Interdiffusion of Exchanging Counterions in Poly(perfluorosulfonic acid

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Interdiffusion of Exchanging Counterions in Poly(perfluorosulfonic acid) Membrane Suparna Sodaye,† G. Suresh,‡ Ashok K. Pandey,*,† and A. Goswami† Radiochemistry DiVision and Research Reactor SerVices DiVision, Bhabha Atomic Research Centre, Mumbai 400 085, India ReceiVed: March 2, 2009; ReVised Manuscript ReceiVed: June 25, 2009

The kinetics of forward (Li+/Na+mem h Mn+aq) and reverse (Mn+mem h Li+/Na+aq) exchanges of metal ions (Mn+ ) Ag+, Cs+, Ba2+, and Eu3+) in the poly(perfluorosulfonic acid) membrane (Nafion-117) equilibrated with a well-stirred aqueous salt solution were experimentally measured using radioactivity tagged counterions. A numerical solution of the Nernst-Planck (N-P) equation for interdiffusion was used to interpret the kinetics of these exchanges. The experimentally measured kinetics of forward and reverse Na+ ion exchanges with Mn+ ions between the membrane and the equilibrating solution were found to be close to that predicted by the N-P equation. The minor differences between the experimental and predicted exchange rate profiles were attributed to change in water content of the poly(perfluorosulfonic acid) membrane in different ionic forms. The kinetics of Li+ ion exchanges with Na+ and Cs+ ions was found to be unusual as the self-diffusion + coefficient Dm ion of Li ions in the membrane were quite different in the forward and reverse exchanges. The + values of Dm for Li ion, obtained by the N-P equation, were found to be 2 × 10-6 and 0.2 × 10-6 cm2 s-1 ion + + + in Li mem h Na /Cs aq and Na+/Cs+mem h Li+aq exchanges, respectively. The drastic differences in diffusion mobility of Li+ ions during forward and reverse exchanges was attributed to weak electrostatic interactions of Li+ ions with the fixed exchange sites as indicated by the reported equilibrium constant Kex of the Li+exchange reaction in Nafion membrane. Dm ion is high in the forward exchange due to minimum retardation in diffusion mobility of Li+ ions by the electrostatic forces in the membrane. In reverse exchange, Li+ diffusion mobility is retarded due to its slow replacement of counterions from the membrane. Introduction Study of ionic transport processes across charged membrane is important for understanding the membrane-based separation process. The transport of ions in ion-exchange membrane is influenced by the chemical and physical structures of the membrane as well as valence and hydration characteristics of the ions.1-4 Self-diffusion coefficients of ions in ion-exchange membranes provide information about the membrane morphology as well as the electrostatic interactions between the fixed charged sites and the mobile counterions. Self-diffusion coefficient of water in the ion-exchange membranes in a particular ionic form indicates the tortuosity factor, which is also related to water channels providing the transport path for the ions in the membranes.5,6 The self-diffusion mobility of water and a variety of counterions have been measured in poly(perfluorosulfonic acid) (Nafion-117) membrane using different techniques such as radiotracer permeation,5,7-11 nonstationary radiotracer sorption/desorption,12-15 ion exchange,16,17 conductivity/impedance measurements,18,19 and pulsed-field gradient nuclear magnetic resonance spectroscopy (PFG-NMR).20,21 While self-diffusion coefficient provides information about the mobility of individual ions, the counterion exchange involves interdiffusion of the two counterions in the membrane. During the ion exchange, the process of self-diffusion is complicated due to coupling of fluxes of interchanging ions. The fluxes of the counterions are governed by the interdiffusion coefficient of the ions, which changes as the ion exchange progresses. As * To whom correspondence should be addressed. Tel.: +91-22-25594094. Fax: +91-22-25505150/-25505151. E-mail: [email protected]. † Radiochemistry Division. ‡ Research Reactor Services Division.

