J . Phys. Chem. 1990, 94, 8741-8744
8741
Intradiffusion Coefficients and Integral Mutual Diffusion Coefficients of Dilute Associating Solutes Are Identical. Caffeine in Water Derek Leaist* and Lu Hui Department of Chemistry, University of Western Ontario, London, Ontario, Canada N6A 5B7 (Rereired: May 14, 1990)
Equations are developed for the intradiffusion coefficient (D*) and the mutual diffusion coefficient (D) of self-associating solutes. At low concentrations the two coefficients are shown to be related as follows: D* = ( 1 /C)s$D dC and D = D* + (CdD*/dC), where Cis the concentration of total solute monomers. The limiting slope of D versus Cis found to be twice thc limiting slope of D* versus C. The relationships between D* and Dare tested by using diffusion data for aqueous caffeine, B stepwise associating solute.
Introduction Associating solutes show a wide range of behavior, from simple dimerization' to stepwise a g g r e g a t i ~ n ~and - ~ the formation of micelles.s The diffusion of these materials has been the subject of a number of studies.',b'8 Both intradiffusion coefficients and mutual diffusion coefficients have been reported. intradiff~sion'~ refcrs to the diffusion of labeled solute molecules in solutions of uniform chemical composition. N M R techniques20q2'and capillary-tube or diaphragm-cell experiments with radiochemical tagging'9*22are often used to measure intradiffusion. Mutual d i f f u s ~ o nthe . ~ ~interdiffusion ~~~ of solutions containing the same solute but at different concentrations, can be measured by tech~ ,the ~ ~ Taylor dispersion niques such as optical i n t e r f e r ~ m e t r y 'or met hod. I 9*23 Whcrcas intradiffusion involves the interchange of tagged and untaggcd solutc molccules, solute and solvent molecules are interchanged during mutual diffusion. In general, there is no direct relationship bctwccn the diffusion coefficients for the two proc e s s ~ ~ . *The ~ - work ~ ~ reported here, however, suggests there is
( I ) Stokes, R. H . J . Phys. Chem. 1965, 69, 4012. (2) Stokes, R. H.; Marsh, K. N . Annu. Reo. fhys. Chem. 1972, 23, 65. (3) Tucker, E. E.; Becker, E. D. J . Phys. Chem. 1973, 77, 1783. (4) Hoiland. H.: Skauge. A.; Stokkeland, 1. J . fhys. Chem. 1984,88, 6350. (5) Mukcrjcc, P.; Mysels, K. J . Critical Micelle Concentrations of Aqueous Surfactant Solutions. Natl. Stand. Re/. Data Ser., Natl. Bur. Stand.
(US.)1971. (6) Miiller, G . T. A.; Stokes, R. H. Trans. Faraday Soc. 1957, 53, 64. (7) Dunn. L. A.: Stokes, R. H . Aust. J . Chem. 1965, 18, 285. (8) Stockmayer, W. H. J . Chem. fhys. 1960,33, 1291. (9) Carman. P. C . J . Phys. Chem. 1967, 71, 2565. (IO) Vitagliano, V.; Sartorio, R. J . Phys. Chem. 1970, 74, 2949. (1 I ) Schonert, H. Z . fhys. Chem. (Munich) 1980, 119. 53. ( I 2) Weinheimer, R. M.: Evans, D. F.; Cussler, E. L. J . Colloid Interface Sei. 1981. 80, 357. (13) Evans, D. F.; Mukherjee, S.; Mitchell, D. J.; Ninham, B. W. J . Collord InrerJace Sei. 1983, 93, 184. (14) Bender. T . M.; Pecora, R. J . Phys. Chem. 1988, 92, 1675. (15) Price, W. E. J . Chem. Soc., Faraday Trans. I 1989,85, 415. (16) Price. W. E.; Trickett. K. A,: Harris, K. R. J . Chem. Soc., Faraday Trans. I 1989, 85. 328 I , (17) Leaist, D. G . J . Colloid Interface Sei. 1988, 125, 327. (18) Leaist, D. G.Can. J . Chem. 1988, 66, 1129. (19) Tyrrell. H. J . V.: Harris, K. R. Diffusion in Liquids: Butterworths: London, 1984. (20) Fabre. H.: Kamenka, N.: Khan, A.; Lindblom, G.; Lindman, 8.; Tiddy. G . 1. T.J . Phys. Chem. 1980, 84, 3428. (21) Stilbs, P. J . Colloid Interface Sei. 1982, 87, 385. (22) Robinson, R. A.: Stokes, R. H. Electrolyte Solutions, 2nd ed.: Academic Press: New York, 1959. (23) Pratt. K . C.: Wakeham, W. A . Proc. R . Soc. London 1975, A342. 401.
