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360
Horio. M.; Wen, C. Y. AIChESymp. Ser. No. 161, 1977, 73, 9. Keto, K.; Wen, C. Y. Chem. Eng. Sci. 1969, 2 4 , 1351. Kobayashi, H.: Aral, F.; Chlba, T.: Tanaka, Y. Chem. Eng. (Tokyo)1969, 3 3 , 274.
Taimor, E.: Benenati, R. F. AIChEJ. 1963, 9 , 536. Viswanathan, K.; Rao, D. S. Proceedings of the 33rd IIChE Conference, New Delhi, India, 1980; Voi. I, 61. Yoshida, K.; Kunii, D. J. Chem. Eng. Jpn. 1968, 1 , 11.
Kunii, D.; Levenspiel, 0. “Fluidization Englneering”; Wiiey: New York; 1969. Mori, S.; Wen, C. Y. AIChE J . 1975, 21, 109. Rowe, P. N.; Partridge, B. A. Trans. Inst. Chem. Eng. 1965, 4 3 , 157. Rowe, P. N.: Wldmer. A. J. Chem. Eng. Sci. 1973, 2 8 , 980.
Received f o r review March 2, 1981 Accepted June 15, 1982
Application of the Stefan-Maxwell Equations to Diffusion in Ion Exchangers. 1 Theory E. Earl Graham’ and Joshua S. Dranoff Department of Chemical Engineering, Northwestern University, Evanston, Illinois 6020 1
Starting with the Stefan-Maxwell equations, general expressions for the ionic flux rates for binary exchange in ionsxchange resins have been developed. These equations have been shown to reduce to the Nernst-Planck equations exactly only as the concentration of either exchanging ion approaches unity. Furthermore, the single-ion diffusion coefficients used in the Nernst-Planck equations are shown to be certain combinations of the StefanMaxwell interaction coefficients. Most importantly, these combinations of the Stefan-Maxwell interaction coefficients are shown to reduce to the tracer diffusion coefficient of each exchanging ion, measured in ion-exchange resin completely in the competing ion form. As these limiting tracer ion diffusion coefficients may be very different from the usual pure selfdiffusion coefficients, this result may explain existing anomalies resulting from the use of the Nernst-Planck equations to describe diffusion in ion-exchange resins and related ion-exchange systems.
Introduction In describing the diffusion of ions within ion-exchange resins and related systems (such as membranes, porous solids, and glasses) the generally accepted expression for the diffusional flux of species i is the Nernst-Planck equation.
In using this relation the resin matrix is generally treated as quasi-homogeneous and an effective diffusion coefficient, di, is taken as equal to the self-diffusion coefficient of species i measured using tracers for a particular resin completely in the pure i form. For the binary exchange of two counterions whose charge is completely balanced by the fixed ionic sites, the electrical potential may be eliminated between the Nernst-Planck equations for counterions 1 and 2.
which defines a single coupled interdiffusion coefficient (Helfferich, 1962). D12 =
didz(zi2Ci + zz2c2) 212Cldl + z22Czdz
(3)
While the equations given above can describe a given ion exchange very well, this is only the case if the effective diffusion coefficients, di are chosen as those that give the “best fit” to the actual exchange data. The value of the best fit coefficient for a given ion depends strongly on the *Address correspondence to this author a t the Department of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802.
