J. Phys. Chem. B 2008, 112, 3013-3018
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Observable Quantities for Electrodiffusion Processes in Membranes Javier Garrido Departamento de Fı´sica de la Tierra y Termodina´ mica, UniVersitat de Vale` ncia, E-46100 Burjassot (Valencia), Spain ReceiVed: February 26, 2007; In Final Form: December 28, 2007
Electrically driven ion transport processes in a membrane system are analyzed in terms of observable quantities, such as the apparent volume flow, the time dependence of the electrolyte concentration in one cell compartment, and the electrical potential difference between the electrodes. The relations between the fluxes and these observable quantities are rigorously deduced from balances for constituent mass and solution volume. These relations improve the results for the transport coefficients up to 25% with respect to those obtained using simplified expressions common in the literature. Given the practical importance of ionic transport numbers and the solvent transference number in the phenomenological description of electrically driven processes, the transport equations are presented using the electrolyte concentration difference and the electric current as the drivers of the different constituents. Because various electric potential differences can be used in this traditional irreversible thermodynamics approach, the advantages of the formulation of the transport equations in terms of concentration difference and electric current are emphasized.
Introduction The study of the equations that describe ionic and solvent transport across a membrane in the presence of electric current and concentration differences is of fundamental importance because of its close relation to electrically driven membrane separation processes.1,2 There are different ways to deduce the transport equations that describe the processes developed in a nonequilibrium system. They can be derived from the dissipation function Ψ, or by considering practical reasons. In the case of a twocomponent solution (solvent and a binary electrolyte), the contribution of electrodiffusion in membranes to the dissipation function is Ψ ) j1∆µ1 + j+∆µ˜ + + j-∆µ˜ -.3 Traditionally, the fluxes (e.g., j1,j+,j-) are expressed in terms of the forces (e.g., ∆µ1,∆µ˜ +,∆µ˜ -) and the transport coefficients are interpreted as “conductances”; otherwise the forces are expressed in terms of the fluxes, and the coefficients are considered as transport “resistances”. The main interest in this approach is to apply the Onsager reciprocal relations (ORR) and therefore to reduce the number of transport coefficients. When practical reasons are brought out, the first step in order to postulate the transport equations is to determine a set of independent observable quantities that characterizes well the nonequilibrium states of the system. In the case of electrodiffusion in membranes, we choose the pair formed by the electric current I and the concentration difference between the subsystems ∆c2. Then the transport equations are postulated in relation to these two quantities. These two ways of postulating the transport equations cannot generate different results. We will spend a section in this paper to deduce the pertinent transformations in order to compare these two approaches. In the literature we find that the dissipation function is expressed in terms of different electric potentials. When the electrostatic potential difference across the membrane ∆φ is used, we have Ψ ) j1∆µ1 + j+∆µ+ + j-∆µ- + I∆φ. Alternatively, the dissipation function can be presented in terms
of the ohmic potential drop across the membrane as4 Ψ ) j1∆µ1 + j2∆µ2 + I∆φohm. However, it should be emphasized that ∆φ, ∆φohm, and their difference, the diffusion potential drop ∆φdif ) ∆φ - ∆φohm, cannot be directly measured experimentally. Hence, a third alternative with significant advantages5 is the use of the electric potential difference directly measured with, e.g., electrodes reversible to the anion in solution, ∆ψ-. In this case, the dissipation function takes the form Ψ ) j1∆µ1 + (j+/ν+)∆µ2 + I∆ψ-. The different choices of the electric potential difference make the comparison between the transport coefficients difficult. In relation to this, the different characteristics of the transport of electrolytes and nonelectrolytes in membrane systems are often rationalized in terms of the electric potential distribution,6,7 but this is not completely satisfactory because such distribution cannot be observed experimentally. Therefore, one of the aims of the present study is to show the advantages of transport equations that relate the fluxes j1 and j+/ν+ to ∆µ2 and I. Note that ∆µ1 can be related to ∆µ2 by the Gibbs-Duhem equation. A second aim of this study is the rigorous derivation of the equations employed for the determination of the fluxes from observable quantities, such as the apparent volume flow and the rate of change of the electrolyte concentration in one compartment. This aim is achieved from the derivation of constituent mass and volume balances. When compared to those used in the literature, it is shown that deviations up to 25% in the evaluation of ionic fluxes can be found. In the last sections, other transport equations deduced from the dissipation function of the membrane system are briefly discussed. Electrodiffusion Processes in Membranes System Description. Consider a system of uniform temperature and pressure formed by a membrane that separates two compartments (I and II) containing solutions of the same binary electrolyte Aν+Bν- (component 2). The electrolyte dissociates into ν+ cations Az+ and ν- anions Bz-, with the stoichiometric
10.1021/jp071578z CCC: $40.75 © 2008 American Chemical Society Published on Web 02/20/2008
3014 J. Phys. Chem. B, Vol. 112, No. 10, 2008
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VjBI dV s )dt z-F
Figure 1. Sketch of the cell. WE ) working electrodes. M ) membrane. X ) stirrers.
