J. SANDBLOM, G. EISENMAN, AND J. L. WALKER, JR.
3862
Electrical Phenomena Associated with the Transport of Ions and Ion Pairs in Liquid Ion-Exchange Membranes. I. Zero Current Properties’.
by J. Sandblom,lb G. Eisenman, and J. L. Walker, Jr.lC Department of Physiology, Unicersity of Chicogo, Chicago, Illinois
60697
(Received April 12, 1967)
A theoretical examination has been carried out of the steady-state properties of a homogeneous, ideally permselective, liquid ion-exchange membrane in which sites and counterions are incompletely dissociated. Dissociated species are assumed to be in chemical equilibrium with neutral ion pairs a t every point in the membrane, their concentrations being interconnected by the law of mass action. The flux equations, which describe the complete behavior of the system, are derived by considering the free ions and their combined forms as separately flowing entities, and the boundary conditions are obtained by assuming the sites to be completely trapped in the membrane phase (although free to move within it) while the counterions are free to undergo ion exchange. In the present paper, under the restriction of zero membrane current, a general expression for the membrane potential is deduced in terms of external solution conditions and membrane parameters (e.g., mobilities, dissociation constants). From this expression, the factors governing the electrode properties of liquid ion-exchange membranes are discussed not only for the steady state but for certain transient situations as well. It is concluded that the steady-state expressions derived for convection-free systems describe the situations usually encountered with membrane electrodes made from liquid ion exchangers where the instantaneous values of successive potentials are measured with electrode systems which are not in the steady state and even when no precautions are taken to avoid convective mixing within the electrode. The parameters controlling electrode specificity are also discussed.
Introduction Systems c 3mposed of a water-immiscible liquid interposed between two aqueous solutions were among the first “membranes” in which electric properties and selective ion permeabilities were When such membranes contain an appreciable concentration of an ionizable species which is preferentially soluble within the membrane phase (e.g., a fatty acid or an aliphatic amine), they function as liquid ion exchangers whose properties are of interest not only because they can be made into electrodes specific for various ions,‘j-1° but also because t,hey constitute a model for one of the conceivable mechanisms of ion permeation through biological membranes. l1 Despite their long history, the theory of such liquid ion-exchange membranes is not as advanced as that for solid ion-exchange membranes, l 2 no doubt because the lack of fixation in a liquid of the ion-exchange sites The Journal of Physical Chemistry
introduces additional complexities in obtaining explicit solutions to the flux equations. Thus, theoretical considerations for liquid ion exchangers prior to that (1) (a) This work was supported by National Science Foundation Grant GB-4039 and USPHS Grant GM 14404-01. It was assisted by USPHS General Research Support Grant FR-5367 and an NIH postdoctoral fellowship t o J. L. Walker, Jr. (b) Institute of Physiology and Medical Biophysics, University of Uppsala, Uppsala, Sweden. (0) Department of Physiology, University of Utah, College of Medicine, Salt Lake City, Utah. (2) (a) W. Nernst and E. H. Riesenfeld, Ann. Phys., 8, 600 (1902); (b) M . Cremer, Z. Biol., 47, 562 (1906). (3) F . G. Donnan and W. E. Garner, J . Chem. Soc., 115, 1313 (1919). (4) R. Beutner, “Physical Chemistry of Living Tissues and Life Processes,” Williams and Wilkins Co., Baltimore, Md., 1944. (5) W. J. V. Osterhout, Cold Spring Harbor Symp. Quant. Bwl., 8 , 5 1 (1940). (6) X. F. Bonhoeffer, M. Kahlweit, and H . Strehlow, Z . Physik. Chem. (Frankfurt), 1,21 (1954). (7) K. Sollner and G . M. Shean, J. Am. Chem. Soc.. 86, 1901 (1964). (8) 0. D. Bonner and 3.Lunney, J . Phys. Chem., 70, 1140 (1966).
