Effect of Film Diffusion on the Ion-Exchange Kinetics of a Tracer Ion in

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Effect of Film Diffusion on the Ion Exchange Kinetics of a Tracer Ion in Nafion-117 Membrane from a Mixture of Salt Solution Apurva N. Naik, Chhavi Agarwal, Sanhita Chaudhury, and Asok Goswami J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b07615 • Publication Date (Web): 03 Oct 2017 Downloaded from http://pubs.acs.org on October 4, 2017

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The Journal of Physical Chemistry

Effect of Film Diffusion on the Ion Exchange Kinetics of a Tracer Ion in Nafion-117 Membrane from a Mixture of Salt Solution Apurva N. Naik Ϯ, Chhavi Agarwal Ϯ, Sanhita Chaudhury Ϯ and A. Goswami $* Ϯ

$

Radiochemistry Division, Bhabha Atomic Research Centre, Trombay, Mumbai-400085, India

Formerly, Radiochemistry Division, Bhabha Atomic Research Centre, Trombay, Mumbai-400085, India *Corresponding Author: Email: [email protected] (A.G.); Tel: +91-22-27814973

Abstract: Ion exchange kinetics of a tracer ion (Cs+&Ba2+) in presence of a bulk ion (Na+/H+) has been measured in Nafion-117 membrane for a range of concentration of NaCl/HNO3 using nonstationary radiotracer method. A systematic increase in the ion exchange rate and decrease in the partition coefficients of the tracer ions between membrane and solution have been observed with the increase in bulk ion concentration. The sigmoidal nature of experimental profiles indicates film diffusion controlled kinetics even for well stirred solutions. In absence of an existing analytical or numerical solution, a simple empirical approach has been proposed to find the variable membrane surface concentration; and has been used to solve the membrane diffusion equation by finite difference method. The fitting of the experimental curves with a single diffusion coefficient for Cs+/Ba2+ has been achieved. The exchange rate has been found to be independent of the stirring speed beyond a limiting speed.

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Introduction Ion exchange membranes are extensively used in electrochemical devices such as batteries, fuel cells etc.; and also in Donnan dialysis and electrodialysis based applications.1-9 Understanding the mechanism governing the ion exchange kinetics in these membranes is of utmost importance for optimizing the operating conditions for these applications. There is substantial amount of literature dealing with the measurement and theoretical analysis of ion exchange kinetics, for both ion exchange resins as well as ion exchange membranes.9-27 Most of the early developments in this area is based on the studies of absorption kinetics in ion exchange resins.

10,21-25

The theory was subsequently extended to ion exchange

membranes,1,6,10,11,28 to elucidate the mechanism of ion transport through the membranes. In ion exchange membranes, the overall rate of mass transfer is governed by two steps: (i) interdiffusion of counter ions in the thin liquid film which adheres to the membrane surface (film diffusion) and (ii) interdiffusion of counter ions within the ion exchange membrane (membrane diffusion). A third possible rate determining step, involving slow complexation - decomplexation reaction at membrane surface, may exist during ion exchange in presence of chelating groups.27,29 Helfferich10 has described the solution of Nernst-Planck (NP) equation for film and particle diffusion for spherical resin beads and discussed the boundary conditions for both infinite and finite solution volume condition. The ion exchange rate is determined by the slower of the two steps. When the rates of both steps are comparable, the overall rate is governed by the sum of the two rates similar to flow of current through two resistances in series. It has also been shown that the film diffusion prevails for dilute solutions, higher film thickness (slower stirring speed), thinner membrane, lower diffusion coefficient in solution, and higher selectivity of the membrane for the entering ions.


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The criteria for film diffusion cited above however, are only approximate. Furthermore, mass transport kinetics through the membrane becomes more complex in presence of more than one counter ion in solution or membrane. A suitable theory describing the diffusion of counter ions in such complex solutions is not well developed. The film diffusion is often ignored assuming thorough agitation15-20,28 of the solution which minimizes the thickness of the film; thus ensuring convective mass transfer to the membrane surface. However, the assumption may not hold good for solutions with lower salt concentration; where the film diffusion may dominate affecting the rate of ion exchange even with high stirring rates. Radiotracers have been extensively used to study the equilibrium and kinetics of ion exchange in beads

30-34

as well as in membranes.15-21,35-39 In their pioneering work,

Boyd et al.21 investigated the absorption kinetics with ion exchange beads using radiotracer ions with a mixture of alkali metal salt solutions under the condition of constant solution and beads composition. It was shown that, at low salt concentration (≤ 1 x 10-3 mol L-1) and low flow rate of the aqueous phase, the rate of ion exchange was determined by film diffusion while, membrane diffusion governed the ion exchange rate at high salt concentration (≥ 0.1 mol L-1) and high flow rate. Many separation processes based on ion exchange membrane may involve feed solution containing ions at trace level in presence of bulk salt solution.17,18,40,41 Transport characteristics of these trace ions may deviate from their usual behaviour in presence of bulk concentration of other ions. There is limited literature on the study of ion exchange kinetics of such systems.1,10,38,39,42 Recently we reported37 the kinetics of exchange of Na+ ions in the Nafion-117 membrane with trace level of Cs+/Ba2+/Eu3+ present in 0.1 mol L-1 NaCl solution. Non-stationary radiotracer technique was used under finite solution volume condition. Results were analyzed using an analytical solution of simplified form of Nernst-Planck equation, and assuming membrane controlled diffusion. Diffusion coefficients of the tracer 3 ACS Paragon Plus Environment


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ion required to fit the ion exchange profile were found to be appreciably lower than the literature reported self diffusion coefficients15,16 indicating much slower rate of exchange. This was attributed to the presence of bulk concentration of NaCl (0.1 mol L-1) in the aqueous solution. However, the fitting of the experimental profile was not satisfactory even with the lower values of the diffusion coefficients. In view of this, we have carried out systematic investigations on the ion exchange of Cs+ and Ba2+ with H+/Na+ in Nafion-117 membrane using non-stationary radiotracer method under finite solution volume condition. In order to study the effect of bulk salt solution, exchange rates have been studied for trace level of Cs+ / Ba2+, present in NaCl or HNO3 solutions of varying concentrations. Radiotracers of Cs+ / Ba2+ have been used to ensure that the membrane composition does not change during the ion exchange so that simplified form of the Nernst-Planck equation is applicable for membrane diffusion. The time varying tracer concentration at the membrane surface has been empirically obtained from the variation of tracer concentration in the solution with time. The effect of film diffusion has been implicitly incorporated in this empirical approach. Experimental exchange rates have been fitted to membrane controlled diffusion equations which has been solved numerically by finite difference method. Diffusion coefficient of tracer ion in the membrane has been obtained from the fitting. Both the effect of stirring speed and salt concentration on the kinetics has been discussed.

