Analysis of Diffusion and Reversible Reaction in Spatially ConfinedSystems

Jerry H. Meldon†

Chemical and Biological Engineering Department, Tufts University, Medford, Massachusetts 02155

Facilitated gas transport in membranes is a diffusion/reaction system in which at least onereactant is spatially confined. Despite longstanding interest in this process, the consequencesof spatial constraint have generally been overlooked. They are examined here in the cases oftransport of the acid gases CO2, H2S, and SO2 in solutions of alkali and of ethylene in solutionsof silver salts. Diffusion potentials are shown to be significant in each case, playing a role thatcomplements spatial constraints. Nonetheless, at least in the ethylene case, fluxes calculatedusing an analysis that enforces spatial constraints but neglects ionic interactions are indistin-guishable from those generated by an analysis that accounts for both.

Introduction

Confinement of reactants or catalysts in membranes,1zeolites,2 and gels3 can promote selective reaction andseparation. When an immobilized liquid membraneconfines nonvolatile solute B that reacts reversibly withdissolved gas A (see Figure 1), B “facilitates” the flux ofA, ΦA, while the global reaction rate remains zero.4Developers of separation processes have sought toexploit the selectivity of facilitated transport.5,6 Othershave elucidated the underlying phenomena7 and deter-mined parameters such as kinetic constants from trans-port measurements.8 Most analyses have neglected thepotentially significant consequences of differences (∆D)between diffusivities of confined reactants. When con-fined reactants are electrolytes, ∆D causes diffusionpotentials9 that ensure the absence of local current inelectrically floating systems. For example, facilitatedcarbon dioxide transport in aqueous media involvescounterdiffusion of bicarbonate and slower carbonateions. Diffusion potentials equalize opposing chargefluxes while forcing inert cations into Nernstian distri-butions.10 Much larger electrical effects, including mea-surable potentials of up to 40 mV,11 mediate bicarbonate-facilitated CO2 transport in protein solutions.12

Confined solutes need not react directly with per-meants or carry charge for ∆D to mediate fluxes.Consider hemoglobin (Hb)-facilitated oxygen trans-port.13 Human red blood cells confine both Hb and themetabolic intermediate 2,3-diphosphoglycerate (DPG).Binding of DPG to Hb reduces the protein’s O2 affinity.Because DDPG > DDPGHb, [DPG]T (≡[DPG] + [DPGHb])increases in the direction of O2 diffusion. On the basisof a theoretical analysis that neglects electrical effects,it has been estimated that skewing of the [DPG]Tdistribution increases O2 fluxes by as much as 50%.14

There has also been considerable interest in usingmembranes containing Ag+ to separate olefins, particu-larly ethylene, from alkanes.15,16 Yet, the open literatureapparently contains no analysis of silver-facilitatedolefin transport that explicitly accounts for diffusionpotentials.

The following section includes an examination ofcoupled spatial constraint and electrical effects on

facilitated transport of CO2, H2S, and SO2. This isfollowed by two analyses of Ag+-facilitated olefin trans-port: one analogous to that applied to acid gas transportand another that neglects electrical effects. Unexpect-edly, the two analyses yield essentially identical fluxestimates.

Facilitated Transport of Acid Gases

We first consider steady-state transport of dissolvedacid gas A ()H2S, CO2, or SO2) in a film of an aqueoussolution that may contain alkali-metal ion M+. Com-plexes such as NaCO3

- are neglected; the metal istreated as a free cation.

The following reactions mediate gas transport:

where n ) 0 when A ) H2S and n ) 1 when A ) CO2 orSO2.

where (B-, C2-) denotes (HS-, S2-), (HCO3-, CO3

2-), or(HSO3

-, SO32-).

† Phone: 617-627-3570. Fax: 617-627-3991. E-mail:Jerry.Meldon@granite.tufts.edu.

Figure 1. Schematic diagram of the membrane including con-centration profiles when the membrane is exposed to gradientsin partial pressure of gas A, the permeation of which is facilitatedvia a reversible reaction with nonvolatile solute B to form AB.