TRANSPORT OF IONS AND ION PAIRS IN
LIQUID
ION-EXCHANGE MEMBRANES
of Conti and Eisenman13 have been restricted to the potentials or fluxes observed under zero current condiI n addition, most treatments have either t i o n ~,14-19 .~ assumed the diffusion potential t o be negligiblels or have made particular assumptions about the concentration profiles within the membrane (e.g., linear mixtures of the Henderson typeI4-l6 or linear concentration profiles1*). However, for the case of a liquid ion-exchange membrane with complete dissociation between sites and counterions, it has been shown recently that closed solutions to the flux equations can be obtained in the steady state without making a priori assumptions about the concentration pr0fi1es.l~ The characteristic features of such a membrane result from the redistribution of sites and are therefore related to such classical concentration polarization phenomena as occur in depletion layers at the interface between an electrode and an aqueous solution.20-22 As was recognized by Conti and Eisenman, the restriction of their treatment to completely dissociated systems constitutes a serious limitation when attempting to apply the results to the usual liquid ion-exchange membranes in view of the relatively low dielectric constant normally characteristic of these. I n the present paper, we shall therefore consider the effects of incomplete dissociation in order to deduce the properties of a liquid ion-exchange membrane having any degree of dissociation. Additional complexities appear in our treatment as a result of association of the sites and counterions to form electrically neutral species. In particular, when an external force is applied to the system so as to perturb the concentration of the dissociated species, the concentration of the associated species is also perturbed by virtue of the dissociation equilibrium which couples the concentrations of dissociated and undissociated species a t all points within the membrane. Despite these complexities, explicit expressions have been obtained for the membrane potential at zero current and also for certain nonzero current properties (in the case of a single counterion species).
Experimental Section Description of the System. The system to be studied (see Figure 1) is isothermal and consists of a membrane separating two homogeneous solution phases (e.g., aqueous) whose electric potentials are #’ and #”. The membrane is composed of a single liquid phase immiscible with the external solutions. In the membrane is dissolved a “site” species s bearing a charge zs = f1 (given a negative sign in Figure ‘1. The sites are assumed to be completely reflected at the boundaries,
3863
SOLUTION (‘I)
I +”
,- x-” d
0
Figure 1. Diagram of the system; I+, S-,and X- refer to the counterion, the site, and the eo-ion species, respectively.
0 and d (as indicated by the arrows), but are free to move within the interior of the membrane. The system also contains n number of permeable univalent counterion species I whose charge zi is opposite to that of the sites and which are free to cross the membrane solution interfaces as indicated by the arrows in Figure 1. Any number of co-ion species may be present in the external solutions although the co-ions are assumed to be completely excluded from the membrane. Even species which behave as strong electrolytes in aqueous solutions will in general not be completely dissociated in a liquid ion-exchange membrane. We shall therefore assume that species I and S are in chemical equilibrium at every point in the membrane with the associated species IS through reactions of the type
I+
+ s- E IS
(1)
indicating that the only reactions assumed to occur in the membrane are those in which electrically neutral (9) J. W. Ross, Science, 156,1378 (1967). (10) Corning CaZ+ Electrode Specification. Corning Glass Works Caa + electrode data sheet No. EL-Ca R P P 2/67. (11) G. Eisenman, J. Sandblom, and J. L. Walker, Jr., Science, 155, 965 (1967). (12) I?. Helfferich, “Ion Exchange,” McGraw-Hill Book Co., Inc., New York, N. Y., 1962,pp 19,20. (13) F. Conti and G. Eisenman, Bwphys. J . , 6 , 227 (1966). (14) K. H. Meyer, H , Hauptmann, and J. F. Sievers, Helv. Chim. Acta, 19,946 (1936). (15) F. M. Karpfen and J. E. B. Randles, Trans. Faraday Soc., 49, 823 (1953). (16) M.Dupeyrat, J . Chim. Phys., 61,306,323(1964). (17) H. L.Rosano, P. Duby, and J. H. Schulman, J . Phys. Chem., 65, 1704 (1964); H. L. Rosano, J. H. Schulman, and J. B. Weisbuch, Ann. N . Y . Acad. Sci., 92,457 (1961). (18) J. T.Davies, J . Phys. Chem., 54,185 (1950). (19) M.Kahlweit, Archiv. Ges. Physwl., 271, 139 (1960). (20) E.Warburg, Ann. Physik Chem., 67,493(1899). (21) P. Delahay, “New Instrumental Methods in Electrochemistry,” Interscience Publishers. Inc.. New York. N. Y.. 1954; I. M.Kolthoff and J. J. Lingane, ”Polarography,” 2nd ed, Interscience Publishers, Inc., New York, N. Y., 1952. (22) F. Helffefich, “Ion Exchange,” McGraw-Hill Book Co,, Inc., New York, N. Y., 1962,pp 360-363.
