L~.TTERE AL NUOVO CIMENTO VOL. 38, N. 5 1 0 t t o b r e 1983

Transport Nilmhers in Ion Exchange Membranes from Membrane Potential.

J. A . I B ~ z

Depa~tamento de Termologla, .Facultad de Cie~cias Universidad Autdnoma de Barcelona - Bellaterra (Barcelona), Espa~a

A. F. TEJERINA

Departamento de Termologta, _Facultad de Ciencias Universidad de Valladolid - Valladolid, Espafia

(ricevuto il 20 Giugno 1983)

PACS. 02.65. - Surface processes.

When a membrane is subjected to various driving forces such as gradients of electric potential, concentration, hydrostatic pressure, etc., a variety of transport phenomena take place through the membrane.

This paper deals with an analysis of an isothermal system consisting of a homo- geneous ion-exchange iaembrane of uniform thickness, bathed by two aqueous binary electrolyte solutions at different concentrations, on the basis of nonequilibrium thermo- dynamics. No gradients of electric potential and pressure exist across the membrane. The final purpose of the work is to establish a method that can be used to determine mean transport numbers of ions, ti, (i = 1 for the counterion and i = 2 for the colon) and the transport number of water, t3, inside the membrane.

The efficiency with which a membrane transports selectively any particular ionic species may be inferred by measuring the transport number of the species in the membrane. There are two methods normally used to determine the transport number (1): i) direct measure of i~ and t3 by Hittorf 's method; ii) indirect e.m.f, method con- sisting in the measurements of the membrane potential (~). The e.m.f, method leads to the apparent t ransport numbers (ti(app)), whereas Hittorf 's method leads to the so-called true transport numbers, i.e. the numbers corrected for the transport of water accompanying the transport of ions. In both methods, different concentrations of electrolyte exist on either side of the membrane and therefore, the transport number values derived by these methods cannot be directly related to a definite concentration

(0 N. LAKSY, Xh'ARAYA.'~AIA~: Transport Phenomena in Membranes ( ~ e w York, N. Y. , 1969), p. 232. (~) J. A. IBASEZ and k . F. TF~J~.ttlNA: Phys . Lett. A, 88, 262 (1982).

145

146 z . A . IBI~V,Z and A. F. TEJ]~RINA

of the ex te rna l solut ions and they will be re fe r red to a mean concent ra t ion . F o r th is reason, the symbols of the, t r anspor t numbers have been indica ted by an overbar .

In the present l e t te r the e.m.f, me thod has been considered and a re la t ionship ob- t a ined be tween tl(app) and tl and ta which al lows us t he de t e rmina t ion of ti (i = i , 2, 3) by s tudying the dependence of tdapp) wi th respect to the concent ra t ion of the solut ions ba th ing the n iembrane.

According to nonequi l ib r ium t h e r m o d y n a m i c s (3), if the grad ien ts of e lec t rochemica l po ten t ia l wi th in the membrane are small and the system is close to equi l ibr ium, t he fluxes and dr iv ing forces are connected by l inear re la t ions as

(1) J i = ~ l i jz i (i = 1, 2, 3),

where J~ is the flux of componen t i r e l a t ive to the membrane ma t r ix , the l~i va lues are the phenomenologica l coeffw.ients and ZJ is the t h e r m o d y n a m i c force on j which con- juga tes wi th the flux J~. The x-axis is ~akeu in the d i rec t ion of the membrane thickncss, and the fluxes are considered to occur only in this di rect ion. Then X~ is equal to the neg- a t ive g rad ien t of e lec t rochemica l po ten t ia l of the species j, - - ~fi~/ex in the membrane .

If the m e m b r a n e is t r ea t ed as d iscont inuous , then Z~ may be wr i t t en as

(2) Z~ . . . . d

= - - ( V i A P q-Att i - z, F A F ) / d ,

where V,, z~, d and ff are the par t i a l molar vo lume, the va lency of ion i, the membrane thickness and the Fa raday constant , respect ive ly . AP, Ay and Ap~ are the differences in pressure, e lectr ic po ten t ia l and chenlical po ten t ia l of ion i, be tween the solutions on opposi te sides of the membrane .

In the case in which an electr ic forc.e is appl ied, but there are no grad ien ts of ehe,,fical po ten t ia l a.ud pressure, thc electr ic cur ren t I through the membrane is re la ted to the ion flows as

(3) iT = ( Z l J 1 -;- z2J2) . s ~ : - - k r a A l ~ d ,

where k,,, is the specific conduc t iv i ty of the m e m b r a n e and, upon subs t i tu t ion of eq. (1) into eq. (3), is g iven by

(4) k,,, = (Z~l l l -}- 2ZlZ2112 q- Z2122).~ 2 .

