ion Transport Across an Ion-ExchangeMembrane
f the Solution-Membrane Interface during Ion Transport Across an Ion-Exchange Membrane Y. C h n ’ and A. Litan Nuchsar Research Center, Negev, Beer-Sheva, Israel (Received December 8, 1972; Revised Manuscript Received May 6, 1974) Pubflcation costs assisted by the Nuclear Research Center-Negev
A ciition-exchange membrane, placed between two identical solutions of Na+ and Ca2+ions and originally equilibrated with them, is found to undergo a significant decrease in its Ca2+ equivalent fraction upon passage of an electric current. The transport number of the two ions also show a strong dependence on current density. A theoretical evaluation of the experimental results, based on the Nernst unstirred layer model, leads to the conclusion that there is no local equilibrium a t the solution-membrane interface. This conclusion contrasts with the common assumption that, a t an interface between two phases, local equilibrium is maintained even when there is matter transport across the interphase boundary. The Butler-Volmer equations which are used in electrode kinetics are shown to be applicable to the solution-membrane interface, especially as a frame for a qualitative discussion of deviations from local equilibrium.
Introduction The system presently studied is, essentially, a cationexchange menibrane placed between two aqueous solutions of the ions ]Vat, Ca2+and C1-. For a case of identical solutions on both sides of the membrane, and when the system is at equilibrium, the ionic composition of the membrane, defined as the equivalent fraction of e a 2 +rounterions, has a typical equilibrium value (exhibiting the well-known preference for the divalent ion). When a steady electric current is passed through the membrane, the ionic composition changes and reaches a steady-state value. Dependences of membrane ionic composition and of ionic transport numbers on current density have already been reported but the theoretical treatments were only qualitative. The presently reported study was undertaken with the aims to expel imen-tally characterize the dependences of membrane composition and of transport numbers on electric current density and to interpret these dependences theoretically. Experimenlta! Section Materials. Measurements were made on an AMF-C-100 cation-exchange membrane (a sulfonated polyethylenestyrm e graft copo1ymer)i with an exchange capacity of 0.9 mery weight.. Ca2* form) and a thickness of 0.01 cm. A mixed NaCI-CaCl2 aqueous solution was used of 0.08 N total concentration and with a Ca2+ to Na+ equivalent ratio of 1:l. Method jar Varying Membrane Composition. Figure 1 represents the apparatus used to produce changes in the ionic composition of a membrane by an electric current. An essential feature of 1,his apparatus is the means for quick removal of the membrane, after the passage of current. In this way the new ionic composition which has been produced is safeguarded from further changes due to reequilibration with the solutions. The apparalus is made out of Perspex. There are two
symmetric units, each of which is composed of an electrode cell (EC1, EC2), a circulation cell (CCI, CC2), and a separating anion-exchange membrane (AE), pressed between two Parafilm “M” rings; the unit is tightly and permanently clamped by means of four stainless-steel screws (S). The anion-exchange membranes prevent products of the electrode reactions from reaching the central membrane which is being studied. Between the two units there is a groove into which is inserted a Perspex plate P (shaded in Figure 1j with a circular hole, 2.4 cm in diameter, over which the membrane to be studied (a disk 3.2 cm in diameter) is placed, between two Parafilm “M” rings, and tightened with a ring screw. The membrane remains planar throughout the experiment because its dimensions do not change, practically, upon changes in Ca2+ content and there are equal pressures on both sides of the membrane. In order to prevent leakage of electric current through interstices between the Perspex plate and the groove, the outer form of the plate is made to fit well. the inner form of the groove and silicone grease is used to secure perfect sealing. The whole assembly is held in a clamping stainless steel frame (F). When the current is passed, the frame is tightened by means of the knob I(;when the current is discontinued, the knob is slightly released and the membranecarrying plate P is quickly pulled out. Efficient stirring and streaming of the solutions over the membrane surface is produced by Mears and Sutton’s method4 of forcing the solutions unto the membrane surface through perforated, porous polypropylene disks (D). Through each circulation cell, separately, about 2.5 1. of solution are circulated by a peristaltic pump, renewing the contents of the cell approximately 40 times per minute. Each electrode cell contains about 90 cm3 of solution, stirred by a micromotor. The electrodes (E) are made of 80 mesh platinum wire gauze. Constant current is supplied by a Regatron Model C630CM constant power supply (Electronic Measurements) and measured by a Hewlett Packard VTVM Model 412-A. The Journalof Physical Chemistry, Vol. 78. No. 18. 7974
Y . Oren
and A. Litan
n
I
I
Figure 1. Apparatus used to produce ionic composition changes in an ion-exchamge membrane by an electric current (see text for details).
