Predictive Theory for Multicomponent Diffusion Coefficients

---

(10) Kang, T. L., Hirth, L. J., Kobe, K. A., McKetta, J. J., J.Chem. Eng. Data 6,220 (1961). (11) Kestin, J., Wang, H. E., Physl’ca 26, 575 (1960). (12) Landolt-Bornstein, “Zahlenwerte and Funcktionen,” springer-Verlag, Berlin, 1950. (13) Mason, E. A., Monchick, L., J . Chem. Phys. 36,2746 (1962). (14) Mason, E. A., Vanderslice, J. T., YOS,J. M., Phys. Fluids 2, 688 (1959). (15) Mavroyannis, C., Stephen, M. J., Mol. Phys. 5,629 (1962). (16) Mpnchick, L., Mason, E. A., J . Chem. Phys. 35, 1676 (1961). (17) 0 Connell, J. P., Prausnitz, J. M., “Advances in Thermophysical Properties at Extreme Temperatures and Pressures,” ASME, p. 19, 1964. (18). Prigogine, I., “The Molecular Theory of Solutions,” Interscience, New York, 1957. (19) Rappenecker, R., Z . Physik. Chem. 72, 695 (1910). (20) Rowlinson, J. s., Trans. Faraday SOC.45, 974 (1949).

(21) Rushbrooke, G. S., Zbid., 36, 1055 (1940). (22) Saxena, S.C., Joshi, K. M., Phys. Fluids 5,1217 (1962). (23) Stewart, W. W., Maass, O., Can. J . Res. 6, 453 (1932). (24) Sutton, J. R., “Progress in International Research Thermo and Transport Prop.,” ASME, 1962, p. 266. (25) Titani, T., Bull. Chem. SOC.Japan 5, 98 (1930). (26) Zbid., 8, 255 (1933). (27) Trautz, M., Heberling, R., Ann. Physik. 10, 155 (1931). (28) Trautz, M., Narath, A., Zbid., 79,637 (1926). (29) Trautz, M., Ruf, F., Did., 20, 127 (1934). (30) Trautz, M., Weizel, W., Zbid., 78, 305 (1925). (31) Trautz, M., Winterkorn, H., Zbid., 10, 511 (1931). (32) Trautz, M., Zink, R., Zbid., 7,427 (1930). (33) Vogel, H., Zbid., 43, 1235 (1914). RECEIVED for review March 21, 1966 ACCEPTED November 7, 1966

PREDICTIVE THEORY FOR MULTICOMPONENT DIFFUSION COEFFICIENTS H A R R Y T. C U L L I N A N , J R . , A N D M . R . C U S l C K Department of Chemical Engineering, State University of New York at Bujalo, Bufalo, N . Y The modified absolute rate theory is extended to multicomponent systems, and expressions for the variation of the multicomponent friction coefficients with composition are obtained. These relationships, in terms of the values of the diffusion coefficients of the various binary pairs a t infinite dilution, are confirmed for some recently studied ternary systems. Ternary diffusion coefficients are calculated on the basis of the predicted friction coefficients. The theory correctly predicts the relative magnitude of the main diffusion coefficients in all cases with an average absolute deviation of 5%. The theory is also shown to predict properly the relative magnitude of the cross diffusion coefficients as well as to reproduce the correct sign on these terms, ECENTLY, a

relationship has been presented which success-

R fully predicts the variation of the binary mutual diffusion

coefficient with composition for a wide variety of liquid and solid systems. With associated systems the apparent sole exception, the binary diffusion coefficient a t any point in the binary composition field is given in terms of a thermodynamic factor and the diffusion coefficient a t the two composition extremes (4, 74). Since fairly adequate semiempirical methods are available for the prediction of the infinitely dilute values of the binary coefficient (75),the utility of this relationship rests entirely on the accessibility of the thermodynamic factor, which is essentially a composition derivative of chemical potential. Unfortunately, methods for the independent prediction of binary liquid and solid phase thermodynamic activity are not very well developed, and one must rely on experiment whenever possible. The situation in the case of multicomponent systems is, of course, even less hopeful. T h e interrelationship between the diffusional behavior and the thermodynamic properties has long been qualitatively evident, yet no real quantitative framework to describe this interrelationship has been forthcoming save for the fairly obvious idea, popularized by the application of the principles of irreversible thermodynamics (7), that gradients of chemical potential are the true driving forces for the diffusion process. T h e real objective in multicomponent systems is a predictive theory for the “practical” multicomponent diffusion coefficients defined on the basis of gradients of composition as driving forces. T h e multicomponent diffusion coefficients defined on a consistent practical basis ( 3 ) are related to the binary coefficients of the various possible binary pairs a t the limits of the composition field (72), and thus ultimately to the infinitely dilute values of the binary 72