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J. SANDBLOM, G. EISENMAN, AND J. L. WALKER, JR.
3864
complexes are formed. For simplicity, we have also restricted considerations to the smallest complex possible (ie., the ion pair). Such behavior is often characteristic of weak electrolytes where the formation of triple ions and higher order aggregates is negligible over wide concentration ranges.23 The chemical potentials corresponding to reaction 1 are everywhere in the membrane related through pa
+
~s
(i = 1, . . , i n )
= pis
(2)
where p a , ps and p t s are the chemical potentials of the counterions, sites, and ion pairs, respectively. For simplicity, we will assume activities to be equal to concentrations. Under this assumption, the concentrations of the species (ct, c,, cis) are related to the dissociation constants K , through a simple law of mass action
K,
ctc,
(i = 1, . . . , n)
= cis
CCi i
(4)
Flux Equations I n describing ionic transport processes in a liquid ionexchange membrane, it is useful to consider separately the flows of associated and dissociated species. The total flux of each species through the membrane can be written as a sum of partial fluxes =
+ J,, + CJts
J,
= Js
(5a) (jb)
1.
where J , * and J,* are the total fluxes of counterions and sites, Ji and J, are the fluxes of the species in their dissociated state, and J,, are the fluxes of the ion pairs. Since counterions, sites, and associated pairs are treated as separately flowing entities, it is possible to write separate linear relationships between forces and fluxes for each of the species. By the Curie principle, the chemical reactions introduce no additional driving forces since the membrane is assumed to be isotropic, and assuming diffusion only in the x direction, we can therefore write the flux equations in the classical manner as
a
In c,
+ ztF$)
c -(RT In c, ”x
+ z,F$)
J , = --u,c,-(RT ax
J,
= --u
where us,ur,and u,, are the mobilities of sites, counterions, and ion pairs, respectively, $ is the electric potential, R is the gas constant, F is the Faraday constant, and T is t.he absolute temperature. We will assume us,u,, and ui, to be constants.
Continuity Equations The law of conservation of mass leads to a set of continuity equations for the total fluxes and concentrations
where the asterisks refer to total quantities (ct*
cs =
Js*
bX
(3)
which, in effect, describes a chemical coupling between the various ijpecies within the membrane. In addition, the concentrations crf the charged species are coupled electrically through the condition of electroneutrality
rJZ*
a
J,, = -uzscrs-(RT In cis)
b
The Journal of Physical Chemistry
(64
=
ct
+
At steady state, the total fluxes are constant as seen from eq 7, a conclusion which is not necessarily valid for the partial fluxes. In fact, as will be seen later, the partial fluxes are generally functions of distance (cf. Figure 3 of part 11, the following article) and the membrane may be thought of as containing loca,l sources and sinks. Because we can invoke the continuity equations only for the total fluxes, the number of variables is increased by n (the number of partial fluxes Jt,)over that in the usual Nernst-Planck treatment. The n additional equations needed to solve the problem are given by eq 2. Boundary Conditions Assuming that boundary processes are not rate limiting, the boundary conditions for the counterions follow from the continuity of electrochemical potentials. Hence we may equate the electrochemical potentials of the counterions across each of the two boundaries 0 and d as
RT In a,’
+ p? + z,FV
+
+
+
+
RT In c t ( 0 ) plo(m) z,F$(O) (i = 1. . . ., n) (sa) =
and
RT In at”
+ p? + ziF$”
RT In c,(d) pt(m) zrF+(d) (i = 1, . . ., n) (8b) =
respectively. The quantities on the left-hand side of eq 8a and 8b refer to the solution phases, while the (23) R. A. Fuoss and C.A. Kraus, J . Am. Chem. Soc., 55, 476 (1933); R. A. Fuoss, ibid., 57,488 (1935); Trans.Faraday Soc., 32,894 (1936).