A + nH2O S B- + H+ (1)

B- S C2- + H+ (2)

H2O S OH- + H+ (3)

456 Ind. Eng. Chem. Res. 2002, 41, 456-463

10.1021/ie0102706 CCC: $22.00 © 2002 American Chemical SocietyPublished on Web 12/27/2001

Reactions (2) and (3) are effectively instantaneous. Inthe absence of a suitable catalyst, reaction (1) kineticsare CO2-flux-limiting.17 For simplicity, we assume sucha catalyst is present and treat all reactions as instan-taneous, i.e.,

We also treat K’s as constants for a given gas andtemperature and implicitly incorporate the activity ofwater in Kw and in K1 when n ) 1.

Steady-state species balances are expressed by

while flux is given a Nernst-Planck formulation, i.e.,

Because alkali-metal ions are both inert and confinedby nonvolatility, eq 7 implies that NM is uniformly zero.In light of eq 8, this further implies that

Thus, metal ions achieve uniform electrochemicalpotentials consistent with their overall concentration,i.e.,

When eq 7 is applied to A, B-, and C2- and the resultsare summed to eliminate indeterminate reaction terms,one obtains

or after integration

Fluxes of B- and C2- sum to zero at the film’sboundaries (with the assumption of equilibrium ofreaction (2) having precluded enforcement of boundaryconditions for both B- and C2-); ΦA may be identifiedwith the transmembrane flux of A.

The remaining constraints are the absence both ofelectrical current:

and of net charge

neglecting minute local net charges responsible fordiffusion potentials.18

Equation 14 expands to

Insertion of eq 8 into eq 11, differentiation of eq 15 toeliminate d[C2-]/dx, insertion of the results into eq 13,and rearrangement yield

Equation 16 explicitly links diffusion potentials todifferences, ∆D, in diffusion coefficients of reactive ions.Together with concentration gradients, ∆D’s determinethe sign of the electrical field, -dV/dx, and thus (a) thatof d[M+]/dx (see eq 9) and (b) whether migration of B-

and C2- promotes or reduces gas transport (see eq 8).Although eq 16 is complicated, generally one or moreconcentration gradients is negligble. Appendix I outlinesan exact solution to the general problem.

At 25 °C and with [M+] ) 2 M, the reactive ions withsignificant gradients in a membrane may be deducedfrom the equilibrium compositions shown in Figure 2a-c, which were derived from the thermodynamic param-eters in Table 1, eqs 4-6 and 16, and the simplesolubility relationship

Parts a and c of Figure 2 elucidate the difficulty ofregenerating hydroxide from solutions preequilibratedwith even minute partial pressures P of either CO2 orSO2. When P is as low as 10-6 atm, the predominantanions are B- and C2- and solutions contain 0.5-1 molof reacted gas/equiv of alkali. On the other hand, thesecond dissociation of H2S is so weak that there is norange of P in which sulfide predominates (see Figure2b). Furthermore, although H2S is more easily stripped,its solutions too require large reductions in P to promoteits release.

To examine facilitated transport of the acid gases,fluxes were calculated over practical ranges of upstreampartial pressure, P0, with [M+] ) 2 M, PL/P0 ) 0.1, andT ) 25 °C. Note that hydrolyzed SO2 is such a strongacid that bisulfite concentrations in alkali-free water

Table 1. Thermodynamic and Transport Parameters at25 °Ca

parametercarbondioxide

hydrogensulfide

sulfurdioxide

pK1 6.3521 7.0522 1.8123

pK2 10.321 14.922 6.9123

pKW 14 same sameR (mol m-3 atm-1) 3424 10022 116025

Da × 109 (m2/s) 1.9224 1.4422 1.7625

DB × 109 (m2/s) 1.19b,26 1.19c 1.19b,25

DC × 109 (m2/s) 0.92b,26 0.82c 0.7223

DH × 109 (m2/s) 9.0b,27 same sameDOH × 109 (m2/s) 5.1b,27 same samea Infinite dilution values, corrected for small temperature

differences as needed. b Di ) RTΛi/F2|zi|, where Λ denotes equiva-lent ionic conductance. c Estimate.

