Predictive Theory for Multicomponent Diffusion Coefficients

---

62.

= concentration a t surface of film with lowest concentra-

tion D1 = true diffusion coefficient D o = diffusion coefficient a t lowest concentration in a n experiment (assumed zero) D = averaged diffusion coefficient F,,, = correction factor to be applied to b according to method used k = see Equation 2 c, = surface concentration c, = initial concentration L’ = film thickness of a film attached to a substrate L = film thickness of free film T = defined by Equation 5 X = defined by Equation 6 c = defined by Equation 7 K = see Equation 8a or 8b D = see Equation 8a or 8b s = defined by Equation 4 Q = mass flux a t steady state t = time x = distance

SPECIAL SUBSCRIPTS ‘/2 ‘/4

a d

based on half times based on quarter times = absorption = desorption = =

literature Cited

(1) Crank, J., “Mathematics of Diffusion,” pp. 186-91, Oxford University Press, Oxford, 1957. (2) Ibid., pp. 207-10. (3) Ibid., pp. 241-8. (4) Hansen, C. M., O$i. Digest 37, 57 (1965). ( 5 ) Kishimoto, A., Matsumoto, K., J . Polymer Sci. A2, 679 (1964). (6) Kokes, R. J., Long, F. A., J . Am. Chem. Soc. 75,6142 (1953). 4,445 (1965). (7) Long, R. B., IND.END.CHEM.FUNDAMENTALS (8) Prager, S., Long, F. A., J . Am. Chem. Soc. 73, 4072 (1951).

CHARLES M. HANSEN Technical University of Denmark Copenhagen, Denmark

RECEIVED for review November 7, 1966 ACCEPTEDApril 14, 1967

~

CORRESPONDENCE PREDICTIVE THEORY FOR MULTICOMPONENT DIFFUSION COEFFICIENTS SIR: I n a recent article Cullinan and Cusick (1967) tried to generalize to multicomponent diffusion coefficients a n empirical law of variation with composition of the diffusion coefficient in a binary system (Vignes, 1966). T h e lack of rigor of their analysis led the authors to a result (Equation 15) which is in fact incorrect and not coherent with the necessary symmetry relations that the friction coefficient must satisfy. So the theory proposed by the authors cannot be considered as a predictive theory for multicomponent diffusion coefficients. This can be shown as follows: T h e authors start their analysis with a n extension of the Stefan-Maxwell equations to nonideal mixtures (Equation 8 of their article) : N

when p i is the chemical potential, C5 the molar concentration, and Vi the velocity of species i. T h e F i j are the multicomponent friction coefficients which satisfy the symmetry relations Fi5 = F5i

(2)

Then, the authors define an activation energy AGlj for each friction coefficient (Equation 9 of their article) (3)

of the multicomponent composition field. Two of these are directly accessible from known binary limits (Equations 13 and 14 of the paper),

RT V ~

lim F i j = -

DYL

X,+l

RTV,

lini Fij = D; j

x,--tl

where Dit is the diffusion coefficient of species j present a t infinite dilution in species i. Limiting the discussion to a ternary system, for the third limit, Cullinan and Cusick proposed the following expression (Equation 26 of their paper)

where the p:j are of the form

I t will be shown that this limiting form (7) is incorrect and has no justification and that it leads to an inconsistency. Following the derivation of the authors, the friction coefficient F 1 2 can be expressed as a function of the practical diffusion coefficients D$ defined in a k-fixed reference frame, by (9)

and they state as a starting assumption that this activation energy is a linear function of the composition of the system, in terms of mole fraction X , (Equation 10 of their article) : N

AGtj =

k

X , lim AGt5

(4)

Xk-1

Then the authors proceed to the evaluation of the limiting values of the friction coefficients a t each of the N corners 614

I&EC FUNDAMENTALS

Now, when Xa+ 1, the limiting values of the practical diffusion coefficients are 071

-+

D;,

Diz

+

Dis

as dG is a total differential, we have the relation

- G3) -

a(&

and the limiting values of the p:j are:

dXZ

a(& - 8,) ax1

(1 9)