All measurements have been carried out a t room temperature with no precise temperature control; heating of the solutions upon passage of current was negligible. In most experiments, all the cells are filled with the same solution, with which the cation-exchange membrane has been equilibrated prior to its mounting on the hole of the central Perspex plate. When the plate is pulled out, after the passage of current, it is immediately rinsed with deionized water, the membrane is removed, dried by Whatman50 filter paper, and analyzed for its Ca2+ content. Determination of Transport Numbers. The transport number of Na+ in the cation-exchange membrane separating cells CC1 and CC2 (Figure 1) can be determined by a tracer technique. For this purpose, means are provided for circulating the solution of each cell through a glass tube spiral mounted on the crystal of a scintillation counter. Holding cell CC2 empty, about 15 $2 of 22NaC1 (supplied by Amc!rsham) i s injected into the solution circulating through cell CC1 and through the glass spiral. After the activity (CPM) is determined, the glass spiral is taken out of the circulation circuit of cell CC1 and rinsed with a nonradioactive solution until the counter indicates the background activity (B). Cell CC2 is then connected to the spiral and filled wifh the same (nonactive) solution with which cell CC1 was previously filled. As soon as the solution begins to circulate, w current i is switched on for an interval of time t, so that cations pass from cell CC1 into cell CC2. Switching the current off, cell 661 is immediately drained, so as to avoid any further change of activity in cell CC2, and the activity in cell CC2 (CPM') is determined. The transport number of Na+, t l , is calculated from the relation
t, =
(CPM' - B)ClVS (CPM - B)it
where C is the Na+ concentration in cell CC1, V is the volume of the solution in cell CC2, and 5 is the Faraday number. Determination ot the Ca2+ Content i n the Membrane. The membrane disk is shaken for 1.5 hr with 100 cm3 of 1 N KCl, removed, dried with a filter paper, and shaken again with another 100-cm3 portion of 1 N KCI for 1 hr. The Ca2+ conceaitrations in both solutions are then determined by titration with EDTA, at pH 12, using Acid Alizarin Black SN us an indicator. The e a 2 + cont,ent in the membrane region which is exposed to electric current flow is calculated from the experimentally determined (:a2+ contents in the whole membrane disk, before and after passage of current. The Journal of Physical Chemistry, Vol. 78, No. 18, 1974
Figure 2. Apparatus for determination of H* transport, due to water splitting, through the membrane separating cells CC1 and CC2 and schematic concentration profiles near central membranes. AE and CE denote anion- and cation-exchange membranes, respectively.