Theoretical The Nernst-Planck equation has been solved separately to obtain ion exchange rates for membrane 13-16,19,20,44,45 and film 23-25 diffusion controlled process. However, exact solution to the problem of calculation of rate of ion exchange, of a tracer ion in a bulk salt solution, in 4 ACS Paragon Plus Environment

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membrane, taking into account both the rate processes viz. film diffusion and membrane diffusion, is not available in the literature. Difficulty arises because of the lack of knowledge of concentration of diffusing counter ion at the interface ion exchange solution in presence of film diffusion. Thus the initial and the boundary conditions at the interface solution cannot be defined explicitly for solution of the diffusion equations. The effect of film diffusion can be implicitly taken into account in the ion exchange rate calculations if the variable membrane surface concentration can be given as an input in the evaluation of simplified form of NernstPlanck equation by finite difference method. This approach has been adopted in this work. + A

+ B

+ A

Consider a solution containing a mixture of two cations AZ and BZ , where AZ is + B

present in bulk concentration and BZ is present only in trace concentration (≤ 10-12 mol L-1) as shown in Scheme 1. Let a piece of ion exchange membrane, of surface area S and thickness + A

2L, in AZ counter ionic form be placed in this solution of volume V while it is continuously stirred. Let the two surfaces of the membrane piece be at L and –L while the centre of the membrane coincides with the origin. This consideration makes the problem symmetrical about the origin. At time t > 0 the ion exchange will take place at the solution membrane interface as follows +

+

AZ A ( m) + B Z B ( s )

+

+

AZ A ( s ) + B Z B ( m)

Where, s and m refer to the solution and the membrane phase respectively. The co-ions are assumed to be completely excluded from the membrane phase due to Donnan exclusion.



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Scheme 1. Ion exchange process between the bulk solution and Nafion-117 membrane. Only half of the membrane is shown (x = -L to 0) for simplicity. + B

With the assumption that the concentration of BZ ion is negligible in the membrane at any + B

time, a simplified form of Nernst-Planck equation for the diffusion of the BZ ion in the plane sheet of membrane is37

∂C B ∂ 2 CB = DB ∂t ∂x 2

(1)

+ B

Where, C B is the concentration of BZ in the membrane at any time t and DB is its diffusion coefficient in the membrane. This simplification amounts to ignoring the interdiffusion of + A

+ B

counter ions AZ and BZ in the membrane. The equation has to be solved with appropriate initial and boundary conditions to obtain the ion exchange rate. 6 ACS Paragon Plus Environment

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+ A

Initial condition: Since the membrane is initially in AZ form, the initial condition is given by,

− L ≤ x ≤ L, t = 0

CB = 0

(2)

Boundary condition: Boundary condition is not well defined when film diffusion occurs + B

since the concentration of BZ at interface ion exchange solution differs from its bulk solution concentration. However, the variable membrane surface concentration can be obtained + B

numerically using the condition that the rate at which BZ leaves the solution is always equal to the rate at which it enters the membrane sheet at x = ±L (surface of the membrane). For this, let us divide half of the thickness of the membrane sheet into N layers, each of thickness

∆x such that ∆x =

L N

(3)

Using the mass balance condition, the equation defining the boundary condition can be written as:

−V

dC B d C B = 2S∆x dt dt

x = ±L , t > 0

(4)

+ B

The right hand side gives the amount of BZ entering through the surfaces per unit time into the volume element S∆x at x = ±L . Here C B is the concentration of BZ at x = ±L and CB is + B

+ B

the concentration of BZ in the bulk solution. The equation can be rearranged as

dCB V dCB =− dt 2S∆x dt

(5)



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If dCB / dt is experimentally obtained, d C B / dt can be evaluated at any time t. Subsequently, the membrane surface concentration at any time step can be obtained by using Taylor expansion of Equation 1 in the time direction, at constant x as:

 ∂C  C B(0,t +∆t ) = C B (0,t ) +  B  ∆t  ∂t 0,t

(6)

Here, 0 refers to the space coordinate of the grid point corresponding to membrane surface at

(

)

x=L and ∆t refers to a small time step. In order to evaluate ∂C B /∂t 0,t the following procedure has been adopted. Tracer counts in the solution at various time intervals have been calculated by subtracting the experimental membrane counts from the initial counts added in the solution. + B

The time dependent total tracer counts of BZ ( c B ( s ) ( t ) ) in the solution have been fitted to the following empirical equation

cB ( s ) (t ) = a + be − kt

(7)

Where, a, b and k are empirical constants which are obtained from the fitting. + B

Since total counts in solution, c B ( s ) (t ) = C B V , the change in concentration of BZ in the ion exchange solution is: dC B kb − kt =− e dt V

(8)

+ B

The change in concentration of BZ at the membrane surface layer has been calculated using Equation 5 and Equation 8, dCB kb − kt = e dt 2 S∆x

(9) 8

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Hence, according to Equation 6 and Equation 9, membrane surface concentration at any discrete time step can be numerically evaluated as

C B (0,t + ∆t ) = C B (0,t ) +

kb −kt e ∆t 2 S∆x

(10)

But at each time step, the diffusing ion will also diffuse out of the surface layer into the second layer and the flux will be given by the Fick’s law:

J = − DB

C B (1,t ) − C B (0,t ) ∆x

(11)

Thus the reduction in concentration at time t due to diffusion is:

∆C B ( t ) = − J

C B (1,t ) − C B ( 0,t ) ∆t ∆t = DB ∆x ∆x ∆x

(12)

This term is to be added to the right hand side of Equation 10 to obtain the concentration in the surface at any time step:

C B ( 0,t + ∆t ) = C B ( 0,t ) +

kb −kt C B (1,t ) − C B ( 0,t ) ∆t e ∆t + DB 2 S∆x ∆x ∆x

(13)

At t=∆t (1st step), the first term is zero (initial condition) and the third term is also assumed to be zero. The surface concentration for the subsequent time steps can be evaluated from Equation 13. + B

The concentration of BZ ion inside the membrane is governed by the Fick’s second law and is given by Equation 1. The boundary condition at the centre of the membrane sheet is defined by the assumption that at x=0 there is no concentration gradient and given by:

∂C B =0 ∂x

x = 0,

t >0

(14) 9

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The time and the thickness of the layer can be expressed in terms of dimensionless quantities as follows

τ =

DB t L2

ε=

x L

(15)

In terms of the dimensionless quantities, Equation 13 can be written as:

kbL ∆τ − C B (0,τ + ∆τ ) = C B (0,τ ) + ( )e 2 SDB ∆ε

kL2τ DB

+ (C B (1,τ ) − C B (0,τ ))

∆τ ∆ε 2

(16)

+ B

Equation 1 can be solved using finite difference method37 to find the concentration of BZ in the membrane at time t using the equation:

C B (ε ,τ + ∆τ ) = C B (ε ,τ ) +

∆τ [C B (ε − ∆ε ,τ ) − 2C B (ε ,τ ) + C B (ε + ∆ε ,τ )] ∆ε 2

(17)

+ B

The total amount of BZ ion in the membrane is then obtained by integrating the membrane concentration at each time step numerically using Simpson’s 1/3rd rule. On the other hand, for membrane diffusion controlled process with finite solution volume condition, rate of attainment of equilibrium is given by 37,47

Mt ∞ 2α (1 + α ) ( − DB qn2t ) = 1 − ∑ n =1 exp M∞ 1 + α + α 2 qn2 L2

(18)

+ B

Where M t and M ∞ are the total amount of BZ that has entered the membrane sheet at time t and after an infinite time. qn s are the non-zero positive roots of the equation tan q n = −α q n

(19)

V vλB

(20)

α=


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+ B

Where, λ B is the partition coefficient of BZ ion and v is the volume of membrane obtained from the known thickness and area of the membrane pieces used.

Experimental Materials Analytical reagent grade sodium chloride (NaCl), and nitric acid (HNO3) from SD Fine Chem. Ltd., Mumbai, India, were used. The radiotracers, 133Ba (t1/2= 10.51 y) and

137

Cs

(t1/2= 30.17 y) were obtained in the form of Ba(NO3)2 and CsNO3 solutions respectively from the Board of Radiation and Isotope Technology Mumbai, India. The radioactivity concentration of each solution was~50 µCi/mL. Nafion-117 membrane (maximum operating temperature 463 K) having equivalent weight 1100 g/equivalent of SO3H, and thickness 178 µm in dry condition was used in this study and was obtained from Ion Power Inc (Germany). The membrane was preconditioned as described elsewhere15 to make it free from any organic impurities. De-ionized water (18 MΩ cm−1) purified by model Quantum TM from Millipore (Mumbai, India) was used during all the experimental procedures.

Ion Exchange Kinetics Study The fractional attainment of ion exchange equilibrium with time was measured using a 2 x 2 cm2 piece of Nafion-117 membrane. The conditioned membrane piece was converted to suitable ionic form by equilibrating it with 0.1 mol L-1 solution of NaCl or HNO3 overnight. A series of aqueous solution of NaCl and HNO3 with varying concentrations ranging from 0.02 mol L-1 to 0.5 mol L-1 were prepared. In a typical experiment, 30 mL solution of known concentration of NaCl or HNO3 was mixed with trace concentration (≤ 1012

mol L-1) of the ion in the form of corresponding radiotracer (133Ba2+ or

137

Cs+). The

solution was stirred at a constant stirring speed of 250 rpm for all the experiments excluding 11 ACS Paragon Plus Environment


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those where it was changed to 150 rpm and 350 rpm to study the effect of stirring rate on the ion exchange kinetics. The membrane piece, in corresponding ionic form as that of the bulk ion in the solution, was then placed in the solution. It was taken out at regular time intervals and subjected to γ-ray counting in a fixed geometry using a well type NaI(Tl) detector coupled with 4K multichannel analyzer. The membrane piece was washed thoroughly with water before counting to remove any solution adhering to the surface of the membrane. The residence time of the membrane in the equilibrating solution was taken as the actual counter ion exchange time and the membrane piece was replaced in the equilibrating solution after counting. This process was continued till equilibrium was attained. The γ-ray energies in the region 75–400 keV were monitored for 133Ba2+ and the 662 keV γ-ray energy of

137

Cs+, was

used.

Determination of Partition Coefficient For the experimental determination of partition coefficient, a filter paper standard was prepared as described elsewhere.48 Briefly, a 2 x 2 cm2 piece of adsorbent sheet was taken to exactly mimic the geometry of the membrane. A known volume (0.2 mL) of the equilibrating solution was transferred on the filter paper and evaporated to dryness. The as prepared filter paper standard was then subjected to γ-ray counting in same geometry as that of the membrane piece to measure the amount of radiotracer initially present in the equilibrating solution. Partition coefficient was then obtained from the ratio of concentrations of radiotracer ion in the membrane and in solution after equilibration.

Results and Discussion In order to study the effect of the bulk ion concentration on the ion exchange kinetics of a tracer ion, experiments have been performed to measure the fractional attainment of equilibrium of the tracer ion (Cs+ / Ba2+) in presence of various concentrations of the bulk ion 12 ACS Paragon Plus Environment

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(Na+ / H+). Figure 1 shows the plot of fractional attainment of equilibrium with respect to square root of time (t1/2) for Cs+ (Figure 1a) and Ba2+ (Figure 1b&c) in presence of varying Na+ or H+ ion concentration. For a membrane diffusion controlled process, the initial rate varies linearly with t1/2. It is seen from the figure that initial rates, in all the cases except at highest concentration, tend to show a sigmoidal variation with t1/2. Such sigmoidal ion exchange sorption curve indicates that the equilibrium at membrane surface with the bulk solution is not achieved instantaneously.47 In the present case, this may be indicative of the effect of film diffusion on the ion exchange kinetics. It is also seen from the Figure 1, that the rate of ion exchange strongly depends on the concentration of the bulk ion irrespective of the tracer ion and co-ions present in the solution. The rate is lowest with pure tracer solution without any added bulk ion, and is seen to be gradually increasing with the increase in bulk ion concentration for all the three systems under consideration. This concentration dependence of ion exchange rate also suggests the influence of film diffusion on the ion exchange kinetics. Thus, the sigmoidal absorption curves together with the bulk ion concentration dependence clearly imply film diffusion prevails during ion exchange for the current systems. The sigmoidal nature of the curves is seen to be less prominent with increase in bulk ion concentration. In fact, the curves tend to be linear at the initial stage of equilibration at highest concentration of bulk ion (Figure 1). This indicates that the ion exchange kinetics tend to become membrane diffusion controlled at higher bulk ion concentration. This is consistent with the general conclusions of Boyd et al.21 and Helferich10 from their studies of ion exchange kinetics using resin beads. The concentration dependence is seen to be more pronounced for bivalent-monovalent ion exchange (Ba2+-Na+/H+) as compared to monovalent-monovalent ion exchange (Cs+-Na+) reflecting the effect of selectivity and mobility of the ions on the exchange rate. In our earlier report,37 the effect of film diffusion was ignored completely assuming the condition of well stirred solution and



