. If the reaction were very slow, each species would behave as an independent component. If the chemical reaction is sufficiently rapid.'5,'6as it appears to be for CD-Phe, the system may be treated as having only two independent solute components. Consequently diffusion may be represented more simply by - J , = D , , grad C, + D,,grad CZ (5a) -J2 = D,, grad C,
+ D??grad C2
(5b)
Here the subscripts 1 and 2 now denote the two independent solute components CD and Phe. respectively. Values of the diffusion coefficients of the independent components (eqs 5) are influenced by the chemical equilibrium. If all the concentrations are expressed in moles per unit volume and if all the flows in eqs 4 and 5 are in moles crossing a unit area on exactly the same frame of reference, then the following relations hold: C, = c,* + c3* (6a) C? = cz*
+ c3*
(6b)
Jl = Jl*
+ J?*
(7a)
J2 = JZ*
+ J3*
(7b)
K, = C3*/ (Cl *C2*)
(8)
If both solute concentrations are low and K, sufficiently large, it is reasonable to make the approximation that the species cross-terms are zero; namely D,* = 0 for i # j (9) When this approximation is made in the equations for the general case,I3*I4the Dil become related to the Dii*, Ci, and K, according to the following equation^:'^^'^ D , ] = y2{(Djl* + D33*)
+ !Dll*- D33*)[I - Kc(C2 - C , ) ] R ) (loa)
D12 = 1/2I(D33* - D I I * ) + ( D I I * - D33*)[1+ K,(C, D21 =
%I(&*
-4
2 * )
+ (&z*
-
C,)lR1 ( 1 Ob)
- D33*)[1 - KC(C2 - C,)IRI
(10c) 4
2
= j/2I(Dzz*
+ D33*) + (D22* - &3*)[1
+ Kc(C2 - C,)lR1 (10d)
where
R
([I
+ Kc(C2- C,)]' -t ~ K , C I ) - " ~
( 1 1)
Experimental values of the four Di, and these four equations are sufficient to calculate the four Dll*, D22*,D,,*, and K , at each concentration by using the assumption of eq 9. Alternatively, if reasonable values of one or more of the Dij.* can be estimated at low concentrations, then K, and the remaining Dii* can also be calculated. A comparison of these two techniques is provided below. Experimental Section Materials. a-CD and L-phenylalanine (Phe) were purchased from Sigma and used without further purification. The water content of air-equilibrated a-CD was determined by drying at 120 OC for several days under vacuum. The initial sample of a-CD was found to be 9.50,% water, in good agreement with the value ( 1 5 ) Equations in ref 16 show that a chemical reaction will cause the value of DAto be dependent on time where DAwill approach the value expected for instantaneous equilibrium of the reacting species. The dependency will become linear in I / f as I / r 0 and should lead to abnormal values of Ar if the reaction cannot be considered instantaneous. The values of AI appeared normal for this set of experiments, indicating that the reaction was sufficiently fast that instantaneous equilibrium can be assumed. (16) Albright. J . G . J. PhyS. Chem. 1963, 67, 2628.
-
calculated from the formula a-CD.6H20. The molar masses used for Phe, a-CD, and water were 165.2, 972.9, and 18.016 g mol-', respectively. Preparation of Solutions. All solutions were prepared by weight with doubly distilled water, and weights were corrected to masses. The densities of solid CD and Phe used for the corrections were 1.42 and 1.2 g/cm3, respectively. The concentration dependence of refractive index of binary solutions of each solute were used to predict concentration differences AC, and I C 2that would give an appropriate number of interference fringes in the ternary experiments. At each of the four sets of mean concentrations, at least four experiments were performed where at least three had significantly different values of the ratio ACl/AC2 In series CP-2 only three experiments could be used for the calculation of the four diffusion coefficients, but there were four usable experiments in the other three series. Density Measurements. Densities of the solutions are needed to convert mass concentrations to volume concentrations. They were measured with single-stem 30-cm3glass pycnometers that were calibrated with doubly distilled water at 25.00 "C. The density of air-saturated water was assumed to be 0.997045 g/cm3 at this temperature. Solution-filled pycnometers were allowed to equilibrate for 1 h in the water bath of the diffusiometer before drawing solution down to the calibration mark and weighing. The density data were fitted by the method of least-squares to the equation d(C,,C2) = d(Cl,c2)
