Ind. Eng. Chem. Res. 1994,33, 750-751
750
Comments on “Negative Maxwell-Stefan Diffusion Coefficients’’ Gerrit Kraaijeveld and Johannes A. Wesselingh’ Department of Chemical Engineering, University of Groningen, Nijenborgh 4, 9747 Groningen, The Netherlands
Gerard D. C. Kuiken Laboratory for Aero- and Hydrodynamics, Delft University of Technology, Rotterdamseweg 145, 2628 AL Delft, The Netherlands
Sir: Our article,Kraaijeveld and Wesselingh (1993),has recently given rise to a discussion between G. D. C. Kuiken and the authors. These have revealed that the constraints presented are incomplete. The main conclusion, that Maxwell-Stefan diffusion coefficients can be negative, remains correct. It is the purpose of this Correspondence to present the complete constraints and a few additional observations on negative Maxwell-Stefan diffusion coefficients. In our previous article (Kraaijeveld and Wesselingh, 1993) we discussed the constraints which were imposed on the Maxwell-Stefan diffusion coefficients by the fact that the total entropy production should always be greater than or equal to zero. For a ternary system the constraints imposed are as before:
a, b = 1, ..., 3 a # b (1)
Although eq 1 appears to represent three different equations, it is fairly easy to show that the equations are in fact identical, and reduce to the condition
Besides eq 2, the diffusion coefficients also need to satisfy the following three equations:
($E)1o
a = l , ...,3
(3)
These equations originate from the fact that the parabolic equation, presented as eq A5 in the previous article, should have an extremum which is a minimum. This condition is satisfied, if the coefficient of the quadratic term in the equation is larger than zero. The generalization of the complete constraints for a multicomponent system thus becomes (Kuiken, 1994)
a, b = 1, ...,n a # b (4)
Having obtained the complete constraints, the argument of the concentration dependence should be reconsidered. The two possible options will be discussed below. If the Maxwell-Stefan diffusion coefficients are taken to be composition dependent (experimental evidence for this option was presented in Figure 1 of the previous article), then eqs 4 and 5 should be satisfied. In this case Table 1 shows the values of the constraints for the NaC1/ KCI system presented in the previous article (where flab is the left-hand side of eq 4 and f a is the left-hand side of eq 51, and clearly the thermodynamic constraints are satisfied. There is only one problem with this option, namely the fact that only electrolyte systems exhibit the phenomena of negative diffusion coefficients, and there is no explanation why nonelectrolyte systems do not. The alternate option, Maxwell-Stefan diffusion coefficients which are composition independent, does offer an answer to the question why negative diffusion coefficients only occur in electrolyte systems. This option would mean that any multicomponent system could be reduced to a number of binary systems (since the diffusion coefficients are taken to be composition independent), with diffusion coefficients which satisfy Dab
10
(6)
The only type of system where eq 6 is not applicable is for multicomponent electrolyte solutions which are described in terms of ions. This is caused by the fact that a cation is always accompanied by an anion, and hence a binary system consisting of e.g. an anion and water is not possible. Hence multicomponent electrolyte solutions cannot be reduced to a set of binary systems, but can instead only be reduced to a set of ternary systems. Of course ternary systems need to satisfy eqs 2 and 3, which do offer the possibility of negative Maxwell-Stefan diffusion coefficients. In the case of the earlier mentioned NaC1-KC1 system, there are two approaches possible. Firstly, if the system is described in terms of neutral components, then this is a ternary system which can be reduced to three binary systems, and eq 6 yields
The first two inequalities in eq 7 are particularly interesting, since these diffusion coefficients are related to the diffusion coefficients of the ionic species by (Lightfoot, 1974)
and Combining the facts that eq 8 must hold for all combinations of cations and anions, that the Maxwell-Stefan diffusion coefficient of an ion is independent of the other 0888-5885/94/2633-0750$04.50/00 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994 751 Table 1. Composition-Dependent Constraints for NaCl-KCl System. Cl
c2
fl
f2
0.25 0.5 0.25 0.5 1.5
0.25 0.25 0.5 0.5 1.5
0.37 0.77 0.36 0.75 2.50
0.25 0.25 0.50 0.50 1.59
0
fs 0.50 0.77 0.76 1.02 3.35
f4
B12
BlS
614
829
0.98 1.54 1.40 1.95 6.18
0.91 1.89 1.83 3.72 39.5
1.82 5.82 2.73 7.54 82.1
2.48 7.12 3.98 10.0 104
1.21 1.87 3.74 4.98 51.9
Bu 1.92 3.26 5.07
B34
3.13 7.96 6.70 13.1 142
1.10
79.2
Subscripts: 1refers to Na+, 2 to K+, 3 to C1-, and 4 to HzO. The concentrations are in mol/L, the f s in 107 s/m2, and the f l s in 1018 s2/m4.
