9096
Ind. Eng. Chem. Res. 2006, 45, 9096-9106
Three Generations of Polystyrene-Type Strong Acid Cation Exchangers: Textural Effects on Proton/Cadmium(II) Ion Exchange Kinetics Ana Maria S. Oancea,*,† Marius Radulescu,† Dumitru Oancea,‡ and Eugen Pincovschi§ Department of Inorganic Chemistry, UniVersity “Politehnica” of Bucharest, Str. Gh. Polizu nr. 1, Bucharest 011061, Romania; Department of Physical Chemistry, UniVersity of Bucharest, Bd. Elisabeta nr. 4-12, Bucharest 030018, Romania; and Department of EnVironmental Protection and Inorganic Technology, UniVersity “Politehnica” of Bucharest, Str. Gh. Polizu nr. 1, Bucharest 011061, Romania
Proton/cadmium(II) ion exchange kinetics on sulfonated polystyrene resins of type gel, macroporous, and hypercross-linked were investigated using a potentiometric method in conditions favoring a particle diffusioncontrolled mechanism. The data were modeled with different quasi-homogeneous resin phase kinetic models and with the Ruckenstein et al. bidisperse pore model. The interdiffusion and self-diffusion coefficients were obtained and reported. The diffusion coefficients on resins with different texture slightly decrease in the series hypercross-linked g gel > macroporous resins. The evaluated ratio between the sulfonic groups grafted in micro- and macropores was 1/60 for the macroporous resin and 1/100 for the hypercross-linked one. The evaluated Cd2+ self-diffusion coefficients were 8.4 × 10-12, 6.4 × 10-12, and 9.1 × 10-12 m2 s-1, while the evaluated H+ self-diffusion coefficients were 1.7 × 10-10, 1.4 × 10-10, and 4.1 × 10-10 m2 s-1 on gel, macroporous, and hypercross-linked resins, respectively. Introduction The strong acid polystyrenic resins are the most used materials in the popular ion exchange technologies: water and wastewater treatment1 (softening, desalination, deionization, in nuclear and fossil power plants, for ultrapure water in electronics and semiconductor industry, to remove and recover metal ions arising from the chemical and hydrometallurgical industries, etc.). Other uses of these resins are as solid catalysts in catalyzed reactions in heterogeneous systems, especially in acid catalysis and reaction catalyzed by transition metal ions.2 Besides the classical gel-type and macroporous resins, a new type of polystryrene hypercross-linked matrix (also called macronet) has become available in the last few years.3 Knowledge of the ion exchange kinetics of different systems is important in the design of an ion exchange technology or of a separation method for a special purpose, because the exchange rate as well as equilibrium processes determine the efficiency of the ion exchange separation. At the same time, the kinetic investigations can offer valuable information about the resin texture and also in the resin quality control after manufacturing.4 The goal of the present work is to investigate the textural effects of a polystyrenic resin matrix of type gel, macroporous, and hypercross-linked on proton/cadmium(II) interdiffusion coefficients. Another aim is to discriminate between different kinetic models. To our knowledge, no ion exchange kinetic data were reported for hypercross-linked (macronet) resins in the open literature. Cadmium removal from polluted waters arising from different sources (electroplating industry, nickel-cadmium batteries, pigments, fertilizers, pesticides, dyes, mining waste, etc.) is important because of its toxic effects, determined mainly by * Corresponding author. Tel.: +40-21-4023986. E-mail: soancea@ yahoo.com. † Department of Inorganic Chemistry, University “Politehnica” of Bucharest. ‡ Department of Physical Chemistry, University of Bucharest. § Department of Environmental Protection and Inorganic Technology, University “Politehnica” of Bucharest.
the replacement of zinc in many enzymes, inducing severe diseases.5,6 Cadmium ion has a large ionic radius both in crystal (92, 101, 109, 117, 124, and 145 pm at a coordination number 4, 5, 6, 7, 8, and 12, respectively)7 and hydrated states (426 pm)8 and can be screened by the small pores of the resin matrix. The selected resins were sulfonated polystyrene-divinylbenzene (DVB) of type gel (∼7.6% DVB), macroporous (∼12% DVB), and hypercross-linked (>100% cross-linking agent). The gellike resins are obtained by suspension polymerization of styrene and divinylbenzene (radical addition), and the resulting spherical beads are sulfonated using different sulfonating agents and conditions. The gel resin bead contains the polymer network distributed in the entire volume. The cross-linking is not uniform, and in the solvated state, the resin contains micropores and small mesopores (1000 m2/g). The obtained resins could have micro- and mesopores or micro-, meso-, and macropores. The SEM picture13 showed a microphase separation for the basic inert matrix used to obtain sulfonated hypercross-linked resins. The property of swelling in both good and poor solvents is also observed for bi- or trimodal porous hypercross-linked resins. The inert resin beads are spherical milky white and opaque, while the sulfonated resin beads are opaque and dark brown colored. Only a few papers characterizing the properties of this new class of resins were reported in the open literature until now. These papers report the results regarding the porous structure of the homogeneous hypercross-linked polystyrenic resins;14 the physical and chemical properties of some inert and weak acid and weak basic hypercross-linked resins;13,15,16 the H+/Cu2+, Zn2+, Ni2+ ion exchange equilibria on a weak acid hypercross-linked resin;15,17 the sorption properties toward aurocyanide of the hypercross-linked resins bearing tertiary amines and quaternary ammonium groups;18 and the sorption capacity for organic compounds of different types of hypercrosslinked polystyrenic resins.13,19 The sulfonated hypercross-linked
