Salt Flux and Electromotive Force in Concentration Cells with

J. Phys. Chem. 1996, 100, 15261-15273

15261

Salt Flux and Electromotive Force in Concentration Cells with Asymmetric Ion Exchange Membranes and Ideal 2:1 Electrolytes Torben Smith Sørensen*,† Physical Chemistry, Modelling & Thermodynamics, DTH, DK 2720, Vanløse, Denmark

Vicente Compan˜ Departamento de Ciencias Experimentales, UniVersitat Jaume I, E 12071 Castello´ n, Spain ReceiVed: April 2, 1996; In Final Form: July 1, 1996X

The previously proposed Nernst-Planck-Donnan description (J. Phys. Chem. 1996, 100, 7623) of the salt flux and the emf in concentration cells with asymmetric ion exchange membranes is generalized to encompass ideal 2:1 electrolytes (doubly charged cation and singly charged anion). Any point in the membrane may be considered to be in Donnan equilibrium with a given external salt concentration. The profile of this salt concentration through the membrane determines the ion concentrations, the local salt flux, the profile of electric field strength, and the immediate value of the emf. The Donnan potential and ion distribution are found as the unique, positive and real root of a third-degree polynomial. In this paper we focus on the stationary state rather than the initial state. For this purpose, the stationary state nonlinear differential equation for the salt concentration profile is solved numerically by the “shooting method”. We consider salt concentration profiles, ion concentration profiles, and field strength profiles in three different cases: (1) a very weak cation exchange membrane (VWC), (2) a weak anion exchange membrane (WA), and (3) a strong anion exchange membrane (SA). The membranes are asymmetric with spatial dependence of the Nernst distribution coefficient for the salt, of the fixed charge density, and of the ion diffusion coefficients. We study both directions of stationary flow through the membranes. The emf is a functional of the salt concentration profile and is found by numerical integration. The overall behavior of the VWC is almost Fickian with respect to diffusion of salt in both direction, and there is practically no diffusion asymmetry. However, there may be considerable differences in the stationary state emf values for the two directions of diffusion. The WA is close to Fickian for the diffusion in one direction, but strongly non-Fickian with reversed diffusion flux. There is a large diffusion and emf asymmetry for stationary state diffusion in the two directions for large differences of concentration. The order of magnitude found for the calculated stationary state emf asymmetry corresponds to observed values for various membranes. The SA is strongly non-Fickian, but there is practically no diffusion asymmetry. The SA is almost an ideal anion exchange membrane because of the Donnan exclusion from the membrane of the doubly charged cation. Thus, the emf measured with electrodes reversible to the anion should be zero, and so it is found within numerical uncertainty.

Introduction study1

In a previous we have investigated the electrodiffusion of ideal 1:1 electrolytes in asymmetric ion exchange membranes as well as the emf values produced by such diffusion. The membranes were asymmetric with spatial variations of the Nernst distribution coefficients for the electrolyte, of the fixed charge density, and of the ion diffusion coefficients. It was seen that the ion concentrations at a given point in the membrane could always be expressed through a Donnan equilibrium with some (fictitious) external salt concentration. At the membrane interfaces, we assume local Donnan equilibrium with the (real) external solutions. Furthermore, it was shown that the local salt flux is proportional to the negative gradient in the “external” salt concentration profile, with a positive coefficient of proportionality. Therefore, salt diffusion is always in accordance with the second law of thermodynamics (in local formulation) in asymmetric membranes. The trick is to incorporate the variation of the standard chemical potential for the ions in the driving forces for the ion fluxes. With the salt concentration profile found or given, the emf of a concentration cell with given electrodes is found as a functional of this salt concentration †Guest X

professor at Universitat Jaume I until April 1997. Abstract published in AdVance ACS Abstracts, September 1, 1996.

S0022-3654(96)00989-6 CCC: $12.00

profile. Thus all the properties to be measured can be found from one single function, namely, the salt concentration profile in the membrane. This is so when the quasi-electroneutrality condition holds, i.e. for membranes of “macroscopic” dimensions. The intimate relation between the electrodiffusion in asymmetric membranes and the Donnan distribution should be expected to hold not only for 1:1 (or symmetric) electrolytes but for electrolytes of all charge types. However, it will only be in the two-ion case that only one salt concentration profile is enough for the analysis of the system. In the present paper we analyze the behavior of an ideal 2:1 electrolyte in an asymmetric membrane. The calculation of the Donnan distribution is somewhat more complicated than the well-known expressions for a 1:1 electrolyte (cubic instead of quadratic equation, see for example refs 2 and 3), but we shall show here that the method presented in ref 1 can be nicely generalized to this case also. In ref 1, the focus was on the investigation of the validity of the so-called “initial time emf method” developed with the purpose of being able to characterize each membrane phase of asymmetric membranes individually by means of emf measurements.4-6 The salt concentration drop is localized near one or another of the interfaces in this method. If the diffusion zone is narrow enough, the membrane properties may be © 1996 American Chemical Society

15262 J. Phys. Chem., Vol. 100, No. 37, 1996

Sørensen and Compan˜

assumed constant, and the transport number appropriate for this zone may be calculated from the emf data. This conclusion was validated in ref 1, but a case was also treated with a salt concentration profile that was assumed linear through the membrane. In the latter case, the emf is a functional of the salt concentration profile, and a direct calculation of the transport number cannot be performed. However, it is more realistic to consider the profile generated by stationary diffusion. From this profile a value of emf is generated that corresponds to the value that is measured in an experiment with large reservoirs after a sufficiently long time for the emf value to be independent of time. It is of considerable interest to calculate the emf as well as the salt flux for different asymmetric ion exchange membranes and to observe whether or not there is any asymmetry with respect to the direction of the flux. Such calculations for a 2:1 electrolyte (with a doubly charged cation and a singly charged anion) will be the object of the present paper. However, as a preparation, in the subsequent section we shall start out somewhat more generally and demonstrate the uniqueness of the Donnan potential and the Donnan distribution from the rule of Descartes in the case of any electrolyte mixture in electrochemical equilibrium with a membrane phase with a fixed charge density. The Uniqueness of the General Donnan Distribution and Potential In this section we consider a completely arbitrary (but electroneutral) mixture of ions with valencies arranged on a charge axis in the following way:

z1 ) zmin, z2, z3 ... zan, 0, zan+1, zan+2, ... zan+cat ) zmax (1) The number of different anions ) an, and the number of different cations ) cat. The mixture is in electrochemical equilibrium with a phase with fixed charge density ωcq, where ω ) 1 for a positive fixed charge (an anion exchange membrane) and ω ) -1 for a negative fixed charge (cation exchange membrane). If the mixture is considered to be ideally dilute, we may express the equilibrium between the two phases as follows (neglecting any influence of pressure differences):