It is clear that a composition change in the exposed region leads to composition changes in the unexposed region, due to radial diffusion in the membrane disk. These changes extend to a radial distance of order (&)1/2 beyond the radius of the exposed region; D is the interdiffusion coefficient for Na+-Ca2+ and t is the time of current flow through the membranea5D is of order to cm2 sec-l and t was usually well below 200 sec. For these values, the error due to radial diffusion is smaller than the overall experimental error. Determination of Coion Penetration into the Membrane. The membrane is gently wiped with Whatman-50 filter paper to remove any superficially adhering solution, and then shaken with 50 cm3 of deionized water for 1.5 hr. The solution thus obtained is analyzed for C1- by precipitation with AgN03 and measurement of optical density a t some wavelength in the range 400-600 nm. Determination of H+ Contribution to Electric Current Through the Membrane. For this purpose, the apparatus described in Figure 1 is modified, as shown schematically, in Figure 2. If concentration polarization near the cation-exchange membrane which separates cells 6C1 and CCZ i s sufficiently large, water splitting will occur on the left side of the membrane and H+ions will penetrate into cell CC2, lowering its pW. The high salt concentration in the salt cell SC warrants negligible water splitting at the AE membrane which, had it occurred, would result in undesirable OWpenetration into CC2. In CC1 the total circulating volume E5 about 2 1. while in CC2 it is kept down to 250 cm3. The pH of the CC2 solution is measured, both before and after passage of the electric current, with a Metrohm precision potentiometer E-353B. Experimental Results The selectivity coefficient of the membrane is defined by12 - 2
cx,
-
K = /x,),,",,/[(~1°~2/c20](2 ) where XI, 2 2 are the equivalent fractions of Na+ and Ca2+ in the membrane, respectively, and ClO,Cz*are the respective concentrations in the solution. The values of K ob-
Ion Transport Across an Ion-Exchange Membrane
1807
TABLE 1: V d u e s of the Selectivity Coefficient, K , Obtained from Equilibrium Measurements 0.9
XCa
Total solution concn, N
0.781 0.572 0.372 0.182 0.090 0.045 0.197 0.194
0.0912 0.0934 0.0966 0.0978 0.0989 0.0995 0.030 0.015
Toa
K
I _
122 124 121 126 157 165 125 108
0.973 0.939 0.893 0.807 0.685 0.565 0.654 0 754 ~
0.7
x* ob
I-
0.4
1
o
TABLE 11: Transport Number of Naf a n d Equivalent Fraction of Cat+ in the Membrane at Steady State at Various Current Densities ~
I, mA cm-2
0 22 1
66,3 110 220
tl(mixed solution)
8 7
0.93 0.87 0.67 0.59 0.59
~
aa
1
1
1
I
40
BO
120
160
l
t
~~
tl(O.08 N NaCl solution)
zoo
l
24~1
I
280
t
320
1
J ~ O
KC.
Figure 3. Experimental results of Ca2+ equivalent fraction in the membrane, X2, as function of time, at various current densities: 0, 16.6 mA cm+; A,31.0 mA cm-2; 80.0 WIAcm-2; A, 111.0 mA M, 221.0 mA cm-2.
*,
(equiv) :k 0.01 :k 0.01 0.01 :k 0 . 0 1
0.25 0.39 0.45 0.60
TABLE 111: Steady-State Values of Anodic aad Cathodic SoIutions
fO.01 f 0.01 f 0.01 f 0.01
1.02 1.03
z2for Differing +
Anodic solution: 0.08 N NaCl CaClz 0.032 Ca2+ equiv fraction Current densty, mA em-*
Cathodic solution: 0.08 N NaCl
0.08 N CaClz
2.2 16.6 177 . Q
0.42 0.06 0.04
0.60 0.20 0.04
tained from equilibrium measurements are given in Table 1. Figure 3 shows the time dependences of when steady currents of various densities are being passed through the membrane. The steady-state values of X z , at the various current densities, are presented in Table 11. The attainment of a steady-state value of x2 a t some current density I was verified by reaching the same value of xp both when the current densky is I from the very beginning of current passage, and when I is applied after a larger current density has been passed for some time. Table I1 also gives the transport number of Na+, tl, in the membrane. The transport number of Ca2+, tz, was calculated from the relation dz =
1 -
tl
In the determinations of tl, the amount of radioactive Na+ diffusing through the membrane, due to the difference in activities on the two sides of the membrane, was found to be negligibly small compared to the amount passing due to the electric current. The values of x2 im Table I1 were obtained with identical solutions on both sides of the membrane; Table I11 shows results obtained with different solutions on both sides. At sufficiently high current densities, the ionic composition of the cation-exchange membrane is seen to be independent of the composition o f the solution on the cathodic side, Le., of the solution into which ions enter, through the membrane.