polystyrenic resin investigated in this work was used for the ammonia removal from wastewater in the presence of organic contaminants20 and in the size-exclusion chromatography of mineral electrolytes in comparison with the nonfunctionalized microporous hypercross-linked polystyrene and activated carbons.21 To our knowledge, no ionic interdiffusion coefficients inside these hypercross-linked structures were reported, justifying the present study. Kinetic Background (i) Quasi-Homogeneous Resin Phase Kinetic Models with Constant Diffusion Coefficient. The diffusion coefficients in the resin phase can be obtained using the quasi-homogeneous resin phase (QHRP) kinetic models22,23 for particle diffusion control. In the first approximation, the Fick’s laws can describe the ionic flux and the time dependence of the concentration, as in the case of the isotopic exchange. The diffusion coefficient is named a self-diffusion coefficient, and it is a constant during the process. The Fick’s second law for systems with spherical geometry and constant diffusion coefficient was integrated for two boundary conditions: infinite solution volume (ISV) and finite solution volume (FSV), respectively. The eq 1 is the analytical solution at ISV,22,23 which supposes that the concentration of the outgoing ion in the external solution tends to zero:
F)1-
∞
6 π
∑ 2 n)1
1
n
2
exp(-n2π2τ)
(1)
Reichenberg24 proposed two simplified eqs 2 and 3 for ISV, one for low fractional attainment of equilibrium (F < 0.85) and the other for high conversions (F > 0.86), respectively, which were derived from eq 1, having the advantage of simplicity and avoiding the problem of the series convergence:
F)
6 2 1/2 3 (π τ) - 2(π2τ) 3/2 π π
F)1-
(2)
6 exp(-π2τ) π2
(3)
Paterson integrated Fick’s material balance equation for spherical particles at FSV, obtaining eq 4,22,23
F)1-
2
∞
∑
exp(-Sn2τ) (4)
3ω n)1 1 + S 2/9ω(ω + 1) n
where Sn are the roots of the equation Sn cot Sn ) 1 + Sn2/3ω. He also proposed22,23 the approximation given by eq 5, for τ < 0.1, because the series in eq 4 converges slowly in this region,
F)
{
ω+1 1 [R′ exp(R′2τ)(1 + erf R′τ1/2) 1ω R′ - β′
}
β′ exp(β′2τ)(1 + erf β′τ1/2)] (5) where R′ and β′ are the roots of the equation x2 + 3ωx - 3ω ) 0 and ω is the ratio of the amount of ingoing counterion in the resin phase and in the external solution at equilibrium. The dimensionless time τ ) D h t/rj02 contains the effective intraparticle diffusion coefficient in the resin, D h , which takes into account the retarding effect of the framework of the resin on the diffusion of a species. Helfferich has shown that “the effective diffusion coefficient is a macroscopic average over a large number of ions in pores of all different sizes, shapes, and directions and
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Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006
Table 1. Correlation between β/r Ratio and the Micropore and Macropore Uptake at Equilibrium, Mi∞/Ma∞ β/R
0.003
0.01
0.03
0.0375
0.05
0.075
0.1
0.2
0.3
0.5
1.0
Mi∞/Ma∞
1/1000
1/300
1/100
1/80
1/60
1/40
1/30
1/15
1/10
1/6
1/3
at larger and smaller distances from the pore walls, expressing the average ability of the species to make headway in any given direction”.23 In the case of isotopic exchange, D h is an intraparticle effective self-diffusion coefficient. The FSV boundary condition is the rigorous one for the batch ion exchange kinetic measurements. If the kinetic experiments are done in a batch reactor for low values of ω, ISV can also be considered as a reasonable assumption. (ii) Bidisperse Pore Kinetic Model with Constant Diffusion Coefficients. Ruckenstein et al.25 proposed a bidisperse pore model for transient diffusion in porous spherical particles, which was used for macroporous ion exchangers.4,26 The model was proposed for isotope exchange at ISV, with constant selfdiffusion coefficients D h a and D h i. A limiting case of the bidisperse pore model (LC-BDM) considering a two-stage ion exchange process with macroporous diffusion being much faster than the diffusion within the micropores is described by the following equation,25 F)
[
Mt M∞
1-
6 2
π
) ∞
∑ n)1
] [
1 β 6 exp(- n2π2θ) + ‚ 1 3 R n2 π2 1
∞
∑n n)1
1 2
exp(-n2π2Rθ)
]
1 β 1+ ‚ 3 R (6)
where
θ)
Mi∞ D h at D h ijr02 β , R ) and ) 3 2 2 R Ma∞ jr0 D h ajri
The Ruckenstein model has two parameters: R and β/R. The physical significance of R is the ratio between the times required for the ion penetration by diffusion in the macrosphere and in the microsphere. In the limiting case considered above,25 R must be equal to or smaller than 10-3. The other parameter β/R is related25 to the ratio of micro- and macropore uptake at equilibrium, Mi∞/Ma∞, i.e., to the ratio of the ion exchange capacity of the resin from the functional groups grafted onto the walls of the micropores and macropores, respectively, accessible to the considered counterions. Table 1 gives this correspondence. (iii) Variable Diffusion Coefficients: Integral Interdiffusion Coefficients. The kinetic equations presented above were derived for isotopic exchange. If eqs 1-5 are used to describe the ion exchange kinetics between two different counterions, it must be pointed out that the contribution of the electrical forces to the ionic interdiffusion is neglected. In these cases, the coefficient D h is considered as a constant for each 0 - t or 0 F interval and is called “the integral interdiffusion coefficient”,27,28 also being an effective diffusion coefficient and varying with the degree of conversion of the resin from one ionic form to another, because the actual F at time t (the measured value) depends also on the electrical transference. The diffusion coefficients obtained according to eq 6 for mutual ion exchange processes are integral interdiffusion coefficients in macropores and micropores, respectively, and vary with the exchanged fraction.