and charge numbers of the ions satisfying the relation z+ν+ + z-ν- ) 0. Stirrers in both compartments keep the solute concentrations uniform, and the quantity ∆c2 ) cII2 - cI2 can be measured. An electric current I is driven through the membrane by electrodes reversible to the anions B/Bz-, where the electrode reaction Bz- a B + |z-|e- is developed. Two horizontal capillaries at the same level ensure negligible pressure difference between the compartments. Figure 1 shows a sketch of this cell. Constituent Mass Balances. The molar fluxes of the solvent (component 1) and cation, j1 and j+, through the membrane can be measured indirectly by using their relation to two observable quantities. These fluxes are measured with respect to the membrane and defined as positive when the constituents go from compartment II to I. The capillaries shown in Figure 1 allow for the measurement of the apparent volume flow rate that exits compartment I. We denote this first observable as dV c/dt. The second observable that is experimentally monitored is the time derivative of the electrolyte concentration in compartment I, dc2/dt. These two observables depend on the values of ∆c2 and I, which characterize our membrane system. Our next objective is to formulate the mass balances that establish the relations between the fluxes j1 and j+ and the observable quantities dV c/dt and dc2/dt. Because the solvent is not involved in the electrode reaction, the change in the amount of solvent n1 in compartment I must be due to the inward flow through the membrane and the outward flow through the capillary. The mass balance for the solvent is then
dn1 dV c ) j1 - c1 dt dt
(1)
where > 0 has been assumed. The quantities j1 and c1 are both functions of time. Similarly, the rate of change of the amount of cations n+ in compartment I is given by dV c/dt
dn+ dV c ) j+ - ν+c2 dt dt
(2)
where c+ ) ν+c2 is the cation molar concentration in compartment I. Equation 2 implies that the mass balance for the electrolyte is
dn2 j+ dV c - c2 ) dt ν+ dt
(3)
The volume of compartment I is fixed, but the volumes of the solution and the electrode in this compartment are not constant because of the electrode reaction, Bz- a B + |z-|e-. By Faraday’s law, this reaction proceeds at a rate dξ/dt ) I/z-F and, therefore, the rate of change of the solution volume Vs is equal to the negative rate of change of the electrode volume
(4)
where VjB is the partial molar volume of species B at the electrode. The molar concentration of solvent and electrolyte are related to their amounts in compartment I by the relation, ni ) ciVs (i ) 1, 2), with n2 ) n+/ν+. Moreover, the equation Vs ) Vj1n1 + Vj2n2, where Vji is the partial molar volume of constituent i, can be written as 1 ) Vj1c1 + Vj2c2 and implies that Vj1dc1/dt + Vj2dc2/dt ) 0. Then, using the relations Vsdci/dt ) dni/dt - cidVs/ dt, the mass balances for the solvent and electrolyte can be transformed into the relations we were looking for:
j1 ) - V s
Vj2 dc2 VjBI dV c + c1 - c1 Vj1 dt dt z-F
dc2 VjBI j+ dV c + c2 - c2 ) Vs ν+ dt dt z-F
(5)
(6)
Phenomenological Transport Equations. Some of the quantities we have used so far may be considered as fluxes (e.g., j1, j+, I) or thermodynamic forces (e.g., ∆c2). These quantities have to fulfill the condition that all of them disappear when the membrane system lies at equilibrium. In order to analyze the irreversible processes, it is very useful to determine a set among these fluxes and driving forces so that the nonequilibrium states are well characterized only by them; these quantities could be considered as the independent variables of the process. In our membrane system we choose the pair ∆c2 and I; we see that, when ∆c2 ) 0 and I ) 0, our membrane system is at equilibrium, and all the fluxes and forces disappear. Then it is natural to describe the deviations from equilibrium and, thus, the irreversible processes relating to them as a first approximation by linear transport equations for the different fluxes and forces on the independent quantities ∆c2 and I.8 Thus, for the fluxes j1 and j+, we have