a result, the rate of the forward exchange (An+mem h Bn+aq) and the reverse exchange (Bn+mem h An+aq) differs due to the unequal mobility (diffusion constant) of the two interchanging counterions An+ and Bn+ in the membrane. Two approaches based on Nernst-Planck (N-P) and Maxwell-Stefan (M-S) theories have been used to analyze the rates of counterion exchange between membrane and electrolyte solution for obtaining the self-diffusion coefficient of ions in the membrane.22-24 The interdiffusion equation obtained from Nernst-Planck theory has been examined to predict the kinetics of forward (H+mem h Mn+aq) and reverse (Mn+mem h H+aq) exchanges in poly(perfluorosulfonic acid) membrane.13 This theory has been found to be reasonably accurate for predicting kinetics of the reverse ion exchange but completely failed for the forward ion exchange. Contrary to the prediction of the theory, the kinetics of forward exchanges of Mn+ ()Cs+, Cu2+, and Eu3+) with H+ ions in the poly(perfluorosulfonic acid) membrane have been found to be independent of the metal ions.13 The deviation of forward counterion exchanges with H+ ions from nonlinear interdiffusion N-P equation was attributed to the hopping mechanism involved in the diffusion of H+ ions. To understand the interdiffusion process involved in counterions exchanges, the rates of forward (Li+/Na+mem h Mn+aq) and reverse (Mn+mem h Li+/Na+aq) exchanges of metal ions (Mn+ ) Ag+, Cs+, Ba2+, and Eu3+) in Nafion-117 equilibrated with a well-stirred aqueous salt solution were experimentally measured using the radioactivity tagged counterions. A numerical solution of the N-P equation for interdiffusion of counterions was used to interpret the experimental exchange rate profiles. Na+ ion was used as a reference counterion with other interchanging ions (Ag+, Cs+, Ba2+, and Eu3+) in forward (Na+mem h Mn+aq) and reverse (Mn+mem h Na+aq) exchanges

10.1021/jp901907v CCC: $40.75  2009 American Chemical Society Published on Web 08/25/2009

Transport of Ions in Poly(perfluorosulfonic acid) Membrane to test the validity of the N-P equation. The choice of Na+ ions as a reference counterion in poly(perfluorosulfonic acid) membrane was based on the following: (i) the membrane in Na+ form has a high water content, and (ii) Na+ ions do not diffuse through the hopping mechanism as in the case of H+ ions. The self-diffusion coefficients of Na+, Ag+, Cs+, Ba2+, and Eu3+ ions in the poly(perfluorosulfonic acid) membrane are reported in the literature.12,13 Okada et al. have reported the mobility of alkali metal cations in the following order: Li+ < Na+ > K+ > Rb+ > Cs+.4 This is in accordance with the observation that the self-diffusion coefficient of the counterions decreases with an increase in the selectivity except for Li+ ions.4 Therefore, the counterion-exchange profiles of Li+ ions with Na+ and Cs+ have also been studied. The results of analyses of experimental counterion-exchange kinetics in terms of a nonlinear diffusion equation of interchanging counterions have been discussed in this paper.

J. Phys. Chem. B, Vol. 113, No. 37, 2009 12483 where

D+ )

2 + b[y(ξ + ∆ξ) + y(ξ)] ; 2 + a[y(ξ + ∆ξ) + y(ξ)]

D- )

2 + b[y(ξ) + y(ξ - ∆ξ)] ; 2 + a[y(ξ) + y(ξ - ∆ξ)]

a)

ZADA - 1; ZBDB

b)

ZA -1 ZB

The value of y(ξ,τ+∆τ) can be evaluated from the values of y(ξ+∆ξ,τ), y(ξ,τ), and y(ξ-∆ξ,τ) with the following initial and boundary conditions:

Theory The details of numerical solution of nonlinear diffusion equation for interdiffusion of two counterions based on Nernst-Planck equation are described elsewhere.13 For a brief description, consider the process of ion exchange between an ion “A” initially present in the plane sheet of the membrane and an ion “B” initially present in the solution as

ZB AzAmem + ZA BzBaq h ZB AzAaq + ZA BzBmem

yA(ξ, 0) ) 1

(0 e ξ < 1)

(5a)

yA(1, τ) ) 0

(5b)

yA(0, τ) ) yA(0 + ∆ξ, τ)

(5c)

A Fortran program was used to evaluate y(ξ,τ). The fraction qA(τ) still present in the membrane at any time τ is given by

∫01 yA(ξ, τ) dξ

where ZA and ZB are the charges on the ions A and B and “mem” and “aq” represent the membrane phase and solution phase, respectively. Above equation represents counterions exchange without considering the charge balance. Electro-neutrality requires that the total positive charge of all cations inside the membrane should be equal to fixed negative sites. The diffusion equation describing the time and space dependence of the ion concentration ci in the membrane can be written as

This integral was numerically evaluated using Simpson’s rule. Finally the fractional attainment of equilibrium is obtained as

[

The quantity F(τ) is experimentally measured and can be compared with the calculated values obtained from F(τ), which depends on the parameters ZA, ZB, and DA/DB.