TRANSPORT OF IONSAND IONPAIRS IN LIQUIDION-EXCHANGE MEMBRANES
quantities on the right-hand side refer to the membrane phase. pi0 and pP(m) are the standard chemical potentials in the solution and membrane phases, respectively, and the superscripts ( I ) and (”) refer to the two solution phases (cj. Figure 1). Subtracting any pair of eq Sa and rearranging gives
ailkt - ajfkj Ci(0) c m
(9)
where the constants k, and IC, are defined as
Equations 9 can be writ,t,enfor all the ions
3865
J,*
=
0
(164
Equation 16a is valid for all values of 5 a t steady state, but in the nonsteady state it is only valid a t the boundaries where
J,*(t,O) = J,*(t,d) = 0
(16b)
Equation 15 expresses the condition that the total number of sites contained in the membrane is constant, regardless of their concentration profile. Expressed per unit area, this number is equal to the average total concentration Ea* times the membrane thickness d. Equation 16a states that the total flow of sites must be zero at all points in the membrane in the steady state, whereas eq 16b indicates that for nonsteady state J,* is zero only at the membrane-solution interfaces. If eq 5b is introduced in eq 16a we get i
where the condition of electroneutrality (4) has been used to obtain the last term. From eq 11the membrane concentrations ~ ~ (can 0 )be expressed in terms of the solution concentration c,’ as
i
Similar expressions hold for the other membrane boundary. Subtracting eq 8a from eq 8b gives the expression for the total membrane potential (V = $ I ’ - $’) in terms of internal potential (A$ = $(d) - $(O)) and the concentrations a t the boundaries as
Combining eq 12 and 13 yields an expression in terms of the activities of the counterions in the solutions and ratio of the site concentrations a t 0 and d
V = A$
+
RT
c,(d) RT - In - + z P c,(O) nl- ziF
-Zat‘k,-
Results Since the electric current I is carried only by the charged species, it is given by
Substituting eq 5 in eq 17 yields i
i
Car”kr (14) i
This equation will be used to express the total potential in terms of external conditions and membrane parameters. The boundary conditions for the sites follow from the assumption that the sites are completely reflected a t the membrane boundaries, which leads directly t o the equations 1
which indicates that a “circulation” of sites exists in the steady state as a result of equal and opposite flows of sites in their dissociated and associated forms. This circulation of material is characteristic of membranes containing carrier^"^^-^^ and the neutral ion pairs may therefore be viewed as ionic carriers. Equations 1-16 describe the total behavior of the system, but we shall only consider special cases: (1) the membrane potential at zero current here and (2) the complete steady-state properties for the case of a single counterion species, in the following paper.
r d
(15)
from which it is seen that the electric current is also given by the sum of the total flows. The potential gradient a t zero current can now be expressed by inserting eq 6a and 6b into eq 17 for I = 0 and rearranging to yield (24) J. .E. Best and J. 2. Hearon, “Minerals and Metabolism,” Vol. lA, C. L. Comar and F. Bronner, Ed., Academic Press, Inc., New York, N.Y., 1960, p 11. (25) A. H. Katchalsky SPd P. Curran, “Nonequilibrium Thermodynamics in Biophysics, Harvard University Press, Cambridge, Mass., 1965, p 204. (26) A. Finkelstein, Bwphys. J., 4,421 (1964).
Volume 71,Number 18 November 1967
J. SANDBLOM, G. EISENMAN, AND J. L. WALKER, JR.
3866
which is recognized as the usual expression for the potential gradient in a “Planck” liquid junction. The difference appears when eq 19 is integrated since the presence of ion pairs will influence the profiles of ct and c,. I n order to perform thicintegration, we shall derive another expression containing bc,/bx. Combining eq 5b, 6b, and 6c, we get
J,*
=
b - u,c,-(RT bX
In c,
+ z,F#)
recombining the various terms in eq 23 to obtain a common denominator.) When eq 23 is integrated between the two boundaries 0 and d , we obtain the internal potential a t zero current, designated as A#o
I n order to arrive at an expression for V o ,the total membrane potential a t zero current, eq 25 is combined with the boundary condit,ions (1 2) and (14) to yield
and inserting the mass-law eq 3, this yields directly
F-Vo zI RT where Equation 21 is now combined with eq 19 to eliminate bc,/bx and solving for the potential gradient, we get
Fzt _ arc._-_ RT bx
-
By rearranging this equation, we finally obtain the expression
cut
i
I=utkial‘
-
s, s, -
(26)
i
are the two integrals appearing in eq 25.