(iv) Evaluation of Self-Diffusion Coefficients. The actual interdiffusion coefficients in the resin matrix could be obtained using the Nernst-Planck model22,23 for particle-diffusion control. This model takes into account the influence of the electric field on the ion interdiffusion. In this case, the interdiffusion coefficients vary with the resin ionic composition and depend on the self-diffusion coefficients of the two exchanging ions:22
D h HCd )
D h HD h Cd(C h H + 4C h Cd) (C h HD h H + 4C h CdD h Cd)
(7)
The Helfferich minority rule23 shows that the actual interdiffusion coefficient tends to the self-diffusion coefficient of the ion present in the resin phase in trace concentration. If the proton is initially in the resin and the cadmium(II) is the entering ion, then the actual interdiffusion coefficient increases as the exchange progresses, because the proton diffuses faster than the cadmium(II) ion. The value of the integral interdiffusion coefficient (obtained using eqs 1-6 for mutual exchange) varies also during the ion exchange process. In the first approximation, one could consider that these coefficients approach the actual interdiffusion coefficients. The evaluation is better for small exchanged fractions. In this case, at very low fractional attainment of equilibrium F, the integral interdiffusion coefficient tends to the self-diffusion coefficient of cadmium ion. The maximum value of the integral interdiffusion coefficient tends to the proton self-diffusion coefficient, but this is a more rough approximation than for cadmium(II) self-diffusion coefficient. Experimental Section Materials. The commercial Purolite C 100 type gel, C 150 macroporous, and MN 500 macronet (hypercross-linked) strong acid resins with polystyrenic networks and -SO3H functional groups were used. The same manufacturer was selected for a better comparison, but no endorsement is implied; other manufacturers produce polystyrenic resins with different textures. The resins were air-dried and sieved to separate several size fractions. Each size fraction was purified in a separate column, in three cycles, with 1 M solutions of HCl and NaOH and deionized water with specific conductivity of 0.05 µS cm-1. For the macronet resin, four supplementary treatments with 1 M KOH and NaOH were accomplished, to eliminate the brown impurities, together with corresponding washing with a large number of bed volumes (BV) of deionized water. Finally, the resins were transformed in the hydrogen form with 300% excess of 1 M HCl and washed until the rinsing water had a specific conductivity < 1 µS cm-1 (∼30 BV of water were used). The purified fractions were air-dried and kept in a desiccator over a saturated solution of NaCl, to become saturated with water vapor until reaching a constant weight. The diameters of the swollen resin beads in hydrogen and cadmium(II) form were measured microscopically, and the average mean radius was obtained from at least 50 measurements. The results together with the confidence limits for a Student distribution and 99.9% probability are given in Table 2. It must be noted that the macronet beads shrunk after purification and had smaller radii than the corresponding size fraction of gel and macroporous resins.
Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 9099 Table 2. Resin Properties and Experimental Parameters resin C 100 matrix functional group sieve diameter/(mm) 1-0.80 0.80-0.63 0.63-0.50 0.50-0.40 0.40-0.315 sieve diameter/(mm) 1-0.80 0.80-0.63 0.63-0.50 0.50-0.40 0.40-0.315 weight capacity (Na+ form) (eq kg-1) specific gravity /(g cm-3) moisture (%) (H+ form) surface area/(m2 g-1) BET pore volume (mL g-1) d50/(nm) meso and macropores d50/(nm) micropores initial concentration of Cd(NO3)2 in the kinetic run/ (M) stirring speed/(min-1) temperature/(K) a
C 150
styrene-divinylbenzene (∼7.6% DVB) type gel -SO3H
4.98 ( 0.36
styrene-divinylbenzene (∼12% DVB) type macroporous -SO3H mean radius of H+ form‚103/(m) 0.487 ( 0.019 0.405 ( 0.022 0.331 ( 0.015 0.269 ( 0.015 0.224 ( 0.013 mean radius of Cd2+ form‚103/(m) 0.444 ( 0.023 0.351 ( 0.024 0.283 ( 0.018 0.227 ( 0.013 0.181 ( 0.015 4.84 ( 0.54
45.81a
51.7a
0.507 ( 0.026 0.406 ( 0.022 0.335 ( 0.017 0.278 ( 0.019 0.452 ( 0.027 0.374 ( 0.021 0.279 ( 0.016 0.231 ( 0.014
0.495
experimental conditions 0.495
MN 500 styrene-divinylbenzene type macronet (hypercross-linked) -SO3H 0.378 ( 0.019 0.308 ( 0.026 0.287 ( 0.017
0.347 ( 0.018 0.274 ( 0.022 0.244 ( 0.016 2.70 ( 0.35 1.04a 52.7a 1100a 0.53a 100a 1.5a 0.695
500 and 600 298 ( 0.5
From ref 3.