∂ci ∂ci ∂ DAB ) ∂t ∂x ∂x

]

(1)

where the interdiffusion coefficient DAB is given by

DAB )

DADB(ZA2CA + ZB2CB)

During the course of ion exchange, CA, CB, and DAB change with time and thus eq 1 becomes nonlinear. For a plain sheet of membrane, using the dimensionless variables,

yA )

ZACA , C

τ)

DAt l2

,

ξ)

x l

(3)

where l is the half-thickness of the membrane and C ) ZACA + ZBCB, the solution of eq 1 can be written in finite difference form as

∆τ [D+{y(ξ + ∆ξ, τ) (∆ξ)2 y(ξ, τ)} - D-{y(ξ, τ) - y(ξ - ∆ξ, τ)}] (4)

y(ξ, τ + ∆τ) ) y(ξ, τ) +

F(τ) ) 1 - qA(τ)

(6)

(7)

Experimental Section

(2)

DAZA2CA + DBZB2CB

qA(τ) )

AR grade chemicals (NaCl, NaNO3, LiCl, BaCl2, AgNO3, CsNO3, and Eu2O3), deionized water (18 MΩ/cm, Gradient A-10 model, Milli-Q, Bedford, MA), and AR grade HCl (Merck) were used in the present study. Poly(perfluorosulfonic acid) membrane, with an equivalent weight of 1100 and thickness of 178 µm (Aldrich), was conditioned with 0.5 mol L-1 HCl and 0.5 mol L-1 NaOH and equilibrated with 0.5 mol L-1 of relevant salt solution for 18-24 h at room temperature (25 °C) for converting the membrane into the appropriate ionic form. Radiotracers 137Cs and 22Na (without carrier) and 110mAg (with carrier) were obtained from the Board of Radiation and Isotope Technology, Mumbai, India. The radiotracer 154Eu was prepared by irradiating a known amount of Eu2O3 (having 99.1% enriched 153 Eu obtained from Euriso-top) for an appropriate time in Dhruva reactor at BARC, Mumbai, India. The 154Eu radiotracer solution was prepared by dissolving the irradiated Eu2O3 in concentrated HCl and evaporating it to near dryness. The acidity of the stock solution of 154Eu was adjusted to pH ) 2 for preventing any hydrolysis during the period of storage. The measurements of counterion-exchange rates were carried out using 2.0 × 2.0 cm2 pieces of the poly(perfluorosulfonic

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Figure 1. Variations of fractional attainment of exchange equilibrium (F(t)) as a function of the square root of equilibration time (t1/2). The symbols represent the experimental data on Ag+mem h Na+aq (O) and Na+mem h Ag+aq (+) exchange. Lines 1-4 represent the calculated profiles of Ag+mem h Ag+aq, Ag+mem h Na+aq, Na+mem h Ag+aq, and m Na+mem h Na+aq exchanges, respectively, based on the values of Dion of counterions in the membrane.