+ H , ( C , - c1) + H2(C2 - c2)
(12)
where d is the density, Ci are the concentrations in mol/dm3, and the bars signify average concentrations. The values of the parameters d(C,,C,), H I , and H , for the four compositions studied are listed in Table 11. Diffusion Experiments. The experiments were performed with the Gosting diffusiometer at the Lawrence Livermore National Laboratory, following the procedure described in more detail by Albright et aI.l7 This instrument was operated in the Gouy configuration. The temperature of the diffusiometer bath was regulated at 25.00 0.01 "C and remained constant during the experiments to within 0.005 OC. The distance from the center of the diffusion cell to the photographic plate ( b distance) was 3.088 852 m. The diffusion cell was a quartz Tiselius cell of extremely good optical quality. Its inside dimension along the optical axis ( a distance) was 2.5064 cm. Initial sharp boundaries were formed by siphoning at the level of the instrument's optic axis. Once a good boundary had been formed, the bottom of the cell was closed, the needle immediately and carefully removed while still siphoning, and the top of the cell then closed. The 6' and 6" photographsI8 were taken on Kodak metallographic plates, while the Gouy patterns were recorded on Kodak 11 G plates. Fringe positions on the photographs were measured with a Gaertner tool-makers' microscope fitted with a scanning device that allowed the determination of fringe minima to within 1 pm. Analysis of Data Fringe position data for each experiment were analyzed as follows: (a) From 6' and 6" photographs, the fractional part of the total number of fringes J, was determined following Gosting.'* The fractional part of J, was also determined by using the F4J or F3JP computer code19 (the choice of code followed the criteria given in ref 19). The agreement between values of J , from direct measurement of the fractional part of a fringe and the value obtained from the FJ4 and F3JP computer codes were within fO.1% except for one experiment that was outside the conditions for gravitational stability. The J , values from the direct Gosting (17) Albright, J. G , ;Mathew, R.; Miller, D. G.; Rard, J. A. J. Phys. Chem. 1989, 93, 2176. (18) Gosting, L. J.; Morris, M.S. J . Am. Chem. SOC.1949, 71, 1998. (19) Albright, J . G.; Miller, D. G. J . Phys. Chem. 1989, 93, 2163.
Diffusion Coefficients
The Journal of Physical Chemistry, Vol. 94, No. 17, 1990 6881
TABLE I: Data from Diffusion Experiments" Series CP-I
C,
c2
ACI AC2 J,(exPt) J,(calc) a1
D, X 109(expt) DA X 109(calc)
QoX IO'(expt) Qo X IO4(calc) &top) d(bottom)
expt 1 0.020 15 0.025 03 0.00000 0.035 34 60.125 60.265 0.0000 0.6245 0.6243 2.2500 3.0520 1.004 745 1.006 299
expt 2 0.020 15 0.025 03 0.001 90 0.028 26 60.532 60.345 0.2014 0.5355 0.5358 51.910 50.91 2 1.004 547 1.006 493
expt 3 0.020 15 0.025 03 0.007 56 0.007 07 60.329 60.41 3 0.8004 0.3606 0.3606 34.490 34.590 1.003 978 1.007 067
Series CP-2 expt 2 expt 3 0.020 14 0.020 14 Cl 0.05006 0.05005 c 2 0.007 57 0.001 89 ACl AC2 0.009 24 0.036 95 J,(expt) 64.330 75.125 J,(calc) 64.308 75.171 ffl 0.7548 0.1612 DA X 109(expt) 0.3101 0.3605 0.5541 DA X 109(calc) 0.3608 0.5517 Qo X 104(exp) -14.58 38.330 47.920 QoX IO4(calc) 38.355 46.327 d(top) 1.004 875 1.005 037 1.005 470 d(bottom) 1.008 385 1.008 220 1.007 787 expt I 0.020 I5 0.05005 0.009 46 0.00002 60.803
ACI
AC2 Jm(exPt) Jm(calc) ffl
D, X 109(expt) DA X 109(calc) Qo X IO'(expt) QoX 104(calc) d(top) &bottom)
Series CP-3 expt 1 expt 2r 0.020 IO 0.020 IO 0.07491 0.07492 0.00000 0.094 20 0.046 08 0.00006 78.792 60.73Ib 78.789 0.0000 0.9983 0.6339 0.2999 0.6329 -1.780 -28.030 -2.2877 1.006 697 1.005 987 1.008 725 1.009 477
expt I 0.02003 0.10006 c2 ACI 0.001 88 AC2 0.01408 J,(expt) 36.186b J,(calc) 36.205 a1 0.3343 D,, X 109(expt) 0.4729 DA X 109(calc) 0.4720 QoX 104(expt) 78.48 QoX IO'(ca1c) 76.16 d(top) 1.008 200 d(bottom) 1.009504
c,
expt 4 0.020 14 0.05005 0.00000 0.046 21 78.82Ib 78.854 0.0000 0.6303 0.6300 -0.320 1.023 1.005 61 7 1.007 637
expt 3 0.020 IO 0.074 9 I 0.001 89 0.036 85 75.16Sb 75.172 0.1618 0.5483 0.5493 47.530 48.094 1.006 567 1.008 882
Series CP-4 expt ZC expt 3 0.02003 0.020 IO 0.10005 0.09991 0.00939 0.00000 0.000 IO 0.046 17 60.450 78.955 79.033 0.9989 0.0000 0.2912 0.6329 0.6330 -34.09 -1.72 0.142 1.007 084 1.007 782 1.010572 1.009822