The t ranspor t number of ion i (i -- 1, 2) is the f rac t ion of the cur ren t carr ied by the i-th ion, and the t ranspor t nunfl)er of water , ta, is the n u m b e r of moles of wa t e r t rans- ferre, d by one farad ' ly of e lec t r ic i ty in the direct ion of the cur rent ,

(5) ti = z i .~J i /1 = (z~lii q- zlzf l12)~'elk, , (i = 1, 2) ,

(6) t 3 - - f f J a / ] - = (Z l l l a @ Zf l2s) l~ '2 /km ,

obta ined from eqs. (1) and (3).

(a) S. R. DE GnOOT a n d P . MAZUR: Nonequilibrium Thermodynamics ( A m s t e r d a m , 1963) , p . 30.

TRANSPORT NUMBERS IN ION EXCHANGE MEMBRANES ETC. 1~7

I f no electr ic field is appl ied ex te rna l ly across t he membrane , w i th the assump- t ion t h a t A P ~ 0, subs t i tu t ion of eq. (2) in to eq. (1) g ives

(7) J~ ~ -- /~l(A/h A- z l F ~ m / d - /~2(AP2 -4- z2 -4- z2-F~P.=/d - l~sAp3 /d ,

where ~m is the m e m b r a n e po ten t ia l , i . e . t he e lec t r i c -po ten t ia l difference be tween the solut ions separa ted by the membrane . I n th is ease, t he e lec t r ic -cur ren t densi ty , I , mus t be zero th rough a cross-sectton of t he membrane , and so

(S) I = ( z , J I A- z 2 J ~ ) t ~ := O .

In t roduc ing eq. (7) into eq. (8) and solving for t he m e m b r a n e po ten t i a l , we ob ta in

1 (9) ~m = - - ~ (h Ap . /Z lV , + A ~ / Z 2 + t8 A~3),

where p. is the chemica l po ten t i a l of t he sa l t defined as po ~ v~pl + v2P~ wi th v~ and v s t he numbers of counter ions and colons, respec t ive ly , pe r molecule of salt .

Given that- p. = go + R T l n a ~ , a , being the sal t a c t i v i t y and po the chemica l po- Yl �9 Y$ t en t i a l in the s t andard s ta te , by in t roduc ing the m e a n ac t i v i t y a:~ ( i . e . a • ~ a 1 a ~ ,

where u = v 1 + v2) and t ak ing into account t ha t a = a~,, i t fol lows t h a t

(10) Ap. = I~T In [ a+(i)-] [a~(o)J '

where i and o re fer to t he inside and outs ide of the m e m b r a n e (we consider as ins ide t he c o m p a r t m e n t corresponding to t he more concen t ra ted solution).

W i t h respect to Ap~, we h a v e

(11) Ap~ -- /~T In [-a~(i) ] . La2(o)J

Usua l ly the ra t i O of single ion ac t iv i t i es in the two solut ions at bo th sides of the membrane is equa ted to the ra t io of the mean ac t iv i t ies , i . e . a~(i) /a2(o) -~ a • and so eq. (11) becomes

l (12) Ap2 = R T In [a~(o)J "

On the o ther hand, the Gibbs-Duhem equa t ion establ ishes

(13) X s d l n a a + ~ . d l n a, = 0 ,

where X s and X . represent the molar f rac t ions of wa te r and salt , respect ive ly . T h e last equa t ion can be wr i t t en in t e rms of mean ac t iv i t ies as follows:

X. ('14) d in a s ---- - - v --- d In a~ .

X s

1 4 8 J. A, IBA~EZ a n d x . r . T]~JERINA

For di lu te solut ions X , I X 3 = mM3/lO00 , M~ being the water molecular weight and m the molal i ty . Subs t i tu t ing in eq. (9) we have

, , , : - - ] 2' La• LZiVi z2 10~0 mMa '

where we have t aken in to account t ha t

(16) ;.~.. = ~ ; : l n b">-] : _ , , ~ _ ; : ii,,.,-,,: In [v<i) 1 La3(o)j 1000 La~(o)J '

m being the mean molal i ty, m = (re(i) q m(o))12. In the case of a l : l c lectrolyte , cq. (15) reduces to

(17) "m =--~(~;,--1 2;:"'<'-":~,nV+(i) 13.00-6-, ~.~j, given tha t r~ = v 2 = 1, z I = 1, z 2 = - - 1 , and where t~ and ta are the mean t r anspor t number s of countcr ions and water, respectively, between the molal i t ies In(i) and In(o).