acid) membrane in the appropriate ionic form. The sorption/ desorption of radiotracer counterions in the membrane samples in contact with the equilibrating solutions were monitored as a function of time to obtain exchange rates. The experimental details are given in our previous publication13 and described briefly below. The membrane samples in appropriate ionic form were placed in 25 mL of equilibrating salt solution (0.15 mol L-1) containing relevant radiotracer. Only one of the radiotracer 22Na, 137Cs, 110m Ag, or 154Eu was used in the experiment. In exchange experiments involving Ag+ counterions, NaNO3 and Ag NO3 salts were used. The solution (at 25 °C) containing membrane sample was stirred vigorously (≈52 rad s-1) to minimize the concentration gradient of counterions in the equilibrating solution. The amount of radiotracer counterions sorbed/desorbed from the membrane sample was monitored by taking out the membrane sample from an equilibrating solution at regular time intervals and counting the radioactivity in the membrane. Before monitoring the radioactivity, the membrane samples were washed with deionized water to remove traces of equilibrating solution adhering to its surface. The membrane sample was again placed in the equilibrating solution after counting. The actual residence time of the membrane in the equilibrating solution was considered as the time for counterion exchange. The radioactivity of the counterions was monitored by γ-ray counting in a fixed geometry using a well type NaI(Tl) detector connected to a single-channel analyzer. Results and Discussion The experimentally measured profiles of forward (Na+mem h Mn+aq) and reverse (Mn+mem h Na+aq) exchange rates F(t) ()n(tk)/n*, where n(tk) and n* are the radioactivity of counterions in membrane/solution at time t ) tk and t) ∞, respectively) of metal ions (Ag+, Cs+, Ba2+, and Eu3+) as a function of the square root of equilibration time t1/2 are shown in the Figures 1-4. The values of self-diffusion coefficients (Dmion) for different counterions in the poly(perfluorosulfonic acid) membrane are given in Table 1.12,13 It can be seen from all of the four figures

Sodaye et al.

Figure 2. Variations of F(t) as a function of t1/2. The symbols represent the experimental data on 2Na+mem h Ba2+aq (×) and Ba2+mem h 2Na+aq (∆) exchanges. Lines 1-4 represent the calculated profiles of Na+mem h Na+aq, 2Na+mem h Ba2+aq, Ba2+mem h 2Na+aq, and Ba2+mem h Ba2+aq m of counterions in exchanges, respectively, based on the values of Dion the membrane.

Figure 3. Variations of F(t) as a function of t1/2. The symbols represent the experimental data on Na+mem h Cs+aq (0) and Cs+mem h Na+aq (∆) exchanges. Lines 1-4 represent the profiles of Na+mem h Na+aq, Na+mem h Cs+aq, Cs+mem h Na+aq, and Cs+mem h Cs+aq exchanges, respectively, based on the values of Dmion of counterions in the membrane.

that the counterion exchange rate is altered by the mobility of the ion entering into the membrane. Thus, the rate of exchange of an ion present in the membrane, which is having higher diffusivity, is retarded by a slower diffusing ion entering the membrane. Similarly, the rate of exchange of a slower diffusing ion present in the membrane is accelerated by a faster diffusing ion entering the membrane. As can be seen from Figure 1, the exchange of Ag+mem h Na+aq is slower than the Ag+mem h Ag+aq isotopic exchange and the reverse exchange Na+mem h Ag+aq is faster than the Na+mem h Na+aq isotopic exchange. Similarly, in Figures 2-4, the forward exchange Na+mem h Mn+aq is slower than Na+mem h Na+aq and Mn+mem h Na+aq is faster than Mn+mem h Mn+aq isotopic exchange. Thus, these counterion exchanges follow the qualitative trend predicted by the N-P equation for the interchanging counterions.

Transport of Ions in Poly(perfluorosulfonic acid) Membrane

Figure 4. Variations of F(t) as a function of t1/2. The symbols represent the experimental data on 3Na+mem h Eu3+aq (O) and Eu3+mem h 3Na+aq (∆) exchanges. Lines 1-4 represent the profiles of Na+mem h Na+aq, 3Na+mem h Eu3+aq, Eu3+mem h 3Na+aq, and Eu3+mem h Eu3+aq m exchanges, respectively, based on the values of Dion of counterions in the membrane. m TABLE 1: Self-Diffusion Coefficients of Counterions (Dion ) m and Water (Dwater) in the Nafion-117 Membrane

counterion

m a Dion (×10-6 cm2 s-1)

m b Dwater (×10-6 cm2 s-1)

water uptakea (wt %)

Li+ Na+ Ag+ Cs+ Ba2+ Cu2+ Eu3+

1.3c 1.03 ( 0.04 1.62 ( 0.05 0.194 ( 0.007 0.146 ( 0.006 0.29 ( 0.1 0.045 ( 0.005

3.17 ( 0.11 2.85 ( 0.20 2.91 ( 0.14 1.67 ( 0.06 2.15 ( 0.30

21.1 17.2 18.8 8.2 15.5 17.9 13.2

2.04 ( 0.40

a

Data taken from refs 12 and 13. b Data taken from refs 14 and 15. c Data taken from ref 26.