j1 ) P1∆c2 +
τ1 I F
t+ j+ ) P2∆c2 + I ν+ z+ν+ F
(7) (8)
where F is Faraday’s constant, P1 is the permeability of the membrane to the solvent, P2 is the permeability of the membrane to the electrolyte, τ1 is the solvent transference number in the membrane, and t+ is the cation transport number in the membrane. Because transport is driven by I and ∆c2, these two flux equations suffice to describe the transport of all solutions constituents. Thus, for instance, the molar flux of the anion can be written as
tjI ) P2∆c2 + νz-ν- F
(9)
but the equation I/F ) z+j+ + z-j- implies that the ionic transport numbers satisfy the relation t+ + t- ) 1, and therefore no additional transport coefficient is needed. An electrolyte flux j2 through the membrane could also be defined, but this is subject to some arbitrariness (unless I ) 0 and j2 ) j+/ν+ ) j-/ν-) and brings no new transport coefficient that could be experimentally measured.
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Apart from the observables dV c/dt and dc2/dt, we can measure the electric potential difference between two electrodes reversible to the anion in solution ∆ψ-. As we can see in eqs 5 and 6, this quantity does not affect directly the values of the fluxes j1 and j+; it provides only complementary information. In the vicinity of equilibrium, we can also postulate a new transport equation:
∆ψ- ) R-∆c2 + RI
(10)
with two new transport coefficients, R- and the electrical resistance R of the membrane system. When choosing the independent quantities of the nonequilibrium states, we could have selected the pair ∆c2 and ∆ψinstead of ∆c2 and I. The electric current was preferred because (i) it appears explicitly in eqs 5 and 6, which evaluate the fluxes j1 and j+ from the observables, and (ii) it provides a clear distinction between diffusion I ) 0 and electrodiffusion I * 0 in membranes. Volume Flow. The phenomenological flux equations and mass balances derived above allow for a complete description of the electrodiffusion process. However, the consideration of the volume flow provides interesting complementary information that enhances our understanding of the process. Consider first the volume balance for compartment I as a fixed control volume. The volume flow entering this compartment through the membrane is related to the solvent flow and ionic fluxes through the membrane by
j+ I + VjjV ) Vj1 j1 + Vj+ j+ + Vj- j- ) Vj1 j1 + Vj2 ν+ z-F
(11)
The outflow rate through the capillary in this compartment is dV c/dt. The rate of volume change in compartment I due to the electrode reaction Bz- a B + |z-|e- is (VjB - Vj-)dξ/dt. Then, the volume balance for compartment I is
j+ dV c I I + VjB ) jV + (VjB - Vj-) ) Vj1 j1 + Vj2 dt z-F ν+ z-F
(13)
Using eqs 7-8 and 12, the phenomenological coefficients in this equation are
PV ) Vj1P1 + Vj2P2 τV ) Vj1τ1 + Vj2
VjB t+ + z+ν+ z-
Experimental Determination of the Phenomenological Transport Coefficients. The phenomenological coefficients PV and τV can be determined from measurements of the apparent volume flow rate dV c/dt under open-circuit conditions (I ) 0) and under isoosmotic conditions (∆c2 ) 0),
PV ) τV ) F
(12)
Noteworthy, and contrarily to the non-observable flux jV, the observable apparent volume flux dV c/dt only depends on partial molar volumes of neutral constituents. Equation 12 illustrates an essential difference for the electrodiffusion processes developed in membranes with respect to those carried out in the bulk of a solution.9 In the latter, the fluxes are related by 0 ) Vj1j1 + Vj2(j+/ν+) + Vj B(I/z-F) and therefore from the four transport coefficients, only two are independent quantities. The phenomenological equation for the apparent volume flux through the membrane is
τV dV c ) PV∆c2 + I dt F
Figure 2. Values of dV c/dt, dc2/dt, and δj+/j+ for the process ∆c2 ) 0 and a current of 10 mA in the membrane system Zeo-Karb 315 and NaBr solutions. Data from Meares et al.14-15 In the calculation of dc2/ dt, the value Vs ) 100 cm3 has been assumed.