Since in the derivation of eq 26 no equations based on the assumption of steady state were used, eq 26 is valid for nonstationary as well as stationary states. The V o of a liquid ion-exchange membrane is therefore seen from eq 26 to be composed of three terms, only the first of which is independent of site distribution. This is in contrast to the situation in a fixed site membrane where us = 0 and J,* = 0, so that, as can be seen from eq 24 and 25, Voreduces to the first term in eq 26 which is the usual equation for such a membrane.27v28 Steady State
-
RT bx usJs*
RT
(us
l,s,
Cu,ktal” = -In
+ Fg et)Cutc, + uscs~u--” Ki i
i
(23)
General Considerations. I n the steady state when, according to eq 16a, J,* = 0, V ois described by the two first terms in eq 26. Examining the parameter t , given by eq 24 and which determines the behavior of the second term in eq 25, it is seen that the degree of dissociation enters through the quantity
uses
Cf
G
where
(27) S
i
use,
(g, + +
t = ,
(24)
The second term is therefore affected by the total concentration of sites, I?,*,since cts and c, do not vary pro-
l)ptcu# ScS
(That eq 23 is identical with eq 22 can be checked by The Journal of Physical Chemistry
(27) F. Helfferich, “Ion Exchange,” McGrsw-Hill Book GO.,Inc., New York, N . Y., 1962, see eq 8-90. (28) F. Conti and G. Eisenman, Biophys. J.,5,247 (1965); see eq 38.
TRANSPORT OF IONS AND ION PAIRS IN LIQUIDION-EXCHANGE MEMBRANES
portionally when the total number of sites is changed. This fact is most easily illustrated by adding the mass laws (3) and inserting the condition of electroneutrality (4). We get
3867
where the condition of electroneutrality has been taken into account and where we have divided through by c2. The only independent variable now appearing is c1/c2, and the integral can therefore be solved explicitly. This gives
i
and from this it is seen that the quantity given by eq 27 varies inversely as the concentration of free sites, c,. As the degree of dissociation is varied, for example by varying cs from zero to infinity keeping all K t constant, it follows therefore that the quantity t varies from zero to t,, where t,, the transference number of free sites, is given by t, =
uscs
Cu1ct i
+ uses
Consequently, the membrane potential at zero current depends on the site concentrations at intermediate degrees of dissociation. However, when the degree of dissociation is small and the quantity given in eq 27 is small compared to unity, the potential becomes independent of the degree of dissociation. This behavior is analogous to that of fixed-site membranes where Vo depends on the fixed-site density for intermediate values (cf. T e ~ r e l leq , ~22-24), ~ but for fixed-site density values high enough to achieve co-ion exclusion, the potential becomes independent of the site concentration (see below, eq 32). Limiting Case of Complete Dissociation. I n this case is zero (t = 0) and since
s,
is also zero in the steady
where /UPS
us \E2 r = (u1
+
UPS us>K2 -
Uls\
-
- (uz + u,) Ki Uls
(31)
Equation SO depends only on the values of ci of ;he boundaries and i s therefore profile independent, which has important consequences to be discussed later. When the boundary conditions (9) are inserted and eq 30 combined with eq 26, taking into account that = 0 in the steady state since
Js* = 0, then we obtain
for the total potential
state, the potential is described by the first term of eq 26
i
which can be seen to be the proper value for this limit from its agreement with Conti and Eisenman’s eq 35.l3 Special Case of Two Counterions and Strong Association. If we confine ourselves to two counterions and strong association, i.e., when eq 27 is much less than unity, the integral
s, 6 =
(u1
us
in eq 26 can be written as
(2+
1)
+ us) c1 + + us UP
X
The form of eq 32 can be illustrated by plotting the exponent of the potential against the mole fraction of one of the counterions in the external solution. The external solution is kept constant on one side of the membrane and varied on the other in such a way that the total ionic strength is always kept constant. Under these circumstances eq 32 can be written in the following way when solution (’) is constant and solution (”) is varied. (29) T.Teorell, Progr. Bwphys., 3,305 (1953).