The weight capacity was determined by displacing H+ with Na+ in a column and titrating potentiometrically the acid with standard 0.1 M NaOH solution. The values are reported for dry resins (9 h at 120 °C) with confidence limits for a Student distribution and 99.9% probability. The external solution was prepared from Cd(NO3)2‚4H2O Fluka p.a., and the concentration was determined by titration with EDTA 0.1 M at pH ) 10, indicator erio-T. The solutions of HCl, NaOH, and KOH were prepared from reagents of analytical grade. Kinetic Measurements. The kinetic measurements were performed in a batch reactor of polyethylene with tronconical geometry (50 mm lower diameter, 60 mm upper diameter, and 60 mm height). A cylindrical magnetic stirrer (Teflon-covered), 24 mm long and 6.5 mm outer diameter, was activated by a magnetic stirrer (Variomag type maxi), controlled by a Telemodule 20C. A known mass of resin saturated with water vapor (∼0.5 g weighted with a precision of (1 × 10-4 g) was introduced in 25 mL of deionized water (with a specific conductivity of 0.05 µS cm-1) and allowed to swell for 24 h at room temperature in the reactor sealed with a lid, and the volume was corrected for swelling before the kinetic run. A combined pH-electrode, provided with a temperature probe, was introduced into the reactor, and the vessel was equilibrated at constant temperature in a Memmert WB 22 temperature-controlled water bath, under constant stirring speed. The sample was left under stirring a longer time than the time necessary to achieve 98% of the ion exchange process, to eliminate most of the sorbed electrolyte during conversion in the hydrogen form. The ion exchange process was started by the fast addition of Cd(NO3)2 solution (25 mL of stock solution), and the variation of the pH in the external solution was monitored at appropriate time intervals using a Mettler Delta 350 pH meter, with an accuracy of (0.001 units. The absolute value of the pH was measured with an accuracy of (0.01 units, limited by the standard buffers. The minimum and maximum pH variation of the external solution during a kinetic run was (0.61-0.95), (0.73-1.17), and (0.76-1.3) pH units, in the pH range (2.29-1.24), (2.49 -1.29),
and (2.86-1.38) for gel, macroporous, and macronet (hypercross-linked), respectively. The pH at equilibrium was verified after at least 24 h under both static and stirring conditions. The ionic strength of the initial solution, I, was 1.485, 1.485, and 2.085 M for experiments on gel, macroporous, and macronet resins, respectively. The maximum variation for different runs was 0.1. Both series are not convergent for all
Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 9101 Table 3. Fitted Curves (eq 9) to the Experimental Kinetic Data (F vs t) for H+/Cd2+ Ion Exchange on Gel, Macroporous, and Hypercross-linked Resins, at 500 min-1 and 298 K jr0 × 103 (m)
no. exp. pts.
a
b
c
d
coeff. detn.
F-statistic
0.507 0.406 0.335 0.278
145 128 86 90
0.527 ( 0.017 0.308 ( 0.063 0.660 ( 0.047 0.404 ( 0.053
gel resin 0.009 87 ( 0.000 21 0.012 40 ( 0.000 81 0.030 4 ( 0.001 3 0.045 6 ( 0.005 4
0.456 ( 0.019 0.694 ( 0.069 0.298 ( 0.052 0.586 ( 0.058
0.208 ( 0.025 0.086 ( 0.020 1.44 ( 0.98 0.173 ( 0.037
0.999 3 0.997 0 0.997 9 0.996 3
72 029 13 854 13 347 7 644
0.487 0.405 0.331 0.269 0.224
145 127 110 77 79
0.642 ( 0.017 0.662 ( 0.031 0.636 ( 0.029 0.675 ( 0.074 0.680 ( 0.068
macroporous resin 0.0106 8 ( 0.000 17 0.343 ( 0.019 0.0151 3 ( 0.000 39 0.313 ( 0.035 0.022 29 ( 0.000 49 0.329 ( 0.031 0.031 6 ( 0.001 2 0.294 ( 0.082 0.044 3 ( 0.001 9 0.273 ( 0.072
0.291 ( 0.047 0.38 ( 0.12 0.61 ( 0.18 0.59 ( 0.42 2.0 ( 2.1
0.999 5 0.999 0 0.998 9 0.998 4 0.995 9
89 298 42 014 31 749 15 126 6 103
0.378 0.308 0.287
223 88 88
0.807 ( 0.019 0.717 ( 0.040 0.828 ( 0.034
hypercross-linked resin 0.0217 5 ( 0.000 42 0.202 ( 0.021 0.0252 8 ( 0.000 74 0.298 ( 0.045 0.0354 4 ( 0.000 83 0.180 ( 0.037
2.1 ( 1.0 0.696 ( 0.300 3.3 ( 3.0
0.998 5 0.999 3 0.999 1
47 375 39 804 31 908
systems when F < 0.1. The number of terms necessary to achieve the convergence, i.e., until the difference between two successive solutions was 0.75 (see Figure 2), not only for F > 0.86. The ISV condition (eqs 1-3) gives slightly higher values for the diffusion coefficients compared with the FSV models (eqs 4-5) for dimensionless equilibrium parameter ω of 0.05 (see Figure 2, gel and macroporous resins), but the agreement increases when ω < 0.02 (hypercross-linked resin). It is important to note that the simplified eqs 2 (ISV) and 5 (FSV) give D h values in very good agreement with the more rigorous eqs 1 and 4, respectively, and can be used at low fractional attainment of equilibrium, the region where the series 1 and 4 are slowly convergent, but cannot be used for very high F values, F > 0.95 for eq 2 and F > 0.97 for eq 5. Equation 2 was proposed for F < 0.85, but the results show that it could be used for an extended domain. The mean radius of the swollen resin particles in H+ form is higher than that in Cd2+ form for all systems. The shrinkage of the resin beads when swelling from a concentrated solution of electrolytes was recognized