Transition State Theory for Diffusion Coefficients in Multicomponent Liquids SIR: Objections have been raised (Carman, 1973) to some aspects of work reported previously (Mortimer and Clark, 1971). In this work, a transition state theory of multicomponent liquid diffusion was presented which included twobody as well as one-body processes. Numerical values of Onsager coefficients for diffusion in the volume frame calculated from the theory were found to agree approximately with values calculated from published data on the system toluene-chlorobenzene-bromobenzene (Burchard and Toor, 1962). Two objections are raised. One is to the assumption that Mortimer and Clark made, that the Kirkendall (lattice) coordinate frame can be identified with the volume frame. This assumption forms the basis of the numerical comparison between theory and experiment which was made. The other objection is the major one, and is Carman's contention that a theory without two-body diffusive processes is sufficient to explain the published data as well as data on two additional systems. The first objection requires only a little discussion. I n the solid state, the Kirkendall frame (or Darken frame) is easily identified, since the lattice structure actually occurs in the physical system in a macroscopic sense and can be located. I n the liquid, the assumed lattice exists only in disordered short-range regions and cannot be macroscopically located. One must decide from whatever considerations are available which assumed identification is appropriate. In the paper of hIortimer and Clark, the Kirkendall frame was taken by hypothesis to be the same as the volume frame. The sixth equation in Carman's communication is equivalent to assuming that the Kirkendall frame is the same as the center of mass frame if proper conjugate forces and fluxes are used. This assignment is certainly reasonable and actually possesses the advantage over the original assumption of Mortimer and Clark that the Onsager reciprocal relations are immediately obeyed using the proper conjugate forces and fluxes. I n order to discuss the second objection of Carman, it is advantageous to follow his assignment and to assume that the Kirkendall frame is the center of mass frame. M'ith this change, the numerical calculations done by Mortimer and Clark on the data of Burchard and Toor have been repeated. In the original paper, sufficient care was not taken to use properly conjugated forces and fluxes, and Onsager coefficients were used with the chemical potential gradients as forces. This is not the proper set of conjugate forces in the volume frame, as pointed out by Miller (1959), and the statement made by Xortimer and Clark that the independent Onsager coefficients are a subset of this dependent set is incorrect. Carman seems to make this same error, since his sixth equation is for the proper independent Onsager coefficients in the number frame, while his fifth equation uses the same symbols for coefficients using the dependent set of chemical potential gradients. This error has been avoided in the calculations reported in this communication by using only the independent Onsager coefficients for the proper conjugated forces in each coordinate frame. It is only for such coefficients that the Onsager reciprocal relations are known to be valid. It is apparent that comparison between theory and experiment in the volume frame leaves something to be desired, since Carman and also Kett and Anderson (1969) have made calculations on the data of Burchard and Toor leading to approximate agreement with the results of one-body or equivalent theories in this frame. I t is difficult to determine whether 492 Ind. Eng. Chem. Fundam., Vol. 12, No. 4, 1973
MOLE FRACTION
Figure 1 . Typical diagonal Onsager coefficient (mole2 cm-1 cal-1 sec-1) as a function of the mole fraction of component 2. Component 1 = toluene, component 2 = chlorobenzene, and component 3 = bromobenzene
the approximate agreement in either of these calculations is better than the approximate agreement which Mortimer and Clark obtained with their theory including two-body processes. The reason for this is that the presence of mechanical coupling is obscured by the presence of formal coupling, due to the change from the lattice frame and the elimination of the redundant force and flux. A definitive comparison must be made in the lattice frame, in which there is no formal coupling. iiny coupling found in the lattice frame must be the result of two-body processes. For this reason, the data of Burchard and Toor were used to calculate Onsager coefficients in the mass frame, which is now, in agreement with Carman, taken to be the Kirkendall, or lattice, frame. I n this frame, if Carman's theory is correct, the diagonal Onsager coefficients should be proportional to the concentration of the component involved, according to his tenth equation, and the off-diagonal Onsager coefficients should vanish. If the theory of Mortimer and Clark is correct, the diagonal Onsager coefficients should have a parabolic dependence on the concentration of the component, as given by eq 34 of $heir paper
(Lit)g
= azt
+ bz?