0022-3654/90/2094-8741$02.50/0
in fact a simple relation between the intradiffusion and mutual diffusion coefficients of dilute associating solutes. The proposed relation is tested for aqueous caffeine, a stepwise associating s ~ l u t e . ~ ' - In ~ ~a 0.10 mol L-' caffeine solution at 25 OC, for example, only about 40% of the total caffeine exists as the monomer. Dimers account for about 30% of the caffeine, and the remainder exists as higher polymers. Recently, a diaphragm cell and I4C tags were used to measure accurate intradiffusion coefficients of aqueous caffeine.l5.l6 To compliment the diaphragm-cell results, we report here mutual diffusion coefficients of aqueous caffeine. The measured mutual diffusion coefficients are compared with values derived from the measured intradiffusion coefficients, and vice versa. lntradiffusion measurements provide information about extents of association and the mobilities of dissolved species. Mutual diffusion has practical applications related to solubilization, detergency, and nucleation processes. An explicit relation between the two coefficients would be useful because it would allow one diffusion experiment to provide both kinds of diffusion data.
Experimental Section Mutual diffusion of aqueous caffeine was measured by the Taylor dispersion (peak-broadening) method.'9*23v3'A dual-piston pulse-free metering pump was used to maintain a steady laminar flow of caffeine carrier solutions in a Teflon capillary tube (inner radius 0.045 15 cm, length 3 1 17 cm). Small volumes (10-25 pL) of solutions containing caffeine at a concentration different from that of the carrier solution were introduced into the carrier stream at the tube inlet through a liquid chromatography injection valve. A deflection-type recording differential refractometer (Waters Model 401, IO-pL flow cells) monitored the dispersion of the injected sample at the tube outlet. The flow rates were chosen to satisfy the criteria developed by Wakeham et al.2333' The refractometer output voltage was displayed on a chart recorder. Mutual diffusion coefficients (D) were calculated from the e x p r e s s i ~ n ~ ~ , ~ ~
where t R is the retention time and W , is the width at half-height of the eluted solute peak. The vaiues of D obtained by this (24) Laity, R. W. J . Phys. Chem. 1959, 63, 80. (25) Vink, H. J . Chem. Soc., Faraday Trans. I 1985, 81, 1725. (26) Dunlop, P. J . J . Phys. Chem. 1965, 69, 1693. (27) Gill, S. J.; Downing, M.; Sheats, G. F. Biochemistry 1967, 6, 272. (28) McCabe, M. Biochem. J . 1972, 127. 249. (29) Ts'o, P. 0. P. Basic Principles in Nucleic Acid Chemistry: Academic Press: New York, 1974. (30) Bothe. H.; Cammenga, H . K. Thermochim. Acta 1983, 69, 235. (31) Alizadeh, A.; Nieto de Castro, C . A,; Wakeham, W. A. Inr. J . Thermophys. 1980, I , 243.