long time ago.29 The average radius for the same size fraction for the hypercross-linked resin is lower than those expected in comparison with gel and macroporous resins. In most cases, the diffusion coefficients are reported in the literature without specifying the mean radius of the resin particles and the ionic form in which were measured. The size of the particles varies during the ion exchange process from the value corresponding to H+ form in water to that of the Cd2+ form in a concentrated solution of cadmium nitrate (see Table 2). The H+/Cd2+ integral interdiffusion coefficients were calculated with eq 5 for all systems and for all size fractions using the mean radii of the swollen resin particles in H+ and Cd2+ forms, being reported in Figures 3-5. It can be observed that the decrease of the radius of the resin beads from H+ to Cd2+ form for the same size fraction produces the decrease of the interdiffusion coefficients, for all resins. Moreover, the interdiffusion coefficients decrease with decreasing size fraction, behavior that was previously recognized by Liberti et al.30 for Cl-/SO42- interdiffusion in different ion exchangers. Our results showed the same trend, with two exceptions that could be due to many causes, like experimental errors in size fraction separation or a different texture of the resin particles of the corresponding size fraction. The H+/Cd2+ integral interdiffusion coefficients in all resins increase with the fractional attainment of equilibrium, in accordance with the Helfferich minority rule for actual interdiffusion coefficients in the Nernst-Planck model. The shape
of D h vs F curves depends on the size fraction of the resin. This behavior could be associated with a particular texture of a size fraction for a certain resin. For gel and macroporous resins, D h increases, reaching a maximum value and, for high F values, decreases, while for the hypercross-linked resin, D h presents a monotone increase. In gel and macroporous resins at high conversions, a nonideal behavior is observed that can be attributed to ion pair formation and/or to the decrease of the selectivity near equilibrium. Another explanation could be associated with the solvent content of the beads. Helfferich23 has shown that the solvent content of beads passes through a maximum during the exchange of H+ with Na+. Supposing a similar behavior for H+/Cd2+ exchange, the maximum content of solvent could be correlated with D h maximum. For the hypercross-linked resin, this behavior was not observed. The macropores of hypercross-linked resin could accommodate a larger quantity of solvent than the mesopores and the micropores of the other two resins; the free solvent content is probably large enough to prevent the decrease of D h. The comparison of the results obtained for the three generations of sulfonic polystyrenic resins, for comparable radii of the swollen resin particles, shows that the interdiffusion coefficients on hypercross-linked resin are close to those on the gel resin, while on the macroporous one, they are lower (see Supporting Information). When compared with the same size fraction, D h are lower for the hypercross-linked resin because the beads shrunk during purification, and their mean radii became comparable with the lower size fraction of gel and macroporous resins. The brown impurities eliminated during the thorough purification were probably sulfonated macromolecular chains with low molecular weight, weakly bounded on the three-dimensional matrix, soluble in alkaline solution. The different shapes of D h vs F curves on the hypercross-linked resin, compared with the gel and macroporous ones, must be noted. The textural effects of the resin framework on H+/Cd2+ interdiffusion appear in the interdiffusion coefficients values and in the shape of D h vs F curves. (ii) Macropore Integral Interdiffusion Coefficients Obtained with a Bidisperse Pore Kinetic Model. A particle of macroporous resin contains mostly micro- and mesopores and only a few macropores having a small contribution to the pore volume. The used hypercross-linked resin beads have macro-, meso-, and micropores. Both porous resins have a trimodal pore distribution, according to IUPAC classification. In the following treatment, the macroporous and hypercross-linked resins will be considered having a bidisperse pore structure. As micropores we designate real micropores (diameter < 2 nm) and small mesopores, and as macropores we designate most of mesopores
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Figure 3. Dependence of the H+/Cd2+ integral interdiffusion coefficient on gel resin on the size fraction and the ionic form of the swollen resin particles; Paterson τ < 0.1 model, eq 5; 298 K.
Figure 4. Dependence of the H+/Cd2+ integral interdiffusion coefficient on macroporous resin on the size fraction and the ionic form of the swollen resin particles; Paterson τ < 0.1 model, eq 5; 298 K.
Figure 2. H+/Cd2+ integral interdiffusion coefficients vs fractional attainment of equilibrium on gel, macroporous, and hypercross-linked resins calculated with different QHRP models; 298 K.