The contribution of H+ ions to total current did not exceed 1%even at the highest current density used (220 mA cmV2) and was therefore disregarded in the subsequent theoretical analysis. Penetration of C1- coions was also negligible, being always smaller than 1.5%of the total exchange capacity. A decrease in the rate of circulation of the solutions bathing the membrane, from 3.5 to 0.8 1. min-l did not affect the results. Hence, within this range, any random change of circulation rate does not affect significantly the hydrodynamic conditions a t the membrane surfaces. Theoretical The main purpose of this section i s to show that the experimental results, when analyzed in terms of the wellknown Nernst unstirred layer model,6 indicate that the electric current causes a deviation from local equilibrium a t the solution-membrane interface (SMI). It is also shown in this section that the Butler-Volmer equation^,^ which are used in electrode kinetics, provide appropriate mathematical expressions for ionic fluxes at a SMT which i s out of equilibrium. According to the Nernst model,6 the membrane is in contact with two layers of totally unstirred solution, often referred to as Nernst diffusion films. Further away from the membrane, the so-called bulk solutions are well stirred and kept at constant composition throughout the experiment. When a steady electric current is passed through the system, the concentrations in the unstirred layers deviate from the bulk values and steady-state profiles are formed as shown schematically in Figure 4. Inside the membrane, the composition changes also and a steady-state composition profile sets in. SchlogP has derived mathematical expressions for composition profiles in a system with any number of ionic species. However, for the specific system presently studied, it is convenient to start from the basic differential equations. The Membrane Phase. Inside the cation-exchange membrane, the flux of cations type i (i = 1 , 2 for Na+ and ea2+, respectively) is assumed to be given by the Nernst-Planck equationg
The Journal
of PhysicalChemistry,
Voi. 78, No. 18, 1974
V . Oren and A.
1808
Litan
I I
I
which is a modified Sherwood number. Equation 8 reveals that if at { = 0, x2(0)> 8, then both X2 and dXzId5 increase with 5; if Xz(0) < 0, then X 2 and dXzld5 decrease with 5. The two types o f X&) profiles are shown schematically in Figure 5. Integration of eq 8 between i = 0 and Ji: = d yields
Figure 4. Steady-state concentration profiles in unstirred layers and
The average value of X 2 is defined as
in cation-exchange membrane for current denslty /(schematic).
ci
where Ji is the flux, f i i is the diffusion coefficient, is the concentration, % is the distance from the left surface of the membrane, ,ti is the valency of the ion, 5 is the Faraday constant, and $ is the electric potential. A bar indicates that the quantity pertains to the membrane phase; corresponding unlbarred quantities will refer to the solution. At steady state, the fluxes J1 and Jz are constant throughout the membrane and the unstirred layers. For subsequent reference, it is convenient to define the total ion flux
J
JZ
+ J2
(4)
(to be expressed in uinits of equivalents per unit area and unit time), and also the transport numbers
The current density is then given by
I = 5J
(6)
The two Nernst-Plainck equations for J1 and Jz are supplemented by an electroneutrality requirement
where C, denotes the total exchange capacity of the membrane. All concentrations are expressed in equivalents per unit volume. Equation 7 implies negligible coion penetration into the membrane, which has been verified experimentally. Thus, there are three equations for the functions Cz(a), and $(%).They are easily reduced to a single equation
el@),
&Qat where 8 2 , as follows
E,
13, and
-=
p(Z2 -
+ 1)
(8
B tire dimensionless quantities defined
This is the Ca2* fraction actually measured experimentally and reported, e.g., in Figure 3 and in Table I1 (where the sign ( )av is omitted for brevity). Substitution of d t in eq 14 according to eq 8 and integration yield