The fractional attainment of ion-exchange equilibrium “F(t)” was calculated as a function of equilibration time using eq 7. The ion-exchange profile thus obtained was compared with the experimental exchange profile. As described in Theory, the input parameters required for calculating F(t) were Dmion (Table 1) and charges of the two interchanging ions involved in the process. As can be seen from Figures 1-4, the experimentally measured forward and reverse exchange rate profiles of Na+ with Ag+, Cs+, Ba2+, or Eu3+ in the membrane are in reasonably good agreement with that predicted by eq 7. Thus, the experimental exchange rates of Na+ ions with Mn+ ions in the poly(perfluorosulfonic acid) membrane follow the general trend predicted m of the interchanging by the N-P equation based on Dion counterions. However, the rate profiles of Na+mem h Mn+aq exchanges (except for Ag+ ions) are slightly faster than the predicted exchange rates, and the reverse exchange profile of Cs+mem h Na+aq shown in Figure 3 has a distinct lag time. This is quite different from our earlier observation of counterion exchange kinetics between H+ and Cs+, Cu2+, or Eu3+ ions.13 It was observed that the forward exchange rates (H+mem h Mn+aq) were independent of the counterion Mn+, while the reverse exchange rate (Mn+mem h H+aq) followed the predictions of the N-P equation. Similar observations were made by Samec m et al.16,17 They found that the Dion of alkali metal cations, obtained from the analyses of forward (H+mem h Mn+aq) and reverse (Mn+mem h H+aq) exchange profiles, differed signifi-

J. Phys. Chem. B, Vol. 113, No. 37, 2009 12485

Figure 5. Variations of F(t) as a function of t1/2. The symbols represent the experimental data on Na+mem h Cs+aq exchange. Solid and broken lines represent expected and fitted exchange profiles, respectively.

cantly. It was also reported that the diffusion coefficients obtained from the reverse exchange process were comparable m to Dion measured by different techniques. This suggests that the high water content of the H+ form of the membrane and hopping of H+ ions modify the ion-exchange kinetics in the forward exchanges. As seen from Table 1, the water uptake capacities in Cs+, Ba2+, and Eu3+ ionic forms of poly(perfluorosulfonic acid) membrane samples are lower than that in the Na+ form of the membrane. This would result in slight variation of the tortuosity factor in these exchanges than the corresponding ionic form whose Dmion values were utilized for predicting F(t). The comparison of self-diffusion coefficients of water in different ionic forms of poly(perfluorosulfonic acid) membrane given in Table 1 also supports the slight variation of the tortuosity factor. In the case of the Cs+mem h Na+aq exchange, the experimentally measured exchange kinetics follows the trend predicted by eq 7 after a period of time (lag time). The initial lag in the exchange kinetics could be related to the hydration kinetics of the membrane in lowest hydration state in the Cs+ form (8.2 wt %) to the higher hydration state in the Na+ form (17.2 wt %). In our previous work, it was observed that the self-diffusion mobility of water did not change even after complete exchange of Li+ ions in the poly(perfluorosulfonic acid) membrane with Cs+ ions.25 This was attributed to a slow change in the physical structure of the membrane as compared to the time required for complete Li+ ions exchange with Cs+ ions in the membrane. The slow change in the water content of the membrane during counterion exchange would enhance or retard the selfdiffusion mobility of incoming Cs+ or Na+ ions in forward and reverse exchanges, respectively. To get a quantitative m estimate of Dion of Cs+ ions during the forward Na+mem h Cs+aq exchange, the experimental data of exchange rates were m of Na+ ions as a fitted with the N-P equation using Dion m + fixed parameter and Dion of Cs ions as a variable parameter. The comparison of the experimental exchange rate profile m for Cs+ ions with the fitted profile is shown in Figure 5. Dion + + during Na mem h Cs aq exchange was found to be 0.30 × m 10-6 cm2 s-1, which is 50% higher than its Dion in the Cs+ form of membrane given in Table 1.