0 1990 American Chemical Society
8142
Leaist and Hui
The Journal of Physical Chemistry, Vol. 94, No. 24, 1990
procedure represent the differential mutual diffusion coefficient at the composition of the carrier system.31 Check calculations showed that the values of D obtained from eq 1 agreed to within 1% with the values obtained by a more elaborate nonlinear least-squares analysis of the peak shapes. Theory I n general, a self-associating solute may diffuse as monomers ( A , ) as well as dimers (A2)*trimers (A3). etc., produced by the multiple association rcactions iA, = Ai ( i = 2, 3, 4, ...)
(2)
Despite the large number of species, mutual diffusion of the solute can be described by a single mutual diffusion coefficient D defined by Fick’s l a d J = -DVC
(3)
J and C denote that molar flux density and the concentration of the total .solute monomers. Similarly, a single intradiffusion coefficient D* describes intradiffusion of the so1ute.l J * = -DVC*
(4)
J * and C* are the molar flux density and the concentration of the total labeled solute monomers. Both D and D* can be measured without knowledge of the state of aggregation of the solute. For dilute solutions, it is a good approximation to write j , = -DiVc,
(5)
for the flux densityji of each solute species ( A l , A?, A,, ...). c, and Di are the concentration and the limiting diffusion coefficient of species i. Equations 3-5 can be used to show (see Appendix) that the intradiffusion cocfficicnt and the mutual diffusion coefficient are respectively the ici-weighted a ~ e r a g e ’ * ~and J I - the ~ ~ i2c,-weighted average9$’*of the diffusion coefficients of the various solute species: m
m
D* = x i c i D i / x i c i /=1
I=
m
I
(6)
m
D = xi2ciDi/xi2c, i=I
/=I
(7)
D* and D arc idcntical at infinite dilution.26 I n this limit, dissociation is complete (ci = 0 for i > I ) , and D*O and Do equal the diffusion coefficient of the monomer, D,. At first glance, eqs 6 and 7 suggest a complicated interrelation of D* and D involving the concentrations and the diffusion cocfficicnts of thc various solute species. (Information that is diffcrent to obtain in practice.) To show that D* and D are in fact simply related. the conditions of chemical equilibrium ci = Kicli ( i
>
I)
(8)
and mass balancc m
C=
ziti i=I
may bc uscd
L O rcwritc
eqs 6 and 7 as follows
D* = (CID, + 2K2cI2D2 + 3K3cI3D, + ...) / C
+
dC dc,
- = C,
+ 4K2cI2+ 9K3cl3+ ...
OC
c/ lo-’ mol L-’ I .oo 2.00 4.00 5.00 6.00 10.0
(13)
c/ 10-1
Dmca6/1o - ~m2 s-, 0.760 0.751 0.742 0.738 0.73 1 0.703
mol L-’ 20.0 30.0 50.0 70.0 90.0
D,,,,/ 1w9 m2 s-l 0.663 0.636 0.595 0.558 0.538
Dividing eq 12 by eq 13 and comparing the result with eq 7 leads to the astonishingly simple relation D = d(CD*)/dC
(14)
and hence D = D*
+ (CdD*/dC)
(15)
integration of eq 14 gives
D* = ( I / C ) S 0c D d C
(16)
which shows that the intradiffusion coefficient of a dilute associating solute equals its integral mutual diffusion coefficient. (Equation 16 bears a striking but apparently coincidental resemblance to the equation used to calculate differential mutual diffusion coefficients from the measured integral values obtained from diaphragm-cell experiment^.