µ°i + RT ln ci + ziFψ ) µ°i,m + RT ln ci,m + ziFψm (i ) 1, 2 ... an+cat) (2) In eq 2, µ°i is the standard chemical potential per mole of ionic species no. 1, R is the gas constant, T is the absolute temperature, ci is the ion concentration, F is the Faraday constant, and ψ is the electric potential. The subscript “m” means “in the membrane”. Introducing the Donnan potential

∆ψD ) ψm - ψ

(3)

PD(R) ≡ z1c1K1 + z2c2K2R|z1|-|z2| + ... + zancanKanR|z1|-|zan| + ωcqR|z1| + zan+1can+1Kan+1R|z1|+zan+1 + ... + zan+catcan+catKan+catR|zmin|+zmax ) 0 (8) The polynomial PD(R) may be called the Donnan polynomial. It is of degree |zmin| + zmax, and the Donnan potential is given by one (or more) of the real positive roots of this polynomial through eq 4. Having found R, the calculation of the Donnan distribution of the various ions between the phases is straightforward through eq 6. We now proceed to prove that there is always one and only one real positive root of the Donnan polynomial. The proof uses the rule of Descartes: Let N be the number of changes of sign of the coefficients of the polynomial, when we proceed systematically from the left to the right. The number of real, positive roots is then either N or N - 2 or N - 4 ..., and so on, down to 1 or 0. It is obvious from eq 8 that there is only one change of sign. For ω ) 1 this takes place before the term ωcqR|z1| for ω ) -1 after. Thus N ) 1, and there is only one real, positive root for any thinkable Donnan polynomial. The Donnan potential and distribution are uniquely defined and bifurcation into several values is an impossibility despite the nonlinear character of the problem. Donnan Distribution and Potential of a 2:1 Electrolyte Although the direct version of the Donnan polynomial, eq 8, may serve well for theoretical analysis, the polynomial can be simplified in specific situations, and the number of independent dimensionless parameters thus can be reduced.2,7 In the case of a 2:1 electrolyte (with a doubly charged cation and a singly charged anion) we have the following Donnan polynomial equation (denoting the cation as ion no. 1 and the anion as no. 2):

2csK1R3 + ωcqR - 2csK2 ) 0

(9)

We have also used here the bulk electroneutrality condition

cs ) c1 ) c2/2

(10)

where cs is the salt concentration. Introducing the new Donnan parameter β by

β ≡ (K1/K2)1/3R

(11)

the Donnan equation eq 9 is reduced to the simple form

β3 + {ωcq/[2Kscs]}β - 1 ) 0

(12)

Here, Ks is the Nernst distribution coefficient for the whole salt defined as

and the definitions

R ≡ exp(-F∆ψD/RT)

(4)

Ki ≡ exp([µ°i - µ°i,m]/RT)

(5)

the equilibrium conditions (2) can be written as

ci,m/ci ) KiRzi (i ) 1, 2 ... an+cat)

(6)

We have electroneutrality in the membrane:

∑ ziciKiRz + ωcq + 1...cat ∑ ziciKiRz ) 0 i

1...an

Multiplying eq 7 by R|z1|, we obtain

i

(7)

Ks ≡ [K1K22]1/3

(13)

It is apparent from eq 12 that the Donnan potential and distribution depend only on a single dimensionless group, Viz., ωcq/[2Kscs]. The solution of a third-degree algebraic equation has been known for centuries.8,9 It is quite complicated, but somewhat simplified in the present case with a third-degree equation without the term of second degree. Following the procedures given in Bartsch,10 for example, we obtain two cases according to the sign of the discriminant:

Salt Flux and emf in Concentration Cells

J. Phys. Chem., Vol. 100, No. 37, 1996 15263

∆ ) (1/4) + {ωcq/[6Kscs]}3

(14)

Case 1 (one real root+ two complex conjugate roots)

∆ g 0 or ωcq/[Kscs] g -3 × 21/3

(15)

β(real) ) [(1/2) + x∆]1/3 + [(1/2) - x∆]1/3 > 0 (16) Case 2 (three real roots)

∆ < 0 or ωcq/[Kscs] < -3 × 21/3

(17)

(ω ) 1 can never be case 2, so we can put ω ) -1)

β(real, positive) ) 2x[cq/(6Kscs)] cos(V/3)

(18)

where V is found from

cos V ) [3 × 21/3Kscs/cq] < 1

(19)

We notice that anion exchange membranes (ω ) 1) always belong to case 1. Only weak cation exchange membranes (ω ) -1) can be of case 1. Strong cation exchange membranes belong to case 2. In Figure 1 the value of β is plotted Vs the parameter ωcq/[Kscs]. It is seen that there is no discontinuity between the two cases. With increasing strength of an anion exchange membrane, the value of β tends toward 0+, and the Donnan potential tends toward +∞. On the other hand, for strong cation exchange membranes β tends asymptotically toward the curve β* ) x[cq/(2Kscs)]. Thus, β tends toward +∞ and the Donnan potential towards -∞. It is seen from Figure 1 that the convergence toward 0 and β*, respectively, is very slow. For the purpose of the present study, it would be interesting to formulate the Donnan problem even more generally in an inverse way. Given any three values in a point in the membrane c1,m, c2,m, and cq, does there exist an electroneutral external solution with salt concentration cs being in electrochemical equilibrium with the electrolyte at the given point in the membrane? Notice that we have not supposed the membrane to be electroneutral until now. Let us assume an arbitrary ∆ψ ) ψm - ψ. We may then calculate the corresponding value of R, and at electrochemical equilibrium we then have

c1 ) c1,m/[K1R2]; c2 ) c2,mR/K2

(20)

Now, whereas it is possible to imagine a local deviation from electroneutrality of a point in the membrane, it is impossible to imagine that the bulk solution should not be electroneutral. Therefore we have

cs ) c1,m/[K1R2] ) c2,mR/[2K2]

β ) (K1/K2) R ) (2c1,m/c2,m)

1/3

(25)

The quantity on the left-hand side of eq 25 is recognized as the mean ionic activity a(,m for an ideal 2:1 electrolyte in the membrane. The quantity on the right-hand side is the Ks multiplied by the mean ionic activity in the bulk a(,m. The role of Ks is therefore seen to be a distribution coefficient for the salt. We have proven that we can find an electroneutral equilibrium salt concentration for all choices of the ion concentrations and the fixed charge concentrations in the membrane. As a special case we can assume membrane (quasi) electroneutrality, i.e.