expt 4 0.020 I 5 0.025 03 0.007 56 0.007 07 60.246 60.41 3 0.8004 0.3607 0.3606 34.490 34.595 1.003 972 1.007 052 expt 5 0.02005 0.049 87 0.007 27 0.008 53 61.138 61.173 0.1760 0.3594 0.3591 36.760 37.007 1.005 071 1.008 144
expt 4 0.020 IO 0.07491 0.007 53 0.009 18 64.140 64.152 0.7553 0.3430 0.3458 38. I20 38.175 1.006 160 1.009 317
expt 4 0.02003 0.10007 0.007 52 0.009 25 64.225b 64.248 0.7535 0.3439 0.3441 32.93 33.88 1.007 232 1.010377
expt 5 0.020 03 0.10004 0.001 88 0.036 9 1 75.395 75.285 0.1608 0.5460 0.5460 52.36 5 1.93 1.007 650 1.009 975
C,,average concentration of component i , mol/dm3. ACi, concentration difference between bottom and top solutions. J,(expt), experimental Gouy fringe number, [ 191. J,(calc), Gouy fringe number calculated from Table I 1 data: J, = RIACl + R2AC2. a, = RIACI/(RIAC,+ R2AC2). DA. reduced height-area ratio, m2/s. Qo,integral of the fringe deviation graph. d, density of solution, g/dm'. bObtained by using F3JP program.19 cRuns occurring outside the computed gravitational stability range.
method were used, except when diagnostics suggested a possible error. In such cases, the computer code results were used (see Table I ) . (b) The appropriate J,,, values, plus the positions of fringe minima, were used to calculate DA and Q ~ o ~ 2with 1 the F2M or (20) Fujita, H.; Gosting, L. J. J . Am. Chem. Sac. 1956, 78, 1099. (21) Fujita, H.; Costing, L. J. J . f h y s . Chem. 1960, 64, 1256.
F2P codelg (the choice between use of the F2M or F2P codes followed the criteria given in ref 19). Values of J,, DA, and Qo for all experiments are listed in Table I. (c) From these data, a preliminary set of D , for each composition was obtained by using the RFG p r ~ g r a m .The ~ ~ range ~~~ of dynamical stability was then c a l c ~ l a t e dfrom ~ ~ these ~ ~ ~ preliminary Dij's and the Hi's. Some experiments fell outside this range and were unstable of the "over stable (diffusive)" type. Data from the remaining stable experiments were then used in a final computation of the Dij (Table 11).
Discussion This paper has provided experimental diffusion data for dilute aqueous solutions of the two nonelectrolytes CD and Phe, which form a complex. In terms of the independent components, Table I1 shows that the main-term coefficients Dii are not very concentration dependent and are not much different than their corresponding binary value. One cross-term, D,,,is small. However, one of the cross-terms, D Z l ,is quite large and negative and increases with Phe concentration. This indicates that there are solute interactions, that a complex is indeed formed, and that cross-terms cannot always be ignored. It is also possible to interpret these same diffusion data in terms of the uncomplexed species C D and Phe and of the complex CD-Phe. In this circumstance, it is reasonable to assume, as mentioned above, that the cross-term D,* for these three nonelectrolyte species are nearly 0. With this assumption, it is possible to determine by iteration from eqs 10 and 11 and the experimental D, the main term Dii* of uncomplexed CD and Phe and the Dii* of the complex, as well as the equilibrium constant for the formation of the complex. The results are in Table 111. Table 111 shows the following: (a) D l l * and Dz2* are close to their values in the binary solutions (and to D l l and D,,,respectively). (b) Dll*and D33*are very nearly the same, which strongly implies that the mobility and size of the hydrated complex are similar to those of uncomplexed CD. This also implies that the complex is an inclusion complex of Phe inside CD rather than CD "inside" Phe. This is, of course, already known from wellestablished structural forms of CD and Phe., However, if the complex were CD "inside" Phe, D33* would be close to DZ2*. Analogously, if the complex were Phe attached to the outside of CD, the larger size of this species would be expected to have a D33*smaller than either D l l * or D2,*. (c) The values of K, and their average of 9.7 dm3/mol are very close to the value of 13 dm3/mol obtained cal~rimetrically,~ which is a very encouraging result. As can be seen from eq lob, the near equality of D l l * and D33* is responsible for the very small cross-term diffusion coefficient 012.