^^^^^ Measured values of D* together with eq 15 may be used to derive values of D. Conversely, eq 16 may be used to obtain D* from D. The conversion procedures apply to solutes that dimerize, form micelles, or undergo stepwise aggregation. Remarkably, the aggregation numbers, concentrations, and diffusion coefficients of the actual diffusing species are not required. Results and Discussion The derived relationship between D* and D rests on several approximations: the association reactions are rapid enough to maintain local equilibrium; the solutions are thermodynamically ideal; the diffusion coefficient of each species is constant; and the fluxes of the species, except for the association equilibria, are noninteracting. Local equilibrium is usually an excellent approximation. The other approximations, however, restrict the validity of eqs 15 and 16 to dilute solutions of nonelectrolytes. Nevertheless, there are a number of important nonionic detergents, dyes, and biochemicals that associate strongly in the region where the eqs 15 and 16 may be expected to hold. To test the accuracy of the proposed relationship between D* and D, we turn now to the example of aqueous caffeine. The Taylor dispersion tube was used to measure mutual diffusion coefficients of aqueous caffeine at 25 OC and concentrations from 0.001 to 0.090 mol L-I. (Caffeine’s limited solubility precluded measurements at significantly higher concentrations.) D was measured 6-8 times at each concentration, and the average was taken. Table 1 gives the results. The values of D were reproducible within 1 %. The measured mutual diffusion coefficients are plotted against concentration in Figure 1. The results are accurately represented by the polynomial Dm,,,/10-9 m2 s-, = Do
(10)
Ki is the equilibrium constant for the formation of aggregate A, from i free monomers (eq 2). Differentiation of eqs 8-10 gives d(CD*) C ) -- c,D, 4K2c12D2+ 9K3cI3D3+... (12) dc, C,
TABLE I: Mutual Diffusion Coefficients of Aqueous Caffeine at 25
+ a , C + a2C? + a3C3
(17)
with the parameter values Do = 0.7644 X m2 s-’, a I = -6.200 L mol-’, a2 = 7 1.096 L2 mol-2, and a, = -337.34 L3 mol” obtained by least squares. The previously reported intradiffusion coefficient^'^ of aqueous caffeine are also plotted in Figure 1. The measured D* values are reproduced by the expression D*,,,,/
1 0-9 m2 s-l = D*O
+ a l * C + a,*@ + a,*C3
(32) Stokes, R. H.J . Am. Chem. SOC.1950, 72, 763. (33) Stokes, R. H. J . Am. Chem. SOC.1950, 72, 2243.
( 1 8)
Caffeine in Water 0.8
I
The Journal o f Physical Chemistry, Vol. 94, No. 24, 1990 8743 I
I
I
I
1
of the relationship between D* and D. Increasing the concentration of solute usually increases the solution viscosity, thereby decreasing each Di. The viscosity ( q ) of 0.10 mol L-I aqueous caffeine,I5 for example, is about 4% larger than the viscosity (7") of pure water at 25 OC. It is w e l l - k n ~ w nthat l ~ ~ (~v~/ v o ) D iis more nearly constant than D,. Therefore, it might be possible to obtain a more accurate relation between D* and D by including the relative viscosity correction factor in eqs 15 and 16 as follows.
-1
( ? / v o w = ( v / v " ) D * + C d[(v/vO)D*l/dC
(22)
(v/vo)D* = ( I / Q ~ ' ( v / v " ) D dC
(23)
-1 0
\
4 0.6
" C 0.5 0
I
I
I
20
40
60
C/IO-~~OI
80
100
L-'
m2 s-l, a,* = -3.452 L mol-', a2* = with D * O = 0.77S2 X 22.85 L2 mol-2. and a,* = -68.84 L3 mol-,. Thcory19s26,34 provides the exact result that the limiting mutual diffusion cocfficient and the limiting intradiffusion coefficient of a nonionic solute are identical. For aqueous caffeine, the limiting m2 value of D obtaincd in the present study ( D o = 0.764 X s-l) is 1.277 smallcr than the reported limiting D* value (D*O = 0.775 X m2 s-'),Is well within the combined 2% precision of the Taylor dispcrsion and the diaphragm-cell results. In addition, thc valuc of Do agrees within 1.8% with the limiting value 0.778 X I 0-9 m2 S-I measured previously by Taylor dispersion.I6 The agreement between the different results suggests that the experimcnts wcrc frcc of significant systematic error. To prcdict intradiffusion coefficients from the measured mutual diffusion coefficients, eq 17 can be integrated according to eq 16 to obtain
+ ( a l / 2 ) C + ( a 2 / 3 ) P + (a3/4)C3 (19)