I&EC FUNDAMENTALS

coefficients. In view of this, the extension of the abovementioned remarkably successful binary theory to multicomponent systems offers a distinct possibility for the construction of a workable predictive framework. I n this paper, then, the modified absolute rate theory as previously applied to binary systems (4) is extended to a general N-component mixture to yield expressions for the concentration dependence of multicomponent friction coefficients in terms of infinitely dilute values of the various binary coefficients and limiting values of thermodynamic factors. Generalization of the Binary Development

Paralleling the previous binary development ( 4 ) , a free energy barrier is postulated for the transport of a molecule of diffusing species between two successive equilibrium positions. Such barriers exist for each of the A’ diffusing species in a multicomponent system and, indeed, persist even in a completely homogeneous (on a macroscopic scale) mixture. When macroscopic chemical potential gradients are imposed on such an equilibrium state, the energy barrier for each species is distorted. The general relationships may be developed by considering the one-dimensional case. Thus, the distortion of the free energy barrier for any species may, to a first approximation, be represented by

where =k refers to the direction of a jump, a is the distance between equilibrium positions, and bpi/bX is the macroscopic gradient of chemical potential. The specific rate for the diffusion of any species is, according to the theory of absolute rates (8),

limiting Values of Friction Coefficients

-

The net diffusional velocity of any species in the direction of the gradient is

Viz = o ( v ~ +-

vi-)

i

= 1, 2 . .

.N

Thus, the limiting values of the i j friction coefficient a t each of the N corners of the multicomponent composition field are required. Two of these are directly accessible from the known binary limit (7, 4)

(3)

The above relations are linearized with the restriction that RT a"'/laXwith the result, generalized to three dimensions,

>>

lim F,,

-

xi+xj+1

%C

This gives directly

(4)

RTVi

lim Fi, = -

q i

Xli'

Again paralleling the modified binary development (4), Equation 4 is used to calculate the velocity of any species relative to each of the other species. Thus a set of '/z N (A' - 1) independent relative velocities is generated which contains all N gradients of chemical potential. I n view of the Gibbs-Duhem relation

R T V,

lim F,, = -X j-1

a,

so that one is left with

n.

2; Ci VPt = 0

(5)

a

one nonindependent gra.dient may be eliminated and the set inverted to yield expressions of the form -vpt

=

hN -Ca2

C,

i

(y, - &)

i

=

1 , 2 . . .N- 1

where

Equations 6 and 7 are the multicomponent generalizations of the previous binary expressions (4). Equation 7 represents a formal recognition of the existence of an activated state (and a net activation energy) associated with each i - j independent frictional interaction. More explicitly, if one compares the above results with the defining equations for the multicomponent friction coefficients ( 7 , 6, 9) =

--E F4'5 3

This completes the definition of the matrix of friction coefficients which, of course, is symmetrical. There are N equations of the form of Equation 8, but in view of Equation 5, only N - 1 are independent. By using Equation l G , an alternate set of relations can be constructed : N

VPi =

N

-Vpi

-

with N 2 remaining limits to be characterized. T o approach the remaining values, consider Equation 8 in which the coefficients of the form F,, do not appear. The apparent arbitrary nature of the matrix of friction coefficients is removed by the requirement that the local rate of free energy dissipation must vanish when all the velocities are equal ( 7 7 ) . O n the basis of the form of the dissipation function, this leads directly to the requirement that (9, 7 7 )

(_Vi

- _Vi)

Fij

c5

- YK)

(17)

j

(8)

where _VK is an arbitrarily selected species velocity. Now clearly

the following identification can be made:

(9) which is formally equivalent to the previous results for binary mixtures (4). Remarkable success has been obtained in correlating binary diffusion data (74) by simply taking the net activation energy for the binary i - j frictional process to be a linear function of mole fraction (4). If onie accepts the F,, as being truly representative of independent interaction effects, then the following generalized mixing rule is indicated:

JiK

=

ci (yt -

VK)

(18 )

is just the diffusional flux of species i in a reference frame fixed with respect to species K-Le., a solvent reference frame with species K the solvent. Thus

Equation 19 may be written for N - 1 independent gradients and if one considers the set such that i F K,then the Fi,are the elements of a square matrix of order N - 1. For the K-fixed fluxes the fundamental phenomenological equations are N

which represents a hyperplane in N-dimensional space. Combination of Equations 9 and 10 yields and the practical formulations are N

VOL. 6 NO. 1

FEBRUARY 1967

73

From Equations 19 and 20 the Ftj are simply the negative of the elements of the inverse of the matrix of LtjK. From Equations 20 and 21, the familar result is N

(

M

where the p15K are of the form, $)cM,

j,K.

(4),

Direct consideration of Equation 21 in the limit gives lim

JtK

=

Xi+XK+l

-

(

lim Dit.)

VCt

(23)

xi+xK+1

where this clearly represents the uncoupled binary i - k limit. The coefficient appearing on the right side of Equation 23 is certainly related to the binary coefficient, D t K . If one now 0, then the volume-average velocity passes to the limit, X t becomes identical to the velocity of species K and it becomes apparent that (72) -+

lim D f t K =

9 0 tK

xK+l

Furthermore, in the same limit (72) lim DtjK = 0 i =F j xK+1

so that the matrix of practical coefficients reduces to diagonal form with elements of the form, 6,5 DpK. In this limit, Equation 22 may be inverted. Recalling that the Ft, are the negative of the elements of the inverse of the matrix of L i j K ,the result is lim Fij =

xK41

1

-lirn 9:k XK+1

(26)

prjK

where, of course, j =tK , since the matrix of the F,,, as defined, does not contain elements of the form F i g . Furthermore, since FtI = F j t , Equation 26 is valid only for ij =k K. Predictive Relationships

The limiting values of the thermodynamic factors appearing in Equation 26 may be rewritten as lim p t j K =

RTVK

--

XK41

where 1

- = ff

tP

lim XK-1

be shown in a more indirect manner by direct substitution of Equation 29 into Miller's equations (70) for the reciprocal relations and subsequent use of inverted forms of Dunlop's (6) equations (see Equation 33). Equation 29 is the multicomponent generalization of the previously binary results ( 4 ) . In that case, no thermodynamic factors appear in the expression for Ft, [there are N - 2 of them in general for each of the l / 2 N ( N - 1) friction coefficients], so that Equation 11 can be used directly to yield

( - 2)

4-

1

1 f j,K

(28)

With Equations 26 and 27, Equation 15 may be cast in the form

which has been confirmed for a large number of binary systems

(74). The equivalent solution for multicomponent diffusion coefficients (6) involves ( N - 1)2 thermodynamic derivatives in addition to the '/zN ( N 1) ( N 2) limiting thermodynamic derivatives appearing in Equation 29. In a completely ideal multicomponent system, each component is similar to each of the others; all binary diffusion coefficients are constant and equal. In this case, the diffusion is uncoupled, only one diffusion coefficient is needed to describe the system @), and it can be shown that all the friction coefficients are of the form

-

F

-

RT

"

--* - CD

when D* is the constant diffusion coefficient for the system. The indication from Equation 29 is that in the ideal case all of the thermodynamic factors must be unity. This is a remarkable result, since it cannot be deduced from a direct consideration of the thermodynamics-that is, even though the derivatives, p c j , vanish for an ideal system within the composition field, the chemical potential of any species approaches minus infinity as that species becomes infinitely dilute. Hence, all limiting values appearing on the left side of Equation 27 are formally indeterminate. In other words, it is not possible to pass from Equation 29 to Equation 32 for the ideal case because the limiting thermodynamic factors cannot be evaluated despite the fact that the thermodynamics are known. One is forced to take an independent route leading to Equation 32 which, upon comparison, results in the restriction on the limiting values of the thermodynamic factors for ideal systems. For nonideal systems, the restriction that the friction coefficients must be real, positive quantities (9) leads to the restriction that the limiting thermodynamic factors must be positive. This also does not appear to be deducible on the basis of thermodynamic reasoning alone. However, unlike the result for ideal systems, this present restriction is independent of the particular model presented here for the diffusion process, since Equation 26 is a general result. Test of Theory