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hence the diffusion coefficients of the ions had to be adjusted to unrealistically low value to fit the experimental ion exchange rate curve with equation 18 for membrane diffusion controlled kinetics. Even then the calculated ion exchange rate did not match the experimental ion exchange profiles37. It is to be noted that, the results shown in Figure 1 have been obtained at same stirring speed (250 rpm) i.e. constant film thickness. Thus, the effect of film diffusion varies with bulk ion concentration and becomes dominant at low salt concentration even for a well stirred solution. The results of variation of stirring speed on the exchange rate at various salt concentrations have been discussed later in this paper.


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1.2

(a) 1.0

0.8

0.6 0.00 M Exp 0.00 M Fit 0.02 M Exp 0.02 M Fit 0.07 M Exp 0.07 M Fit # 0.10 M Exp 0.10 M Fit

0.4

0.2

Fractional Attainment of Equilibrium

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0.0 1.2

(b) 1.0

0.8

-1

0.00 mol L Exp -1 0.00 mol L Fit -1 0.05 mol L Exp -1 0.05 mol L Fit -1 0.10 mol L Exp# -1 0.10 mol L Fit -1 0.20 mol L Exp -1 0.20 mol L Fit -1 0.50 mol L Exp -1 0.50 mol L Fit

0.6

0.4

0.2

0.0

1.0

(c)

0.8

0.6

0.00 M Exp 0.00 M Fit 0.08 M Exp 0.08 M Fit 0.10 M Exp 0.10 M Fit 0.20 M Exp 0.20 M Fit 0.50 M Exp 0.50 M Fit

0.4

0.2

0.0 0

20

40

60

80

100

120

1/2 1/2 time (sec ) Figure 1. Experimental points and fitted curves for fractional attainment of equilibrium of (a) Cs+ ion in presence of various concentrations of NaCl (b) Ba2+ ion in presence of various concentrations of NaCl and (c) Ba2+ ion in presence of various concentrations of HNO3. 15 ACS Paragon Plus Environment


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(# data taken from reference 37.) Figure 2 shows the plot of variation of relative concentration of tracer ion (Ba2+) with time in 0.5 mol L-1 NaCl solution. The relative concentration have been obtained from the tracer ion counts in the solution; which in turn have been calculated by subtracting the membrane counts at any time ‘t’ from the initial counts present in the equilibrating solution. Non linear least square fitting of the experimental data using Equation 7 is also shown (solid line) in Figure 2. Similar plots are shown in Figure S1 of supporting information for all the other systems. The excellent fitting confirms the validity of empirical Equation 7 for the variation of solution concentration of the tracer ion with time, irrespective of whether the rate is governed by film or membrane diffusion. The empirical parameters a, b and k, obtained from the fitting for all the systems are given in Table 1 along with the fitting errors. The partition coefficients (λcal) of the tracer ion have been calculated from the fitting parameters ‘a’ and ‘b’ as given by Equation 21.

λ=

bV av

(21)

The experimental and calculated values of λ are also given in Table 1 and are found to be matching within ~5% except for two cases.


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Relative Conc. of Tracer (Arbitrary Units)

Page 17 of 33

1.00

0.99

experimental fit

0.97

0.96

0.94

0.93 0

500

1000

1500

2000

2500

Time (sec)

Figure 2. Experimental points and fitted line showing exponential decrease in normalized solution concentration with time for Ba2+ in presence of 0.5 mol L-1 NaCl solution stirred at 250 rpm speed.

Table 1. Parameters a, b and k and partition coefficients (λ) for the bi-ionic systems obtained in the present work. Bulk ion Concentrat ion (mol L1 )

Stirring speed (rpm)

a (x 10-1)

b (x 10-1)

k (x 10-3) (s-1)

λcal

λexp

(I) Cs+-Na+ system

0

250

-0.90 ± 0.10

11.00 ± 0.06

0.80 ± 0.02

-

∞

0.02

250

1.91 ± 0.07

8.19 ± 0.08

1.15 ± 0.03

1610

1563

0.07

250

5.33 ± 0.09

4.68 ± 0.10

1.84 ± 0.07

329

329

0.1

250

5.80 ± 0.05

4.21 ± 0.06

2.26 ± 0.09

272

261

0.2

250

8.41 ± 0.02

0.16 ± 0.02

3.20 ± 0.16

71

70



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(II) Ba2+-Na+ System

0

250

-0.12 ± 0.06

10.01 ± 0.07

0.7 ± 0.01

-

∞

0.05

250

0.93 ± 0.09

9.18 ± 0.10

0.82 ± 0.02

3710

4400

0.1

250

3.51 ± 0.05

6.52 ± 0.05

1.30 ± 0.03

696

698

0.2

250

6.67 ± 0.04

3.31 ± 0.04

1.71 ± 0.07

186

187

0.5

250

9.36 ± 0.01

0.62 ± 0.01

4.75 ± 0.17

25

26

0.05

150

0.81 ± 0.07

9.10 ± 0.08

0.50 ± 0.01

4120

4050

0.5

150

9.36 ± 0.01

0.62 ± 0.01

5.43 ± 0.21

22

23

0.05

350

0.67 ± 0.05

9.43 ± 0.07

0.82 ± 0.01

5280

4480

0.5

350

9.33 ± 0.01

0.65 ± 0.01

6.05 ± 0.21

26

27

(III) Ba2+-H+ system

0

250

-0.94 ± 0.04

11.00 ± 0.05

0.8 ± 0.01

-

∞

0.1

250

0.98 ± 0.06

9.09 ± 0.07

0.66 ± 0.01

3470

3467

0.2

250

2.93 ± 0.04

6.93 ± 0.04

1.70 ± 0.03

885

875

0.5

250

8.95 ± 0.02

0.99 ± 0.02

5.02 ± 0.35

41.5

44

Figure 3 shows the variation of λcal with bulk ion concentration. Decrease in the value of λ with increase in the bulk salt concentration follows from the law of mass action. At same concentration of the bulk ion, λ for Ba2+ is higher for H+ form of the membrane compared to Na+ form, indicating higher selectivity of Ba2+ over H+ than Na+. Also, at same salt concentration, the higher values of λ for Ba2+ as compared to Cs+ indicate higher selectivity