If we compare the last equa t ion wi th this one g iv ing the diffusion po ten t ia l , ~d, between two solut ions i and o (*)

R T [a+(i) ] (18) ~'d = ~ - ~ - - (2 t l - - 1) In [a+(o)J '

wi th t~ the t r anspor t n u m b e r of cat ion in aqueous phase, we may conclude t ha t the membrane poten t ia l is cquiwflent to a diffusion potent ia l , bu t wi th a so-called apparen t t ranspor t number , g iven by

(19) tl(app) = tl tamM3 -- t l - - O.O18mia. lO00

E q u a t i o n (19) may be used to de te rmine the t r anspor t num ber s tl and t3 ins ide the membrane from the values of apparen t t r anspor t num ber s ca lcula ted as a func t ion of the concen t ra t ion m.

If to ~m we add the Ncrns t i an con t r ibu t ion due to the measur ing electrodes (2,4), and in ('as(; of reversible electrodes with respect to the anion, the following expression for the direct ly measured potent ia l , ~,, is ob ta ined

(20) ,:ro-,<',-] -= 7 ' - ( t l - lO-ZiamM3) [.a• "

Equa t i on (20) is the integrate,] form wi th in nar row l imi ts of the equa t ion

(21)

I]

2R T f = - - - - ( i l - - I0-3i3mM3) d l n a . ,

I

~') J . A. IBA.~EZ, •. F . T~.~JERINA, ,]'. (]ARRIDO a n d J . PELLICER'- J. Non.Equilib. Thermodyn., 5, 313 (1980) .

TRANSPORT NUMBERS IN ION EXCHANGE MEMBRANES ETC. 149

obtained for 1:1 electrolyte applying the Scatchard equation for the measured poten- t i a l (5,s)

II "f (22) ~ ----- i~ d In a , ,

!

for al l the components of the system (counterions, coions, water and fixed charges of the membrane matr ix) and where I and I I refer to the boundaries of the membrane.

Expression (17) is val id when the ra te of s t i r r ing in solutions is such as is not necessary to consider the presence of diffusion boundary layers adhered to the mem- brane (a,7). In other cases, the presence of diffusion potent ials in these layers must be

bulk solution

diffusion membrane cb'ffuslon layer t ayer

I a~(i)

i I I ,/ ~

~ I membt'ane sysCem

bu lk solution

Fig . 1. - F o r m a l s t r u c t u r e of t he m e m b r a n e s y s t e m COrresponding to a m e m b r a n e w i th n e g a t i v e c h a r g e s u n i f o r m l y d i s t r i b u t e d . The c o n c e n t r a t i o n profl l is i n d i c a t e d . Ins ide t h e m e m b r a n e , t he sol id l ine r e fe r s to t he a n i o n s a n d t h e d o t t e d l ine r e fe r s to t he ca t ions . The d i scon t inu i t i e s on the m e m b r a n e s u r f a c e s are d u e to D o n n a n equ i l i b r i a .

considered; these potentials correspond to the l iquid junction between the solutions corresponding to the boundaries of the diffusion layers (fig. 1), remaining the membrane potent ia l in the form

R T [a (il l . f l '

+ ( 1 - 2 ~ ( 0 ) ) ~ In [a-~~J -F

RT ra . ( i ) l + (1 - - 2i~(i)) - ~ In - -

where tl(J), j = o, i, are the mean t ranspor t numbers of ions in the aqueous phases cor. responding to the diffusion layers.

(s) G. SCATCHARD: J. Am. Chem. Sor 75, 2883 (1953). (a) G. SC,LTCHA~D: Ion Transport Across ~lembranes (l~ew Y o r k , N. Y. , 1954) ; ~Tec~rochemistry i~ Biology and Medicine (New Y o r k , N . Y . , 1955), p . 18. (7) J , A . IBA~EZ, A. F . TEJERINA, J . GARRIDO a n d J . PELLICER: J . Non-Equflib. Therraodyn., 5, 379 ( 1 9 8 0 ) .

1~0 J . A . I B X ~ Z a n d A. F. TF, J~RINA

The activities corresponding to the surfaces of the membrane inay be ca]eulatecl from tile molar concentration C[ and C: which are related with the sa.lt flux, J , , in the form

5 (24) (:~ = C , - - J . ~ ) , Cto = Co + J .~ ) ,

where C i and Co are the bulk concentrations, 5 is the thickness of the diffusion layers and D the diffusion coefficient of tile electrolyte in aqueous phases. The use of cqs, (24) requires the knowledge of the parameter b/D (,,s), so as the values of the solute flux, which can be measured by different methods (4,9,~0). With respect to the act ivi ty coef- ficients and the io~l tr~lnsport numbers in aqueous phases, they can be directly obtained from the literature.

(*) J . . k . IBA.~EZ a n d zk. F. TEJERINA: J . Non-Equi l ib . Thermodyn. , 6, 85 (1981). (o) M. DELMOTTE a n d J . CaANU: J . Electrochim. _4cla, 18, 963 (1978). (ao) K . NOML'RA, A, ~IATSUBARA a l ld H . KINIZUKA: B ill[. Chem. Soc, Jpn . , 51, 1037 (1978).