(X2)8v = 0
C,b)
The Journal of Physical Chemistry. Vol. 78, No. 18, 1974
+
C,(4 = c,(4
(16)
where C3 refers to the anion. In the unstirred layer from which cations enter the membrane (see Figure 4),the range of variation of x is from x = 0, where the concentrations have their bulk values, Cl0, CzQ, CSo to x = 6 at the SMI. For the second layer the range is chosen as x = -6, at the interface, to x = 0. Solving the four equations (see Appendix), the following expressions are obtained for the concentrations at the interface x = 6 Ci(6) = 2(w0 - Ab)/[4
+ 3~(6)]
C2(6) = Y(6)C*(S)
(17) (18)
xihere A
and
-t
Consider the case @ >> 1. According to eq 12 and 6, this is realized for a given membrane when the current density is sufficiently large. Equation 15 shows that, in this case, ( x2)av is very close to 0 andindependentof Xzbd). Figure 5 indicates that a corollary to (X2)Bvx 0 is ( 8 2 ) 8 v = Xz(0). The independence of ( X Z ) ~on " X z ( d ) , at sufficiently high current densities, implies its independence of the composition of the solution on that side of the membrane. This lack of dependence was also found experimentally (see Table 111). The Unstirred Layers. Here, again, the ion fluxes are assumed to be given by Nernst-Planck equations, i.e., eq 3 with the bars omitted. In contradistinction to the cationexchange membrane, now there is a flux equation for the anion also; at steady state its flux is zero. The three flux equations are supplemented by the condition of electroneutrality
(10)
d being the thickness of the membrane and, therefore, 0 S { 51
( 1 / 2 P ) [ X , ( d )- z 2 ( 0 ) ] [ 2 -I-j?,(O)
x',(d)] (15)
-
5 = x/d
+
a0
2c10
5
J[(ti/D,)
+
(3/2)C,O
+
(f2/2@.)]
(19) (20)
and the ratio of concentrations at the interface, r(6),satis-
Ion Transport Across an Ion-Exchange Membrane
1809
TABLE IV: Calculated Membrane Composition at Various Current Densities, Assuming Equilibrium at the SMI, Compared to Experimental Values
22.1 66.4 110
0.93-0.94 0.93-0.94 0.92-0.93
0.91-0.93 0.88-0.92 0.71-0.92
0.87 f: 0.01 0.67 4 0 . 0 1 0.59 I 0.01
a The values of Z2(0) and z ~ ( dare ) the extremes obtained for every current density, I , using the values 0.58,0.61 for Dz/Di; 110, 165 for K; 221, 300
m~ cm-2 for 11im.
0
--z-
d
Figure 5. The two possible forms of )7,(,$)profiles in the membrane together with corresponding values of (X2)avand the value of 19 (schematic).
fies the relation
wo 1 $. 2o! 4a - 3 4 6 ) --.In w0 - A6 3 .t Bo! In 4a - 3YO
(21)
where (t2/2D,)/(ti/Dl)
(22)
= czo/c*o
(23)
and CaC12.11 In the subsequent calculations a value of 0.58 was also used in order to check the influence which the value of DalD1 has on the final result. The highest current density used in the experiments, 220 mA cmW2,is definitely lower than the limiting current density (see Experimental Results for transport numbers and for H+contribution to the current). This value i s therefore a lower bound for Ilim. Analysis of Experimental Results. Assumption of Local Equilibrium ut the SMZ. The ionic concentrations a t the two SMI's are calculated for various current densities, using the experimental values for the transport numbers and for several values of the parameters 0 2 / 0 1 and I/IIim (see preceding paragraphs). For these concentrations, the membrane ionic compositions at the SMI's are then calculated on the assumption that there is local thermodynamic equilibrium at the SMI's, Le., that12
and Yo
Corresponding expressions are obtained for C1(-6) and C2(-8) a t the second interface. Equations 17, 18, and 21 reveal that the total flux, J, has a limiting value, Jlim, at which
and These relations and eq 6 lead to
Itim2 SJ'"
= 5(2D1Ci0
+ 302cz0)/6 (26)
i.e., the limiting current density is equal to the sum of limiting current densities in the corresponding pure salt solutions. From eq 20,26, and 6
With this expression for AS, eq 17, 18, and 21 indicate that Cl(S) and C2(S) can be expressed in terms of the bulk concentrations, thc ratio of diffusion coefficients, 02lD1, the transport numbers, and the normalized current density, IIZlim.