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Sodaye et al. and the corresponding DB/DA can be read from the dF(τ)/dτ1/2 vs (DB/DA)1/2 plot. Then for a known value of DA, DB can be obtained or vice versa from the measured ion exchange rates. For very large values of DB/DA, dF(τ)/dτ1/2 attains a constant value, and in that case



dF(t) ) constant dt1/2

Figure 6. Variation of slope (dF(τ)/dτ1/2) of counterions exchange rate profiles as a function of (DA/DB)1/2 for exchanges of monovalent, divalent, or trivalent ions in membrane with monovalent ions.

DA l2

(9)

The values of the constant are 1.556, 1.76, and 1.94 for ZB/ ZA ) 1, 2, and 3, respectively. So, in these cases, DA can be obtained from eq 9 without knowing the values of DB. Following the above procedure, DA was obtained for different counterions (Cs+, Ba2+, and Eu3+) from the reverse exchange data of Na+ ions (Mn+mem h Na+aq) obtained in the present work, and for Cs+, Cu2+, and Eu3+ ions from the reverse exchange data of H+ ions (Mn+mem h H+aq) reported in our previous publication.13 The values of Dmion obtained from the above analysis of reverse m and forward ion exchange rates are listed in Table 2. The Dion values obtained from the fitting of forward ion-exchange processes involving H+ ions (H+mem h Mn+aq) is higher than m values obtained from isotopic exchange rates given in the Dion Table 1. This observation is similar to that obtained by Samec m data obtained from the reverse exchange rates et al.16 The Dion n+ + (M mem h H aq) are close to the isotopic exchange Dmion values, m values 3 times higher whereas Samec et al. had obtained Dion m than the isotopic exchange Dion values. The data involving ion exchange with Na+ ions show a different trend. In the case of m Ag+ ion exchange, the Dion value of Na+ ion obtained from the initial slope analysis of the reverse exchange rate (Ag+mem h Na+aq) was found to be 1.56 × 10-6 cm2 s-1, which is slightly higher than that obtained from isotopic exchange rates. In the forward exchange rate (Ag+mem h Na+aq), the experimental data m value of Ag+ ion from isotopic were best fitted with the Dion exchange rates. Unlike exchange involving H+ ion, the forward (Na+mem h Mn+aq) and the reverse (Mn+mem h Na+aq) exchange data for Cs+, Ba2+, and Eu3+ ions give comparable values of m Dion for exchange process involving Na+ ions; see Table 2. But m values are higher than the corresponding in general, the Dion

Figure 7. Variations of F(t) as a function of t1/2. The symbols represent the experimental data on Li+mem h Na+aq ([), Na+mem h Li+aq (]), Li+mem h Cs+aq (b), and Cs+mem h Li+aq (O) exchanges. Solid and broken lines represent Na+mem h Na+aq and Cs+mem h Cs+aq exchanged m of Na+ and Cs+ ions, respectively. based on Dion

Samec et al. evaluated the Dmion from the ion-exchange kinetics of H+ ion with alkali metal ions by analyzing the initial slope of the plot of F(τ) vs τ1/2.16 It was shown that the F(τ) vs τ1/2 plots show good linearity at low and medium values of F(τ), and the slope of the linear portion varies systematically with the ratio (DB/DA)1/2. In the present case, the variation of the initial slope (dF(τ))/(dτ1/2) as a function of (DB/DA)1/2 for different values of ZA/ZB is shown in Figure 6. The initial slopes (dF(τ)/dτ1/2) were obtained from the numerical solution of the N-P equation. It can be seen from Figure 6 that the slope rises fast for DB/DA e 1 (forward exchange), while it saturates at higher values of DB/DA (reverse exchange). The experimental values of dF(t)/dt1/2 were converted to dF(τ)/dτ1/2 using the following equation:

dF(τ) dF(t) ) 1/2 1/2 dτ dt

(√ ) l

DA

(8)

Figure 8. Variations of F(t) as a function of t1/2. The symbols represent the experimental data on Li+mem h Na+aq ([), Na+mem h Li+aq (]), Li+mem h Cs+aq (b), and Cs+mem h Li+aq (O) exchanges. Solid and broken lines represent fitted data for obtaining Li+ self-diffusion coefficients from forward and reverse exchanges with Na+ and Cs+ ions.