and real macropores (diameter > 50 nm). According to IUPAC classification, the mesopores have diameters between 2 and 50 nm. The limiting case of the Ruckenstein bidisperse pore model seems to be an adequate approximation of the ion diffusion in the macroporous and hypercross-linked resins. A computer program previously written26 to numerically solve eq 6 was used. The input data are a t - F file and the mean radius of the swollen resin particles, R, β/R; the output data are given in a multicolumn file, t, F, θ, and D h a. A good convergence (0.1% differences in D h a) was obtained for F > 0.1 when 10 terms n were taken into account. The parameter R was 10-3, as for the considered limiting case. The parameter β/R is unknown, and
consequently, the computations were done for its different values and the results were screened, comparing the data with the following: • H+/Cd2+ integral interdiffusion coefficients vs F in the gellike resin for the size fraction with the mean radius close to the investigated system; • Cd2+ self-diffusion coefficient in dilute aqueous solution31 0.719 × 10-9 m2 s-1 at 298 K; • H+ self-diffusion coefficient in dilute aqueous solution31 9.311 × 10-9 m2 s-1 at 298 K; • H+ self-diffusion coefficient in a sulfonated phenolformaldehyde matrix 2.4 × 10-9 m2 s-1 at 298 K, reported by Helfferich32,33; and • H+/Cd2+ integral interdiffusion coefficients vs F for the same system calculated previously with QHRP model eq 3-ISV with n ) 10. The H+/Cd2+ macropore integral interdiffusion coefficients D h a, calculated with eq 6 for different β/R on the macroporous and hypercross-linked resins, are given in Figures 6 and 7,
Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 9103
Figure 5. Dependence of the H+/Cd2+ integral interdiffusion coefficient on hypercross-linked resin on the size fraction and on the ionic form of the swollen resin particles; Paterson τ < 0.1 model, eq 5; 298 K.
Figure 7. Dependence of the H+/Cd2+ macropore integral interdiffusion coefficient on the hypercross-linked resin on fractional attainment of equilibrium, obtained with LC-BDM for different β/R ratio; 298 K.
Figure 6. Dependence of the H+/Cd2+ macropore integral interdiffusion coefficient on the macroporous resin on fractional attainment of equilibrium, obtained with LC-BDM for different β/R ratio; 298 K.
respectively. When β/R is too high, the obtained diffusion coefficient at high conversions F is greater than the H+ selfdiffusion coefficient in dilute aqueous solutions, and these values are without physical significance and are rejected. For the macroporous resin, for low β/R values, the results for D h a are
lower than for gel resin and increase with increasing β/R. For β/R ) 0.05, D h a vs F curve in the macroporous resin seems to be the most plausible curve because D h a values at high F do not rise above the value of the Cd2+ self-diffusion coefficient in dilute aqueous solution, as it does for β/R ) 0.075 (see Figure 6). These D h a coefficients are still lower than in the gel resin. The QHRP-ISV-eq 1 with n ) 10 gives lower values than eq 6 for β/R ) 0.05 and n ) 10, but at limit could also be considered an acceptable result. The most plausible values for the H+/Cd2+ macropore integral interdiffusion coefficients in the hypercross-linked resin could be considered those calculated for β/R ) 0.03, because higher β/R values give for D h a at high F values higher than Cd2+ or even H+ self-diffusion coefficients in dilute aqueous solution (see Figure 7). The Ruckenstein model indicates that the macroporous resin seems to have one -SO3H group accessible for Cd2+ ions grafted on micropore walls for 60 groups grafted in macropores, while for hypercross-linked resins, the ratio is 1/100 (see Table 1). (iii) Evaluation of Self-Diffusion Coefficients. Table 4 gives the H+/Cd2+ integral interdiffusion coefficients in the gel,
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Table 4. Integral Interdiffusion Coefficients of H+/Cd2+ Pair on Sulfonic Polystyrenic Resins Type Gel, Macroporous, and Hypercross-linked, Calculated with Eq 5 at a Low Fractional Attainment of Equilibrium F ) 0.01, at Half Exchange F ) 0.5, and at a Degree of Exchange Corresponding to the Maximum Value of the Coefficients
a
resin
radius × 103 (m)
D h F)0.01 × 1010 (m2 s-1)
D h F)0.5 × 1010 (m2 s-1)
D h max × 1010 (m2 s-1)
gel H+ gel Cd2+ gel H+ gel Cd2+ gel H+ gel Cd2+ gel H+ gel Cd2+ macroporous H+ macroporous Cd2+ macroporous H+ macroporous Cd2+ macroporous H+ macroporous Cd2+ macroporous H+ macroporous Cd2+ macroporous H+ macroporous Cd2+ hypercross-linked H+ hypercross-linked Cd2+ hypercross-linked H+ hypercross-linked Cd2+ hypercross-linked H+ hypercross-linked Cd2+
0.507 0.452 0.406 0.374 0.335 0.279 0.278 0.231 0.487 0.444 0.405 0.351 0.331 0.283 0.269 0.227 0.224 0.181 0.378 0.347 0.308 0.274 0.287 0.244
0.0973 0.0773 0.0582 0.0494 0.132 0.0915 0.0481 0.0332 0.0768 0.0638 0.0609 0.0457 0.0689 0.0504 0.0418 0.0297 0.0708 0.0462 0.120 0.101 0.0617 0.0488 0.0915 0.0661
2.13 1.70 1.85 1.57 2.16 1.50 1.80 1.24 1.68 1.40 1.48 1.11 1.52 1.11 1.23 0.876 1.29 0.844 1.85 1.56 1.53 1.21 1.65 1.19
2.17 1.72 1.96 1.66 2.46 1.70 2.19 1.52 2.01 1.67 1.83 1.38 1.77 1.29 1.59 1.13 1.53 0.998 3.27a 2.76a 2.56a 2.03a 3.00a 2.17a
Values calculated with eq 4 at F ) 0.99.