The cross-term Dzl tends to be large and negative because the diffusion coefficient Dzz* of the faster moving Phe is much larger than D33* for the CD-Phe complex. In this case, if there is a gradient of CD and no gradient of Phe, the negative Dzl causes Phe to move "uphill" toward the higher concentration of CD. This is true because a higher concentration of CD means a lower (22) R e v ~ i developed n~~ a program to use values of AC,, J,, DA,Qo,and/or
Q1to calculate the four diffusion coefficients based on the method given by Fujita and Gosting20.2'(also see ref 18). In this paper we refer to the program as the RFG program. Our experience has shown that the use of Q,alone or in combination with Qo does not yield any improvement in the D,, calculations. Consequently we use only Q* (23) The quantity Qo is .fQ df(r). The notation Qo was introduced to distinguish it from Q,which is identified as the first moment of the area under the Q graph, SQ f(z) df(r). The first moment is used in a least-squares analysis procedure developed by Revzin to calculate D,y24That procedure, in turn, is based on an unpublished derivation by Fujita. (24) Revzin, A. Ph.D. Thesis, University of Wisconsin, Madison, 1969. (25) Vitagliano, P. L.; Della Volpe, C.; Vitagliano, V. J. Solufion Chem. 1984, 13, 549. (26) Miller, D. G.; Vitagliano, V. J. f h y s . Chem. 1986, 90, 1706. (27) Miller, D. C.;Sartorio, R.: Paduano, L.. unpublished results.
Paduano et ai.
6888 The Journal of Physical Chemistry, Vol. 94, A'o. 17, 1990 TABLE 11: Diffusion Coefficients and Related Data for the System (a-CD)-(L-Phe) in Water at 25 OC" series CP- 1 CP-2 CP-3 0.020 I 5 0.020 1 2 0.020 IO C, 0.074 91 0.025 03 0.050 0 1 36.8 f 0.8 36.7 f 3.0 37.0 f 2.3 H I X 10' 4.4 f 0.2 4.4 f 0.7 4.4 f 0.5 H , x 103 1.007 726 1.005 519 I .006 624 d 6436.1 f 0.5 6396.3 f 0.1 6412.2 f 0.7 R, 1710.0 f 0.1 1705.3 f 0.5 1706.4 f 0.8 R2 3.177 f 0.006 3.276 f 0.005 3.245 f 0.008 D , , x 10'0 0.005 f 0.001 0.007 f 0.002 -0.002 f 0.003 D , , x 1010 -1.220 f 0.025 -0.951 f 0.050 -0.41 1 f 0.030 D2, X 1OIo 6.298 f 0.016 6.287 f 0.01 1 6.315 f 0.018 D2, X 1OIo
CP-4 0.02005 0.100 20 36.8 f 0.6 4.4 f 0.1 1.008 820 6438.0 f 0.9 1711.8 f 1.0 3.157 f 0.016 0.000 f 0.004 -1.630 f 0.101 6.332 f 0.024
c,
O C i , averagc concentration of component i. mol/dm3 (average of the mean concentrations of the individual experiments given in Table I ) . Hi, differential increment of density (eq 12). d, density at the average concentration, g/cm3. R,, differential increment of refractive index in terms of fringe number, J,: An = XJ,/a = 2.1787 X IO-'J, (for mercury green light). D , . diffusion coefficients, m2/s. TABLE 111: Diffusion Coefficients of Individual Species and Binding Constants ComDuted bv Solvine Eas 10 and 11
- .