Transport of Ions in Poly(perfluorosulfonic acid) Membrane

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TABLE 2: Self-Diffusion Coefficients of Ions Obtained from the Analyses of Experimental Counterions Exchange Kinetics m Dion (×10-6 cm2 s-1) +

exchange involving H+ ions

exchange involving Na ions ion Ag+ Cs+ Ba2+ Cu2+ Eu3+

forward process (Na+mem h Mn+aq)

reverse process (Mn+mem h Na+aq)

forward process (H+mem h Mn+aq)

reverse process (Mn+mem h H+aq)

1.62 0.30 0.25

0.30 0.14

2.0

0.27

5.8 0.09

0.06

0.26 0.035

Dmion from the isotopic exchange rates by a factor of 1.5-2. This difference may arise due to a change in water content during the ion exchange process. A large difference in the hydration characteristics of the ions involved in the exchange process may affect the ion exchange kinetics than that in pure ionic form. Membrane thickness was also found to vary from 185 to 200 µm in different ionic forms, but the N-P calculations were carried out using an average value of half-thickness (l ) 100 µm). For example, this may cause ∼5-10% error in the calculations for Cs+ ion exchange. Another feature responsible m values could be the electroosmotic for the difference in Dion flow of water during the course of ion exchange.16 The experimental forward (Li+mem h Na+/Cs+aq) and reverse (Na+/Cs+mem h Li+aq) exchange rate profiles are shown in Figure 7. Unlike H+ ions, it was observed that forward and reverse exchange profiles changed depending upon the counterions. In the exchange involving Na+ ions, the kinetics of exchange follow the order Li+mem h Na+aq > Na+mem h Na+aq > Na+mem h Li+aq. On the basis of the nonlinear interdiffusion equation, Li+ mobility in the forward and reverse exchanges seems to be different as the isotopic exchange profile Na+mem h Na+aq is in between the forward and reverse exchanges. The self-diffusion coefficient of Li+ ions appears to be considerably higher in Li+mem h Na+aq exchange than that in Na+mem h Li+aq exchange. The kinetics of exchange profiles involving Cs+ ions varied in the following order: Li+mem h Cs+aq > Cs+mem h Cs+aq ≈ Cs+mem h Li+aq. This seems to indicate that the self-diffusion coefficient of Li+ ions in the reverse exchange is quite slow and close to that of Cs+ ions in the membrane. The theoretical prediction of the exchange rate involving Li+ m ions was not attempted as a reliable value of Dion of Li+ ions in the Nafion-117 membrane was not available. Stenina et al. have m of Li+ ions in the Nafion-117 membrane as reported the Dion 1.3 × 10-6 cm2 s-1 from conductivity as well as diffusion m permeability measurements.26 They have also reported a Dion + -6 2 -1 7 value of Li ion as 1.1 × 10 cm s from Li NMR spectroscopy. They have observed that Na+ ion has higher mobility than Li+ ion in Nafion-117 membrane. But our experimental exchange rate (Li+mem h Na+aq) is faster than Na+ ion self-diffusion. Hence, the above-mentioned Dmion values, when used in the N-P calculations, did not give a good agreement m of Li+ ions in the to the experimental data. Therefore, Dion membrane was obtained by the fitting of the experimental data of Li+ ion exchanges with Na+/Cs+ ions in the N-P equation m m of Cs+/Na+ ions from literature,12 and Dion of Li+ using Dion ions as a variable parameter. The comparison of experimental and fitted rate profiles of forward and reverse exchanges is shown in Figure 8. The self-diffusion coefficients of Li+ ions deduced from these analyses were found to be 2 × 10-6 and 0.2 × 10-6 cm2 s-1 for forward (Li+mem h Na+/Cs+aq) and reverse (Na+/Cs+mem h Li+aq) exchanges, respectively. This indicates that the Li+ self-diffusion mobility is quite high when Li+ ions diffuse from membrane to aqueous phase, but quite