where z t is the mole fraction of component i, and a and b are parameters related to the free energies of activation of the one-body and two-body processes. Also, the off-diagonal coefficients should not vanish, but should be proportional to the product of the concentrations of the two components involved, as given by eq 35 of Xortimer and Clark
Furthermore, the parameter b must the same in both these relations. The results of the calculations are shown in Figures 1 and 2. Figure 1 shows LZZin the center of mass frame, where chlorobenzene is chosen as component 2. The experimental errors given by Burchard and Toor were propagated through the calculation and are shown. The curve is a parabola which was fit to the data with a least-squares procedure. The data for toluene correspond to a very similar figure with slightly less curvature. Inspection of this figure shows that the parabola is a better fit to the data than any straight line passing through the origin as predicted by the theory of Carman. This indicates that two-body processes are indeed important in this system. Figure 2 shows Lu. The abscissa is 51x2, the product of the two mole fractions. The error limits shown are
0 .05 .IO .I5 MOLE FRACTION PRODUCT Figure 2. Off-diagonal Onsager coefficient (mole2 cm-' cal-l sec-l) as a function of the product of the mole fractions of component 1 and component 2. Component 1 = toluene, component 2 = chlorobenzene, and component 3 = bromobenzene
again obtained from the estimated experimental errors given by Burchard and Toor. The straight line shown was not obtained by fitting the LIZdata but was obtained from the parameter b given by the least-squares fit shown in Figure 1. There was thus no adjustable parameter to make the line fit the data, so that the agreement with experiment is significant. The one-body theory of Carman requires that LIZvanish for all compositions, which is not in agreement with the data. In fact, for L12, none of the error bars overlaps the abscissa. A plot for L1is very similar, since the Onsager reciprocal relations are obeyed by the data of Burchard and Toor to within experimental error, but the error limits are slightly different, and two of the error bars in the LU plot do cross the abscissa.
Thus, the data of Burchard and Toor seem unambiguously to indicate that the two-body processes postulated in the theory of Mortimer and Clark are necessary to explain the data for this system, toluene-chlorobenzene-bromobenzene. The theory of Mortimer and Clark predicts the correzt composition dependence of both the diagonal and off-diagonal Onsager diffusion coefficients, and the correct relative size. The question of whether there exist many multicomponent systems in which two-body processes must be invoked must of course be answered by further experiments. Carman refers to results of diffusion experiments on the solid system ironcobalt-nickel, in which no two-body processes are deemed to be important. If this is correct, it is a reasonable result, since in a solid system the free energy of activation of a two-body process would be rather large compared to a one-body process due to the rigidity of the lattice. In a liquid, where the disordered lattice present is much less rigid, one would expect two-body processes to have a larger relative importance. Experiments are now in progress on additional liquid systems. literature Cited
Burchard, J. K., Toor, H. D., J . Phys. Chem 66, 2015 (1962). Carman, P. C., IND.ENC.CHEM.,FUNDAM. 12,484 (1973). Kett, T. K., Anderson, D. K., J . Phys. Chem. 73, 1268 (1969). Miller, D. G., J . Phys. Chenz. 63, 570 (1959). Mortimer. R. G., Clark. N. H., IND.CHEM.ENG.,FUNDAM. 10, 604 (1971).
'
Robert G. Mortimer Department of Chemistry Soulhwestern ut Memphis Memphis, Tenn. 381 12 This research was supported in part by Grant from the Research Corporation.
8
Cottrell College Science
Corrections STUDIES OF THE LOSCHMIDT DIFFUSION EXPERIMENT. II. A N IMPROVED INTERFEROMETRIC METHOD
Table IV in the paper by S. Gotoh, M. Manner, J. P. Spkensen, and W. E. Stewart [IND.ENG.CHEM.,FUNDAM. 12, 119 (1973)l needs clarification. The predicted constant e ~ ~ =/ 174.26"K k in set 2 is the geometric mean of the purecomponent values e ~ / kand e g / k predicted for Nt and nC4Hl~by correlation iii of L. S. Tee, S. Gotoh, and W. E. Stewart [IND.ENG.CHEM.,FUNDAM. 5, 356 (1966)l. The predictions in set 3 are based, as stated, on Lennard-Jones constants fitted to viscosity data for the two pure gases. If E A B / ~is predicted as in set 3, and UAB is fitted t o the observations of DAB,we get the additional results (set 4) for Table IV. This is a good fit of the data, though not as good as set 1 or 2.
The confidence intervals here and throughout Table IV are calculated with t = 2 in Student's t distribution. The symmetric 95'% confidence intervals are wider by a factor of (4.303/2) for set 1, and (3.182/2) for sets 2 and 4.
Set
UAB,
A
4 4.463 + 0.010 (fitted)
~AB/L,O K
Av abs dev,
162.5 (predicted)
0.35
%
Residual sum of Degrees of squares freedom
48.9 X 10-6
3
COMMENTS O N TRANSITION FROM LAMINAR TO TURBULENT FLOW
The author of this article [H. Dennis Spriggs, IND.ENG. FUNDAM. 12, 286 (1973)l wishes to add the following acknowledgment: The experimental phase of this work was
CIIEM.,
conducted a t the Chemical Engineering Department of the University of Virginia. The author also wishes to thank Dr. L. U. Lilleleht for his assistance throughout the project.
Ind. Eng. Chem. Fundam., Vol. 12, No. 4, 1973
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