(29) Of course, this result is subject to the reciprocal relations. In particular, it must be that Ft, = Fjt in the limit as X K -+ 1, K # i,j. Thus, from Equation 26 one has 1 1 - lim p r j K = - lirn p j i K D;K

XK41

D°K x K 4 1

(30)

so that in view of the definition of the a t j Kgiven by Equation 27, the symmetry of Equation 29 is confirmed. This may also 74

l&EC FUNDAMENTALS

The direct evaluation of Equation 29 presents considerable difficulty, not because of the lack of reliable diffusion data, but rather because very accurate thermodynamic data are required. Considerable data for the diffusional behavior of ternary liquid systems are available (2, 5, 72). However, in the absence of experimental data on t t z required thermodynamic properties, the only alternative is to make estimates of the ternary thermodynamics based on the known binary properties. The resulting estimates of the four thermodynamic derivatives at any point within the ternary composi-

tion field, from which the three friction coefficients could be calculated using the four diffusion coefficients, may be rather crude. T h e one limiting thermodynamic factor appearing in Equation 29 cannot be evaluated by these methods with any accuracy a t all, so that no meaningful comparison is possible by this route. However, a definitive test of the theory is possible using the data of Shuck and Toor (72). I n that work, the four ternary diffusion coefficients, dehned with respect to a volume average velocity, for the system methanol-isobutyl alcohol-n-propyl alcohol were reported a t four composition points within the ternary field. Also, excellent agreement with Miller’s (70) equations for testing the reciprocal relations was obtained by neglecting the change of activity coefficients with composition. Thus, for this system it appears that very good estimates of the actual friction coefficients can be obtained by using the equations of Dunlop (6) with the same thermodynamic assumption. For this system the various infinitely dilute values of the binary coefficients are also reported (72). According to the ternary form of Equation 29 there should be one thermodynamic factor, a, for each of the three independent friction coefficients, which holds throughout the entire composition field. Since the ultimate aim of the theory is the prediction of the diffusion coefficients which are inversely proportional to the friction coefficients, a n a for each friction coefficient was determined so as to minimize the sum of squares of deviations of the inverse of the F,, calculated from Equation 29 from the inverse of the actual F1.jgiven by the equations of Dunlop (6). This was done on a digital computer using a simple Kewton’s iterative procedure. For the diffusion coefficients as reported by Shuck and Toor (72), the equations of Dunlop (6) give two independent estimates of FhfB-i.e., F, and F B M where FMrr= FBni. Even though the reciprocal check was excellent, it was not exact, so the actual FMB for this system was taken as the arithmetic average of the two independent estimates. In Table I, the resulting values of the three a’s for Shuck and Toor’s (72) system are given, and the actual friction coefficients are compared with those calculated using Equation 29 a t each of the four composition points. As can be seen, the agreement is excellent with a n average absolute error of less than 5%. T h e factors, (3,are close to unity which, according to the above arguments, is precisely what one would expect in a system which is not far removed from thermodynamic ideality. Although the agreement of the theory with respect to the friction coefficients in this system is remarkable, the practical test of the utility of the theory is in regard to the predictability of the diffusion coefficients themselves. T h e equations of Dunlop (6) are explicit expressions for the friction coefficients in terms of the diffusion coefficients for a ternary system. These may be inverted to give explicit expressions for the diffusion coefficients (defined in a volume reference frame) in terms of the friction coefficients. T h e results may be summarized as follows:

Table II.

c1

a

C2 Ca C4 All D’s in sq.

CornposilionC CI

Predicted Friction Coefficients for System MethanolIsobutyl Alcohol-n-Propyl Alcohol ( 1 3 ) ffhfBa = 7.078 f f M p = 7.233 f f g p = 0.762 _______ FMB~ Calcd., Actual Eq. 29

3.735 4.995 5.531 C.j 1.888 a M = methanol; B = A l l F’s are in cal. sec.,J(g. Cp Cs

3.770 4.801 5.978 2.050

FMP Calcd., Eq. 29

FBP

Calcd., Eq. 29 3.647 3.516 4.758 4,609 4.459 4.286 5.980 5.758 5.619 5.825 7.059 7.430 1.887 1.769 2.286 2.320 isobutyl alcohol; P = n-propyl alcohol. mole)2 sq. cm. X 70-8. See (73) for Actual

Actual

actual compositions.