(14) (15)
Therefore, four transport coefficients have a fundamental value (e.g., P1, P2, τ1, t+), and two more (e.g., R-, R) play a complementary role.
( )
(16)
dV c/dt I
(17)
dV c/dt ∆c2
I)0
( )
∆c2)0
Similarly, by measuring the two observables dV c/dt and dc2/ dt under these conditions, the coefficients P2 and t+ can be determined from eqs 6 and 8 as
(
)
V sdc2/dt + c2dV c/dt P2 ) ∆c2 t + ) z + ν+
[
c2VjB F s (V dc2/dt + c2dV c/dt) I z-
(18)
I)0
]
(19)
∆c2)0
Finally, the other transport coefficients can be evaluated from eqs 13 and 14. Thus, for instance, the solvent transference number is
τ1 )
τV Vj2 t+ VjB Vj1 Vj1 z+ν+ z-Vj1
(20)
Comparison with Previous Work. In refs 6 and 10-13, the volume balance in eq 12 is used in combination with a simplified form of the electrolyte mass balance
dc2 j+,app ) Vs ν+ dt
(21)
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Figure 3. Approximate and true values of the transport number for a poly(styrenesulfonic acid) membrane with aqueous solutions of BaCl2. The values τ1,app and t+,app are from Figures 4 and 5 of ref 12.
which should be compared with eq 6 above. The relative deviation δj+/j+ ) 1 - j+,app/j+ has been estimated, and the results obtained are presented in Figure 2. From the transport numbers given in Figure 5 of ref 14 and Figure 4 of ref 15 corresponding to a Zeo-Karb 315 membrane and aqueous solutions of NaBr, we have evaluated dV c/dt, dc2/dt, and δj+/ j+ for the process with ∆c2 ) 0 and I ) 10 mA. The current density does not exceed 10.40 mA cm-2. Pt/Ag/AgBr electrodes reversible to the anions Br- are used. It is concluded that the apparent cation flux estimated from eq 21 cannot be accepted when c2 > 0.1 mol dm-3. These relative deviations depend on both volume rates, inside the capillary dV c/dt and that of the electrodes VjBI/z-F, that is, δj+/j+ ) ν+c2(dV c/dt - VjBI/z-F)/j+. For processes ∆c2 ) 0 and I * 0, it can be expressed as a function of the transport numbers δj+/j+ ) [c2(t+Vj2 + ν+z+τ1Vj1)/t+]. In the case shown in Figure 2, we see that the concentration c2 varies from 10 to 103 mol m-3, the quantity t+Vj2 + τ1Vj1 varies from 10.7 × 10-4 to 2.0 × 10-4 m3 mol-1, and the transport number t+ varies from 1.00 to 0.71. In the work of Kumamoto et al.12 on BaCl2 solutions and a poly(styrenesulfonic acid) membrane, Ag/AgCl electrodes reversible to Cl- were used. The cation transport number and the solvent transference number were approximately evaluated as
( )
dc2 F t+,app ) z+ν+ V s I dt τ1,app )
( )
F dV c Vj1I dt
∆c2)0
-
∆c2)0
Vj2 VjB t+,app + 2Vj1 Vj1
(22)
(23)
which should be compared with eqs 19 and 20 above. In Figure 3 the values τ1,app and t+,app are compared with the true numbers τ1 and t+. The approximate values were taken from Figures 4 and 5 of ref 12. Then we calculated the observables (dV c/dt)∆c2)0/I and (dc2/dt)∆c2)0/I from eqs 22 and 23. Finally the true numbers were evaluated from eqs 19 and 20. The current density does not exceed 5 mA cm-2. The deviations are negligible for the solvent transference number, but they can be as high as 25% for the cation transport number at c2 ) 1 M. The deviation between the two values of the cation transport number depends on both volume ratessthat inside the capillary