For D2/Dl a value of 0.594 is obtained from limiting equivalent and a value of 0.61, from selfdiffusion measiirernents in 0.1 N solutions of pure NaCl
where K is the selectivity coefficient (see Table I). The results of the calculations are summarized in Table IV. For all the numerical values chosen for the parameters DzlD1, Zlim, and K , the calculated x~(0) and X2(d) are significantly higher than the experimental average (x2)av, i.e., the experimental results are inconsistent with an assumption of local equilibrium a t the SMI's. Kinetic Treatment for the S M I . There is some analogy between a cation crossing (i.e., being reduced at) a solution-metal interface and a cation crossing the solutionmembrane interface. It is therefore suggestive to describe the kinetics of solution-membrane interface crossing by an adaptation of the Butler-Volmer equations which are used in electrode kinetics.7 The solution-membrane interface is, in fact, an electric double-layer several tens of lngstroms thick.13 Let the electric potential drop across this layer be denoted by A$. At equilibrium, when there is no current flow through the system, A+"q is the so-called Donnan potential a t the interface.14 For a cation of species i, there is, a t every moment, a flux Jiin crossing the double layer into the membrane and a flux JioUt in the opposite direction. The net flux at the left-hand side SMI is Jj
-
-
jfin
-
jiOut
(29)
The inward and outward fluxes are expressed as follows Jiin
= k,'"C,(6) exp[aiZiSAt)/,/RT]
(30)
and The Journal of Physical Chemistry, Vol. 78, No. 18, 1974
Y . Oren
9810
JioUtc k,""t;?i(0) exp[- (1 - a i ) Z i S A $ / R T ] (31) where kiin and hLoUtare rate constants, Ci(S) is the concentration in solution at the interface (see Figure 4 and eq 17 and 18), Xi(0) i s the equivalent fraction of species i ions in the membrane a t the interface, and at is the so-called symmetry factor7 which is bounded, 0 5 ai 5 1 (azshould not be confused with a in eq 21). At present there is too little experimental data to allow a calculation of all the parameters in eq 30 and 31 and these equations m e therefore only useful for a qualitative analysis of ionic fluxes (see Discussion). The following calculation shows that, with a proper choice of parameters, eq 30 and 31 are constant with the experimental results, in contrast to the inconsistency of the equilibrium equations, eq 28 (see preceding subsection). Let 01
= ff2
21:
'/2
(32)
d
kl*"qeq CT 6 . 6 3 x
cm sec"
i:2ihq,(12 .= 1 ~ x 3
i o 4 c m sec"
-
= 3.79 x
IO . 7 equiv cm
k20ut~acl- 2
= 5.64 x
10-8
jil0utqsl
-'sec
m i
equiv cm-' s e c - l (33)
where
qw
= exp(SAdP/2RT)
(34)
It is noted i n passing that, for these values, the equilibrium relations (c7iin)w
are satisfied klinq,Cio
zz
=
(35)
(Ji"Ut)ss
-
kjoutqw-i(Xi)ea= 2 . 6 5 -
10-8
X
equiv c m - 2 see-'
k2in~,a2C20 = k~ou1~es-2jX~)ea = 5.24 x
10
-
equiv c m - 2 sec
(36)
and the selectivity coefficient (see eq 2) is given by
With these values for the kinetic parameters and the ionic concentrations in the solutions at the SMI's, calculated for 1 = 22.1 mA cmW2and the experimental tl = 0.25 (with &/Dl = 0.59 and Jlim = 220 mA cmW2),eq 29-31 yield Xz(0) 0.866 anid X z ( d ) = 0.901. In contrast to the preceding calculation (see Table IV), the obtained values for x2(0)and x 2 ( d ) are consistent with the experimental (x2)av = 0.87 (with X z ( 0 ) = ( X Z ) ~ see ", the paragraph following eq 15).If eq 13 and 15 hold for the membrane phase, these values imply D1 = 3.66 X cm2 sec-l and & = 0.85 X cm2 sec-l which are reasonable values for the diffusion coefficients of Na+ and Ca2+ in the membrane (whose water content is approximately 15% by weight). Since I< depends on the composition of the solution (see Table I), hLinand K i O U t depend on the concentra2-
The Journalof PI;ysk:a/ Chemistry, Vol. 78, No. 18, 1974
and A. Litan
tions at the interface and therefore on the current density. As we have no information on this dependency, we have limited ourselves to the calculation of membrane composition a t one current density only, which serves to illustrate the use of the kinetic equations.