Now 83 may be eliminated by means of the Gibbs-Duhem equation Xld8i

+ XzdG2 + x3dC3 = 0

(20)

to give

So that 1 lirn F1? = ___ [ lim Xs-1 Di3Di3 Xa-+l

(%) -

lim (D;]plz)]

(1 - X2)-ad,

(13)

Xs-1

This Relation 13 shows the assumption not explicitly stated that the authors had to make in order to get the proposed limiting value given by Equation 7. This assumption is:

There is absolutely no reason for assuming Equation 14 unless we know that D:2 N kCy \There n > 1, xvhich no result shows until now. Furthermore, this unfounded assumption leads to a result lvhich is inconsistent. From their three limiting values of the friction coefficients F P I (Equations 5, 6, and 7), the authors get their final result (Equation 29 of their article)

Xk-il

Dg,

lim Xk+l

‘I

=

1

- _ lim p$ %3

Xk-11

lirn F j ,

=

(17)

Xk+l

Furthermore, they claim that this relation can be obtained from the limiting form of Miller’s equations for the Onsager reciprocal relation (Miller, 1959). First, the limiting form of Miller’s equation when X k + 1 is indeterminate (De Groot and Mazur, 1962). Furthermore, certain symmetry re1at:ions exist between the elements of the matrix (Equation 8). This can be shown in the following Jvay. Under isothermal and isobaric conditions, the total differential of the specific Gibbs function G can be written dG = d i d x i

+ 8zdXz + G3dX3

(GI - 83)dXi $- ( 8 2 -

83)dXz

ax1

(21)

-

(1

- C2Vz)p12 X3

-

+ (1 - ClVJ1121 Cz81pzz = 0

(24)

1, Ci + 0, Cz -+ 0

So in vieiv of this last relation, if Equation 17 stated by the authors was nevertheless correct, combination of Equations 17 and 25 xvould lead to the result D;k

=

TI;/,

(26)

So in order to preserve the symmetry of Equation 15 the limiting values of the diffusion coefficients of the tcvo dilute species a t each corner of the composition diagram should be equal. But in that case, it can be shown easily that only one diffusion coefficient is needed to describe diffusion processes in a ternary system. So the theory proposed by the authors cannot be considered a valid description of multicomponent diffusional processes. Furthermore, the comparison with experiment made by authors cannot be considered as a check of the validity of Equation 14. I n that part of the work, it \vas only shown that the experimental data could be fitted with the form of Equation 14 treating each a as a parameter. From a more general point of view, it seems very difficult to predict the ternary diffusion coefficients in all the composition diagram from only the limiting values of these coefficients a t each corner of the diagram. Nomenclature

where the 81 are the corresponding partial molar quantities. Taking component 3 as the solvent, this relation may be writ ten dG

XI)-

\\.‘ith respect to the elements of the matrix p t j (Equation 8), Relation 22 may be \vritten as shown by De Groot and hlazur (1962) :

So that when

1 --

ax,

This relation between the activity coefficients of the solutes is the condition of thermodynamic consistency. From this relation, it is easy to see that when

where

lim F 13. . =

ax,

This expression becomes in terms of the activity coefficients

ClVzpll

As noted by the authors, the result is not symmetric in spite of the fact that F t j must be equal to F j l . So the final result is not coherent. As the symmetry of the friction coefficient, Pi,, must be valid a t each point of the coniposition diagram in order to preserve the symmetry, the authors state from Equation 7 the following relation (Equation 30 of their article)

+ x2-aez = x1-d e l + (1 -

ax2

(18)

= distance between equilibrium positions = total molar concentration C, = molar concentration of species i D , ~ = binary mutual diffusion coefficients Dtj = multicomponent diffusion coefficients

a c

VOL. 6

NO. 4

NOVEMBER 1 9 6 7

615

F,j = friction coefficient G = specific Gibbs function AG,j = activation energy for i - j frictional interact:ion h = Planck constant N = Avogadro’s number Vi = velocity vector of species i T’; = partial molar volume of i X, = mole fraction of species i yi = activity coefficient of species i p, = chemical potential of species i p,j = chemical potential derivative

literature Cited

Cullinan, H. T., Cusick, M. R., IND.ENG.CHEM.FUNDAMENTALS 6, 72 (1967). De Groot, S. R., Mazur, P., “Non-Equilibrium Thermodynamics,” p. 262, North-Holland Publishing Co., Amsterdam, 1962. Miller, D. G., J . Phys. Chem. 63, 570 (1959). Vignes, A , , IND.ENG.CHEM.FUNDAMENTALS 5 , 189 (1966). Alain Vignes Uniclersity of dVancj *\‘a ncy , France