c2,m ) 2c1,m + ωcq

(26)

Introducing eq 26 into eq 23, we obtain

β ) (2c1,m/[2c1,m + ωcq])1/3

(27)

(22)

c1,m ) Kscsβ2; c2,m ) 2Kscsβ-1

(23)

Thus, the external salt concentration in electrochemical equilibrium with the (nonelectroneutral) membrane point considered is quite generally given as

cs ) (c1,m)1/3(c2,m)2/3/[22/3Ks]

(c1,m)1/3(c2,m)2/3 ) Ks(cs)1/3(2cs)2/3

This shows that we can find one and only one solution for β for any given values of c1,m and cq. On the other hand, the ion concentrations in the membrane can be found from eq 6 or from the equivalent equations

or 1/3

which is equivalent to

(21)

This is only possible when we have

R ) (2K2/K1)1/3(c1,m/c2,m)1/3 > 0

Figure 1. Donnan parameter β as a function of the dimensionless fixed charge (the ion exchange strength). The line corresponds to case 1 for the ion exchange strength > -3 × 21/3 and to case 2 for the ion exchange strength < -3 × 21/3. The diamonds correspond to β* ) x[cq/(2Kscs)].

(24)

(28)

Inserting the first of these expressions in eq 27, we easily rederive the third-degree Donnan equation, eq 12. Local Salt Flux in the Membrane As in ref 1, we consider the one-dimensional Nernst-Planck equations for a region with space dependent parameters. The equations take into account the variation of the Nernst distribution coefficients for the ion as a “driving force”:

15264 J. Phys. Chem., Vol. 100, No. 37, 1996


J1(x)/D1(x) ) c1,m(d ln K1/dx) - ∂c1,m/∂x - 2c1,m ∂φm/∂x (29) J2(x)/D2(x) ) c2,m(d ln K2/dx) - ∂c2,m/∂x + c2,m ∂φm/∂x (30) In eqs 29 and 30, J1(x) and J2(x) are the fluxes of the doubly charged cation and the singly charged anion, respectively, as a function of the distance through the membrane (x). Similarly, D1(x) and D2(x) are the spatial dependent diffusion coefficients for the two ions. The first terms on the right-hand sides of eqs 29 and 30 simply state that the diffusion of ions need not be zero even in situations with no concentration gradients and no electric field. When the Nernst distribution coefficient increases in one direction, a uniformly distributed ion will tend to move toward positions with higher Nernst distribution coefficients. The dimensionless potential in the membrane is defined by

φm(x) ≡ (F/RT)ψm(x)

(31)

The fixed charge density cq(x) is also a function of the distance, but this quantity does not appear explicitly in the Nernst-Planck eqs 29 and 30. However, we may ask for the salt concentration cs(x) in an “external” solution in electrochemical equilibrium with the membrane at the coordinate x. In the last section we saw that such a salt concentration is always uniquely defined, and since we assume quasi-electroneutrality, the Donnan formalism applies. Note: This does not mean that the membrane is in overall equilibrium. On the contrary, we have diffusion going on! From eq 28 we now have

c1,m(x) ) Ks(x) cs(x) β(x)2; c2,m(x) ) 2Ks(x) cs(x) β(x)-1 (32) These expressions may be inserted into eqs 29 and 30. Using the relations for the salt flux (Js)

Js(x) ) J1(x) ) J2(x)/2

(33)

3/Ds ) (1/D1) + (2/D2)

(39)

which is the classical expression for the salt diffusion coefficient in a 2:1 electrolyte. Electric Field in the Membrane and emf Alternatively we may rewrite eqs 29 and 30 in terms of the salt flux using eq 33 and express the salt flux through eqs 36 and 37. We obtain

-∂ ln(c1,m/K1)/∂x - 2 ∂φm/∂x ) -{Ks(x) cs(x) Ds(x)/[D1(x) c1,m]} ∂ ln cs/∂x -∂ ln (c2,m/K2)/∂x + ∂φm/∂x ) -{2Ks(x) cs(x) Ds(x)/[D2(x)c2,m]} ∂ ln cs/∂x Subtracting the latter equation from the former and isolating -∂φm/∂x, we obtain after the use of eq 28

-∂ φm/∂x ) -(1/3)∂[ln(c2,mK1/{c1,mK2})]/∂x + [{1/[2 + D21(x) β-3(x)]} - {1/[1 + 2D12(x)β3(x)]}] ∂ ln cs/∂x (40) Dij(x) ≡ Di(x)/Dj(x)

(41)

The observable variable is not the local electric field strength but the emf over the membrane measured by specific electrodes. Therefore we integrate ∂φm/∂x over the membrane to get the membrane diffusion potential, and then we add the two Donnan potentials at the interfaces and the two Nernst electrode potentials. For the present purpose we derive an alternative equation for the Donnan potential in a form identical to the first term on the right-hand side of eq 40. The electrochemical equilibrium of the two ions at the two interfaces is expressed as follows:

2∆φD ) ln K1 - ln(c1,m/cs); ∆φD ) -ln K2 - ln(c2,m/cs) Adding these two equations, we obtain

the transformed equations may be written as

Js(x)/[D1(x) Ks(x) cs(x) β(x)2] ) d ln K1/dx -d ln Ks/dx ∂ ln cs/∂x - 2 ∂ ln β/∂x - 2 ∂φm/∂x (34) Js(x) β(x)/[D2(x) Ks(x) cs(x)] ) d ln K2/dx -d ln Ks/dx ∂ ln cs/∂x + ∂ ln β/∂x + ∂φm/∂x (35) Eliminating ∂φm/∂x from eqs 34 and 35, we obtain as a final result for the local salt flux

Js(x) ) -Ds(x) Ks(x) ∂cs/∂x

(37)

Since β(x) is always positivesand the ion diffusion coefficients tooswe see that

Ds(x) > 0 (all x)

An equation with the same form may be established between the Donnan potential at a given internal point in the membrane and the corresponding “external” solution:

∆φD(x) ) (1/3)[ln(c2,mK1(x)/{c1,mK2(x)})]

(43)

Now it is clear that if we integrate

-(1/3)∂[ln(c2,mK1(x)/{c1,mK2(x)})]/∂x

(36)

where the local salt diffusion coefficient has been defined as

Ds(x) ≡ 3{[β(x)2 D1(x)]-1 + 2β(x)/D2(x)}-1

∆φD ) (1/3)[ln(c2,mK1/{c1,mK2})] (at interfaces) (42)