Discussion In a cation-exchange membrane placed between two solutions of Na+ and Ca2+, the cation composition and the transport numbers are found to depend on current density (see Table 11). A theoretical analysis of this phenomenon, based on the well-known Nernst unstirred layer model, shows that the experimental results cannot be explained if one makes use of the very common assumption,15 t,hat local thermodynamic equilibrium is maintained at a solution-membrane interface (SMI) even when an electric current i s being passed. There is therefore a deviation from equilibrium at the SMI's which is already borne out a t a current density of 22.1 mA cmd2 and becomes more distinct at higher current densities (see Table IV). It is noteworthy that the above conclusion is reached without requiring the Nernst-Planck equations (eq 3) to hold true in the membrane phase; it is merely required that the ionic composition profile in the membrane be a monotonic function of the distance from a surface (see eq 5). The fact that the transport numbers keep on changing with current density even when the membrane composition has reached a limiting value (see Table 11) points to a possible occurrence of electroosmosis and consequent inapplicability of the Nernst-Planck equations. The conclusion that there is a deviation from equilibrium at the SMI, upon passage of an electric current through the membrane, was reached on the basis of the Nernst unstirred layer model. Since the model is :ipoor representation of the present experimental system, the conclusion is only tentative and indicates the need for a more direct experimental check on the state of the interface. The possibility of deviation from local equilibrium at a SMI has been noted in the past in very few studies.16J7 Degree of Deviation from Equilibrium a t the SMI. The degree of deviation from equilibrium at the $MI, for a given current density, necessarily depends on the kinetics of the processes associated with the interface crossing by the different ionic species. This is somewhat analogous to the dependence of overvoltage on electrode kinetics in the case of a metal electrode in contact with a solution of cations of the same metaL7 The kinetics of SMI crossing has, therefore, been expressed (see eq 30 and 31) j.n the terms commonly used in electrode kinetics. These equations can explain the experimental results, as illustrated in the preceding section. Presently, eq 30 and 31 are shown to yield a measure for the degree of deviation from local equilibrium a t the interface. Local equilibrium at the interface, expressed by eq 28, is but a limiting case of eq 30 and 31 when, for every ionic species i, the net flux Ji is negligibly small compared to both J :n and Jcout. In this case, eq 29 indicates that J F J l o U t and eq 30 and 31 then give
-
which, when written out for both i = 1 (Na') and i = 2
Ion Transport Across an Ion-Exchange Membrane
(Ca2+), yields eq 213 At equilibrium JLln= J,out and this quantity shall be referred to as the "exchange flux," in analogy to th; quart1ity called "exchange current" in electrode kinetics (see eiq 36). The magni ades of the exchange fluxes are measures of the rates ait which the ions cross the interface as a result of the random thermal motion which tends to establish the thermodynamic equilibrium a t the interface. Hence, the physical meaning of the above stated condition for local equilibrium at the interface is that for every ionic species the magnitude of the net, directional flux J,be negligibly small compared to the corresponding "random" exchange flux. Speaking loosely, one says that the equilibrium process is then fast compared to the interfering transport process. If, for even one ionic species i, the net flux is not negligible compared to both Jtnand Jzout, then a substantial deviation from equilibirium at the interface is to be expected. The above analysis is a more exact formulation of Ishibashi'sl qualitative theory of membrane composition changes caused by electric current, As to the mechanism of interface crossing by a cation, which determines the magnitude of the exchange flux, it may be speculated that partial dehydration of the ion upon entering the membrme, and a fluctuation of the membrane skeletal chainq, to alow the passage of an ion in either direction, might be important factors. A Note on Electrical Properties. The electrical resistance along a band of membrane was found to depend on the ionic composition of the membrane (Ca2+ to Na+ ratio). It is expecte therefore, that the membrane composition changes observed upon passage of electric current through the membrane will lead to a nonlinear relation between the current density and the voltage drop across the membrane. Acknowledgments. We are indebted to Professor Ora edem for suqgesting the problem and for her continued interest. We are also very t,hankful to Dr. A. Soffer, from our department, to Dr. C. Forgacs of the Negev Institute for Arid Zone Research, BeerSheva, and 1,0 Professors S. Lifson and I. R, Milo (Miller) of the Weizmann Institute of Science for helpful discussions and suggestions. The technical assistance of Mr. S. Shultz and Mrs. B. Ben-Ami was extremely helpful. The perfect construction of the special cell, by the workshop of the Nuclear Research Center, is gratefully acknowledged.
