Ind. Eng. Chem. Res. 1987,26, 2331-2336
233 1
Characterization of Ionic Diffusivities in Ion-Exchange Resins Neville G. Pinto* and E. Earl Graham' Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
The physical diffusion model developed earlier for multicomponent electrolyte solutions has been extended to ion exchange. The effects of ionic interactions, obstruction by the resin matrix, viscosity, and solvation on diffusivity have been incorporated, and an equation for characterizing ion-fixed site interactions has been developed. The method has been applied t o isotopic exchange of Na+ in the presence of Na+, K', and Li+ in Dowex 50W-X8.Experimental diffusivity data have been obtained as a function of concentration by using the shallow-bed technique. It has been shown that interactions within the resin can be characterized effectively.
It has been recognized for some time that ion-exchange rates are, in almost all cases, governed by diffusion within the ion exchanger and/or diffusion from the solution to the ion-exchange surface (Helfferich, 1966). For systems with rates controlled by diffusion within the exchanger, the generally accepted expression for diffusional flux of a counterion is
where pi' = pi
+ qFr$
Theory While the utility of eq 1 in describing ion exchange is well established, this equation is not appropriate for evaluating ionic interactions; all the interaction effects are lumped into one parameter, bi,and thus, it is not possible to separately quantify different types of interactions. It is well-known that interaction effects are best established by using the phenomenological flux equations obtained from the theory of nonequilibrium thermodynamics. Of the different formulations available, Graham and Dranoff (1982) have suggested that the Stefan-Maxwell equations
(2)
In using this relation, the resin is treated as a quasi-homogeneous phase, and thus, an effective diffusivity can be defined for each counterion. In general, effective diffusivities are strongly dependent on resin composition and capacity. For example, Hering and Bliss (1963) have observed variations of the order of 2-3-fold for a number of binary ion-exchange systems. These variations occur, in part, due to interactions between counterions, between counterions and solvent, and between counterions and fixed sites. Unfortunately, there is, as yet, no method available for quantifying interaction effects within ion-exchangeresins. Thus, previous studies of ion-exchangekinetics have used various concentration-independent coefficients-either tracer coefficients or best-fit coefficients (Drummond et al., 1983; Streat and Takel, 1981; Gaus and Lutze, 1981; Gupta et al., 1979; Bajpai et al., 1974; Lutze and Miekeley, 1971; Sharma et al., 1970). While this approach may describe a particular ion-exchange system well, it has no predictive value. A more general approach requires the development of a method for estimating the composition and concentration dependence of ionic diffusivities within the resin. This paper presents results of a fundamental study of interactions within ion-exchangeresins. A physical model developed for multicomponent diffusion in concentrated electrolyte solutions (Pinto and Graham, 1986, 1987) has been extended to ion-exchange resins. This has enabled the characterization of interactions within the resin for binary and ternary ion-exchange systems. *Presentaddress: Department of Chemical and Nuclear Engineering, University of Cincinnati, Cincinnati, OH 45221. 'Present address: Department of Chemical Engineering, Cleveland State University, Cleveland, OH 44115. 0888-5885/87/2626-2331$01.50/0
are most appropriate. From diffusion data for electrolyte solutions, they have shcwn that the coefficients of eq 3 are weakly concentration dependent, and more importantly, these coefficients are pseudobinary in nature; i.e., a particular aij is dependent only on the properties of the ions i and j . These properties of the Stefan-Maxwell flux equations have been recognized by other investigators and have been used previously to characterize transport in ion-exchange membranes (Spiegler, 1958; Willis and Lightfoot, 1966; Bennion and Rhee, 1969; Meares et al., 1972; Pintauro and Bennion 1984). While the separate investigators have incorporated different degrees of complexity in their studies, in general, the approach has been to utilize the binary nature of the Stefan-Maxwell coefficients in combination with appropriate experimental measurements of transport coefficients within the membrane to obtain the concentration dependence of the coefficients of eq 3. This approach works well but is based on the determination of all interactions from rather complex experimental measurements within the membrane. It should be possible, due to the binary nature of the coefficients, to evaluate interactions between mobile ions and between mobile ions and solvent independent of the membrane or resin. This will considerably simplify the characterization, since interactions independent of the membranelresin can be obtained from transport data in free solution (pore solution in the absence of membranelresin matrix). These data are generally available in the literature for all commonly used electrolytes and will reduce the experimentation involved. Incorporation of free solution data into eq 3 requires the ability to characterize the concentration dependence of the Stefan-Maxwell equations in free solution. We have achieved this characterization earlier (Pinto and Graham, 0 1987 American Chemical Society