Table 5. Integral Interdiffusion Coefficients of H+/Cd2+ Pair on Sulfonic Polystyrenic Resins Type Macroporous and Hypercross-linked Calculated with Equation 6 at Half Exchange F ) 0.5 and at a Degree of Exchange Corresponding to the Maximum Value of the Coefficientsa resin
radius × 103 (m)
× 1010 D h F)0.5 a (m2 s-1)
× 1010 D h max a (m2 s-1)
macroporous H+ macroporous Cd2+ macroporous H+ macroporous Cd2+ macroporous H+ macroporous Cd2+ macroporous H+ macroporous Cd2+ macroporous H+ macroporous Cd2+ hypercross-linked H+ hypercross-linked Cd2+ hypercross-linked H+ hypercross-linked Cd2+ hypercross-linked H+ hypercross-linked Cd2+
0.487 0.444 0.405 0.351 0.331 0.283 0.269 0.227 0.224 0.181 0.378 0.347 0.308 0.274 0.287 0.244
1.87 1.55 1.64 1.24 1.69 1.23 1.36 0.965 1.43 0.933 1.95 1.64 1.63 1.29 1.74 1.26
2.27 1.89 2.06 1.54 1.97 1.44 1.77 1.26 1.70 1.11 5.80 4.89 4.57 3.62 5.33 3.85
a R ) 0.001; β/R ) 0.05 for macroporous resin; and β/R ) 0.03 for hypercross-linked resin.
macroporous, and hypercross-linked resins, calculated with QHRP-FSV-eq 5, at low fractional attainment of equilibrium F ) 0.01, at half-exchange F ) 0.5, and at a degree of exchange corresponding to the maximum value of the coefficients. For the hypercross-linked resin, the maximum value was calculated with eq 4, because eq 5 is valid only for F < 0.98. The numerical values of the H+/Cd2+ macropore integral interdiffusion coefficients in the macroporous and hypercross-linked resins, calculated with eq 6, are given in Table 5 at halfexchange and the maximum value. Because of the low convergence of series in eq 6, D h a at F ) 0.01 were not obtained with this model. If one accepts that the variation of the integral interdiffusion coefficients with F is in agreement with the Helfferich minority rule, D h at F ) 0.01 tends to Cd2+ selfdiffusion coefficient and D h maximum tends to the H+ selfdiffusion coefficient in the corresponding resins.
The evaluated Cd2+ self-diffusion coefficients in gel, macroporous, and hypercross-linked resins taken as the mean value for different size fractions, for radii in H+ form, are 8.4 × 10-12, 6.4 × 10-12, 9.1 × 10-12 m2 s-1, respectively; the Cd2+ self-diffusion coefficient on the hypercross-linked resin is the highest, close to the gel resin, while on the macroporous resin, it is the lowest. The evaluated H+ self-diffusion coefficients as a mean value for different size fractions for radii in Cd2+ form are 1.7 × 10-10, 1.3 × 10-10, 2.3 × 10-10 m2 s-1, for the gellike, macroporous, and hypercross-linked resins, respectively; the H+ self-diffusion using mean coefficient in macropores evaluated from D h max a 2+ -10 radius for Cd form is 1.4 × 10 and 4.1 × 10-10 m2 s-1 for the macroporous and hypercross-linked resins, respectively. The analysis of the data presented in Tables 4 and 5 shows the following results: • The mean radius of the resin particles decreases with 8-17%, 9-19%, and 8-15% when passing from H+ to Cd2+ form for gel, macroporous, hypercross-linked resins, respectively; the variation is greater for the smaller size fractions for all resins • The H+/Cd2+ integral interdiffusion coefficients obtained with the QHRP model decrease with 15-31%, 17-35%, and 16-28% with the variation of the mean radius of the swollen particles from H+ to Cd2+ form, when the size fraction decreases, for gel, macroporous, and hypercross-linked resins, respectively. A certain percentage decrease in the swollen resin bead radius produces a decrease around twice greater for the diffusion coefficient, for all resins, and for all fractional attainment of equilibrium; the same effect was observed for the macropore integral interdiffusion coefficients calculated with LC-BDM: 17-35% and 16-29% decrease for macroporous and hypercross-linked resins, respectively. The effective interdiffusion coefficient in bidisperse porous resins with moderate selectivity for an ion was interpreted by Yoshida and coworkers34,35 and Koh et al.36 in terms of parallel diffusion in macropores and micropores, showing its dependence on concentration of the external solution. Li and SenGupta10 have shown that the interdiffusion in macropores is the main process
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for very selective resins, with the effective interdiffusion coefficient also being dependent on the total concentration of the external solution. In terms of these models, the effective integral interdiffusion coefficients reported in this work are valid at high external solution concentrations. • The ratio between the evaluated H+ and Cd2+ self-diffusion coefficients is around 20 for the gel and macroporous resins and 25 for the hypercross-linked one, in accordance with data on other resins,23 for uni-divalent exchange. • The ratios between the self-diffusion coefficient in dilute aqueous solution and the evaluated effective self-diffusion coefficient in the gel, macroporous, and hypercross-linked sulfonic polystyrenic resins are 86, 112, and 79 for Cd2+ and 58, 72, and 40 for H+, respectively. The ratios between the H+ self-diffusion coefficient in dilute aqueous solution and the evaluated H+ macropore self-diffusion coefficient are 62 and 23 for macroporous and hypercross-linked resins, respectively; this ratio has the physical meaning of a tortuosity factor.10 The results presented above show the resin texture influence on the H+/Cd2+ intraparticle interdiffusion. The interdiffusion coefficients on the three generations of polystyrenic resins decrease in the following series: hypercross-linked (macronet) > gellike > macroporous. For each resin, the