Volume 71, Number 1.9 November 1967
3868
J. SANDBLOM, G. EISENMAN, AND J. L. WALKER,JR.
n
+ c v)
where XI” is the mole fraction of species 1 in the external solution on the side which is being varied. This equation is represented in Figure 2 where the exponent of [(Fz,Vo/RT) - constant] is plotted against XI” for various values of 7 . Experimental data can be represented directly by such a plot, but since it can be seen from Figure 2 that the curves are approximately straight lines regardless of the value of 7 , eq 32a might alternatively be represented to a good approximation by a single logarithmic term, defining a set of “average ionic selectivities.” I t should be emphasized that eq 39 has been derived using only the condition that J,* = 0 and does not require the counterions to be in a steady state. T h i s important fact will be used in the following section. All the parameters appearing in eq 32 are measurable by classical methods except uls,the mobilities of the ion pairs. These can be calculated from the current-voltage characteristics as will be shown in the following paper. An interesting limit of eq 32 is obtained by putting all ut = us. In this case 7 = 0.5, and from eq 19 it is apparent that the internal potential vanishes. Although V ois then due to the boundary potentials alone, its value can be seen to depend not only on the expected equilibrium parameters kt and K ibut also, somewhat surprisingly, on the mobilities u , of~ the ion pairs.
0 0 I
// I
//
X‘;
Figure 2. The form of eq 32. Equation 32 is represented by plotting exp[(FziVo/RT) - constant] as a function of mole fraction ( X I ) of species 1 in the external solution. The parameters are chosen arbitrarily so that eq 32 may be written as exp[(FziVo/RT) constant] = exp[(l - 7)In (1 0.9X1”) T In (1 o.lX~”)]. This function is plotted for the following values of 7 : 0, 0.2, 0.5, 0.8, and 1.0.
+
+
+
the present results are directly applicable to a number of experimental situations, not restricted to the steady state. For a more thorough discussion of the different experimental situations and their implications for the observed electrode potential properties, the reader is Transient State referred to Eisenman.SO Despite the many studies of the electrode properties Case of Simple Dilution. It is seen from eq 11, 15, of liquid ion exchangers, none of the measurements puband 16 that a simple dilution of the external solution lished to date appears to correspond to the convectionon one side of the membrane (i.e., all the ai” are changed free steady state to which we have restricted conby the same proportionality factor) will not alter the siderations to this point. This is because the membrane interior concentrations. (The concentration ratio of phase has either been deliberately ~ t i r r e d ~ *or~ J ~counterions is not changed by dilution (cf, eq 11); the because precautions to prevent convective mixing in its remaining two boundary conditions, (15) and (16), interior have not been taken4t5J5J6(the membrane needed to solve for the individual concentrations are phase being so thick that steady states have generally independent of the external solution conditions.) not been reached and because moreover, stable conThe first two terms in eq 14 are therefore unaltered by centration gradients could only be established in the such a dilution, and the potential will vary with the relatively static layers just internal t o the solution natural logarithm of the concentration with a Nernst interfaces). We will show here that the transient slope (Le., RT/ziF) as given by the third term. This behavior of V odepends not only on the membrane pabehavior is a consequence of assuming complete co-ion rameters (particularly the degree of dissociation) but also on the experimental arrangement. Nevertheless, (30) G. Eisenman, Anal. Chem., in press. The Journal o j Physical Chemistry
TRANSPORT OF IONS AKD
ION P+41RS IN
LIQUIDION-EXCHANGE JIEMBRANES
exclusion and completely trapped sites. Since eq 14 is valid at all times, this has the important practical consequence that the membrane potential will respond in a step fashion to a step change in solution conditions whenever this corresponds to a simple dilution. More General Considerations. Except for the above case, however, a step change in external solution conditions will be accompaniedby a redistribution of the sites until a new steady state is reached, which has the following consequences for the behavior of electrode potentials. When a step change is made from a previous steady state, the profiles are initially unaltered. J,* is therefore zero in the interior of the membrane at zero time. At the boundaries J,* is zero at all times by virtue of the boundary conditions (16b). The last integral in eq 26,
l,
is therefore zero instantaneously following a
step change in solution conditions. This i s due to the fact that the initial ion exchange takes place across one of the membrane solution interfaces where the sites cannot move. Since in the limit of complete dissociation /of
eq 26
1
is zero while in the limit of strong association
s,
is a