2332 Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 1986, 1987). By isolating ion-solvent from ion-ion interactions, concentration-dependent equations have been developed for the Stefan-Maxwell coefficients. For ionsolvent interactions, the equation is given as aio= kTxo(w:
+ ~3~)(71~/71)'.~
(4)
This equation incorporates the effects of viscosity and electrophoretic mobility on limiting ionic mobilities. The electrophoretic mobility appears as the term si in eq 4. Explicit expressions of are given in the original reference (Pinto and Graham, 1987). Interactions between ion pairs occur due to electrical forces. These interactions can be evaluated from the equation ~ i =j
gijk T f ( w i j ) ~ $ / T~O (/ 4)0.7
(5)
The function f(wij) assigns a characteristic mobility to the ion pair. This mobility is a function of individual ionic mobilities. For example, for ions of similar sign, the characteristic mobility is
It should be noted that eq 3-5, as written, do not include the effect of ionic solvation. This effect can be incorporated by defining all concentrations in these equations in terms of solvated species. The pseudobinary nature of the Stefan-Maxwell coefficients allows the extension of the model developed for electrolyte solutions (eq 4 and 5) to ion-exchange processes. Particle-diffusion-controlled ion exchange can be modeled as diffusion in an equivalent electrolyte solution into which the resin matrix has been introduced. Since the value of the interaction coefficients is unaffected by the presence or absence of other ions, the exclusion of co-ions from within the resin does not alter the value of ai, in the resin from its value in free solution. Thus, interactions between diffusing ionic species within the resin can be evaluated from analogous electrolyte cases. The most appropriate ion-exchange process for characterization of interactions within the resin is isotopic ion exchange. For this process, activity coefficient gradients are negligible, and there is no appreciable solvent flux. The equivalent electrolyte diffusion process is isotopic diffusion in free solution. For isotopic diffusion in a solution containing n ionic species, eq 1 and 3 can be simplified to Ni = -diVci
i = 1,...,
(7)
and n
- V C ~= NiCxj/ai, j=O
i = 1,...,n
(8)
respectively. Equation 8 can be rearranged to n
Ni = -[ Cxj/aij]-lVci j=O
i = 1,...,n
(9)
From a comparison of eq 7 and 9, it is clear that
The introduction of the resin matrix into the solution has two major effects. First, the polymeric matrix excludes part of the volume from diffusion and thus lowers the ionic diffusivity. Second, since the diffusing species is charged and the pore walls contain fixed charges, wall-solute electrostatic interactions will be present (Bungay, 1986).
The importance of these electrostatic interactions can be established from the ratio of the Debye length to the pore radius (Smith and Deen, 1980). For resins under consideration in this study, pore sizes range from 5 to 50 A. Also, from resin capacities, it can be shown that the Debye length is approximately 3 A. Under these conditions, strong electrostatic interaction effects can be expected. A number of investigators have developed models to relate diffusivities in porous media to diffusivities in free solution (van Brake1 and Heertjes, 1974; Mackie and Meares, 1955; Wheeler, 1951). These models account for excluded volume effects and are of the form di' =
diT
(11)
where T is generally expressed as a function of porosity. The isotopic diffusivity in the presence of an inert matrix can now be related to the free solution Stefan-Maxwell coefficients. Combining eq 10 and 11 gives
The final step in relating free solution coefficients to isotopic diffusivities within the resin is the incorporation of electrical interactions between fixed sites and diffusing ions. This is easily achieved if it is realized that fixed sites are ionic species with zero mobilities. The inclusion of an electrically interacting matrix into solution thus introduces an additional ionic species into the system. This adds a coefficient, an ion-fixed site coefficient, to eq 8 and will modify eq 12 to
Notice that all concentrations and concentration-dependent coefficients are defined to include the fixed site concentration (i.e., based on concentration within the ionexchange resin). The ion-fixed site coefficient, djf,of eq 13 represents an ion-ion interaction. In principle, this interaction between a diffusing and a nondiffusing ion should have the same concentration dependence as that between diffusing ions. Thus, if it is realized that the characteristic mobility for an ion pair made up of a nondiffusing and a diffusing ion is simply the mobility of the diffusing ion, from eq 5 , d i f should be given by dif = gifkT(w:
+ 6i).fif2/3(770/71)0.7
(14)
It is possible to verify the validity of eq 14 with eq 13 and appropriate isotopic diffusivity data. Such an experimental verification has been conducted and will be described in the next two sections.