[M+] + [H+] ) [B-] + 2[C2-] + [OH-] (15)

FRT

dVdx

) [(DB - DC)d[B-]

dx+ (DOH - DC)

d[OH-]dx

-

(DH - DC)d[H+]

dx ]/[DB[B-] + DC(4[C2-] + [M+]) +

DOH[OH-] + DH[H+]] (16)

[A] ) RP (17)

[B-][H+][A]

) K1 (4)

[C2-][H+]

[B-]) K2 (5)

[H+][OH-] ) Kw (6)

dNi

dx) ri (7)

Ni ) -Di(d[i]dx

+ zi[i]F

RTdVdx) (8)

d[M+]dx

) -[M+] FRT

dVdx

(9)

∫0

L[M+] dx ) CML (10)

ddx

(NA + NB- + NC2-) ) 0 (11)

NA + NB- + NC2- ) ΦA (12)

∑ziNi ) 0 (13)

∑zi[i] ) 0 (14)

Ind. Eng. Chem. Res., Vol. 41, No. 3, 2002 457

promote substantial facilitation.19 With CO2 and H2S,alkali must be present for there to be significantfacilitation of transport.

Diffusivities used in the calculations are listed inTable 1. Results are expressed in terms of enhancementfactor E, the ratio of fluxes with and without reaction.Maximal transmembrane potential differences were oforder 1 µV. Values of E0, calculated with all ionicdiffusivities set at DC (which eliminates electrostaticeffects), are depicted in Figure 3. Enhancement is greatwhen P is small, reflecting relatively large contributionsof carrier-mediated transport. As P0 increases, E0 valuesfor SO2 and H2S approach unity more rapidly than that

for CO2. The total amounts of SO2 and H2S level off atlower partial pressures, because of the very highsolubility of SO2 and negligible second dissociation ofH2S. Because of the latter limitation, at practical partialpressures a second weak acid must be present for thereto be significant facilitation of H2S transport.20

Electrical effects are depicted in Figure 4 as ratios ofenhancement factors (and therefore fluxes) calculatedby including and neglecting them. For SO2, the effectsare modest because similarly mobile HSO3

- and SO32-

predominate over the entire range of P. Large effectsfor CO2 are attributable to highly mobile OH-. In thecase of H2S, such effects are first seen at intermediateP values. In general, ion fluxes are determined primarilyby the diffusivity of the more dilute of two ions.10,28

Facilitated C2H4 Transport in AgNO3 Solutions

We next consider transport of ethylene (A) in solutionscontaining silver ions (S+) that complex C2H4 via thereaction

Proceeding as before, the governing relationships areeq 8 plus

with counterion Ì- typically nitrate. A solution isoutlined in appendix II.

In an attempt to decouple the consequences of electri-cal and spatial constraints and thereby compare impacts

Figure 2. Liquid-phase equilibrium composition vs gas-phasepartial pressure of acid gas at 25 °C and [M+] ) 2 M: (a) CO2, (b)H2S, (c) SO2.

Figure 3. Enhancement factor calculated assuming diffusivitiesof all reactive ions equal to DC vs upstream partial pressure ofpermeating acid gas; T ) 25 °C.

A + S+ S C+ (18)

[C+]

[A][S+]) KE (19)

d[X-]dx

) [X-] FRT

dVdx

(20)

∫0

L[X-] dx ) CML (21)

NA + NC+ ) ΦA (22)

[S+] + [C+] ) [Ì-] (23)

FRT

dVdx

)DS - DC

DS([X-] + [S+]) + DC[C+]

d[C+]dx

(24)

458 Ind. Eng. Chem. Res., Vol. 41, No. 3, 2002

upon ethylene transport, an analysis was derived thattreats all species as electrically neutral. Thus, eq 8reduces to

and eqs 19 and 22 remain unchanged while eqs 20, 23,and 24 are not applicable.

Applying eq 7 to S and C, we obtain

and so

Equation 21 is replaced by

Insertion of eq 25 into eqs 22 and 26 and integrationyield

where Φ2 and Φ3 are constants. The solution is outlinedin appendix III.

Both analyses were used to calculate ethylene fluxesin membranes containing aqueous AgNO3 at 25 °C, withPL ) 0. Teramoto et al.16 measured ethylene fluxesunder these conditions with CM ) 1 M and reported thatKE ) 0.11 m3/mol, DA ) 1.87 × 10-9 m2/s, DS ) 1.66 ×10-9 m2/s, and DC ) 1.1 × 10-9 m2/s.

We have calculated enhancement factors using theseparameters and plotted them versus P0 in Figure 5a.Remarkably, results based on a rigorous accounting forelectrical and spatial constraints (solid line) differimperceptibly from those derived from the nonionicanalysis (circles) that accounts only for the latter.