profile independent (being given by eq 30), we reach the important practical conclusion that the instantaneously observed potential is given by the steady-state expressions (26a) for complete dissociation and (32) for strong association in these two limiting cases, respectively. It should be ernphasized that this conclusion is true not only for convection-free systems but also for stirred systems because
s,
is independent of the concentration
profiles, Equations 26a and 3%' therefore can be used to describe measurements carried out by holding solutions ( I ) constant and comparing the instantaneous values of successive potentials observed for various conditions of solutions ( ' I ) even when no precautions are taken to avoid c ~ n v e c t i o n ~or~ when ~ * ~ ~the , ~ membrane ~ phase is deliberately stirred.'s8I1' In addition, if the membrane interior is deliberately stirred, a quasi-steady state of potential will be rapidly established (again described by eq 26a and 32) since in this case L i s zero because the uniform distribution of sites in the stirred interior reduces J,* to zero there long before the interior concentrations attain their steady-state values. This situation is analogous to that existing in a co-ion excluding fixed-site membrane where V Ois time independent, although the profiles are not in a steady state.Z8
3869
For systems of intermediate dissociation, it is difficult to ascertain the behavior of V oeven when
s,
is zero since
the quantity t (eq 24) cannot be expressed in terms of cI/cz in this case. Liquid Junctions wilhin the Membrane Phase. By virtue of the boundary condition (16b). the second 1
, /
was shown to be zero both in a membrane in a stirred interior and instantaneously following a step change in external solution conditions. I n certain other experimental conditions,lgthe membrane phase has been preequilibrated with different solutions, and the 170 is measured across the liquid junction which is formed when two such membrane phases, each in equilibrium with their respective solutions, are brought in contact. I n this case the sites are not constrained by any boundary arid J,* (and thereforeL) is not necessarily zero. In fact, the membrane potential in these cases has been calculated from eq 19 assuming linear mixtures of free sites and counterions.16
The Parameters Controlling Electrode Specificity We shall here only discuss the parameters as they appear in eq 32 for strongly associated systems. This equation applies instantaneously and at qteady state to convection-free membranes as well as in stirred membranes. Each logarithmic term appearing in eq 32 defines a set of ion selectivities. I n the first logarithmic term, the only parameter depending on the properties of the sites is seen t o be us. This affects the selectivity in a particularly simple way since the sum (ui u,)is given directly by the limiting value at infinite dilution of the equivalent conductance A!:. On the other hand, the parameters k , are related to the limiting values at infinite dilution of the distribution coefficients Size of the salt between the solution phase and the pure solvent of the membrane. If the same anion Xis used in measuring the distribution coefficients of two salts, IX and JX, the ratio [S,,0/Sjzol2 is equal to k , / k , , regardless of the anion species. The standard chemical potentials determine S,," according to the treatment of Shedlovsky and Uhlig,31which has been carried out under the same assumptions used in the present paper, and the square appears because ki/ki is the ratio of the distribution coefficients of the counterione while Stzo/ S,zo is the ratio of the distribution coefficients of the salt.32
+
(31) T. Shedlovsky and H. H. Uhlig, J. Gen. Physwl., 17, 563 (1933). (32) Cf. p 161 of L. W. Holm, Arkit Kemi,5 , 151 (1956).
Volume 71, Number 12
iVouember 1967
J. SANDBLOM, G. EISENMAN, AND J. L. WALKER, JR.
3870
It follows then that the selectivity between I and J is given by
Notice that since the parameters on the right-hand side of eq 33 are all defined for infinite dilution, they are completely independent of ion-site interactions, and the first logarithmic term of eq 32 therefore depends only on the properties of the solvent despite the fact that we are considering a system which is strongly associated. On the other hand, the selectivity between I and J for the second logarithmic term of eq 32 is given by
(34) where K,, is the ion-exchange equilibrium constant for the reaction J+(aqueous)
+ IS(membrane) JS(membrane)
The Journal of Phgsical Chemistry
+ I+(aqueous)
(35)
This result was deduced by combining eq 2 and 3 and introducing the expressions for k, and IC, from eq 10, to yield
From eq 34, the ion-exchange equilibrium constant K u is seen to be the product of a term [(kj/k,) = (S5,0/ Si,0)2] dependent on the properties of the solvent and a term (K,/K,) dependent on ion-site interactions. Finally, it should be noted that if the mobilities of the associated pairs are approximately the same for all counterion species (as seems likely to be approximately true for the usual long-chain liquid ion exchangers), the selectivity of eq 34 reduces to Kif and is therefore the same as that characteristic for equilibrium ion exchange. Thus the selectivity of the second logarithmic term of eq 32 is characteristic of the properties of the site as well as of the solvent.