dV c/dt and that of the electrodes VjBI/z-Fsthat is, t+ - t+,app ) ν+z+c2F[(dV c/dt)∆c2)0 - VjBI/z-F]/I. As we can see in Figure 3, this quantity varies significantly with concentration. Nevertheless, the deviation between the two values of the solvent transference number, given by τ1 - τ1,app ) Vj2(t+,app - t+)/ ν+z+Vj1, is a small quantity not affected by the change in concentration. These two examples indicate the importance of evaluating accurately the different contributions to the mass and volume balances even in moderately concentrated solutions. Dissipation Function. The dissipation function is usually the basis to select the independent fluxes and forces for describing an irreversible process in order to apply the ORR. In previous work,5,9,16-20 we developed a formulation of ionic transport processes in terms of an observable electric potential. Thus, different nonequilibrium systems have been studied. Now, we are going to present the case of electrodiffusion in membranes. For a binary solution at constant temperature and pressure across a membrane system, the dissipation function is3,21,22
Ψ ) j1∆µ1 + j+∆µ˜ + + j-∆µ˜ -
(24)
The differences are defined as the value in compartment II minus that in compartment I. The observable electric potential, for the case of electrodes reversible to the anion under consideration in the present study, is defined as
∆ψ- ≡ ∆µ˜ -/z-F
(25)
and represents the potential difference measured between the terminals of two identical electrodes reversible to the anions. Then, the dissipation function can also be presented as
Ψ ) j1∆µ1 +
j+ ∆µ2 + I∆ψν+
(26)
where we have used the relation µ2 ) ν+µ˜ + + ν-µ˜ - between the chemical potential of the electrolyte and the electrochemical potentials of the ions. When the fluxes j1, j+/ν+, and I are expressed in terms of the forces ∆µ1, ∆µ2, and ∆ψ-, the linear phenomenological transport equations are
( )(
)( )
l11 l12 l13 ∆µ1 j1 j+/ν+ ) l12 l22 l23 ∆µ2 l13 l23 l33 ∆ψI
(27)
and the transport coefficients are interpreted as “conductances”. In eq 27, the cross coefficients have been assumed to be equal, i.e., we have accepted the validity of the ORR. The relations between these transport coefficients and the six defined by eqs 7, 8, and 10, i.e., P1, P2, τ1, t+, R-, and R, are shown in the Appendix. We have emphasized before the advantages of the electric current being an independent quantity in the transport equations. Now we need to express j1, j+/ν+, and ∆ψ- in terms of ∆µ1, ∆µ2, and I.2,13 A straightforward algebraic manipulation would lead to the phenomenological transport equations
( )(
)( )
j1 L11 L12 L13 ∆µ1 j+/ν+ ) L12 L22 L23 ∆µ2 ∆ψ- L13 - L23 L33 I
(28)
which also involves six transport coefficients. Although the dissipation function involves three fluxes and three forces, it should be noticed that the state of the solutions
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in both compartments is only a function of the electrolyte concentration (at fixed temperature and pressure), and hence a relation exists between ∆µ1 and ∆µ2, given by the integrated Gibbs-Duhem equation cj1∆µ1 + cj2∆µ2 ) 0, where cj1 ≈ (cI1 + cII1 )/2 and cj2 ≈ (cI2 + cII2 )/2. This implies that eq 28 can be transformed to
( ) ( )(
j1 L′12 L13 ∆µ2 j+/ν+ ) L′22 L23 ∆ψ-L′23 L33 I
)
(29)