Additional results depicted in Figure 5a as the brokenline and squares were respectively calculated by assign-ing DC the value of DB and vice versa. In each such case,mobility equality eliminates electrical effects as well asthe basis for nonuniformity in the total silver concentra-tion.

More generally, in an analysis that neglects electro-static interactions but will, nonetheless, accuratelypredict fluxes, the effective diffusion coefficient thatshould be assigned to each member of a pair of reactiveions varies between the respective D values.10,28 Underthe conditions of Teramoto and co-workers,16 most silverions were not complexed. Consequently, the effective Dfor S+ and C+ was ≈DC, consistent with the resultsdepicted in Figure 5a, where results obtained with DSset equal to DC (squares) lie only slightly below therigorously derived curve. Figure 5b compares E valuescalculated the same three ways: rigorously and withboth ions assigned either DS or DC, as reaction equilib-rium constant KE (see eq 19) was hypothetically variedfrom its reported value to much higher values such thatnearly all of the silver is comlexed with ethylene; tofurther promote the latter condition, PL/P0 was set at0.1 instead of zero. Even with the relatively small ∆D,it is clear that the effective D varies from DC to DS asKE increases.

Interestingly, Teramoto et al. recognized the signifi-cance of electrical effects, yet they developed an accurate

Figure 4. Calculated ratio of rigorously calculated enhancementfactor and E0 from Figure 3 vs upstream partial pressure ofpermeating acid gas; T ) 25 °C.

Ni ) -Did[i]dx

(25)

ddx

(NS + NC) ) 0 (26)

NS + NC ) 0 (27)

∫0

L([S] + [C]) dx ) CML (27)

DA[A] + DC[C] ) -ΦAx + Φ2 (28)

DS[S] + DC[C] ) Φ3 (29)

Figure 5. (a) Theoretical enhancement factors vs upstreampartial pressure of ethylene; T ) 25 °C. Comparison of resultsobtained rigorously with those calculated by setting DS equal toDC, setting DC equal to DS, and neglecting electrical effects. (b)Calculated enhancement factors vs KE; T ) 25 °C and P0 ) 1 atm.Comparison of the results obtained as in Figure 5a (results ofrigorous and nonionic analyses again overlap).

Ind. Eng. Chem. Res., Vol. 41, No. 3, 2002 459

analysis in which inequality of DB and DC was retained,without including electrical terms. Applying instead thesimplifying approximation of linear [S] and [C] profiles,they calculated fluxes with P0 ) 1 atm that wereindistinguishable from what they obtained from anexact numerical analysis of a set of equations equivalentto those solved in our nonionic analysis. The value theyreported for DC was obtained from a fit to theirexperimental data.

In an attempt to determine why there is essentiallyexact agreement between the results of rigorous and“nonionic” analyses, we calculated ratios of up- anddownstream total silver concentrations, [S+] + [C+].This ratio deviates from unity whenever DC * DS,whether or not species are charged.

Because DS > DC, a membrane’s downstream surfaceactually assumes a relatively negative potential. Ac-cordingly, [X-] is greater at x ) 0 than at x ) L, andelectroneutrality requires the same of total silver.Enhanced silver concentrations at x ) 0 translate intolarger d[C+]/dx values. In addition, electrical migrationcomplements C+ transport driven by its concentrationgradient.

Migration naturally vanishes without electrostaticinteractions. However, the total silver concentrationsremain higher at x ) 0 because DS > DC (as DSincreases, the [S+] profile flattens while d[C+]/dx re-mains negative). Having eliminated one of the positiveeffects on ethylene transport, one might anticipate lowerethylene fluxes, which is contradicted by the behaviorin Figure 5a. The explanation, shown in Figure 6, is thelarger silver gradients calculated using the nonionicanalysis; in fact, they are just large enough to compen-sate for the absence of migration.

Conclusions

Electrical effects and spatial constraints are poten-tially important determinants of acid gas fluxes inmembrane-immobilized solutions of alkali-metal ions.Calculations indicate modest effects when the conjugatebases formed by first and second acid dissociations arethe predominant anions. Effects are considerably largerwhen OH- diffusion is significant.

The same phenomena mediate ethylene transport insilver salt solutions. An analysis that treats all species

as electrically neutral but enforces spatial constraintscorrectly predicts transmembrane fluxes. This unan-ticipated result is attributable to the larger transmem-brane gradients in the total carrier concentration thatare calculated in the absence of electrical interactions.Why the mathematics is so forgiving remains a mystery.