slow when Li+ ions diffuse from aqueous to membrane phase. It is interesting to note that Li+ self-diffusion coefficient did not change in its exchanges with Na+ or Cs+ ions. Okada et al. have reported the equilibrium constant Kex of the exchange reaction involving competition of Li+ and H+ ions for available exchange sites in poly(perfluorosulfonic acid) membrane in the following order: Li+ > Na+ > K+ > Rb+ > Cs+.4 They observed that the Li+ is a special ion and almost had no preference over H+ ions in the membrane. In our previous studies, the self-diffusion coefficient of counterion was found to increase with a decrease in its selectivity coefficient in the poly(perfluorosulfonic acid) membrane.12 This is in accordance with the self-diffusion coefficient of Li+ ions obtained in the forward exchange (2 × 10-6 cm2 s-1), which is higher than the self-diffusion coefficient of Na+ ions (1.03 × 10-6 cm2 s-1). The very slow mobility of Li+ ion in the reverse exchange could be attributed to its weak electrostatic interactions with the fixed charge sites in the membrane as indicated by the reported equilibrium constant Kex of the exchange reaction involving competition of Li+ and H+ ions.4 The weak electrostatic interactions of Li+ ions with the fixed negative charge sites in the membrane would retard the displacement of the interchanging counterions. As fluxes of incoming and outgoing counetrions are coupled to maintain the electrical neutrality in the membrane, the retardation of displacement of the counterions from the exchange sites by Li+ ions would lower the overall kinetics of the reverse counterion exchange. It appears from the present work that the self-diffusion mobility of ions having equilibrium constant Kex with respect to H+ ions more than unity was influenced by different factors during the forward and reverse exchanges as in the case of Li+ ions. However, only the diffusion coefficients govern the counterion exchange rates involving ions having equilibrium constant Kex less than unity. Conclusions The experimental counterion exchange kinetics in the poly(perfluorosulfonic acid) membrane were analyzed in terms of a numerical solution of the Nernst-Planck equation based on the self-diffusion coefficients of interchanging counterions. It was observed that the interdiffusion kinetics of the exchanging counterions is dependent on their self-diffusion coefficients that increase with a decrease in their selectivity in the membrane. m values of different ions (Mn+) obtained from the The Dion analyses of forward and reverse exchange kinetics data were found to be different from each other in the case of exchanges involving H+ ions and close to each other when Na+ ion was m values were the exchanging ion. But in both the cases the Dion higher than that predicted from isotopic exchange kinetics. If the selectivity of the exchanging counterions is negligible in the membrane, the forward and reverse exchange kinetics do not follow the expected trend. This is due to the drastic change

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in the self-diffusion coefficients of the weakly interacting counterions in its forward and reverse exchange as observed in the case of Li+ ions. The self-diffusion coefficient of weakly interacting counterions is high in forward exchange due to negligible retardation of its diffusion mobility by the electrostatic interactions. On the other hand, the self-diffusion mobility is retarded in the reverse exchange due to its inability to replace the electrostatically strong interacting counterions in the membrane. The change in water content of the membrane in different ionic forms was also found to give slight deviation in the interdiffusion kinetics of the exchanging counterions from that expected from the N-P equation. Acknowledgment. Authors thank Dr. V. K. Manchanda, Head, Radiochemistry Division, Bhabha Atomic Research Centre for his constant support and encouragement of this work. References and Notes (1) Heitner-Wirguin, C. J. Membr. Sci. 1996, 120, 1–33. (2) Mauritz, K. A.; Moore, R. B. Chem. ReV. 2004, 104, 4535–4585. (3) Steck, A.; Yeager, H. L. Anal. Chem. 1980, 52, 1215–1218. (4) Okada, T.; Satou, H.; Okuno, M.; Yuasa, M. J. Phys. Chem. B 2002, 106, 1267–1278. (5) Yeager, H. L.; Kipling, B. J. Phys. Chem. 1979, 83, 1836–1839. (6) Suresh, G.; Sodaye, S.; Scindia, Y. M.; Pandey, A. K.; Goswami, A. Electrochim. Acta 2007, 52, 5968–5974. (7) Yeager, H. L.; Steck, A. J. Electrochem. Soc. 1981, 128, 1880– 1884. (8) Herrera, A.; Yeager, H. L. J. Electrochem. Soc. 1987, 134, 2446– 2451.

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