Experimental Methods In order to experimentally verify the validity of eq 14, it is necessary to show that the binary constant gifis indeed a constant independent of the total ionic concentration and composition. This is achieved by observing the isotopic exchange of an ion at various equilibrium resin compositions. The system selected for this work was isotopic exchange of Na+ in Dowex 50W-X8, with the resin in equilibrium with various composition mixtures of aqueous NaCl, KC1, and LiCl. The resin particle size range used was 0.7-0.85 mm (H+ form), and the external solution concentration was 0.2 M. Under these conditions, the exchange process was particle diffusion controlled. All measurements were made at 25 f 0.1 "C.
Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 2333 Table I. Equilibrium Properties of Dowex SOW-XS Ion-Exchange Resin in 0.2 M Solutions of NaC1-KC1-LiCl at 25 "C XNaCI xKCl rpt cm Yo, g/g X N d g K f CI,mequiv/cm3 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 0.25 0.25 0.25
1.00 0.75 0.50 0.25 0.00 0.00 0.00 0.00 0.00 0.50 0.375 0.25
0.0356 0.0358 0.0361 0.0363 0.0366 0.0373 0.0371 0.0370 0.0370 0.0358 0.0359 0.0370
0.4072 0.4143 0.4219 0.4342 0.4680 0.5052 0.4779 0.4753 0.4691 0.4243 0.4282 0.4364
0.000 0.116 0.265 0.402 1.000 0.000 0.694 0.867
0.941 0.200 0.221 0.333
1.000 0.884 0.735 0.598 0.000 0.000 0.000 0.000 0.000 0.750 0.694 0.555
3.47 3.41 3.33 3.27 3.19 3.01 3.06 3.09 3.09 3.41 3.38 3.09
Table 11. Equipment Specifications for Kinetic Measurements
11 D5
I
j , '
Section p. NITROGEN PRESSURE HEAD
I
Section B
I
SOtUTlON CONDITIONING
I
ISHALLOW-I 1 BED 'ASSEMBLY
Section D DETECTION
~
Figure 1. Experimental apparatus for the determination of tracer diffusion coefficients in ion-exchange resins.
A. Equilibrium Measurements. The equilibrium measurements involved the determination of resin composition, bead size, and water content. In all cases, the equilibration was achieved by passing approximately 2200 cm3 of solution of known composition at a drop-wise flow rate through a packed bed of resin. Bead sizes were determined by direct measurement with a calibrated moving vernier scale microscope. The water content was obtained by drying the resin under vacuum (2628 in.Hg) at 125 "C for 24 h. The difference between dry and wet weights permitted the calculation of water content. The resin capacity and composition were obtained by using the conventional displacement method (Helfferich, 1962). Hydrochloric acid (1.0 M) solution was used for the displacement, and effluent compositions were analyzed on a Perkin-Elmer 703 atomic absorption spectrometer. Details of the experimental procedures are reported elsewhere (Pinto, 1985). The equilibrium measurements were made at nine binary and three ternary solution compositions. The results are summarized in Table I. B. Kinetic Measurements. Tracer-diffusion coefficient measurements of Na+ were made by using the shallow-bedtechnique. A schematic representation of the experimental setup is shown in Figure 1,and the major components are listed in Table 11. The shallow-bed assembly (Cl) consisted of six beads placed in a fritted glass tube. The beads were selected such that the maximum variation in size was no more than &3%, and the mean radius of the six beads coincided exactly with the mean radius of beads used in the equilibrium measurements, when both sets were in the Na+ form. The beads were first equilibrated with a solution of known composition containing a trace amount of 22Na isotope. Sufficient activity was introduced into the solution to ensure good counting statistics. Once equilibrated, the radioactive beads were rinsed with deionized water and centrifuged, to remove adhering radioactive solution, and placed in the fritted glass tube. The glass tube was placed in a lead well and connected to a reservoir (Bl) through
key (Figure 1) A1 A2 A3 A4 A5 B1 B2 B3 B4 B5 B6 B7 B8 C1 C2