Page 19 of 33

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of Ba2+ for the ion exchange sites in the membrane as compared to Cs+ due to its higher charge state.15 Table 1 also gives values of the empirical parameter ‘k’. This can be shown to be the rate constant for absorption of tracer ion in the membrane by rearranging Equation 8 to

−

dC B = k (C B − C B (eq ) ) dt

(22)

+ B

Where C B ( eq ) refers to equilibrium concentration of BZ in solution. The rate constant ‘k’ therefore determines the time of equilibration irrespective of the mechanism of absorption. If the kinetics is membrane diffusion controlled, membrane surface will be instantaneously in equilibrium with the bulk solution, and then mass balance demands:

 dC B  V  dC B   = − D B     2 S  dt   dx

 V  = −k  (C B − C B ( eq ) )   2S   x=0

(23)

In this case, the rate profile can be used to obtain value of DB using Equation 18. On the other hand, if the kinetics is film diffusion controlled, then mass transfer of the tracer ion in the film can be equated with:

D'  dC   V  dC B  V   = − DB'  B  = B (C B − C B' ) = −k  (C B − C B ( eq) )    2S  dt   2S   dx  film δ

(24)

+ B

Here, D B' , δ , C B' refer to solution diffusion coefficient of ion BZ , film thickness and membrane liquid interface solution concentration. In that case, initial rate (C B' = 0) can be used to obtain DB' / δ . No information on the membrane diffusion constant can be obtained in that case. However, membrane surface concentration, in the present case, has been calculated with the assumption that it is determined by the net flux in the surface layer and does not need any specific assumption about the mechanism. The values of ‘k’, as obtained from the 19 ACS Paragon Plus Environment


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fitting of solution counts with time, are plotted against the concentration of bulk ion in Figure 3. It is seen that, ‘k’ increases systematically for Cs+, indicating higher rate of exchange with increase in bulk ion concentration. However, for Ba2+, initial the values of ‘k’ tend to remain nearly constant and then it increase sharply with the bulk ion concentration. Qualitatively, it can be said that, the higher concentration of bulk ion (higher ‘k’) favors membrane controlled diffusion kinetics; but the transition from membrane controlled diffusion to film controlled diffusion is continuous. It is also seen from the figure that, for Ba2+ at any given concentration, the values of ‘k’ are lower compared to Cs+ and the film diffusion is dominant. This difference is probably due to its higher selectivity and lower solution diffusion coefficient of Ba2+ in the membrane compared to Cs+ which favors the film diffusion. These observations are consistent with the earlier literature observations49-51. G. Kraaijeveld and J. A. Wesselingh49 have studied the kinetics of ion exchange of Na+ / Ca2+ with H+ in Lewatit S100 ion exchanger at a high stirring speed of 900 rpm and analyzed the data with Maxwell Stefan transport equations. It has been concluded that, even at such a high stirring speed, film diffusion dominates the ion exchange rate in the concentration range of 1 x 10-3 - 0.1 mol L-1. The present data also shows, for tracer ion, film diffusion not only dominates ion exchange rates at lower bulk ion concentration but it also persists at higher concentrations of the bulk ion.


Page 21 of 33

(a)

-3

3.5x10

3

2x10 -3

3.0x10

k

3

1x10

-3

2.5x10

-1

k (s )

Partition Coeff. (λ)

-3

2

2.0x10

8x10

-3

1.5x10

2

4x10

Partition coeff. λ

-3

1.0x10

0 -4

5.0x10

0.00

0.05

0.10

0.15

0.20

Concentration of Na+ (mol L-1) k* Ba2+- Na+ system -3

Partition coeff. (λ*) Ba2+- Na+ system k* Ba2+- H+ system

5.0x10

3

4x10

Partition coeff. (λ*) Ba2+- H+ system -3

-1

k (s )

4.0x10

3

3x10

-3

3.0x10

3

2x10 -3

2.0x10

(b)

3

1x10

Partition Coeff. (λ)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-3

1.0x10

0 0.0 0.0

0.1

0.2

0.3

0.4

0.5

-1 Concentration of Na+/H+ (mol L )

Figure 3. Variation of k and the partition coefficient (λcal) with salt concentration (a) for Cs+Na+ system and (b) for Ba2+-Na+ and Ba2+-H+ systems. (Solid lines are eye guides only.) In the present approach, the rate of decrease in concentration of tracer ion in solution has been directly correlated with the increase in its concentration at the surface layers of the membrane, with the use of mass balance condition. Subsequently, the concentration of the 21 ACS Paragon Plus Environment


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Page 22 of 33

tracer ion at any time ‘t’, at the surface layer of the membrane has been calculated from Equation 13 and thus, the effect of film diffusion has been taken into account without rigorous calculations of diffusion kinetics in the film itself. Figure 4 shows the calculated variation of concentration of the tracer ion (Ba2+) at the surface layer of the membrane with time for different concentrations of NaCl in the solution. The exponential rise in the concentration of Ba2+ ion is clearly seen from the Figure 4. In addition, the rate of equilibration is seen to be faster for higher concentrations of NaCl solution representing the decreasing effect of film diffusion with increase in salt concentration. Interestingly, such change in concentration with time at the surface of the membrane sheet has been predicted by Crank47 where the surface concentration is changed rapidly but not instantaneously, when an instantaneous change is attempted in an experiment. A general analytical solution of the diffusion equation for absorption of a diffusing ion has been given where the variable membrane surface concentration has been represented by Equation 25 which well represents the curves of Figure 4. C B = C B ( eq ) [1 − e − β t ]

(25)

Thus, this serves as an evidence of persistence of film diffusion at virtually all concentration in the present case, though it is more prominent for dilute solutions.