CPhe?
mol/L
Dll* x m2js 3.285 3.251 3. I49 3.157
0.02503 0.05001 0.07491 0.10020
4
loko,m2/s
lolo, m2/s
x
Kc. L/mol
Cme. mol/L
6.608 6.618 6.520 6.510
3.21 1 3.229 3.223 3.157
6.9 10.1 9.8 11.7
0.02503
6.973
0.05001
6.894
0.07491
6.823
0.10020
6.757
022:
IOIO,
TABLE IV: Alternative Computation of Diffusion Coefficients of Individual Species and of lnclusion Constantsb
x
033:
1010,
X
m*/s
Dii' X IOIO,
m*/s
3.351 3.395 3.312 3.395 3.271 3.395 3.233 3.395
022: X
Djj'
X
10'0, m2/s
1oIo, m2/s
6.705 6.705 6.630 6.630 6.559 6.705 6.493 6.630
2.928 2.900 3.081 2.900 3.001 2.900 3.080 2.900
D,, L/mol 6.90 IO 9.44 10 8.61 IO 11.28 10
case" a b a b a b a b
"For case a and b, see text. bD, from ref 7. D, of Phe as a function of concentration taken from unpublished measurements.
I
\
\
3.201 3.151
i ,
,
c,,,
, mWL
e\*
1
0.08 0.12 Figure 1 . Main diffusion coefficients of a - C D as a function of Phe conccntration: ( + ) data from Table 111; (0)data from Table IV, case 0.04
3.
concentration of uncomplexed Phe. The calculated values of D,* are essentially unaffected by errors in the D,.. However K, can vary by as much as IO-15% for 1 standard error. The effect is most sensitive to changes of D,,, whose standard error is also the largest. I t is expected that D l l *of Table 111 will have a trend as concentration of Phe goes to 0 and should be consistent with the value of D , of CD at 0.02 M . This is roughly true, as shown in Figure 1.
The above speciation calculation is the most logical use of the experimental data. However, the similarity of D,, and D2, to the corresponding values D , and D2 in their binary solutions suggests some alternative speciation calculations, as mentioned earlier. D, can yield D3)* and K,. These can be compared with each other and with the direct species calculation given in Table 111. A number of variations have been tried including no viscosity correction. Although viscosity corrections are hardly universally valid, the results of various such corrections are more or less similar, with Dii* and K, more or less similar to the direct calculation results in Table 111. Two of the simpler and best examples are given in Table IV. Case a assumes that C D can be viewed as diffusing in a mixed solvent Phe-H20 of viscosity q2 so that D l l * = D I ( q / q 2 )and , the Phe diffuses in a mixed solvent of CD-H20 of viscosity q , so that DZ2* = D2(?/ql). With these assumptions, Dj3* and K, can be calculated for each CD-Phe composition. The values of K , are
more scattered than the direct calculation, but similar. Case b assumes that because D,, is not very dependent on the Phe concentration, D, I * = D,. Thus only D22*needs to be corrected, as in case a, so that DZ2*= D 2 ( q / q l ) . In addition, D33* and K , are assumed to be constant over all four CD-Phe compositions. Trials to get the best overall values of the experimental D,,yield a K , of I O L/mol, which is in good agreement with the average value from Table 111 ( K , = 9.7 L/mol). Conclusions
The consistency of the direct speciation calculations and the good agreement of K, with the calorimetric measurements suggests that the assumption of 0 cross-term between species is indeed a good one. The validity of this assumption here also has implications about the validity of models of other nonelectrolyte systems that neglect cross-term diffusion coefficients bewteen species, e.g., models for diffusion of solutes in systems where micelles are formed. In this study, the value of K , was of the order of 10 dm3/mol. If the value of K, had been much smaller, e.g., 1-2 dm3/mol, this type of interpretation would have been less valid. It may be noted that as K, approaches infinity, the term in eqs 10 [ I - K,(C2 - C , ) ] Rbecomes + I if C2 < C, and -1 if C2 > C,. Thus for sufficiently large K,, it would become difficult to determine K, but the system could still be interpreted in terms of eqs 10 and 1 1. In this case, there would be the associated complex plus monomer species of the excess component. Acknowledgment. The subject of this research is part of a communication presented at the Aachen Meeting on Transport Processes in Fluids and Mobile Phases, Sept 25-27, 1989. We express our appreciation to Dr. Joseph Rard for measuring the fractional water content of the a-CD. L.P., J.G.A., and J.M. express their appreciation to the Lawrence Livermore National Laboratory for making their facilities available. Part of the research was supported by the Italian Minister0 della Pubblica lstruzione and by the Italian C.N.R. Portions of this work were carried out under the auspices of the USDOE at LLNL under Contract No. W-7405-ENG-48. Registry No H,O, 7732-1 8-5; L-phenylalanine, 63-91-2