equipment name nitrogen gas cylinder buffer tank high-pressure regulator low-pressure regulator safety valve solution reservoir water bath cooling water coil thermoregulator stopper assembly vent valve mercury thermometer solution outlet tube fritted glass tube solenoid valve
D1
NaI(T1) detector
D2 D3 D4 D5
high-voltage supply external amplifier multichannel analyzer printer
equipment specifications 1500 psig 2800 cm3 1500/10 psig 2010 psig 3-15 psig thick-walled jar, 2300 cm3 10 L Cu tube, l f 4 in. Fischer, Model 73 SS" plates, O-ring seal gate valve, in. 0-100 f 0.1 " C glass tube, 1/4 in. 10-mm i.d., extra coarse Skinner, V5 Series, 100 psi, in., 120 V/60, 10 W Harshaw, integ. scint. type, preamp, Model DS-13 Fluke, Model 418B Ortec, Model 485 The Nucleus, Model 2048 Radioshack, Model TRS-80
"SS = stainless steel.
a solenoid value ((22). The reservoir was mounted in a constant-temperature bath (B2). About 2300 cm3 of nonradioactive equilibration solution were placed in the reservoir. A nitrogen head applied to the upstream side of the reservoir provided energy for solution flow. The pressure head was maintained constant, at 8.5 psig, with a two-stage pressure reduction, similar to the system used earlier by Graham (1970). This arrangement gave a constant flow rate (&0.5%). The change in radioactivity of the beads was observed continuously with a NaI (Tl) detector (Dl). The multichannel analyzer (D4) was operated in the multichannel scaling mode with a window of approximately 1.00-1.50 MeV. The principle y radiation of 22Naat 1.2746 MeV was used for detection. The kinetic runs were initiated by turning on the timer of the multichannel analyzer. Before fluid flow through the bed was started, two 10-s counts were made to determine initial activity of the beads. In all cases, approximately 2 x lo3 counts/s were obtained. This corresponds to an error of less than 2% in number of counts, with 99% confidence. Exactly 20 s after the timer was activated, the solenoid valve was opened and nonradioactive equilibration solution was allowed to flow through the bed. At the flow rate used ( ~ 1 cm3/s), 6 a kinetic run lasted approximately 2 min and 20 s. Activity changes of the beads were recorded with contiguous 10-s counts. Only the first 1 2 data points were used in the computation of diffusivities, to avoid error due to fluctu-
2334 Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 For the ternary system being considered, this equation gives dNaw =
Table 111. Experimental Isotopic Diffueivities of Na+ in Dowex 5OW-Xt?in the Presence of Ns+. K+.and Li+
+-+,-+?Na 0.00 0.25 0.50 0.75 1.00 0.00
1.00 0.75 0.50 0.25 0.00 0.00
0.25 0.50 0.75 0.25 0.25 0.25
1.92 1.88 1.82 1.84 1.73 1.54
0.00 0.00 0.00 0.50 0.375 0.25
1.62 1.66 1.69 1.83 1.79 1.67
tiNa*O
aNa*Na
) A]-' +
2K aNa*K
aNa*Li
(16)
aNa*f
Thus, in order to calculate the diffusivity of Na+, it is necessary to express the equilibrium compositions of Table I in terms of solvated species. This can be achieved once ionic solvation numbers are known. Glueckauf and Kitt (1955) have obtained cation hydration numbers in sulfonated styrene-type exchangers from swelling behavior. For ions Li+, Na+, and K+, they have reported values of 3.3,1.5, and 0.6, respectively. From a study of single-electrolyte diffusion, we (1987) have reported hydration numbers of 3.7 for LiC1,2.2 for NaC1, and 1.1for KC1. Since the electrolyte model is being extended to ion exchangers, it is important to establish the consistency of these two seta of hydration numbers. Unfortunately, Glueckauf and Kitt (1955) have not reported the ionic solvation number for C1-. It is, however, possible to estimate this value if it is assumed that the hydration number is proportional to the surface charge density. In this case, a plot of ionic hydration number versus inverse square crystallographic radius should yield a straight line. This is indeed the case for the Li+, K+, and Na+ hydration numbers of Glueckauf and Kitt (1955). Thus, the hydration number for C1- can be extrapolated from a least-squares fit and was estimated as 