other ion present and is not the same as the pure selfdiffusion coefficient. The variation of a given effective single ion-diffusion coefficient is often of the order of two to threefold as shown clearly in the studies of Hering and Bliss (1963) and Morig and Rao (1965). It is also well established that the self-diffusion coefficient (or more correctly the tracer diffusion coefficient) itself varies with composition and type of other ions present. Variations in the tracer ion-diffusion coefficients in resins of as much as sixfold have been reported by Boyd and Soldano (1953). The situation is much the same for systems similar to ion-exchange resins such as ion-exchange membranes, ionic porous solids, and glasses subjected to ion exchange. Thus for ion-exchange membranes experimental studies (Lopez-Gonzalez and Henry, 1959; Spencer and Ellison, 1965; Tombalakian et al., 1967; Tombalakian, 1974; and Huang and Lian, 1979) show variations in single ion coefficients of greater than threefold, both with composition and type of other ions present for the same membrane. For porous solids a similar dependence has been reported (Duffy and Rees, 1974; Lutze and Miekeley, 1971; Pehkonen, 1973; and Yu and Davis, 1979) while for molten metals and glasses variations in a given tracer ion diffusion coefficient are often as great as one-hundredfold (Fleming and Day, 1972; Frischat and Kirchmeyer, 1973; McVay and Day, 1970; May and Vollast, 1974; Scheidecker and Berard, 1973, 1976, 1979; Stiglich et al., 1973; Visser et al., 1975; Wei and Wuensoh, 1973; Moynihah et al., 1980). While it is likely that some of these anomalies are the result of steric effects or structural changes especially in the case of the glasses (Doremus, 1974), it is equally likely that they reflect also changes in ionic interactions not accounted for in eq 1-3. This would be expected from the high ionic concentrations of these systems, typically 3-4 M for the ion-exchange resins, membrane, and porous solids and perhaps an order of magnitude greater for the ionic glasses (Doremus, 1966). For such systems it is well known that significant inter-
0196-4313/82/1021-0360$01.25/0 0 1982 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982 361
actions exist and may be accounted for by one of the formulations of irreversible thermodynamics. It is proposed therefore to review these formulations for the general case of ionic diffusion and then apply the most promising to ion exchange in greater detail. Theory The usefulness of eq 2 in describing diffusion of ions in dilute solutions has been well established. However, for many ion-exchange systems where the concentration of counterions (exchanging) is very high its use is questionable. Furthermore, the dependence of di on the ion pair studied (even where macroscopic gradients of activity coefficients are eliminated by use of radioactive tracers) strongly suggests some sort of ionic interaction. The theory of nonequilibrium thermodynamics suggests that for simple electrolytes at uniform temperature and pressure the diffusional flux should be written (DeGroot and Mazur, 1962; Fitts, 1962). n
n
(i = 0, ..., n) (4) The summations in eq 4 are taken over the solvent species o and n chemically independent species. For systems in mechanical equilibrium and for which the solvent velocity is zero, Miller (1960) has shown that the summation may be taken over only the independent (neutral) species. Furthermore, this experimental work (Miller, 1960,1966,1967) has shown that the summation may be taken over n 1 ionic species and the Onsager reciprocal relations will hold even though the ionic species are no longer strictly independent (due to electroneutrality requirements).
+
(9
J? =
(8)
j=1
1ij”Vp; = j=1
concentration or the relative amounts of the various species. Secondly, the coefficients of one formulation may make more sense physically and be directly related to other electrochemical parameters. Finally, for a particular system or experimental study one formulation may be considerably easier to use, whether because of the form of the equations or because of the number of independent coefficients needed to describe the system. In choosing a particular formulation it should be noted first that by inversion and/or algebraic manipulation the coefficients of any one formulation may be expressed in terms of those of any other formulation. A summary of the relationships has been given elsewhere (Graham, 19701, and it may be noted that in general, if the coefficients of one formulation are “constant” those of the others cannot be. As an example, Miller (1967) has evaluated the 1,” of eq 5 from the data available for mixed chlorides at 25 OC. He has assumed the Onsager reciprocal relations, having shown they hold in a previous work (Miller, 1966). By using the relationships between the various coefficients, these data provide a good comparison of the different formulations. A comparison of the coefficients of the two formulations based on his data is presented in Tables I and 11. These data show the Stefan-Maxwell relations for ionic species yield coefficients that are reasonably constant especially as the type of ionic species is varied as shown in Table 11. Furthermore, for dilute solutions where xi 0 for all solutes and x , 1, the Stefan-Maxwell relations (6) for ionic species reduce to the Nernst-Planck equations and finally to the Fick’s law expression with a single diffusion parameter. Equation 6 may thus be reduced as follows
-
-
- lij”(Vpj + zjFV4)
(i = 1, ..., (*)) ( 5 ) 1.9 = 41
l..” 11
(*) where the summations may be taken over either n neutral species or n 1 ionic species and the flux is solvent-fixed flow of species i with the stipulation that the solvent velocity is zero. It should be noted that while the electrostatic potential #I defined in (4) and (5) is not actually measurable (Denbigh, 1981; Newman, 1973), it is correct to use in a formal treatment of electrolytes (Denbigh, 1981). In this treatment for ion-exchange resins, 4 will be eliminated by the requirement of electroneutrality and no electric current. Equation 5 may be written in many forms consistent with the theory of nonequilibrium thermodynamics. Of particular interest is the form of (5) called the StefanMaxwell relations and derived by Stefan and Maxwell from the classical treatment of kinetic theory to describe diffusion in mixture of ideal gases (Curtiss and Hirschfelder, 1949).