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Concentration (x1012)(arbitrary units)

Page 23 of 33

-1

+

0.00 mol L Na

5

-1

+

0.05 mol L Na

4

3

2 -1

+

0.20 mol L Na 1

0 0

1000

2000

3000

4000

5000

6000

t (sec)

Figure 4. Concentration of Ba2+ ion at the surface layer of the membrane during the equilibration for various concentrations of NaCl as obtained from the fitting of experimental data. (Solid lines are eye-guides only). The concentration of the tracer ion in the interior layers of the membrane at any time step has been calculated using Equation 17 by finite difference method. The total amount of the tracer ion in the membrane has been obtained by integrating Equation 17 over all the layers in the membrane at each time step. Fractional attainment of equilibrium has been obtained by dividing the total amount of tracer ion by the amount at equilibrium in the membrane and has been plotted as a function of time. The fitted ion exchange profiles, obtained for Cs+ and Ba2+ ions, in presence of various concentrations of Na+ or H+ ions, are shown in Figure 1 along with the experimental data. It can be seen that in all the systems, good fitting of the experimental profile has been achieved with a single value of the diffusion coefficient (DB) which is ~3 x 10-11 m2s-1. This value of diffusion coefficient is higher than that of the literature reported self diffusion coefficients for Cs+ and Ba2+ ions.15,16 It is to be



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Page 24 of 33

mentioned that, at lower concentration ≤ 0.1 mol L-1, obtaining a unique value of diffusion coefficient is not possible as the mechanism is predominantly film diffusion controlled. The values of diffusion coefficients obtained by neglecting the film diffusion,37 were found to be much lower than the self diffusion coefficients of the literature reported values, incorrectly suggesting a much slower rate of diffusion of tracer ion in the membrane, in presence of bulk concentration of Na+ ion in the ion exchange solution. Moreover, the fitting of ion exchange profiles at the initial time of the ion exchange process was observed to be very poor.37 Furthermore, the value of DB would have to be changed continuously with the concentration of bulk ion to fit the curves of Figure 1 using Equation 18. In the present work, it has been possible to reproduce the sigmoidal nature of the ion exchange absorption curve at all the concentrations with a single value of diffusion coefficient for Cs+ and Ba2+ indicating film diffusion effect on the ion exchange rate has been appropriately taken into account. Figure 5 shows the concentration of Ba2+ ion across the thickness of the membrane at various time steps during equilibration with 0.05 mol L-1 (Figure 5a) and 0.5 mol L-1 (Figure 5b) solution of NaCl. The non-steady nature of the flux in the membrane at initial stage is clearly visible from the figure. It can be seen that, equilibration within the membrane is established almost simultaneously with the outside solution in case of 0.5 mol L-1 NaCl (~300 sec). In the case of 0.05 mol L-1 NaCl, equilibration within the membrane (~300 sec) is established well ahead of equilibration with the outside solution (~4000 sec), indicating film diffusion controlled ion exchange rate. Since the diffusion coefficient of the tracer ion in the membrane (DB) is same for both the cases, it can be concluded that the ion exchange kinetics in case of 0.5 mol L-1 NaCl is mostly membrane diffusion controlled.


Page 25 of 33

5 sec 10 sec 15 sec 20 sec

Concentration (Arbitrary Units)

11

10

10

10

30 sec 40 sec 50 sec 65 sec

9

10

80 sec 110 sec 150 sec 300 sec 600 sec

8

10

7

10

900 sec 2000 sec 4000 sec 6000 sec

6

10

5

10

(a) L

0.5 L

0

Membrane Thickness (Arbitrary Units)

Concentration (Arbitrary Units)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


11

5 sec 10 sec 15 sec

10

10

10

20 sec 30 sec 40 sec 50 sec 65 sec 80 sec

9

10

8

10

110 sec 150 sec 300 sec 600 sec 4000 sec

7

10

6

10

5

10

6000 sec

(b) L

0.5 L

0

Membrane Thickness (Arbitrary Units)

Figure 5. Concentration of Ba2+ ion across the thickness of the membrane during equilibration with (a) 0.05 mol L-1 NaCl and (b) 0.5 mol L-1 NaCl. (Solid lines are eye-guides only). In order to confirm the presence of film adherent to the surface of the membrane, the ion exchange experiments has been carried out at three different stirring speeds for Ba2+-Na+ system consisting of trace concentration of Ba2+ and two different bulk concentrations of Na+. 25 ACS Paragon Plus Environment


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Page 26 of 33

The experimental ion exchange profiles are shown in the Figure 6. For 0.05 mol L-1 NaCl solution, the experimental ion exchange rate is seen to be slower at a stirring speed of 150 rpm while, ion exchange rates are same at stirring speed of 250 and 350 rpm. Although, the rate of ion exchange appears to be saturated above a limiting stirring speed of 250 rpm, the sigmoidal nature of the exchange rate curves is visible which suggests the film diffusion controlled mechanism. Thus, it can be concluded that the effect of film diffusion cannot be completely eliminated by increasing the stirring speed alone. For 0.5 mol L-1 NaCl solution, the experimental ion exchange profiles are seen to be independent of the stirring speeds studied in the present work. Also, the experimental ion exchange rates appear to be approximately linear with respect to t1/2 at the initial stage of the ion exchange. Both the results show the effect of film diffusion is not very prominent on the ion exchange rate at this concentration of Na+. The difference in stirring speed dependence of rate of ion exchange in 0.05 mol L-1 and 0.5 mol L-1 solutions indicate additional effect of salt concentration. As pointed out by Helferich,10 there may be a shift in co-ion concentration at the film due to interdifusion of counter ions of unequal mobility. If the ion present in the membrane is faster moving, it may lead to developing a positive potential in the bulk solution side of the film, thereby pushing the co-ions away from the film surface. Since the total concentration of the co-ion should be equal to the concentrations counter ions in the film, this reduces the concentration of the counter ions in the film and hence the flux across the film. The effect of interdiffusion in the film will be most prominent when there is no added salt as the composition of the solution will change during ion exchange. At higher salt concentration, the electric potential due to interdiffusion will be screened by the excess co-ions and counter ions present in the film, possibly reducing the effect of film diffusion. This can account for the near absence of film diffusion at 0.5 mol L-1 bulk ion solution.


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Fractional Attainment of Equilibrium

Page 27 of 33

1.0

0.8

0.6

*Ba2+- Na+ system 0.4

-1

150 rmp 0.05 mol L -1 250 rmp 0.05 mol L -1 350 rmp 0.05 mol L -1 150 rmp 0.50 mol L -1 250 rmp 0.50 mol L -1 350 rmp 0.50 mol L

0.2

0.0 0

20

40

60

80

100

120

Time1/2 (sec1/2)

Figure 6. Effect of stirring speed on the ion exchange rate of Ba2+ in two different concentrations of NaCl.