(38)

and the local salt flux will always be in the direction of decreasing “external” salt concentration. Furthermore, in a system with no fixed charge and no spatial variation we have that β ) 1 (see Figure 1), and therefore

over the membrane, we get the total contribution from the two interfacial Donnan potentials to the emf. Therefore the above term has to be added to ∂φ/∂x and then integrated to obtain the potential difference ∆φ between the right-hand bulk solution and the left-hand bulk solution:

∆φ(t) ) φ(right) - φ(left) ) ∫x)0 G(c1,m,c2,m,x) dx (44) x)d

G(c1,m,c2,m,x) ≡ [{1/[1 + 2D12(x) β3(x)]} {1/[2 + D21(x) β-3(x)]}] ∂ ln cs/∂x (45) It is seen that in general the potential difference ∆φ is a function of time (t), since it is found by spatial integration of a


J. Phys. Chem., Vol. 100, No. 37, 1996 15265

Figure 2. Visual description of the linear profiles of the membrane parameters for the three membranes considered and the two membrane orientations. The salt flux is always directed from left to right. The calibration diffusion coefficient D0 is always taken to be the diffusion coefficient of the divalent cation in the middle of the membrane, D1(y)1/2).

functional G of the ion concentration profiles which are in general time dependent. In the stationary stateswhich is the only state investigated in this papersthe ∆φ becomes indepen-

dent of time, but it is still a functional of the stationary ion concentration profiles. Now, the emf measured with electrodes reversible to the anion (for example Ag/AgCl electrodes for

15266 J. Phys. Chem., Vol. 100, No. 37, 1996


Cl-) is the sum of the ∆ψ and the difference between the two ideal Nernst electrode potentials, or in dimensionless form

(F/RT)emf(t) ) ∆φ(t) + ln(cs,L/cs,R)

(46)

Calculation of the Stationary State Concentration Profiles We calculate the salt concentration profile in the stationary state numerically using eqs 36 and 37 in the dimensionless form

dC/dy ) H(C,y)J

(47)

The dimensionless distance through the membrane with thickness d is defined as

y ≡ x/d

(48)

and the dimensionless salt concentration at position y as

C(y) ≡ cs(y)/cs,L

(49)

where cs,L is the external salt concentration in the left-hand electrode chamber. The dimensionless salt flux is defined as

J ≡ Jsd/(D0cs,L)

(50)

In eq 50 the “calibration” diffusion coefficient D0 is taken as

D0 ≡ D1(y)1/2)

(51)

Finally, in eq 47 the dimensionless function H(C,y) is defined as

H(C,y) ≡ -1/[Ks(y) ds(C,y)]

(52)

ds(y) ≡ 3{[β2(C,y) D10(y)]-1 + [2β(C,y)/D20(y)]}-1 (53) Di0(y) ≡ Di(y)/D0

(54)

To integrate the nonlinear, first-order ordinary differential equation eq 47, we have used the method of “shooting” starting always with a dimensionless salt concentration CL ) 1 at y ) 0 and selecting for each “shot” a value for the dimensionless salt flux J, until the desired value for CR at y ) 1 is arrived at. For the integration two fourth-order Runge-Kutta procedures were selected11 with N ) 40 and 60 intervals. We have also tried with Runge-Kutta methods of second and third order and a smaller number of intervals. In all cases the two different methodssreferred to as method A and method Bsstated in ref 11 were used. One clear sign of an insufficient number of intervals or too low an order is that the two methods differ by more than 1-2% for the CR. Shortly “after” such a deviation is seen (varying the J), method A (for fourth order, section 25.5.10 in ref 11) breaks down somewhat “before” method B, giving for example negative or complex concentrations. Generally, the results of method B (for fourth order, section 25.5.11 in ref 11) should be preferred, but the result given by method B is only approximately correct when methods A and B coincide within a few percent. For numerical differentiation (calculation of the electric field) we have used the equidistant five-point ordinate weighting method based on local least squares polynomials of second order; see the monograph of Lanczos, ref 12, Chapter V, section 8. For the numerical integration (calculation of the G-integral and the emf), the trapezoidal rule was generally found too crude with 40 and 60 intervals. However, the trapezoidal rule with end corrections (TE) (see ref 12, Chapter VI, section 7), the Simpson rule (S) (ref 26, Chapter VI, section 4), and the

Figure 3. Some nondimensional “external” salt concentration profiles for a very weak cation exchange membrane in the “normal” direction. The values of the nondimensional salt fluxes (J) are indicated.

Simpson rule with end corrections (SE) (ref 12, Chapter VI, section 17) are generally quite satisfactory (at least for 60 intervals). The precision increases in the direction mentioned. According to Lanczos (p 390), the estimated error of the TE method is one-fourth of the estimated error of the Simpson rule and of opposite sign. Indeed, the formula (6-17.9) in ref 26 for the SE quadrature may alternatively be derived by taking 20% Simpson and 80% TE. For the end corrections, the derivative of the integrand G is needed in the end points. For these the numerical four-point formula (5-8.6) in ref 26 for y ) 0 and its analogue for y ) 1 have been used. For the strong anion exchange membrane the G-integrals using the three different methods of integration are quite close to each other, but since emf is calculated as a difference between this integral and an almost canceling electrode potential difference, there is considerable uncertainty with respect to the precise values of emf in that case. However, the emf values are trivially close to zero from an experimental point of view except for the highest fluxes in the reverse direction. In the present study we shall consider three examples of asymmetric ion exchange membranes: (1) a very weak cation exchange membrane (VWC), (2) a weak anion exchange membrane (WA), and (3) a strong anion exchange membrane (SA). The words “weak” and “strong” refer to the magnitude of the absolute value of the fixed charge density (in equivalents per volume) relative to the external salt concentration. In a strong ion exchange membrane this ratio has a high value. The membranes studied have linear variations of the parameters D1, D2, Ks, and cq as shown in Figure 2. The membranes are studied in a “normal” position relative to the salt flux and in the “reverse” position, as is also shown in Figure 2. Example 1: An Asymmetric Very Weak Cation Exchange Membrane Figure 3 shows the dimensionless salt concentration profile in the VWC membrane for three different values of J. As J increases, the right-hand side concentrations hit by the “shot” decrease in value, of course. The direction of diffusion is from left to right, and the membrane is in the “normal” position. Figure 4 shows the salt concentration profiles for the same three values of the dimensionless flux with the membrane in “reverse” position. Whereas in Figure 3 the steeper part of the concentration profile is positioned in the left half of the membrane, in Figure 4 it is positioned in the right half of the membrane.


J. Phys. Chem., Vol. 100, No. 37, 1996 15267

Figure 4. Same as in Figure 3, but with “reverse” membrane orientation.