effective interdiffusion coefficients decrease with the decrease of the swollen resin bead size. In the QHRP models, the decrease of the effective diffusion coefficient is associated with a higher “excluded-volume”,23 a greater occupancy of volume and cross section by the polymeric framework. The measured intraparticle interdiffusion coefficients are macroscopic properties, with their variation from one resin to another being induced by the differences in the texture of the resins, as an ensemble. In addition, different texture determines different swelling. The hypercross-linked resin seems to have the larger free-water content, taking into account the effective diffusion coefficients values and the shape of their variation with the ionic composition of the resin. The retarding effect of the framework is probably due more to the resin rigidity than to the pore dimension.23 The macroporous resin has the more rigid polymeric network and the lowest interdiffusion coefficients. The dependences of the diffusion coefficients on the size fraction of the resin particles and, for the same size fraction, on the variation of the radius of a resin particle during conversion from one to another ionic form, are very important in comparing the results obtained for different systems, in different laboratories. Conclusions The H+/Cd2+ intraparticle interdiffusion coefficients on three generations of sulfonated polystyrenic resins, of type gel, macroporous, and macronet (hypercross-linked), were investigated in order to evaluate the magnitude of the textural effects on mutual ion exchange kinetics. The ion exchange rates were measured at 298 K in a batch reactor using a potentiometric method, in conditions favoring a particle-diffusion controlled mechanism, supported by the experimental data. Quasihomogeneous resin phase kinetic models at finite and infinite solution volume were used to obtain the integral interdiffusion coefficients, and the results were used to compare the models and the textural effects. The Ruckenstein bidisperse pore model was employed to obtain the macropore integral interdiffusion coefficients in the macroporous and hypercross-linked resins. The evaluated ratio between the sulfonic groups grafted in micro- and macropores available to Cd2+ ions was 1/60 and 1/100 for macroporous and hypercross-linked resins, respectively. The H+/Cd2+ intraparticle interdiffusion coefficients on the inves-
tigated resins decrease in the series hypercross-linked (macronet) > gel > macroporous. The interdiffusion coefficients vary with the size fraction of the resin and with the ionic form of the swollen resin particles. The evaluated Cd2+ self-diffusion coefficients were 8.4 × 10-12, 6.4 × 10-12, and 9.1 × 10-12 m2 s-1, while the evaluated H+ self-diffusion coefficients were 1.7 × 10-10, 1.4 × 10-10, and 4.1 × 10-10 m2 s-1 on gellike, macroporous, and hypercross-linked resins, respectively. The higher retarding effects of the macroporous framework on the ion diffusion seem to be due more to its rigidity than to the pore diameters. Acknowledgment This work was supported by CNCSIS, Romania, under Grant No. 1379/2005. The authors acknowledge Purolite International Ltd. for supplying the resins. Supporting Information Available: The Supporting Information includes the following: (a) experimental data regarding the lack of influence of the stirring speed on the rate of H+/ Cd2+ ion exchange process on gel, macroporous, and macronet resins (Figure 1); (b) the linear dependence of the half-time of the same processes on the square of the radius of swollen resin beads (Figure 2); (c) the comparison of the variation of H+/ Cd2+ integral interdiffusion coefficients in gel, macroporous, and hypercross-linked (macronet) resins with the exchanged fraction, for particles with similar size in the swollen state (Figure 3). Nomenclature Ci ) molar concentration of ion i in the external solution (M) C h H, C h Cd ) molar concentration in the resin phase of the proton and cadmium ions, respectively (M) D h ) effective intraparticle diffusivity; self-diffusion coefficient for isotopic exchange; integral interdiffusion coefficient for mutual ion exchange (m2 s-1) D h a ) effective macropore diffusivity; macropore self-diffusion coefficient for isotopic exchange; macropore integral interdiffusion coefficient for mutual ion exchange (m2 s-1) D h i ) effective micropore diffusivity; micropore self-diffusion coefficient for isotopic exchange; micropore integral interdiffusion coefficient for mutual ion exchange (m2 s-1) D h HCd ) actual effective intraparticle diffusivity for H+/Cd2+ exchange (m2 s-1) h Cd ) effective intraparticle self-diffusion coefficients of D h H, D proton and cadmium ions, respectively (m2 s-1) ej, e ) number of ion equivalents at equilibrium in the resin and solution phases, respectively (eq) F ) fractional attainment of equilibrium (dimensionless) I ) ionic strength, I ) (1/2)∑iCizi2 (M) Ma∞ ) macropore uptake at equilibrium (eq/kg) Mi∞ ) micropore uptake at equilibrium (eq/kg) n ) number of terms in a series pH0, pH∞, pHt ) pH of the external solution at initial, equilibrium, and time t jr0 ) mean radius of the swollen beads of the resin (m) jri ) microsphere radius (m) Sn ) roots of eq: Sn cot Sn ) 1 + Sn2/3ω t ) time (s) zi ) valence of ion i Greek Symbols R ) (D h ijr02)/(D h ajri2) ) dimensionless rate parameter β/R ) Mi∞/Ma∞ ) dimensionless equilibrium parameter
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ReceiVed for reView February 15, 2006 ReVised manuscript receiVed October 5, 2006 Accepted October 5, 2006 IE060186W