Conclusion Ion exchange kinetics of monovalent (Cs+) and bivalent (Ba2+) tracer ions in the presence of Na+/H+ ions have been measured. The experimental ion exchange profiles have been found to be sigmoidal in nature. This has been attributed to mainly film diffusion controlled kinetics of ion exchange of tracer ion. The effect of film diffusion has been observed to decrease with the increase in concentration of the bulk ion and above a certain concentration of the bulk ion, the kinetics has been found to become mostly membrane diffusion controlled. The film diffusion has been found to prevail even in well stirred solutions. Owing to the lack of exact method to calculate the rate of ion exchange in presence of both, film diffusion and membrane diffusion, an empirical method is adopted in the present work to incorporate the effect of film diffusion in the mathematical approach without explicit calculations for rate of diffusion in the film itself. The current approach has been found to reproduce well the experimental ion exchange profiles in all the cases. A common value of diffusion coefficient 27 ACS Paragon Plus Environment


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Page 28 of 33

(~3 x 10-11 m2s-1) for both the tracer ions has been satisfactorily used for the calculations of the ion exchange rates.

Supporting Information The experimental profiles and the corresponding fitted curves depicting the exponential decrease in relative concentration of tracer ion (Cs+/Ba2+) in solution with time in presence of various concentrations of NaCl/HNO3 solutions, at various stirring speeds are given in supporting information.

Acknowledgement A.N. Naik thanks Department of Atomic Energy, India for providing financial assistance under Collaborative scheme between University of Mumbai and BARC. The authors thank Dr. A.K. Pandey RCD, BARC for providing Nafion-117 membrane. Authors thank Dr. P.K. Pujari, Head, RCD for his keen interest during the course of work.

References 1. Sata, T. Ion-Exchange Membranes-Preparation, Characterization, Modification and Application; Royal Society of Chemistry: London, 2004.

2. Eisenberg, A.; Yeager, H.L.; (Eds.). Perflourinated Ionomer Membranes; ACS Symposium Series 180; American Chemical Society: Washington, DC, 1982, p. 41.

3. Cao, C.; Liu,W.; Tan, L.; Liao, X.; Li, L. Sodium-ion Batteries using Ion Exchange Membranes as Electrolytes and Separators. Chem. Comm. 2013, 49, 11740- 11742. 4. Li, X.; Zhang, H.; Mai, Z.; Zhang, H.; Vankelecom, I. Ion Exchange Membranes for Vanadium Redox Flow Battery (VRB) Applications. Energy Environ. Sci., 2011, 4, 11471160. 5. Liu, Y.; Cai, Z.; Tan, L.; Li, L. Ion Exchange Membranes as Electrolyte for High Performance Li-ion Batteries. Energy Environ. Sci., 2012, 5, 9007- 9013. 28 ACS Paragon Plus Environment

Page 29 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


6. Strathmann, H. Introduction to Membrane Science and Technology; John Wiley& Sons: Rende (CS), Italy, 2011. 7. Wang, Y.; Chen, K.S.; Mishler, J.; Cho, S.C.; Adroher, X.C. A Review of Polymer Electrolyte Membrane Fuel Cells: Technology, Applications and Needs on Fundamental research. Appl. Energy, 2011, 88, 981- 1007. 8. Appleby, A.J.; Beresovski, T.; (Eds.). Fuel Cells: Trends in Research and Applications: Trends in Research & Applications; Taylor and Francis, USA, 1987.

9. Kusoglu, A.; Weber A.Z. New Insights into Perfluorinated Sulfonic-Acid Ionomers. Chem. Rev., 2017, 117, 987−1104.

10. Helfferich, F. Ion Exchange; McGraw-Hill Book Company, New York, 1962. 11. Lakshminarayanaiah, N. Transport Phenomena in Membranes; Academic Press. New York, 1969. 12. Fimrite, J.; Struchtrup, H.; Djilali, N. Transport Phenomena in Polymer Electrolyte Membranes I. Modeling Framework. J. Electrochem. Soc., 2005, 152, A1804-A1814. 13. Verbrugge, M.W.; Hill, R.F. Ion and Solvent Transport in Ion-Exchange Membranes I. A Macrohomogeneous Mathematical Model. J. Electrochem. Soc., 1990, 137, 886-893. 14. Pintauro, P.N.; Tondon, R.; Chao, L.; Xu, W.; Evilia, R. Equilibrium Partitioning of Monovalent/Divalent Cation-Salt Mixtures in Nafion Cation-Exchange Membranes. J. Phy. Chem. B, 1995, 99, 12915-12924.

15. Goswami, A.; Acharya, A.; Pandey, A.K. Study of Self-Diffusion of Monovalent and Divalent cations in Nafion-117 Ion-Exchange Membrane. J. Phy. Chem. B, 2001, 105, 91969201. 16. Suresh, G.; Scindia, Y.M.; Pandey, A.K.; Goswami, A. Isotopic and Ion-Exchange Kinetics in the Nafion-117 Membrane. J. Phy. Chem. B, 2004, 108, 4104- 4110. 17. Chaudhury, S.; Agarwal, C.; Pandey, A.K.; Goswami, A. Self-Diffusion of Ions in Nafion117 Membrane having Mixed Ionic Composition. J. Phy. Chem. B, 2012, 116, 1605 –1611.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 33

18. Chaudhury, S.; Agarwal, C.; Goswami, A. Transport Properties of Multivalent Cations in Nafion-117 Membrane with Mixed Ionic Composition. J. Phy. Chem. B, 2015, 119, 10566 – 10572. 19. Agarwal, C.; Chaudhury, S.; Pandey, A.K.; Goswami, A. Kinetic Aspects of Donnan Dialysis through Nafion-117 Membrane. J. Membr. Sci., 2012, 415, 681 -685. 20. Agarwal, C.; Goswami, A. Nernst Planck Approach Based on Non-Steady State Flux for Transport in a Donnan Dialysis Process. J. Membr. Sci., 2016, 507, 119 -125. 21. Boyd, G.E.; Adamson, A.W.; MyersJr., L.S. The Exchange Adsorption of Ions from Aqueous Solution by Organic Zeolites. II. Kinetics. J. Am. Chem. Soc., 1947, 69, 2836-2848. 22. Richenberg, D. Properties of Ion-Exchange Resins in Relation to their Structure. III. Kinetics of Exchange. J. Am. Chem. Soc., 1953, 75, 589-597. 23. Smith, T.G.; Dranoff, J.S. Film Diffusion Controlled Kinetics in Binary Ion Exchange. Ind. Eng. Chem. Fundam., 1964, 3, 195-200.