Figure 6. Some dimensionless electric field profiles for a VWC membrane in “normal” direction. Rectangles on line: Electric field for J ) 0.004 207 4. Rectangles without line: Electric field for J ) 0.002 631 2. Line without symbols: Equilibrium (J ) 0).

Figure 5. Some ion concentration profiles in the VWC membrane with “normal” orientation. Rectangles on line: Cation (++) for J ) 0.004 207 4. Diamonds on line: Anion (-) for J ) 0.004 207 4. Rectangles without line: Cation for J ) 0.002 631 2. Diamonds without line: Anion for J ) 0.002 631 2. Lines without symbols: Cation and anion at equilibrium (J ) 0).

Figure 7. Dimensionless salt flux Vs dimensionless salt concentration drop for a VWC membrane.

From the salt concentration profile, the ion concentration profiles can be calculated. Figure 5 shows these ion profiles for two different fluxes and for membrane equilibrium (J ) 0). The curves fall in two groups of three curves. The two groups correspond to cations (rectangles) and anions (diamonds). The lines correspond to equilibrium profiles. We observe that when the salt flux diminishes in value (from symbols with lines to symbols without lines), the ion profiles approach the equilibrium profiles. The reason for the existence of equilibrium profiles for the two ions is to be found in the spatial variation of the membrane parameters K1 and K2. In Figure 6 some dimensionless electric field strength distributions are shown as a function of the position in the membrane. The field strength is calculated from eq 40. Notice that to perform the calculation, we have to make assumptions concerning the Nernst distribution coefficients K1(y) and K2(y) for the ions individually. In Figure 6 (as well as in the other

figures showing the field distribution), we have assumed

d ln K1/dy ) d ln K2/dy ) d ln Ks/dy

(55)

This corresponds to the assumption that the ratio K1(y)/K2(y) is fixed and independent of y. However, to calculate the salt concentration profile, the ion concentration profiles, the salt flux, and the emf of a concentration cell, it is not necessary to know anything else than Ks(y), so these calculations are independent of the assumption eq 55. The curves in Figure 6 correspond to the same situations as in Figures 3 and 5 (normal direction of membrane). The rectangles on a line correspond to J ) 0.004 207 4 and the rectangles without a line to J ) 0.002 631 2. The latter is closer to the equilibrium field distribution (solid curve without symbols). Figure 7 exhibits the dimensionless salt flux as a function of the dimensionless concentration drop over the membrane. The overall diffusion is almost Fickian (J proportional to -∆cs/cs,L), although there is a very slight increase in the slope with increasing concentration drop. It is noteworthy that the salt

15268 J. Phys. Chem., Vol. 100, No. 37, 1996


Figure 9. Some dimensionless salt concentration profiles for a weak anion exchange membrane with “normal” direction. Figure 8. Dimensionless values of emf Vs the dimensionless “driving force” for a VWC membrane. Rectangles: “Normal” orientation. Diamonds: “Reverse” orientation. The two lines have slopes 1 and 1.13, respectively. See text.

In eq 57, t+ is some “average” transport number of the cation in the (uncharged) membrane given by the usual expression

t+(2:1) ) 2D1,av/[2D1,av + D2,av] ) 2(D1,av/D0)/[2(D1,av/D0) + (D2,av/D0)] (58)

diffusion flux is the same without regard to the orientation of the highly asymmetric membrane. The concentration profiles and the field strength distribution are very different for the “normal” and the “reverse” membrane orientation, however. The constant of proportionality between J and -∆cs/cs,L is very close to 5 × 10-3. Consequently, we have

Js(VWC,stat) ≈ -0.005(D0/d)∆cs

(56)

Thus, the “efficient” value of Ks for overall diffusion in the membrane is 0.005, and this corresponds to an efficient overall salt diffusion coefficient taken as the diffusion coefficient of the doubly charged cation in the middle of the membrane. It is seen that the interpretation of the membrane behavior behind the innocent looking “Fickian diffusion” is not possible without detailed calculations. Whereas no asymmetry could be discerned for the salt flux, there definitely is an asymmetry in the emf results; see Figure 8. The emf asymmetry between the two membrane orientations increases with increasing “driving force” ln(cs,L/cs,R). For a driving force equal to 3, the dimensionless emf asymmetry amounts to ca. 0.4, corresponding to 10 mV at 25 °C. The membranes with “reverse” direction have increasingly lower values of emf as the driving force increases. For comparison, two straight lines have been inserted in Figure 8, the first with slope 1 and the other with slope 1.13. The first one is approximately the asymptotic line to which the emf values approach for small driving forces. It could be called the “Onsager line”, because it corresponds to the “global linearization” intended in the original approach to the nonequilibrium thermodynamics of discrete systems put forward by Onsager.13,14 Of course, a membrane asymmetry will not show up in this approach since the transport properties are averaged over the membrane. However, the Onsager approach is valid only up to chemical potential differences of ∼0.8RT, as seen in Figure 8. (Locally, the Onsager linearity is completely valid in this Nernst-Planck model and the Onsager reciprocity relations are valid, since the cross coefficients are zero). If we assume that the VWC membrane is virtually uncharged and has constant transport numbers throughout the membrane, we obtain as a crude approximation for the emf

(F/RT)emf ≈ (3t+/2) ln(cs,L/cs,R)

(57)

The Onsager line then corresponds to t+ ≈ 2/3 and D2,av/D1,av ) 1. In the membrane the values of D2(y)/D1(y) vary between 0.5263 and 3, and 1 is in this interval, but it would be difficult to argue for precisely this value without performing the detailed calculations made in this paper. Alternatively, we may insert the mean values of (D1/D0) and (D2/D0) for the values (D1,av/ D0) and (D2,av/D0) in eq 58. These mean values are equal to 1 and 0.65, respectively. From these values the average transport number is evaluated as 0.755, from which follows a slope in Figure 8 ≈ 1.13. The real emf values seem to lie between the two straight lines for the higher values of the driving force. Example 2: An Asymmetric Weak Anion Exchange Membrane Figure 9 shows some salt concentration profiles for the WA membrane in the “normal” direction. For the same concentration differences the fluxes are not very much different from the ones observed for a VWC membrane (see Figure 3), and the profiles are similar, too. However, the absolute values of the fixed charge densities for the WA membrane are 10 times the values for the VWC membrane. The Donnan exclusion of the doubly charged cation in an anion exchange membrane should be much more pronounced than the corresponding exclusion of the singly charged anion in a cation exchange membrane with the same absolute value of the fixed charge density. Since the difference in the fluxes is only slight, it must mean that the Donnan effects are still weak in the WA membrane in the “normal” position. Figure 10 shows some examples of salt concentration profiles for the WA membrane in “reverse position”. Compared to the similar Figure 4 for a VWC membrane, the fluxes are of the same order of magnitude, but the fluxes are somewhat less for the same concentration differences for the larger values of these differences. Figure 11 shows the ion concentration profiles in a “normal” WA membrane for three different fluxes. For the lowest flux (lines without symbols) the ion concentration profiles are close to the ones for equilibrium, cf. the corresponding ones for a VWC membrane in Figure 5. For the highest flux (symbols on lines) the anion concentration profile (diamonds on lines) is completely reverted compared to equilibrium, whereas the cation concentration profile (rectangles on line) has an inverse slope


J. Phys. Chem., Vol. 100, No. 37, 1996 15269

Figure 10. Same as Figure 9, but with “reverse” orientation of the membrane.