Table 2. Composition-Independent Constraints for NaCl-KCl-H20 Systems. NaC1-KCl system
c1 0.25 0.5 0.25 0.5 1.5 0
c2 0.25 0.25 0.5 0.5 1.5
NaCl-H20 system
KCl-HzO system
fl
f2
f3
8123
fl
f3
f4
BlU
f2
0.3 0.8 0.3 0.7 2.6
0.2 0.2 0.6 0.5 2.1
0.8 1.5 1.3 2.0 8.0
0.01 0.01 0.01 0.02 0.26
3.8 7.9 3.9 7.9 27
4.6 7.3 6.7 9.3 30
7.6 13. 9.5 15. 48
1.7 5.7 2.6 7.2 78
2.6 2.7 5.2 5.3 18
fs 4.5 6.7 7.0 9.1 29
f4
8234
6.4 8.5 11. 13. 39
1.2 1.8 3.6 4.8 50
Subscripts: 1refers to Na+, 2 to K+, 3 to C1-, and 4 to HzO. The concentrations are in mol/L, the f s in 106 s/m2, and the f l s in 1013 sYm4.
Table 3. Composition-Independent Constraints for NaOH-Nafion 120-H~OSystems. NaOH-Nafion system mlalt
f+
0.1 0.25 0.5 1.0 2.0 4.0
11.3 11.8 12.9 13.0 14.8 19.7
0
Subscripts:
1014 s2/m4.
f-0.03 0.03 0.12 0.42 0.75 3.04
NaOH-H20 system
fm
8+-m
f+
11.4 12.1 13.3 13.2 14.3 13.2
-0.3 0.32 1.56 5.47 10.7 37.0
14.1 13.6 13.7 14.8 22.3 43.4
f0.01
14.3 13.8 14.0 15.2 23.1 41.9
-0.04
0.01 0.61 2.13 8.04
+ refers to Na+, - to C1-, w to H20, and m to Nafion.
ions present, and that the largest known diffusion coefficients of ions in water (those of the hydrogen ion and the hydroxide ion) are both positive leads to the conclusion than the Maxwell-Stefan diffusion coefficient for an ion in water should always be positive definite. Secondly, if the NaCl/KCl system is described in terms of ionic species, then this is a quaternary system which can be reduced to three ternary systems, and these need to satisfy eqs 2 and 3 (Table 2 illustrates that the constraints are satisfied). On the basis of the arguments presented above, it may be concluded that ion-ion Maxwell-Stefan diffusion coefficients in electrolyte systems can be negative. Further, on the basis of the evidence presented above, the authors feel that the compositionindependent approach is the preferred approach. The NaOH-Nafion system presented in the previous article should of course also satisfy the compositionindependent constraints. However, as can be seen in Table 3, at low external salt concentrations some of the thermodynamic constraints are not satisfied. It must however be borne in mind that membrane diffusion coefficients are usually a lot more difficult to measure and may involve a significant experimental error (especially so at low external salt concentrations, when the co-ions (anions) are almost completely excluded from the ion exchange membrane). As can be seen in Table 3, at higher external salt concentrations the constraints are indeed satisfied. Hence although no guaranty can be given about the existence of negative diffusion coefficients in ion exchange membranes at low external salt concentrations, in the light of the other evidence presented above, the authors feel that it is not unlikely that these diffusion coefficients can be negative. Another interesting detail for the NaOHNafion system is the fact that the composition-dependent constraints are satisfied, whereas the composition-independent constraints are not. This indicates that the latter are a more stringent set of constraints. Possibilities to
fw
Nafion-HzO system
B+-W 0.16 -0.7
0.14 9.04 47.1 326.
f+
fw
fm
B+rm
25.6 25.6 26.8 27.6 35.8 53.5
23.7 24.5 25.7 29.3 43.4 67.7
21.0 22.7 24.9 27.6 35.9 43.9
407 440 499 594 1080 2130
The molality is in mol/kg, the f s are in 10' s/m2, and the f l s are in
evaluate the quality of experimental data using these thermodynamic constraints seem likely as well. As before, the same general conclusion can be drawn: negative Maxwell-Stefan diffusion coefficients do exist and are consistent with the theory of irreversible thermodynamics. We should however add to this that these negative diffusion coefficients only exist in electrolyte systems, and are unlikely to occur in nonelectrolyte systems.
Nomenclature = left-hand side of eq 4 for species i and j , sVm4 = left hand side of eq 2 for species i, j , and k , s2/m4 Bij = Maxwell-Stefan diffusion coefficient for species i and j , m2/s fi = left-hand side of eq 3 or 5 for species i, s/m2 n = number of components x i = mole fraction of species i zi = charge number of species i Bi,
Bijk
Subscripts
+ = cation - = anion 0 = solvent a, b, c , i, j = species a, b, c , i, j
Literature Cited Kraaijeveld,G.; Wesselingh, J. A. Negative Maxwell-StefanDiffusion Coefficients. Ind. Eng. Chem. Res. 1993,32,73&742. Kuiken, G. D. C. Thermodynamics of irreversible processes with application to diffusionand rheology;Wiley & Sons: New York, 1994; in press. Lightfoot, E. N. Transport Phenomena and Living Systems; Wiley: New York, 1974.