+
Either of the formulations (5) or (6) are “equally valid” and should give a complete explanation of the general diffusion problem. However, they are not equivalent and the coefficients of one formulation may be “more constant” than the others. In more meaningful words, the coefficients of one may be less dependent on either the total
= -ai,VCi
-
(for V # I
0)
(10)
Therefore, in the limit of infinite dilution, where the Nernst-Einstein relation holds
RT
ai, = -u?(o) F where u: is the ionic mobility of species i in solvent o at infinite dilution. It should also be noted that the Stefan-Maxwell relations are reference frame independent so that the coefficients will not depend on the reference velocity used as is the case for the Onsager coefficients. A comparison similar to this one was made by Lightfoot et al. (1962) but for a system water, glycine, and potassium chloride. Only formulations using neutral species were compared in their work. The coefficients of the StefanMaxwell relations (neutral species) were again reasonably concentration independent. Since there were only two ionic species (K+and Cl-), electroneutrality considerations would guarantee that (K+ + Cl-), would behave like a neutral species. For the case where three or more ionic species are present (as in ion exchange), it may be shown (Graham, 1970), that the Stefan-Maxwell coefficients based on ionic species are more constant than those based on neutral species.
362 Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982 Table I. Comparison of Onsager and Stefan-Maxwell Coefficients for Isothermal, Isobaric, Biionic Diffusion (Effect of Ionic Composition) systema molZ J 1 m-l
I
S-I
I-A
I-B Onsager, Ionic 1.22 2.34 1.82 1.81 3.65 5.31 -0.05 -0.08 0.22 0.46 0.26 0.28 Stefan-Maxwell. Ionic
loNaNa x 10" PKKx 10" lOClC1 x 1 O ' O ~ " N ~ xK 10" loNaCl x 10" l°KCl x 10''
I-c
I-D
I-E
1.23 3.59 5.44 -0.08 0.24 0.56
2.36 3.59 7.06 -0.14 0.48 0.60
6.03 9.57 18.44 -0.47 1.41 2.00
1.28 1.92 2.06 -0.18 0.12 0.15
1.22 1.88 2.04 -0.20 0.14 0.17
0.97 1.58 1.71 -0.38 0.31 0.35
m a s-l
lo9 aclH,o x l o 9 U N ~ Kx lo9 aNaCl x lo9 aKC1 x i o 9 aNaH,O x ~ K H , O x IO9
1.33 1.95 2.03
1.28 1.93 2.08 -0.15 0.09 0.11
1.28 1.88 2.05 -0.18 0.11 0.14
a Systems: I (calculated from ionic mobility data a t infinite dilution); I-A (0.25 M NaCl, 0.25 M KC1); I-B (0.50 M NaCl, 0.25 M KCI); I-C (0.25 M NaCl, 0.50 M KCl); I-D (0.50 M NaCl, 0.50 M KCI); I-E (1.5 M NaCI, 1.5 M KC1).