In Figure I , the predicted D* values obtained in this manner (upper solid curve) may be compared with the measured D* values. Good agrccment is obtained up to 0.06 mol L-I. Above 0.06 mol L-I? the predicted values are slightly (-2%) higher than the measured values. Mutual diffusion coefficients can be predicted from the measurcd intradiffusion cocfficicnts by combining eqs 15 and 18 to obtain Dprcd/1 0-9 m2 s-I = D*O
+ 2a ,*C + 3a2*C2 + 4a3*C3
+ b,C + b2P + b,C3 (24) = D*O + b,*C + b2*Q + b3*C3
(t7/7°)Dmeas/10-9m2 s-' = Do
Figure 1. Measurcd intradiffusion coefficients ( 0 ,ref 15) and measured mutual diffusion cocfficicnts (0, this work) of aqueous caffeine at 25 "C. Predicted intradiffusion coefficients: upper solid curve, eq 19 (no viscosity correction); upper dashed curve, eq 26 (viscosity correction included). Prcdictcd mutual diffusion coefficients: lower solid curve, eq 20 (no viscosity correction); lower dashed curve, eq 27 (viscosity correction included).
D*p,,d/10-9 m2 s-I = Do
To test this refinement, the equations
(v/v0)D*m,,s/10-9 m2 s-'
(25) were fitted to the measured diffusion coefficients ( D o = 0.7644 X m2 s-I, b, = -5.996 L mol-', 62 = 71.054 L2 mol-2, 6, = -335.40 L3 mol-,, D*' = O.77S2 X mz s-l, b l * = -3.231 L mol-', b2* = 22.860 L2 mol-', b3* = -66.51 L3 mol-,). The relative viscosities used in the calculations were taken from ref 15. Proceeding as before, the expressions D*pred/ Dpred/1
= ( v o / v ) [ D o+ ( b l / 2 ) C + (b2/3)Q + (b3/4)C31 (26) Ill'
S-I
m2 S-I = (v0/v)[D*O
+ 2b1*C + 362*Q + 463*C3] (27)
are obtained for the predicted diffusion coefficients. The dashed curves in Figure 1 give the values of D* and D that are predicted with the help of the viscosity correction. Excellent agreement with the measured values (2% or better) is obtained over the complete concentration range. In conclusion, the laws governing dilute solutions lead to a simple relation between the intradiffusion and mutual diffusion coefficients of dilute associating solutes which has been successfully tested for aqueous caffeine at concentrations up to 0.10 mol L-I. The relation between the two coefficients may be used to check the consistency of diffusion data or to predict intradiffusion coefficients from measured mutual diffusion coefficients, and vice versa. Acknowledgment. This work was supported by the National Sciences and Engineering Research Council. Appendix Mutual Diffusion. Substituting the mass balance relation m
(20)
J = xiji i= I
(lower solid curve in Figure I ) . Satisfactory agreement with the mcusured D values is obtained up to 0.03 mol L-I, but the predicted valucs arc too low at higher concentrations. Study of eqs 17-20 reveals the unexpected result that the limiting slopc of D against C is twice the limiting slope of D*, i.e.. o1 = 2a,* or dD dD* lim - = 2C-o d C dC
The approximations leading to eq 2 1 become exact as the concentration drops to zero. Therefore, eq 21 is an exact limiting relation. For aqueous caffeine, the present mutual diffusion coefficient data yield al = -6.20 L mol-', while 2a1* = -6.90 L mol-', in good agreement with predicted behavior. Thc assumption that the species have constant diffusion coefficicnts is probably the poorest approximation in the derivation
and eq 9 into the mutual diffusion eq 3 gives m
i= I
(A21
i= I
Combining the derivative of eq 8 Vci = iKicli-lOcl (i with eqs 5 and A2 gives
+
>
1)
('43)
+
D I V c l + 4K2clD2Vcl 9K,cI2D3Vc, ... = D(OC, + ~ K ~ c I V+C~ ~K ~ c I ' V C+I ...) (A4) The common factor Oc, can be eliminated from eq A4. Multiplying the result by cl, and noting that ci = K,cli for i > 2, provides m
m
x i 2 c i D i = D x i 2ci
i= I
(34) Bearman, R. J . J . Phys. Chem. 1961, 65, 1961.