24. Copeland, J.; Marchello, J.M. Film Diffusion Controlled Ion Exchange with a Selective Resin. Chem. Eng. Sci., 1969, 24, 1471-1474. 25. Bunzl, K. Kinetics of Film Diffusion Controlled Ion Exchange Processes, Theory and Applications. Isr. J. Chem., 1973, 11, 1-5. 26. Bunzl K. Ion-Exchange Kinetics in Heterogeneous Systems. Ion Exchange and Solvent Extraction Vol. 12; Marcel Dekker, Inc., New York, NY, 1995.

27. Kumar, S.; Jain, S. History, Introduction, and Kinetics of Ion Exchange Materials J. Chem., 2013, 2013, 13.

28. Baker R.W. Membrane Technology and Applications; John Wiley & Sons Ltd, England, 2004. 29. Hatanaka, M.; Hoshi, M. Ion Exchange Kinetics Based on Film Theory. J. Appl. Polym. Sci., 2014, 131, 39358.

30. Boyd, G.E.; Schubert, J.; Adamson, A.W.; The Exchange Adsorption of Ions from Aqueous Solution by Organic Zeolites. I. Ion Exchange Equilibrium. J. Am. Chem. Soc., 1947, 69, 1947-. 30 ACS Paragon Plus Environment

Page 31 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


31. Singare, P.U.; Lokhande, R.S.; Patil, A.B. Application of Radioactive Tracer Technique for Characterization of Some Strongly Basic Anion Exchange Resins. Radiochim. Act. 2008, 96, 99-104. 32. Bachet, M.; Jauberty, L.; Windt, L.D.; Tevissen, E. Dieuleveult, C.D.; Schneider, H. Comparison of Mass Transfer Coefficient Approach and Nernst–Planck Formulation in the Reactive Transport Modeling of Co, Ni, and Ag Removal by Mixed-Bed Ion-Exchange Resins. Ind. Eng. Chem., 2014, 53, 11096-11106. 33. Singare, P.U.; Lokhande, R.S. Studies on Ion-Isotopic Exchange Reactions using Nuclear Grade Ion Exchange Resins. Ionics. 2012, 18, 351-357. 34. Singare, P.U. Studies on Kinetics and Thermodynamics of Ion Adsorption Reactions by Applications of Short-Lived Radioactive Tracer Isotopes Ionics. 2016, 22, 1433-1443. 35. Lopez, M.; Kipling, B.; Yeager H. Ionic Diffusion and Selectivity of a Cation Exchange Membrane in Nonaqueous Solvents Anal. Chem., 1977, 49, 629. 36. Russell, R.L.; Ion Exchange Kinetics Testing with SRF Resin. Pacific Northwest National

Laboratory, 2014, PNNL-21645 Rev 1, WTP-RPT-223, Rev 1. 37. Naik, A.N.; Agarwal, C.; Chaudhury, S.; Goswami, A. Non-Stationary Radiotracer Method for Diffusion Coefficients of Cs+, Ba2+, Eu3+ Tracers in Nafion-117 Membrane. Sep. Sci. Technol. 2017, 52, 2379-2384.

38. Mackay, D.; Meares, P. On the Correction for Unstirred Solution Films in Ion-Exchange Membrane Cells. Kolloid-Zeitschtift, 1959,167, 31-39 39. Blaedel, W. J.; Haupert, T.J.; Evaenson, M.A. Mechanism of Trace Counterion Transport through Ion-Exchange Membranes. Anal. Chem., 1969, 41, 583-590. 40. Davison, W.; Zhang, H. In Situ Speciation Measurements of Trace Components in Natural Waters using Thin-Film Gels. Nature, 1994, 367, 546-548. 41. Temminghoff, E.J.M.; Plette, A.C.C.; Eck, R.V.; Riemsdijk, W.H.V. Determination of the Chemical Speciation of Trace Metals in Aqueous Systems by Wageningen Donna Membrane Technique. Anal. Chim. Acta, 2000, 417,149-157.



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42. Mafe, S.; Aguilella, V.M.; Pellicer, J. Film Control and Membrane Control in Charged Membranes J. Membr. Sci., 1988, 36, 497-509. 43. Jan, D.S.; Ho, C.C.; Tasi, F.N. Combined Film and Membrane Diffusion-Controlled Transport of Ions through Charged Membrane. J. Membr. Sci., 1994, 90,109-115. 44. Schögl, R.; Helfferich, F. Comment on the Significance of Diffusion Potentials in Ion Exchange Kinetics. J chem. Phy. 1957, 26, 5-7. 45. Sodaye, S.; Suresh, G.; Pandey, A.K.; Goswami, A. Interdiffusion of Exchanging Counterions in Poly(perfluorosulfonic acid) Membrane. J. Phys. Chem. B, 2009, 113, 12482–12488. 46. Liberti, L.; Helfferich, F. Mass Transfer and Kinetics of Ion Exchange; Martinus Nijhoff Publishers, The Hague, the Netherlands, 1983. 47. Crank, J. The Mathematics of Diffusion; Oxford: Clarendon Press, London. 1964. 48. Sodaye, S.; Agarwal, C.; Goswami, A. Study on Multicomponent Diffusion of Ions in Poly(perfluorosulfonated) Ion-Exchange Membrane using Radiotracers. J. Membr. Sci., 2008, 314, 221-225.

49. Kraaijeveld G.; Wesselingh J.A.; The Kinetics of Film Diffusion Limited Ion Exchange. Chem. Eng. Sci., 1993, 48, 467-473.

50. Gorka, A.; Bochenek, R.; Warchol. J.; Kaczmarski, K.; Antos, D. Ion Exchange Kinetics in Removal of Small Ions. Effect of Salt Concentration on Inter- and Intraparticle Diffusion. Chem. Eng. Sci., 2008, 63, 637-650.

51. Ostroski, I.C.; Borba, C.E.; Silva, E.A.; Arroyo, P.A.; Gulrardello, R.; Barros, M.A.S.D. Mass Transfer Mechanism of Ion Exchange in Fixed Bed Columns. J. Chem. Eng. Data, 2011, 56, 375–382.


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TOC graphic


Effect of Film Diffusion on the Ion-Exchange Kinetics of a Tracer Ion in

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