Table 11. Comparison of Onsager and Stefan-Maxwell Coefficients for Isothermal, Isobaric, Biionic Diffusion (Effect of Ionic Species) system a I1 Onsager, Ionic 1.22 lONaNa x 10'' PKKx 10" 1.82 1.46 P L & i x 10'O 0.90 l0LiCl x 1O'O 0.18 l°KCl x 10'' 0.26 0.21 0.22 loNaCl x 10" Stefan-Maxwell, Ionic
I11 0.97 0.88 0.17 0.18
mz s - ' aNaH20 ~ K H , Ox aLM20 x aNaCl x
aKCl x aLicl x UClCl
x
lo9
1.28 1.93
109
lo9 io9 109 lo9
0.09 0.11 2.05
1.25 1.90 0.93 0.094 0.068 209
0.90 0.07 0.069 2.09
a Systems: I-A (0.25 M NaC1, 0.25 M KCl); I1 (0.25 M LiCl, 0.20 M KCl); I11 ( 0 . 2 5 M LiCl, 0.20 M NaCI).
Before proceeding to the development of the StefanMaxwell relations for ion exchange, it should be noted that the ai;s are similar to the friction coefficients of the model of Spiegler (1958) developed for transport processes in ionic membranes. This treatment is based on the law of friction which states that the friction force that impedes the motion of object i (in this case say a hydrated Na+ ion) gliding on another object j (say a water molecule) is proportional to the relative velocity of i with respect to j . A set of force balances then gives his transport equations which are essentially the same as the Stefan-Maxwell relations. The following development of the Stefan-Maxwell relations is specifically for ion-exchange resins but could be applied to other ion-exchange systems. It should be noted that for ion-exchange membranes a formulation based on the 1,s' is often used as recently reviewed by Bennion and Pintauro (1981). To develop the Stefan-Maxwell relations (6) for diffusion within the exchanger matrix (intraparticle diffusion) it is necessary to consider the exchanger as essentially one continuous cross-linked polystyrene matrix into which fixed ionic charges have been introduced (active sites).
Associated with these fixed ionic charges (usually by weak electrostatic forces) are mobile ions of opposite charge (counterions). Mobile ions of the opposite charge to the counterions are assumed completely excluded from the ion exchanger as a result of the Donnan potential. This assumption is considered good only if the ionic strength of the bulk solution surrounding the exchanger is less than one- or two-tenths normal (Gregor and Gottlieb, 1953; Gregor and Tetenbaum, 1954). Thus, the water-swollen resin matrix may be considered a single continuous species "through which" the counterions diffuse; the process considered, therefore, is interdiffusion of counterions 1 and 2 through the solvent o in the presence of active sites a. In addition it is also assumed that: (a) the flux of solvent o and any transient pressure gradients may be neglected; (b) the active sites, a are uniformly distributed; (c) the ai; are constants independent of the relative amounts of the different ionic species. These assumptions are discussed in detail elsewhere (Graham, 1970), but it should be noted that due in part to the assumption of a quasi-homogeneousmedium, there is a certain arbitrariness in what should be considered the solvent species o. Two obvious possibilities suggest themselves. All ionic species may be considered hydrated and the solvent may be considered to be free-water plus resin matrix, or alternately the ionic species may be taken as unhydrated and the solvent considered to be total water plus resin matrix. Although the latter case makes more sense thermodynamically (since the distinction between free and bound water is thermodynamically ill-defined), the Fist case makes more physical sense and may well lead to more "constant aij's". Furthermore, to the extent that the flux of free water is zero, as would be the case only if the resin "swelled" exactly enough to compensate for the influx of a larger (more hydrated) ion, the flux of total water cannot be zero. These matters will be discussed further in light of experimental results in a subsequent paper. The Stefan-Maxwell relations for intraparticle diffusion for the two counterions may now be written as
L,
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982 363
The condition of no current requires that z ~ N , 22N2 = 0 so that (12) and (13) become
+
(14)
where the Y~~ are activity coefficients for neutral species defined as follows (Denbigh, 1963)
--)..(
-
yia = T ~ (
2,
-x2Vpfz =
(17)
(18)
RT X o + xa( -2, 1 + $)IN2
(19)
-[c
+ x , -- + & ) I N l
a10
I( - :)(-)I 1
21 a21
or
If “effective” Nernst-Planck diffusion coefficients are defined as 1 (22) 4 2 , =
d2(l) =
- 2,
(i = 1, 2) (27)