Figure 12. Some dimensionless electric field profiles for a WA membrane in the “reverse” direction. Rectangles on line: Electric field for J ) 0.001 813. Diamonds on line: Electric field for J ) 0.001 370. Crosses on line: Electric field for J ) 0.000 462.

Figure 11. Some ion concentration profiles in the WA membrane with “normal” orientation. Rectangles on line: Cation (++) for J ) 0.003 771 5. Diamonds on line: Anion (-) for J ) 0.003 771 5. Rectangles without line: Cation for J ) 0.002 667 5. Diamonds without line: Anion for J ) 0.002 667 5. Lines without symbols: Cation and anion for J ) 0.000 231 5.

only in the left-hand side of the membrane. Compared to Figure 5 for a VWC the equilibrium Donnan exclusion of the doubly charged cation (rectangles) is approximately doubled, whereas the increase in the concentration of the singly charged anion (diamonds) is increased by ca. 50% in the left side of the membrane. Some examples of electric field strength distributions for the WA membrane are shown in Figure 12, but this time for the “reverse” orientation. In the left-hand side of the membrane the field strengths tend toward zero, but there is no sign reversal as in a VWC membrane (Figure 6). The leftmost part is not shown to see more clearly what happens in the right end of the membrane. Here, with increasing flux, a minimum in the negative field strength is produced, followed by a sharp increase resulting in a sign reversal in the very rightmost part of the membrane. This might seem to be numerical difficulties, but it is not. The reason is to be sought in the fact that the field strength is calculated by eq 40 as a difference between a “Donnan term” and the negative of the emf integrand -G(c1,m,c2,m,y). The first term decreases sharply when the righthand interface is approached, whereas the second term increases even more sharply very close to the right interface. This overcompensates the decrease of the “Donnan term”. Since the

Figure 13. Dimensionless salt flux Vs dimensionless salt concentration drop for a WA membrane.

charge density is related to the derivative of the field strength through the equation of Poisson, we see that a dynamical electric double layer is created by diffusion near the right interface in “reverse” WA membranes. Exactly the same is seen in “reverse” VWC membranes and in “reverse” SA membranes. In the latter case the field variations are extreme at the higher fluxes. (In ref 1 we saw that, for membranes of macroscopic dimensions, the charge densities calculated by the differentiation of the field strength are so small that they have no influence on the assumption made concerning quasi-electroneutrality). “Dynamic double layers” induced by electrodiffusion are also of importance for measurements of impedance and complex dielectric permittivity in membranes and polymer films.15-21 Figure 13 shows that the Donnan effects are indeed visible for the salt fluxes through the WA membrane compared to the VWC membrane (Figure 7). The curvature is downward and not slightly upward. Fickian behavior is limited to small concentration differences, where the salt fluxes in the two directions coincide. Now, there is a marked asymmetry in the salt flux. The salt flux is much smaller in the “reverse direction” than in the “normal direction” at higher values of the concentration differences. Indeed, the salt flux in the reverse direction seems to saturate at a maximum salt flux. The asymmetric membrane functions as a kind of “diffusional diode”. The reason is that, in the reverse, direction, the membrane face with

15270 J. Phys. Chem., Vol. 100, No. 37, 1996

Figure 14. Dimensionless values of emf Vs the dimensionless “driving force” for a WA membrane. Rectangles: “Normal” orientation. Diamonds: “Reverse” orientation.

the high charge density turns toward the low-concentration side. Then, the effective ion exchange strength is much stronger, and the Donnan exclusion of one ion is much more pronounced in the rightmost part of the “reverse” membrane. This tends to block the salt diffusion and also creates great gradients in the emf integrand and in the electric field. Therefore, we would also expect a very great asymmetry in the emf values, and this is indeed found, as seen in Figure 14. The emf in the “normal” direction is crudely proportional to the driving force as in Figure 8 for a VWC membrane, but the curvature is greater and opposite (downward). Indeed, if the membranes had been homogeneous, the slope of the emf Vs driving force curve is equal to the transport number of the cation at the variable salt concentration at the right-hand side. This salt concentration decreases for increasing salt flux and driving force. Thus, the ratio between the fixed charge density and the salt concentration increases with increasing driving force at the right interface, and so does the Donnan exclusion of the cation. Therefore, the transport number of the cation at the right-hand interface diminishes with increasing driving force, and the curvature of the emf/driving force curve has to be downward. Exactly the same argumentation can be performed for a homogeneous cation exchange membrane. Therefore, if an upward curvature is seensas in Figure 8 for a VWC membranesthe curvature is due to membrane asymmetry and not to homogeneous Donnan effects. The downward curvature is much more pronounced for a WA membrane in the “reverse direction”; see Figure 14. This is clear, since in this case the membrane face with the higher value of the fixed charge turns toward the low external salt concentration. Therefore, the emf values are much lower with the WA membrane in the “reverse” direction than the corresponding emf values for the “normal” direction. The emf asymmetry of the WA membrane is very pronounced, indeed. Example 3: An Asymmetric Strong Anion Exchange Membrane In Figure 15 some salt concentration profiles are shown in a SA membrane. Compared to Figure 3 for a VWC membrane and Figure 4 for a WA membrane, we notice in Figure 15 the extended regions in the right three-quarters of the membrane with virtually constant salt concentration. This part is in diffusional equilibrium. All the production of entropy will take place in the leftmost part of the membrane, where the salt concentration profiles are steep. In contrast for the reverse SA membranesFigure 16sthe concentration drop is located in an even more narrow zone in


Figure 15. Some dimensionless salt concentration profiles for a strong anion exchange membrane with “normal” direction.

Figure 16. Same as Figure 15, but with “reverse” orientation of the membrane.