(33) where

(334

(334

Equation 33 was used to calculate the ternary diffusion coefficients a t the four composition points of Shuck and Toor (72), using the calculated values of the friction coefficients from Equation 29 which appear in Table I. These results are compared with the actual measured diffusion coefficients in Table 11. As can be seen, the theory is in very good agreement. In all cases the relative magnitude of the main diffusion coefficients is properly predicted and the average absolute deviation for the main coefficients is less than 5y0. T h e absolute deviation for the cross diffusion coefficients is larger, percentagewise, but the magnitude of these coefficients is relatively small in this system. T h e fact that in all cases the relative magnitude of the cross coefficients is properly predicted, and, in every case save for a single marginal one, the correct sign on the cross coefficients is predicted, seems to indicate substantial agreement with the theory. A comprehensive verification of the theory, analogous to that of Vignes (74) in the binary case, is not possible in the multicomponent case, primarily because of the lack of reliable thermodynamic data. On the other hand, it appears expedi-

Predicted Diffusion Coefficients for System Methanol-Isobutyl Alcohol-n-Propyl Alcohol (13)

L)MM. Composition

Table I,

-

DBB

Actual

Calcd., Eq. 33

1.04 0.909 0.765 1.51

1.06 0.936 0.810 1.61

Actual

Calcd., Eq. 33

Actual

0.875 0.721 0.624 1.38

0.903 0.736 0.650 1.46

0.032 0.030 0.027 0.211

DMB

DBM

Calcd., Eq. 33

0.057 0.049

-0.001 0.181

Actual

-0.023 -0.009 -0.039 -0.004

Calcd., Eq. 33 -0.023 -0.002 -0.062 -0.011

cm./sec. X 705.

VOL. 6

NO. 1

FEBRUARY

1967

75

ent simultaneously to extend the verification of the theory as well as to test its present practical applicability. In this regard, Cullinan and Toor (5) have reported a comprehensive study of the diffusional behavior of the nonideal ternary liquid system acetone-benzene-carbon tetrachloride. Although reliable thermodynamic data on the three binary pairs are available, no ternary thermodynamic data have been reported. Since this is a fairly typical situation in the case of multicomponent liquid systems, it appears worthwhile to test the above theory of diffusion within the framework of the present state of knowledge with respect to the thermodynamic properties of solutions. For the ternary system acetone-benzene-carbon tetrachloride, the available binary activity coefficient data (73) were fitted with three suffix Margules equations, and the extensions of Wohl (76) to ternary systems were used to estimate the activity coefficients within the ternary field. By differentiating the resulting expressions with respect to composition, the various composition derivatives of chemical potential were estimated; and these results were used in the equations of Dunlop (6) to calculate friction coefficients from reported diffusion coefficients a t four concentrated ternary composition points reported by Cullinan and Toor (5). Since binary diffusion coefficients were also reported in this study (5), an cy for each of the three friction coefficients was determined so as to minimize the sum of squares of deviations of the inverse of the F i j calculated from Equation 29 from the inverse of the “actual” F,, given by the equations of Dunlop ( 6 ) . Once again the two independent estimates of one of the three friction coefficients were normalized by a simple averaging procedure. In Table 111, the resulting values of the three CY’S for Cullinan and Toor’s (5) system are given and the actual friction coefficients are compared with those calculated using Equation 29 a t each of the four composition points. Once again the overall agreement is remarkable, with a n average absolute deviation of less than 5%. Finally, Equation 33 was used to calculate the ternary diffusion coefficients a t the four concentrated composition

Table 111.