m
x i j i = -DVCici
and hence eq 1 I .
i= I
('45)
8744
The Journal of Physical Chemistry, Vol. 94, No. 24, 1990
Intradiffusion. The flux density and the concentration of the total tagged monomers are sums of the flux densities Ji* and the concentrations C,* of tagged monomers held by each of the A, species
Additions and Corrections Since the tagged and untagged monomers are randomly mixed at each point in the solution, the fraction of the total tagged monomers held by species A, equals the fraction of total monomers held by that species.
m
J* =
cJi* I=
C,*/C* = i c , / c
I
During an intradiffusion experiment, both Cand ic,/C are uniform throughout the solution, and hence VC,* = (iC,/OVC*
Thc ratc of transport o f tagged monomers as A, is governed by thc diffusion coefficient of that species, and hence
Substitution of eq AI I into eq A9 gives
J,* = -D,TC,* (A81 Subqtitution of eqs A6-A8 into the intradiffusion eq 4 gives
~ i i c , D , *= D * Z i c ,
m
m
~ D , V C , *= D * Z V C , * I=
I
(A101
,=I
(‘491
m
I=
m
I
from which eq 6 is obtained. Registry No. Caffeine, 58-08-2: water, 7732-18-5.
ADDITIONS AND CORRECTIONS Roger S . Grev,* Ian L. Alberts, and Henry F. Schaefer 111: C3+ Is Bent. Page 338 I . Due to an error in the input data for the computer program, the higher excitation results in Table I 1 are in error. Table I 1 should be replaced by the following table. With the corrected data, and employing reasoning similar to that originally given, we estimate the actual difference in energy between the *Xu+and *B, states of C3+to be around 4 kcal/mol with the DZP basis set, instead of 7 kcal/mol estimated from the erroneous data. This value is probably accurate to f4 kcal/mol. Given the huge changes in barrier height with electron correlation method, C3+ is clearly one of the “problem molecules” in computational quantum chemistry. Acknowledgment. We gratefully acknowledge correspondence with J. M. L. Martin, J . P. Francois, R . Gijbels, and P. R. Taylor concerning this problem. TABLE 11: Total Energies (hartrees) and Relative Energies (kcal/mol) of the *Cy+and ‘B, States of C3+ Obtained with the DZP Basis Set at the CISD/TZZP Optimized Geometry Employing CISD, ClSD+Q, CISDTQ, and FULL CI Energies with Various Choices of Actite Space Orbitalso
I 2 orbitalh ClSD CISD+Q CISDTQ FULL CI 2 1 orbital?, ClSD ClSD+Q CISDTQ 3 5 orbitals ClSD CISD+Q CISDTQ 45 orbital!, ClSD CISD+Q 4 S e c text for dctails.
,=I
- I 13.04280 -113,05366 - I 13.06858 - I 13.06823 - 1 13.06950 - I 13.06973 - I 13.072 19 -1 13.07056 - 1 13.14591 -1 13.16670 -113.18694 -113.19696 - 1 13.202 89 - 1 13.208 29 -113.21383 -113.23692 -113.26488 -113.27775 -113.28696 -113.29354 -113.23245 -113.25745 -113.28563 -113.30054
6.8 -0.2 0. I -1 .o 13.0 6.3 3.4 14.5 8.1 4.1 15.7 9.4
(AI 1)