Appendix I: Solution to Eqs 4-6, 8-10, 12, 15,and 16

For conciseness, dimensionless variables are definedas follows:

It follows from eqs 4-6 that

where

Equation 15 becomes

from which b may be calculated, given a and m.Equations 9 and 10 become

while eq 16 transforms to

where

Differentiation of the second and third terms of eqA1-3 and combination of the results with eqs A1-5 andA1-6 yield

where

Figure 6. Ratio of up- and downstream total silver concentrationsvs DC/DS, calculated rigorously and via nonionic analyses.

i ≡ [Izi][A]0

i ) a, b, c, ... and I ) A, B, C, ... (A1-1)

y ≡ xL

; ψ ≡ FRT

dVdy

(A1-2)

c )θ2b

2

θ1a, h )

θ1ab

, oh )θwbθ1a

(A1-3)

θ1 ≡ K1

[A]0, θ2 ≡ K2

[A]0, θw ≡ Kw

[A]02

(A1-4)

m +θ1ab

) b + 2θ2b

2

θ1a+

θwbθ1a

(A1-5)

dmdy

) -mΨ (A1-6)

∫0

1m dy ) F (A1-7)

ψ )(λB - λC) db

dy+ (λOH - λC) doh

dy- (λH - λC) dh

dyδ

(A1-7)

λi ≡ DiFDA

, F ≡ CM

[A]0,

δ ≡ λBb + λC(4c + m) + λOHoh + λHh (A1-8)

dbdy

) 1f1

(f2 - mabg5)dady

(A1-9)

Ψ ) g5dady

(A1-10)

460 Ind. Eng. Chem. Res., Vol. 41, No. 3, 2002

Combination of eqs 8 and 12, conversion to dimen-sionless variables, and insertion of eqs A1-9 and A1-10yield

or

where

Finally, combination of eqs A1-6, A1-10, and A1-17yields

Equations A1-17 and A1-22 are easily solved (e.g.,using Runge-Kutta methods) to obtain ú(a) and m(a).The value of ú where a ) 1 is 0; that of m requires atrial-and-error experiment. Because the y value wherea ) γ must be 1, φ may be equated with ú(γ), after whicheach y(a) may be obtained by dividing ú(a) by φ. Thecorrect initial m ensures satisfaction of eq A1-7.

Appendix II: Solution to Eqs 8 and 19-24

When dimensionless variables are defined as in eqsA1-1 and A1-2, eq 19 becomes

where θE ≡ KE[A]0.

Equations 20 and 21 become

while eq 23 becomes

From eqs A2-1 and A2-4, it follows that

Equation 24 becomes

where

Proceeding as in appendix I, we obtain the followingdifferential equations:

where

With a guess of the ø value where a ) 1, we mayproceed to solve eqs A2-8 and A2-9; dimensionlessethylene flux φE ) ú(γ); y(a) ) ú(a)/φE. Equation A2-3tests the guess of the initial ø.

Appendix III: Solutions to Eqs 19 and 27-29

Upon transformation to dimensionless variables asbefore, eq A2-1 remains unchanged, while eq A2-3 isreplaced (see eq 27) by

Equations 28 and 29 become

where φ2 and φ3 are constants.Combining eqs A2-1 and A3-3, we obtain

where

f1 ≡ m - 2ab -6θ2b

2

θ1-

2θwbθ1

(A1-11)

f2 ≡ b2 - mb - 2θ1a (A1-12)

g1 ≡ f1

f2, g2 ≡ mab

f1(A1-13)

g3 ≡ (λB - λC) + ohb

(λOH - λC) +θ2

c(λH - λC) (A1-14)

g4 ≡ oha

(λOH - λC) +θ1

b(λH - λC) (A1-15)

g5 ≡ g1g3 - g4

∆ - g2g3(A1-16)

dady

) - φ

g6(A1-17)

dúda

) -g6 (A1-18)

φ ≡ ΦAL

DA[A]0, ú ≡ φy (A1-19)

g6 ≡ 1 + λBg7 + λCc(2g7

b- 1

a) (A1-20)

g7 ≡ g1 + g5(g2 - b) (A1-21)

dmda

) -mg5 (A1-22)

c ) θEas (A2-1)

dødy

) øψ (A2-2)