where L′i2 ) Li2 - (cj2/cj1)L1i (i ) 1, 2) and L′23 ) L23 - (cj2/ cj1)L13. The relations between these coefficients and those of eqs 7, 8, and 10 are shown in the Appendix. Potentiometric Determination of the Ionic Transport Numbers. As we have shown above, the description of electrodiffusion processes in membranes does not require the evaluation of electric potential differences and can be accomplished in terms of the electric current only. However, it does not escape our attention that ionic transport numbers are often determined potentiometrically. Since the potentiometric definition of the cation transport number is
t1+ ) -
ν+z+F(∆ψ-)I)0 ∆µ2
role and can be computed from measurements of another observable, the electric potential difference ∆ψ-. Finally, we verify that there are two ways of determining the cation transport number t1+: (i) from potentiometric measurements as a consequence of the ORR, and (ii) from the values of t+ and τ1, both computed from measurements of dV c/dt and dc2/dt. The first is the usual method applied by the authors at the present time. The second is recommended in this paper. Unfortunately, we have not been able to find in the literature enough data to carry on a comparison between the two ways. Appendix Relations between the Transport Coefficients. Using ∆µ1 ) -(cj2/cj1)∆µ2 and ∆µ2 ) (∂µ2/∂c2)∆c2, from eqs 7, 8, 10, 27, and 29, we deduce
( (
) )
l13l23 cj2 cj2 l213 ∂µ2 ∂µ2 P1 ) l12 - l11 + ) L′12 l33 cj1 cj1 l33 ∂c2 ∂c2
(A1)
l232 cj2 cj2 l13l23 ∂µ2 ∂µ2 - l12 + ) L′22 P2 ) l22 l33 cj1 cj1 l33 ∂c2 ∂c2
(A2)
(30)
where (∆ψ-)I)0 is the membrane potential, eq 29 leads to the conclusion that this cation transport number is t1+ ) t+ (ν+z+cj2/cj1)τ1.23 Note that t1+ is a transport number in a reference frame linked to the solvent because the flux equation for the cation in this frame j1+ ) j+ - (ν+cj2/cj1)j1 can be shown to be j1+/ν+ ) P12∆c2 + t1+I/(ν+z+F) with P12 ) P2 - (cj2/cj1)P1. Unfortunately, we have not been able to find in the literature enough data to carry on a comparison of the results obtained potentiometrically for the transport numbers with those obtained from the experimental determination of t+ and τ1 and the use of the relation t1+ ) t+ - (ν+z+cj2/cj1)τ1. Conclusions Electrodiffusion processes in membranes have been described in terms of the independent quantities ∆c2 and I. The advantages of the electric current with respect to the electric potential difference in order to characterize the transport through the membrane system have been emphasized. On one hand, the ion transport numbers and solvent transference number appear explicitly in the transport equations. On the other hand, there is no need to discuss diffusion potential, ohmic potential, electrostatic potential, observable electric potential, and so forth, thus indicating that the electric potential plays a complementary role in this description. In the frame of this formulation, six transport coefficients have been characterized. Four of them determine the solvent and electrolyte fluxes in electrodiffusion processes in membranes: the solvent and solute permeabilities P1 and P2, the solvent transference number τ1, and the cationic transport number t+. These can be computed from measurements of dV c/ dt and dc2/dt. New relations between the fluxes and the observables dV c/dt and dc2/dt have been rigorously derived from constituent mass and volume balances. These relations improve the results for the transport coefficients up to a 25% with respect to those obtained using simplified expressions common in the literature. The two remaining transport coefficients R- and R complete the description of the process. They play a secondary