0.4. Assuming ionic hydration numbers are invariant, our electrolyte values give hydration numbers of 3.3 for Li+, 1.8 for Na+, and 0.7 for K+. These are in excellent agreement with the numbers of Glueckauf and Kitt (1955) and have been used to obtain the solvated compositions of Table IV. In writing eq 16, it has been implicitly assumed that co-ions are completely excluded from the exchanger. For Dowex 5OW-X8, this assumption is justified. Thus, the only free solution interaction coefficients that appear are the ion-solvent coefficient and the ion-ion coefficients for ions of similar sign (i.e,, counterion pairs). These coefficients can be evaluated from eq 4-6, with all concentrations based on hydrated species. Also, the ?iiterms in these equations can be neglected, since, for isotopic diffusion, the electrophoretic effect is negligible. For eq 5, the binary constant gijis determined by using our earlier approach (1987). The value of g N & has already been determined as 1.97 (Pinto and Graham, 1987). From tracer diffusion data for Na+ in NaCl and LiCl solutions (Robinson and Stokes, 1968), the values of g,,, and gN&j were calculated to be 1.92 and 2.46, respectively. The viscosity correction factors in eq 4 and 5 were obtained experimentally at pore solution concentrations. A
ations in flow observed toward the end of the experiment.
Results and Discussion From the shallow-bed kinetic data, the isotopic diffusivities of Na+ were calculated by using the combined film-particle diffusion-controlled equations (Helfferich, 1962). Graham and Dranoff (1982) have determined that it is important to include the effects of film diffusion control even in cases when particle diffusion is predominant, since the very initial stages of exchange are film diffusion controlled. Thus, it is necessary to determine the thickness of the Nernst liquid film for the flow conditions in the bed. A single kinetic measurement was made at a low solution concentration (0.002 M) using exactly the same flow conditions as the 0.2 M runs. A t this low concentration, the exchange is film diffusion controlled, and the f h thickness can be calculated from the film diffusion equation for isotopic exchange (Helfferich, 1962). The film thickness was determined to be b = 1.46 X lo4 cm. Once the film thickness had been estimated, the kinetic rate data were used in conjunction with the equilibrium data of Table I and the film-particle equation to determine isotopic diffusivities. Since the transcendental film-particle equation requires a numerical solution, the NewtonRaphson method was utilized, and it was determined that seven roots guaranteed convergence with the desired degree of accuracy. The experimental diffusivities obtained are shown in Table 111. The estimated accuracy of these diffusivities is *3%. From Table 111 it is clear that ionic interactions and resin solvent content have a significant effect on isotopic diffusivities. In order to characterize these effects, it is necessary to define the concentrations and concentration-dependent terms of eq 13 in terms of solvated species. Conventionally, in treating tracer diffusion, solvation is neglected. However, Pinto and Graham (1987) have shown that for electrolyte solutions this assumption may not be valid. In terms of solvated species, eq 13 takes the form
Table IV. Hydrated Compositions of Dowex 5OW-XS in 0.2 M Solutions of NaCl-KCl-LiCl at 25 OC at Equilibrium" XNaCl
XKCl
x^O
fNa
fK
x^Li
x^f
0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 0.25 0.25 0.25
1.00 0.75 0.50 0.25 0.00 0.00 0.00 0.00 0.00 0.50 0.375 0.25
0.788 0.790 0.794 0.796 0.798 0.782 0.798 0.802 0.800 0.788 0.788 0.804
0.000 0.012 0.027 0.041 0.101 0.000 0.070 0.086 0.094 0.021 0.023 0.033
0.106 0.093 0.076 0.061 0.000 0.000 0.000 0.000 0.000 0.079 0.074 0.054
0.000 0.000 0.000 0.000 0.000 0.109 0.031 0.013 0.006 0.005 0.009
0.106 0.105 0.103 0.102 0.101 0.109 0.101 0.099 0.100 0.106 0.106 0.098
nIonic hydration numbers: Li+ = 3.3, Na+ = 1.8, K+ = 0.7.