∫0

1ø dy ) F (A2-3)

s + c ) ø (A2-4)

s ) ø1 + θEa

(A2-5)

ψ ) µ1dcdy

(A2-6)

µ1 ≡ λS - λC

λS(ø + s) + λCc(A2-7)

dúda

) -µ3 (A2-8)

døda

) øµ1µ2 (A2-9)

µ2 ≡ θEs

1 - θEa(µ1ø - 1)(A2-10)

µ3 ≡ 1 + λCµ2(1 + µ1c) (A2-11)

∫0

1(s + c) dy ) F (A3-1)

a + λCc ) -φEy + φ2 (A3-2)

λSs + λCc ) φ3 (A3-3)

c )θEφ3a

u(A3-4)

Ind. Eng. Chem. Res., Vol. 41, No. 3, 2002 461

Equations A3-2-4 transform eq A3-1 to

Combining eqs A3-2 and A3-4, evaluating the resultat y ) 0 and 1, and subtracting one from the other yield

where

Differentiating eq A3-6 and the relationship derivedfrom eqs A3-2 and A3-4 and combining the results yield

After inserting eqs A3-8 and A3-9 into eq A3-6 andintegrating, one may solve for φ3 and, in turn, φE.

Nomenclature

a, b, c, h, m, oh, s ) defined by eq A1-1CM ) overall metal ion concentration (mol/m3)Di ) diffusion coefficient of species I (m2/s)F ) Faraday’s constant (96 500 J V-1 equiv-1)f1, f2 ) defined by eqs A1-11 and A1-12g1, g2 ) defined by eq A1-13g3, g4, g5, g6, g7 ) defined by eqs A1-14-16, A1-20, and

A1-21K1, K2 ) equilibrium constants of reactions 1 and 2

(mol/m3)KE ) equilibrium constant of reaction (18) (m3/mol)Kw ) equilibrium constant of reaction (3) (mol2/m6)L ) membrane thickness (m)P ) partial pressure (atm)r ) rate of production by recation (mol/m3/s)R ) gas constant (8.314 J mol-1 K-1)S+ ) silver ionu ) defined by eq A3-5V ) electrical potential (V)X- ) anion in the silver saltx ) distance from the upstream membrane surface (m)y ) defined by eq A1-2zi ) charge of species i

Greek Symbols

R ) solubility coefficient (mol m-3 atm-1)δ ) defined by eq A1-8ς ) defined by eq A1-19θ1, θ2, θw ) defined by eq A1-4θE ) defined following eq A2-1F ) defined by eq A1-8λ ) defined by eq A1-8µ1, µ2, µ3, µ4 ) defined by eqs A2-7, A2-10, A2-11, and A3-8ΦA ) transmembrane flux of A (mol m-2 s-1)Φ2, Φ3 ) constants of integration (mol m-1 s-1)φ ) dimensionless flux defined by eq A1-19

φE ) dimensionless ethylene fluxφ2, φ3 ) constants of integrationø ) [X-]/[A]0

ψ ) defined by eq A1-2

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u(a) ≡ λS + λCθEa (A3-5)

φ3

λC∫0

1[u + λC - λS

u2 ] dy ) 1 (A3-6)

φE ) (1 - γ)[1 +µ4φ3

u(1) u(γ)] (A3-7)

µ4 ≡ λSλCFθE (A3-8)

dy ) - 1θEφEλC

(1 +µ4φ3

u2 ) du (A3-9)

462 Ind. Eng. Chem. Res., Vol. 41, No. 3, 2002

(24) Meldon, J. H.; Smith, K. A.; Colton, C. K. The Effect ofWeak Acids upon the Transport of Carbon Dioxide in AlkalineSolutions. Chem. Eng. Sci. 1977, 32, 939.

(25) Hikita, H.; Asai, S.; Nose, H. Absorption of Sulfur Dioxideinto Water. AIChE J. 1978, 24, 147.

(26) Perry, J. H., Ed. Chemical Engineers’ Handbook, 4th ed.;McGraw-Hill: New York, 1963.

(27) MacInnes, D. A The Principles of Electrochemistry; Do-ver: New York, 1939.

(28) Newman, J. S. Electrochemical Systems; Prentice-Hall:Englewood Cliffs, NJ, 1973.

Received for review March 23, 2001Revised manuscript received October 19, 2001

Accepted October 31, 2001

IE0102706

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