PREDICTIVE THEORY FOR MULTICOMPONENT DIFFUSION COEFFICIENTS SIR: Vignes has unfortunately overlooked an important detail which I believe invalidates his contention that our result is inconsistent. His claim that our result is generally in violation of the Onsager relations is erroneous. This is because his Equation 25 is incorrect, except in the special case of constant total molar concentration. To demonstrate this briefly, if the first two of Equations 12 of Vignes are used in his Equation 24, the correct version of his Equation 25 is found to be RTfl

+ lim

p;2 =

RTF2

X3+0

+ x3lim pil +0

(1)

I n other ivords, inspection of the latter tIvo of his Equations 12 shows that his incorrect Equation 25 follows from his Equation 23 (which is, of course, correct) only in the special case of constant total molar concentration. I t is true that our previous work (Cullinan and Cusick, 1967b) does not confirm Equation 14 of Vignes. The point to be emphasized here is that, even if the first term on the right of Vignes’ Equation 13 does not vanish, the following combination of Equations 26 and 27 of our paper (Cullinan and Cusick, 196713)

along with Equation 14 of Vignes may still be taken simply as a unique definition of the factor, &. This is a valid interpretation, provided that the friction coefficient remains finite. This, in turn, is implied by the linear mixing rule given by our Equation 10 (Cullinan and Cusick, 1967b), since an infinite activation energy for a n i - j interaction in dilute solution is not expected. With this alternate, less restrictive interpretation the final form is the same as Equation 15 of Vignes. Thus, our \vork (Cullinan and Cusick, 1967b) may be interpreted as confirming the application of the linear mixing rule for activation energies, but not the interpretation of the a: factors given by our Equation 27. I t is known that, independent of their interpretation, the CY factors must all be unity for an ideal liquid system. If the

616

l&EC FUNDAMENTALS

interpretation given by our Equation 27 is not valid, little hope exists for independent estimation of the CY factors for nonideal systems. O n the other hand, if our Equation 27 is valid, then recent work has shown independent estimation to be possible. Thus, if Equation 1 above is combined with Equation 17 of Vignes, the following is obtained (Cullinan and Cusick, 1967a) : (3) This equation has already been confirmed for some dilute ternary systems (Cullinan and Cusick, 1967a). Also, when used for the concentrated systems previously studied, results (Cullinan, 1967) were obtained which were only a few per cent different from our optimum-fit analysis (Cullinan and Cusick, 1967b). This can only be interpreted as confirmation of Equation 14 of Vignes for the particular systems studied. According to the statistical mechanical theory of transport in liquids (Bearman, 1961), Equation 3 does become unity for ideal systems. The generality of Equation 14 of Vignes, as applied to liquid systems, remains open to question. Some theoretical support has been advanced (Cullinan, 1967), but no definitive statement concerning the types of liquid systems expected to behave in this way is yet available. In conclusion, our theory, rather than being inconsistent, offers a rational framen.ork for the direct prediction of multicomponent diffusion coefficients. Although it is impossible to generalize Lvhen the number of measured systems available for testing is so small, all systems tested to date follow the linear mixing rule for activation energies as well as the “unfounded assumption” referred to by Vignes. literature Cited

Bearman, R. J., J . Phys. Chern. 6 5 , 1961 (1961). Cullinan, H. T., Can. J . Chem. Eng. to be published (1967). Cullinan, H. T., Cusick, M. R., A.Z.Ch.E. J . to be published (1967a). Cullinan, H. T., Cusick, M. R., IND.EKG.Cmm. FUNDAMENTALS 6, 72 (196713). Harry T . Cullinan, J r . State University of New York Buffalo, A’.Y.