Agpendix Solution of the Mernst-Planck Equations in the Unstirred Layer." Denoting the three ionic species Na+, Ca2+, and 61- by 1, 2, and 3, respectively, the Nernst-Planck equations in an unstirred layer adjacent to a cation-exchange membrane are, at steady state (for notation, see Theoretical section) - J f i / D , z= dC,/dx (PC, (A1 )
+
1811
0 = d%/dx - @G3
where
@(x)
5
($/RT)d$//dx
The electroneutrality condition is C,(x)
+
C,(X) =
G3(x)
(A4)
(concentrations are expressed in equiuaknts per unit volume). Addition of eq Al, A2, and A3, substitution of CB, according to eq A4, and integration from x = 0 yields 261
+ -32 C,
= w0 - A x
(A5
where uoand A are constants, defined in eq 3.9 and 20. Next, eq A1 and A2 are divided by C1 and @2, respectively. The difference of the resulting equations is multiplied by 2C12/C2giving
Equation A5 can be rewritten as
When this expression is substituted for CI2/C2 in eq A6, there results a differential equation for the ratio r ( x ) = C2(x:)/CI(x).Integration of the equation from x: = O to x: = 6 yields eq 21. References and Notes (1) N. Ishibashi, Kogyo Kagaku Zasshi, 61, 798 (1958). (2)Y. Onoue, Y. Mizutani, R. Yamane, and Y. Takasaki, J. Nectrochem. Sac. Jap., 29, E-155,E-229(1961). (3)H. P. Gregor and D. M. Wetstone, Discuss. Faraday Sac., 21, (1956). (4)P. Meares and A. H. Sutton, J. Colloid hferface Sci., 28, 118 (1968). (5)This estimate is based on Einstein's formula for the displacement of a particle undergoing random thermal motion. (6)F. Helfferich, "Ion Exchange, McGraw-Hili, New York, N. Y.. 1962,p
253. (7)J. O'M. Bockris and A. K. N. Reddy, "Modern Electrochemistry," Vol. 2, Plenum Press, New York, N. Y., 1970,Chapter 8. (8)R. Schloql, Z.fbys. Chem. (Frankfurtam Main), 1, 305 (1954). (9) Reference 6,p 288. (IO) R. A. Robinson and R. H. Stokes, "Electrolyte Solutions," Butterworths, London, 1968,p 463. (11) H. S. Harned and B. B. Owen, "The Physical Chemistry of Electrolyte Solutions," 3rd ed, Reinhold, New York, N.Y., 1958,pp 259 and 260. (12)Reference 6,p 153. (13)A. Mauro, Biophys. J., 2, 179 (1962). (14)Reference 6,pp 134 and 372. (15) See, e.& J. G. Kirkwood in "ion Transport Across Membranes," H. T. Clark, Ed., Academic Press, New York, N. Y., 1954. (16)T. Teorell, frogr. Biophys. Biophys. Cbem., 3, 305 (1953).and references cited therein.
(17)R. McCllntock. R. Neihof, and K . Soilner, J. E/ectrochem.Soc,, 107,315 (1960).
The Journal of Physical Chemistry, Vol. 78, No. 18, 1974