Figure 17. Some ion concentration profiles in the SA membrane with “normal” orientation. Rectangles on line (lower): Cation (++) for J ) 9.9781 × 10-7. Rectangles without line: Cation for J ) 8.8012 × 10-7. Rectangles on line (upper): Cation for J ) 1.434 × 10-7. Line without symbols: Cation at equilibrium (J ) 0). Diamonds on line: Anion (-) in all the above cases.

the outer rightmost part of the membrane. We notice that the order of magnitude of the salt fluxes in Figures 15 and 16 is a factor of 104 less than for the two previous membranes. Figure 17 shows some ion concentration profiles in the membrane with “normal” orientation. The anion concentration profiles (diamonds) are all on the equilibrium curve without


Figure 18. Some ion concentration profiles in the SA membrane with “reverse” orientation. Rectangles on line: Cation (++) for J ) 9.787 × 10-7. Rectangles without line: Cation for J ) 6.608 × 10-7. Line without symbols: Cation for J ) 1.434 × 10-7. Diamonds on line: Anion (-) in all the above cases.

regard to the salt flux. The concentration profiles of the doubly charged cation (rectangles) are generally many orders of magnitude less than the anion concentration. They deviate very much from the equilibrium line (without symbols) for the higher values of the salt flux (even if those salt fluxes are very small compared to the other cases). The SA membrane is very nearly an ideal anion exchange membrane with almost complete Donnan exclusion of the cation and with the anion in perfect electrochemical equilibrium everywhere. Figure 18 exhibits the ion profiles with “reverse” orientation of the SA membrane. Also in this case, the anion (diamonds) is in perfect electrochemical equilibrium for all the salt fluxes studied. In addition, the cation (rectangles) is also in electrochemical equilibrium (line without symbols) for the salt fluxes, except in the outer rightmost part, where the salt concentration drop is located. In Figure 19 the field strength profile is shown for the “normal” SA membrane. There is no field reversal, but there is a minimum field strength approximately in the middle of the membrane. This corresponds to an electric double layer extending over the whole membrane. The electric field strengths (and so the double layer) are very close to the equilibrium field distribution and some deviation for the higher fluxes in the rightmost part of the membrane. The overall diffusion is highly non-Fickian; see Figure 20. It is remarkable, thatswith the same difference in salt concentrationsthere is no difference in salt flux with the membrane in “normal” or “reverse” position, even though the salt and ion concentration profiles are very different. The salt fluxes are very small. The low values indicate that diffusion of the doubly charged cation is the rate-limiting step. The “effective” Ks value for overall diffusion is therefore related to the smallest concentration of this ion in the membrane. The emf of the SA membrane is extremely close to zero, except for the higher flows in the reverse direction. The emf is determined by numerical integration, and the addition of the electrode potentials (with electrodes reversible to the anions) virtually cancels the values. A precise determination of the very small (and in practice uninteresting) emf values would require a much greater numeric precision than has been used. That the emf should be very close to zero is seen very easily in the following way: An ideal anion exchange membrane lets

J. Phys. Chem., Vol. 100, No. 37, 1996 15271

Figure 19. Some dimensionless electric field profiles for a SA membrane in the “normal” direction. Rectangles on line: Electric field for J ) 9.9781 × 10-7 and J ) 8.8012 × 10-7. Line without symbols: Electric field at equilibrium and for J ) 1.434 × 10-7.

Figure 20. Dimensionless salt flux Vs dimensionless salt concentration drop for a SA membrane.

pass only anions through the membrane. Sending a (virtual) Faraday through the membrane, 1 mol of anions pass the membrane from the right electrode chamber to the left electrode chamber. However, simultaneously 1 mol of anions is absorbed at the left electrode and produced at the right. Thus, chemically nothing happens, and the changes in Gibbs’ free energy ∆G for this process is zero. Consequently the emf is zero even for different salt concentrations in the two chambers. The fact that we find zero emf in the quite complex calculations performed can be seen as a successful test of the consistency of the model. However, a strictly ideal anion exchange membrane has zero salt flux, but in the present case the salt fluxes can be evaluated exactly. They are small, but not zero. Conclusions and Discussion We have demonstrated the same close connection between the Donnan formalism and the salt transport and emf for a 2:1 electrolyte in asymmetric ion exchange membranes as was shown previously for a 1:1 electrolyte.1 Instead of solving an equation of the second degree, in the 2:1 case we have to solve an equation of the third degree for each point in the membrane.

15272 J. Phys. Chem., Vol. 100, No. 37, 1996 There is only one real, positive root of this equation from which the local Donnan potential and the local Donnan distribution of ions relative to a fictitious “external” salt solution is determined. It is demonstrated that for different mixtures of ions of different valencies, the degree of the equation to be solved is varying, but there is always only one positive real root which is related to the Donnan potential. In passing we notice that the Donnan distribution may be given by the root of a third-degree polynomial even in 1:1 electrolyte mixtures, if one ion titrates the group carrying the fixed charge, for example when H+ titrates the glucuronic acid groups in cellulose acetate; see ref 17. The fictitious external salt concentration is the “driving force” for the salt diffusion and also determines the electric field distribution, the emf, and the ion concentration profiles in the membrane. At the two membrane/solution interfaces the fictitious salt concentration coincides with the real external salt concentrations, and the Donnan potential is the real interfacial Donnan potential. We have studied asymmetric membranes in a stationary state of diffusion, the membranes having spatial variation of the fixed charge density, the salt distribution coefficient, and the two ion diffusion coefficients. The information contained in these four functions of the position is sufficient for the calculation of the distribution of ions, the salt flux through the membrane, and the emf of a concentration cell with the membrane as a diaphragm. Solely for calculating the distribution of the electric field strength in the membrane is it necessary to know the individual distribution coefficients for the two ions. The electric field strengths in the present paper are calculated under the assumption that there is a constant ratio between these individual distribution coefficients throughout the membrane. Three specific asymmetric membranes have been studied: a very weak cation exchange membrane (VWC), a weak anion exchange membrane (WA), and a strong anion exchange membrane (SA). For the same concentration differences the salt flux in one direction and in the reverse are practically identical for the VWC and the SA membranes. For the WA membrane there is considerable flux asymmetry, however. The flux direction, in which the low, external salt concentration is facing the membrane face with the higher fixed charge, is generally called the “reverse” direction of diffusion. The salt fluxes in this direction are lower than in the “normal” direction for the WA membrane because of the increased Donnan exclusion of the divalent cation in the former case. The salt flux through the SA membrane is independent of direction, and the fluxes have very low values. The rate-limiting step in the SA membrane is the diffusion of the Donnan excluded cation, and it is only the cation concentration profile that deviates significantly from electrochemical equilibrium for the higher salt fluxes. In the “normal” direction these deviations appear throughout the membrane, whereas the deviations are located near the low concentration membrane face in the “reverse” direction. The overall membrane diffusion is almost Fickian in the VWC membrane, but non-Fickian in the two other membranes. The emf’s of concentration cells with the membranes as separators and electrodes reversible to the monovalent anion are calculated as a function of the “driving force” (the difference in the chemical potential of the salt at the two sides of the membrane). For the VWC membrane there is almost proportionality between the emf and the driving force, but the apparent transport number of the cation calculated from the slope does not precisely correspond to the mean values for the membrane. In fact there is a slight increase in the transport number with