R- )
τ1 l13 ) ) L13 F l33
(A3)
t+ l23 ) ) L23 z+ν+F l33
(A4)
∂µ2 ∂µ2 1 jc2 l - l23 ) - L′23 l33 cj1 13 ∂c2 ∂c2
(
)
(A5)
1 ) L33 l33
(A6)
R) Nomenclature
ci, molar concentration of constituent i ) 1, 2, +, -; mol m-3 F, Faraday’s constant I, electric current; A ji, molar flux across membrane of constituent i ) 1, 2, +, -; mol s-1 jV, volume flow entering compartment I through the membrane; m3 s-1 lik, phenomenological coefficients of membrane system i,k ) 1,2; mol2 J-1 s-1. i ) 1,2 and k ) 3; mol V-1 s-1. i ) k ) 3; S Lik, phenomenological coefficients of membrane system i,k ) 1,2; mol2 J-1 s-1. i ) 1,2 and k ) 3; mol C-1. i ) k ) 3; ohm ni, amount of substance of constituent i ) 1, 2, +, -, in Vs; mol Pi, membrane permeability to constituent i ) 1,2; m3 s-1 R, electrical resistance; ohm t, time; s ti, transport number in the membrane of constituent i ) +, VjB, molar volume of species B of the electrode B/Bz-; m3 mol-1 Vji, partial molar volume of constituent i ) 1, 2, +, -; m3 mol-1 V c, volume of the solution inside the capillary I; m3
3018 J. Phys. Chem. B, Vol. 112, No. 10, 2008 Vs, volume of solution submitted to agitation in compartment I; m3 zi, charge number of constituent i ) +, R-, transport coefficient; V m3 mol-1 µi, chemical potential of constituent i ) 1,2; J mol-1 µ˜ i, electrochemical potential of constituent i ) +, -; J mol-1 νi, stoichiometric number of constituent i ) +, ξ, extent of reaction; mol τ1, solvent transference number in the membrane Ψ, dissipation function; J s-1 ψ-, observable electric potential with electrode reversible to anions; V References and Notes (1) Kedem, O. J. Membr. Sci. 2002, 206, 333. (2) Tanaka, Y. J. Membr. Sci. 2004, 235, 15. (3) Demirel, Y.; Sandler, S. I. J. Phys. Chem. B 2004, 108, 31. (4) Kontturi, K. Acta Polytech. Scand. 1983, 152, 1. (5) Garrido, J. J. Phys. Chem. B 2004, 108, 18336. (6) Tasaka, M.; Kondo, Y.; Nagasawa, M. J. Phys. Chem. 1969, 73, 3181. (7) Tasaka, M.; Aoki, N.; Kondo, Y.; Nagasawa, M. J. Phys. Chem. 1975, 79, 1307.
Garrido (8) Haase, R. Thermodynamics of IrreVersible Processes; Dover: New York, 1990; p 88. (9) Garrido, J. J. Phys. Chem. B 2006, 110, 3276. (10) Tourreuil, V.; Dammak, L.; Bulvestre, G.; Auclair, B. New J. Chem. 1999, 23, 173. (11) Va´zquez, M. I.; Benavente, J. J. Membr. Sci. 2003, 219, 59. (12) Kumamoto, E.; Kimizuka, H. J. Phys. Chem. 1981, 85, 635. (13) Scibona, G.; Radatti, N.; Botreˆ, C.; Botreˆ, F.; Gavelli, G. Ber. Bunsen-Ges. Phys. Chem. 1989, 93, 766. (14) McHardy, W. J.; Meares, P.; Sutton, A. H.; Thain, J. F. J. Colloid Interface Sci. 1969, 29, 116. (15) Meares, P.; Sutton, A. H. J. Colloid Interface Sci. 1968, 28, 118. (16) Garrido, J.; Compan˜, V.; Lo´pez, M. L.; Miller, D. G. J. Phys. Chem. B 1997, 101, 5740. (17) Garrido, J.; Mafe´, S.; Aguilella, V. M. Electrochim. Acta 1988, 33, 1151. (18) Garrido, J.; Manzanares, J. A. J. Phys. Chem. B 2000, 104, 658. (19) Garrido, J.; Compan˜, V.; Lo´pez, M. L. Phys. ReV. E 2001, 64, 016122. (20) Garrido, J. J. Electrochem. Soc. 2003, 150, E-567. (21) Katchalsky, A.; Curran, P. F. Nonequilibrium Thermodynamics in Biophysics; Harvard U. P.: Cambridge, MA, 1975; p 151. (22) Lakshminarayanaiah, N. Transport Phenomena in Membranes; Academic Press: New York, 1969; p 182. (23) Larchet, C.; Dammak, L.; Auclair, B.; Parchikov, S.; Nikonenko, V. New J. Chem. 2004, 28, 1260.