0.011
,
Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 2335 Table V. Experimental Pore Viscosity Correction Factors at 25 OC for NaCI-KCl-LiCl Solutions XNaCl
XKCl
(7/7°)0'7
XNaCl
XKCl
(7/7°)0'7
0.00 0.25 0.50 0.75 1.00 0.00
1.00 0.75 0.50 0.25 0.00 0.00
1.035 1.061 1.090 1.123 1.267 1.349
0.25 0.50 0.75 0.25 0.25 0.25
0.00 0.00 0.00 0.50 0.375 0.25
1.282 1.264 1.260 1.092 1.112 1.129
calibrated Cannon-Fenske viscometer was used. The results are reported in Table v. The parameter r in eq 16 has been related to the porosity by two general classes of equations: empirical and theoretical. The empirical equations include interaction as well as excluded volume effects and are not suitable for characterization of the ion-fiied site term. The theoretical equations have been derived, to various degrees of complexity, for porous media in which the solid phase is dispersed in the form of regularly shaped bodies. These equations account only for excluded volume effects and are more suitable for this analysis. Six theoretical equations have been considered: (a) Wheeler, 1951 7
= E/2
(17)
=
(18)
(b) Marshall, 1957 tv2
(c) Weissberg, 1963 r = t/(1 - f/* In
(19)
e)
(d) Millington, 1959 7
=
(20)
e413
(e) Neale and Nader, 1973 r = 2 t / ( 3 - t)
(21)
(f) Mackie and Meares, 1955 7
= ( € / ( 2 - t))2
(22)
Porosities in these equations were calculated from equilibrium water contents (Graham, 1970). Equations 17-22 do not all predict the same effect of excluded volume. Equations 18 and 19 give very similar r values and so do eq 20 and 21. However, these two sets differ by about 10-15%. Furthermore, eq 17 and 22 give, respectively, r values between 50-60% and 100-120% smaller than those obtained from the other equations. Because of this wide variation, all six equations were used in the analysis.
The experimental data of Tables 111, IV, and V were used in conjunction with eq 4-6 and 16 and each of the excluded volume factor ( 7 ) models to verify the validity of eq 14. Since, for eq 14 to be consistent with the physical model, gifmust be positive and composition independent, eq 16 should fit all the diffusivity data of Table I11 with a single positive value of giF The linear least-squares technique was used to determine the best-fit values of glP These values and the corresponding best-fit diffusivities are shown in Table VI. All six excluded volume models give positive values of g,p Also, in five of the six cases, the concentration dependence of the diffusivities is well represented. In the best case, eq 20,lO out of 12 calculated values are within the experimental error ( f 3 % ) . Of the six equations used for r , four, eq 18-21, give essentially the same valuejor g,, and comparable accuracy in the calculation of dNa. These four equations appear to account suitably for the effects of excluded volume. Equations 17 and 22, on the other hand, overcorrect for the excluded volume. This is especially evident with eq 22, which gives a poor representation of the experimental diffusivities and a high value of gbF In the past, eq 22 has been used most frequently to relate diffusivities within the resin to diffusivities in free solution (Helfferich, 1962). From the results of this work, it is clear that the success of this equation has been fortuitous in that the overcorrection for excluded volume has, in a number of cases, compensated adequately for electrical interactions within the resin. Conclusions and Significance A method based on the Stefan-Maxwell flux equations has been developed to enable the incorporation of diffusivity data obtained in free solution to ion-exchange calculations. Interactions that occur independent of the resin matrix are evaluated from free-solution data, and interactions between the matrix and diffusing ions are obtained from isotopic ion-exchange data. This approach is particularly advantageous when applied to complex multicomponent systems. For these systems, it is possible, due to the properties of the Stefan-Maxwell coefficients, to calculate most interactions