Predicted Friction Coefficients for System AcetoneBenzene-Carbon Tetrachloride (5) t y ~ ~= ’ 0.969 = 1.816 age* = 0.787

FAC FBC FAB‘ Calcd., Calcd., Calcd., Actual Eq. 29 Actual Eq. 2 9 Actual Eq. 29 C1 1.935 2.126 1.937 1.903 2.499 2.688 2.590 3.057 3.336 3.043 2.605 C2 2.930 1,986 1.648 1.671 2.827 2.703 CB 2.337 1,429 1.504 1.485 2,050 1.984 C4 1.429 a A = acetone; B = benzene; C = carbon tetrachloride. All F’s See ( 5 )for actual compositions. in cal. sec./(g. mole)%sq. cm. X 10-9. CompositionC

Table IV. Comp osition c1

C2 Ca C4 a

76

Conclusions

T h e verification of the multicomponent theory as presented here is hardly as substantial as that for the corresponding binary theory (74). However, the gap lies in the area of thermodynamic information, not only for the further testing of the range of applicability of the theory, but also (and potentially far more important) for the practical application of these results. I t is hoped that these promising preliminary successes will stimulate efforts in this direction. From a more fundamental viewpoint the application of the modified absolute rate theory to the diffusional behavior of multicomponent liquid systems should lead to a more substantial understanding of the liquid state ; or, more specifically, to a better characterization of the relationship between the transport phenomena and the thermodynamic properties in such systems. In the light of the range of validity of the corresponding binary theory (4, 7 4 , coupled with the present demonstrated success with ternary systems, it might be expected that the multicomponent theory is valid for all completely miscible multicomponent liquid and solid mixtures save for associated systems. Of course, confirmation of this is not a t present possible, as indicated previously. T h e total practical applicability of these results must await more detailed characterizations of multicomponent solution thermodynamics. In lieu of this, available estimating procedures (76) similar to those used here appear to be adequate, provided that the limiting thermodynamic factors appearing in Equation 29 are known. According to the procedure presented here, a t least one multicomponent diffusion data point (for a ternary system) is required for these. For systems of more than three components, each of the limiting thermodynamic factors can be obtained from ternary diffusion experiments on the appropriate limiting ternary trio. Of course, the required limiting values of the various binary diffusion coefficients can be estimated adequately (75) if data are lacking. Finally, if the system in question is not far removed from thermodynamic ideality, the assumption of limiting thermodynamic factors of unity should give adequate estimations of multicomponent diffusion coefficients.

Predicted Diffusion Coefficients for System Acetone-Benzene-Carbon Tetrachloride (5)

DAA~ Calcd., Actual Eq. 33

1.89 1.60 1.96 2.33

points of Cullinan and Toor (5) using the calculated values of the friction coefficients from Equation 29 which appear in Table 111. These results are compared with the actual measured diffusion coefficients in Table IV. Again, the relative magnitude of the main diffusion coefficients is properly predicted in all cases, and the average absolute deviation for the main coefficients is about 5%. For the cross diffusion coefficients, the relative magnitude is properly reproduced in all but one case (in this one case the values are almost indistinguishable) and the correct sign is predicted in all but one case [in this case, the 9595 confidence limits, as reported (5), overlap zero].

1.84 1.63 2.30 2.35

All D’s in sq. cm./sec. X IO6.

l&EC FUNDAMENTALS

DBB Actual

2.26 1.81

1.93 2.97

DAB Calcd., Eq. 33

Actual

2.08 1.65 2.04 3.04

-0.213 -0,058 0.013 -0.432

DBA Calcd., Eq. 33

Actual

-0.144 -0.084 -0.057 -0.351

-0,037 -0.083 -0.149 0.132

Calcd., Eq. 33 -0.018

-0.060 -0.244 0.140

Nomenclature

factors defined by Equation 33 distance between equilibrium positions C = total molar concentration Ci = molar concentration of species i ~ i j = binary mutual diffusion coefficients DijK = multicomponent diffusion coefficients defined by Equation 21 D i j v = multicomponent diffusion coefficients defined in a volume average reference frame D* = diffusion coeffic,ient defined by Equation 32 Fi; = friction coefficients defined by Equation 8 fij = factors defined by Equation 33 AGi* = free energy barrier for diffusion of species i AGi, = free energy ba.rrier for diffusion of species i in a homogeneous mixture AGij = net activation energy for the i j frictional interaction h = Planck constanr = diffusional flux of i relative to K JiK k = Boltzmann constant L . .K = phenomenological coefficients defined by Equation 20 = Avogadro’s number R = gas constant T = absolute temperature Vi, = diffusional velocity of i in x direction = velocity vector of species i Vi = molar volume of pure i Vi = partial molar vlolume of i X = distance coordinate Xi = mole fraction of species i aij

a

= activity coefficient of species

yi

= =

-

i’

vi

i

Kronecker symbol = chemical potential of species i = chemical potential composition derivative defined by Equation 22 = frequency of jumps of species i =