Sørensen and Compan˜ increasing driving force, and this is a phenomenon that can only be produced by the asymmetry, not by the ion exchange properties. Also, there is a considerable asymmetry in emf between the two directions of salt flux through the VWC membrane. For the WA membrane, this emf asymmetry is even larger. The emf measured with the SA membrane as a separator is found to be very close to zero (except for high values of the driving force in the reverse direction). This is so, since the almost complete Donnan exclusion of the divalent cation makes the transport number of the cation virtually zero. Thus, using the classical procedure of Helmholtz22 sending a (virtual) Faraday through the cell, no chemical changes will be noticed, and the emf’s have to be zero for an ideal anion exchange membrane. The emf steady state asymmetry up to ca. 10 mV at 25 °C found for the VWC membrane corresponds very well to the values found experimentally for cellulose acetate membranes; see for example ref 2, Figures 1 and 2, and ref 23, Figure 1. In ref 24 the stationary state emf asymmetries were found to be up to 7 mV for various cellulose acetate membranes deliberately cast as asymmetric. Cellulose acetate membranes are weak cation exchangers due to the dissociated glucuronic acid groups. For a phenol sulfonic acid cation exchange membrane, a stationary state emf asymmetry of ca. 2 mV was found in ref 6. Many membranes designed for water desalination of purification seems to act by the principle of ion exclusion by a dense, polymeric layer with a low dielectric constant, and ions of higher valencies are excluded to a much higher degree than monovalent ions, because the free energy of charging of the ions depends quadratically on the ion charge and is inversely proportional to the dielectric permittivity of the medium.25,26 The present paper suggests that the fixed charge can be used as another mechanism for retention of ions, especially those of higher valencies. In cellulose acetate membranes for reverse osmosis for example, a compromise is made to produce the highest possible salt rejection in the dense skin layer without lowering the water permeability too much; see the articles by Merten, Loeb, and Lonsdale in ref 27. The addition of additional fixed charge in the skin layer of desalination membranes might considerably shift the position of this compromise. Acknowledgment. The authors are grateful for the financial support provided by the Comisioń Interministerial de Ciencia y Tecnologıá, DGCYT (Ministry of Education and Science of Spain), under project PB92-0773-C03. One of the authors, T.S.S., is indebted to the Ministry of Education and Science for a grant for a sabbatical year stay at Universitat Jaume I of Castelloń, Spain. References and Notes (1) Sørensen, T. S.; Compan˜, V. J. Phys. Chem. 1996, 100, 7623. (2) Sørensen, T. S.; Jensen, J. B.; Malmgren-Hansen, B. J. Non-Equilib. Thermodyn. 1988, 13, 57. (3) Sørensen, T. S.; Rivera, S. R. Mol. Simul. 1995, 15, 79. (4) Garrido, J.; Compan˜, V. J. Phys. Chem. 1992, 96, 2721. (5) Compan˜, V.; Lo´pez, M. L.; Sørensen, T. S.; Garrido, J. J. Phys. Chem. 1994, 98, 9013. (6) Compan˜, V.; Sørensen, T. S.; Rivera, S. R. J. Phys. Chem. 1995, 99, 12553. (7) Sørensen, T. S.; Jensen, J. B. J. Non-Equilib. Thermodyn. 1984, 9, 1. (8) Cardano, G. Artis magnae siVe de regulis algebraicis; Treatise; Nu¨rnberg, 1545. (9) Vie`te, F. De emendatione; Treatise; 1615 (1591); Chapter VI. (10) Bartsch, H.-J. Matematische Formeln; VEB Fachbuchverlag: Leipzig, 1974; section 2.1.2.3.

Salt Flux and emf in Concentration Cells (11) Abramowitz, M.; Stegun, A., Eds. Handbook of Mathematical Functions; Dover Publications: New York, 1972; section 25.5.1025.5.11. (12) Lanczos, C. Applied Analysis; Dover Publications: New York, 1988. (13) Onsager, L. Phys. ReV. 1931, 37, 405. (14) Onsager, L. Phys. ReV. 1931, 38, 2265. (15) Malmgren-Hansen, B.; Sørensen, T. S.; Jensen, J. B.; Hennenberg, M. J. Colloid Interface Sci. 1989, 130, 359. (16) Sørensen, T. S. In Capillarity Today, Lecture Notes in Physics, Vol. 386; Pe´tre´, G., Sanfeld, A., Eds.; Springer-Verlag: Berlin, 1991. (17) Plesner, I. W.; Malmgren-Hansen, B.; Sørensen, T. S. J. Chem. Soc., Faraday Trans. 1994, 90, 2381. (18) Sørensen, T. S. J. Colloid Interface Sci. 1994, 168, 437. (19) Sørensen, T. S.; Compan˜, V. J. Chem. Soc., Faraday Trans. 1995, 91, 4235.

J. Phys. Chem., Vol. 100, No. 37, 1996 15273 (20) Sørensen, T. S.; Compan˜, V. J. Colloid Interface Sci. 1996, 178, 186. (21) Sørensen, T. S.; Compan˜, V.; Diaz-Calleja, R. J. Chem. Soc., Faraday Trans. 1996, 92, 1947. (22) von Helmholtz, H. Ann. Phys. 1878, 3, 201. (23) Jensen, J. B.; Sørensen, T. S.; Malmgren-Hansen, B.; Sloth, P. J. Colloid Interface Sci. 1985, 108, 18. (24) Malmgren-Hansen, B.; Sørensen, T. S.; Jensen, J. B. J. Non-Equilib. Thermodyn. 1988, 13, 193. (25) Scatchard, G. J. Phys. Chem. 1964, 68, 1056. (26) Sørensen, T. S.; Jensen, J. B.; Malmgren-Hansen, B. Desalination 1991, 80, 293. (27) Merten, U., Ed. Desalination by ReVerse Osmosis; M.I.T. Press: Cambridge, MA, 1966.

JP9609895

Salt Flux and Electromotive Force in Concentration Cells with

Recommend Documents