that occur within the resin from appropriate free-solution single-electrolyte data. This considerably simplifies the characterization, since these data are available in the literature for most common electrolytes. The method has been applied to the ternary exchange system Na+-K+-Li+. For this system, the method gives consistent values of solvation numbers in free solution and within the resin. It also enables the distinction between various models that exist for estimating the excluded volume effect within ion-exchange resins. It has been shown that the model of Mackie and Meares (1955), often
Table VI. Comparison of Experimental and Calculated Isotopic Diffusivities [ ;i~,(CalCd)/a~,(eXptl)]of Na+ in Dowex 50W-X8 equation for T , g, (best fit) XNaCl XKCl eq 17, 0.081 eq 18, 0.064 eq 19, 0.063 eq 20, 0.062 eq 21, 0.062 eq 22, 0.310 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 0.25 0.50 0.25
1.00 0.75 0.50 0.25 0.00 0.00 0.00 0.00 0.00 0.50 0.375 0.25
0.95 0.99 1.01 0.99 0.92 0.96 1.01 1.00 0.98 1.03 1.04 1.12
0.95 0.98 1.01 0.98 0.93 0.96 1.01 0.99 0.97 1.01 1.02 1.10
0.96 0.99 1.01 0.98 0.92 0.96 1.00 0.99 0.97 1.01 1.02 1.10
0.97 0.99 1.02 0.98 0.93 0.97 1.01 1.00 0.98 1.02 1.03 1.10
0.97 0.99 1.02 0.99 0.93 0.97 1.01 0.99 0.98 1.02 1.03 1.11
0.81 0.90 0.95 0.97 0.94 1.04 1.09 1.07 1.02 0.99 1.02 0.90
2336 Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987
used in ion-exchange calculations, is inappropriate when electrical interactions within the resin are incorporated. In this study, the method has been applied to ion exchange in which co-ions are excluded from the resins; it is by no means limited to such systems. The method can be applied to any number and type of ions as long as the binary constants for each ion pair are known. In fact, the equations are expected to work better for weak-acid and weak-base exchangers since the physical process is more accurately represented. Acknowledgment We gratefully acknowledge the financial support of the Department of Chemical Engineering, The Pennsylvania State University. Nomenclature a, = ionic Stefan-Maxwell coefficients, cmz/s c = total concentration when electrolyte concentrations are expressed in terms of individual ionic concentrations, gmol/cm3 c, = concentration of ionic species i, g-mol/cm3 CI = total ionic concentration excluding fixed sites, mequiv/cm3 d, = tracer diffusivity of species i, cm2/s d,’ = tracer diffusivity of species i in solution in the presence of an inert matrix, cm2/s D,= effective diffusivity of species i, cmz/s F = Faraday constant, esu/g-mol g, = binary constant, eq 5 J, = flux of species i referred to molar average velocity, gmol/(cm2.s) k = Boltzmann constant, erg/(Kdon) N , = flux of species i referred to a fixed frame, g-mol/(cm2.s) r = radius of ion-exchange particle, cm 8 = gas constant, erg/(g-mo1.K) T = absolute temperature, K x , = mole fraction of species i = effective mole fraction of species i and j = mole fraction of neutral species yo = weight fraction of internal water in resin (backbone and internal water) z, = valency of ionic species i
2,
Greek Symbols 6 = film thickness, cm 6, = correction for electrophoretic mobility of species i
(cm.ion)/(s.dyne)
= porosity of resin 9 = viscosity, g/(cm.s) p, = chemical potential of species i, erg/g-mol t
= electrochemical potential of species i, erg/g-mol r = excluded volume parameter, eq 11 4 = electric potential, erg/esu ut = ionic mobility of species i, (cm.ion)/(dyne.s)
p,’
Subscripts f = fixed ionic site in resin i = species i
j = speciesj 0 = solvent (free solvent when ionic species considered sol-
vated)
Superscripts 0 = property at infinite dilution * = tracer ionic species
= property in the ion-exchange resin = property within resin when ionic species are solvated Registry No. Na+, 17341-25-2; K+, 24203-36-9; Li+, 17341-24-1; Dowex 50W-X8, 11119-67-8.
*
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Received for review October 15, 1986 Revised manuscript received July 6, 1987 Accepted July 27, 1987