&j

pi pij vi*

SUPERSCRIPT 0

=

infinite dilution of indicated species

literature Cited (1) Bearman, R. J., J . Phys. Chem. 65, 1961 (1961). (2) Burchard, J. K., Toor, H. L., Zbid., 66,2015 (1962). 4, 1 3 3 (3) Cullinan, H. T., IND. ENG. CHEM.FUNDAMENTALS 11965) ,- -- ,. (4) Zbid., 5 , 281 (1966). (5) Cullinan, H. T., Toor, H. L., J . Phys. Chem. 69, 3941 (1965). (6) DunloD. P. J.. Zbid.. 68. 26 (19641. (7) Fitts, D: D., “Non-Eq;ilib;ium Thermodynamics,” McGrawHill, New York, 1962. (8) Glasstone, S. K., Laidler, K. J., Eyring, H., “The Theory of Rate Processes,” McGraw-Hill, New York, 1941. (9) Laity, R. W., J . Phys. Chem. 63, 80 (1959). (10) Miller, D. G., Zbid., 63, 570 (1959). (11) Onsager, L., Ann. N . Y. Acad. Sci. 46,241 (1945). (12) Shuck, F. O., Toor, H. L., J . Phys. Chem. 67, 540 (1963). (13 ) Timmermans, “Physico-Chemical Constants of Binary Systems in Concentrated Solutions,” Vol. I, Interscience, New York, 1959. (14) Vignes, A., IND. END.CHEM. FUNDAMENTALS 5,189 (1966). (15) Wilke, C. R., Chang, P., A.Z.Ch.E. J . 1, 264 (1955). (16) Wohl, Trans. A.Z.Ch.E. 42, 215 (1946). ~

GREEKLETTERS = factor defined b’y Equation 28

RECEIVED for review March 28, 1966 ACCEPTED September 19, 1966

ai/

ESTIMATING LIQUID DIFFUSIVITY K . A . R E D D Y A N D L. K . D O R A I S W A M Y National Chemical Laboratory, Poona, India

The Wilke-Chong correlation for predicting liquid diffusivities has been modified by replacing the association parameter of this equation by the square root of the solvent molar volume. Based on this, equa1.5, which represent the experimental data with tions have been proposed for VZ/VI 5 1.5 and V~/VI average deviations of 13.5 and 18%) respectively. VI and Vz represent the molecular volumes of the solute and solvent, respectively.

>

THE most widely used equation for estimating liquid diffusivity is that of Wilke: and Chang (6), which is based on the Stokes-Einstein equation. I n this equation the so-called solvent factor of the earlier Wilke equation (5) was replaced by a n association parameter, which could be taken as unity for unassociated solvents. Scheibel (3) proposed another equation in which the solvent factor was eliminated by the introduction of the molal volumes of the solvent and the solute. Subsequently Kuloor and coworkers (4) replaced the association parameter of the Wlke-Chang equation by the latent heats of the solvent and solute. I n the present note new equations are proposed for estimating liquid diffusivity, which are largely free from the limitations of the Wilke equations. I n the Scheibel correlation the exponent of the solute molal volume is as against 0.6 in the Wilke-Chang equation. This exponent has been modified to 0.455 by Kuloor et al. By retaining the exponent of and introducing the solvent molal volume, also raised to the same power, the association parameter of the Wilke-Chang equation can be replaced, thus

eliminating the only drawback of this useful equation. [In Eyring’s equation also ( I ) , when the liquid free volume is expressed in terms of the liquid molal volume, a n exponent of is obtained for V Z . ] This results in a n equation of the form

T h e constant of Equation 1 has been found to depend on the relative molecular volumes of the solvent and solute. Thus: CASE1

v-z 5

1.5

v 1

VOL. 6

NO. 1 F E B R U A R Y 1 9 6 7 77