LIQUIDION-EXCHANGE MEMBRANES
WITH WEAKLY IONIZED
249
GROUPS
Liquid Ion-Exchange Membranes with Weakly Ionized Groups.
I.
A Theoretical Study of Their Steady-State Properties1
by John Sandblom2 Department of Physiology and Committee on Mathematical Biology, University of Chicago, Chicago, Illinois 60637 (Received September 1, 1967)
The behavior of liquid ion-exchange membranes with weakly ionized groups is examined by applying previous theories of mobile-site membranes to the case of strong association, Le., when the number of ions is small compared with the number of ion pairs. By introducing a number of idealizing assumptions, the flux equations are integrated for a multicomponent system of univalent electrolytes, and the profiles, fluxes, and membrane resistance are expressed in terms of the electric current. It is shown that the current-voltage relationship is practically linear for intermediate fields, whereas higher fields give rise to a nonlinearity which depends not only on the mobilities of the counterions but also on their dissociation constants. Comparisons are also made between membranes with weakly and strongly ionized groups, between liquid and solid ion-exchange membranes, and between heterogeneous and homogeneous membranes. Finally a special case is considered, namely, the simultaneous presence of completely dissociated and strongly associated ions, which is shown to possess “apparent” limiting conductances for intermediate fields.
Introduction
theory for the single-counterion case was also verified experimentally. The present paper continues the treatment by Sandblom, et aZ.,17and considers a multicomponent system with the purpose of presenting a complete steady-state description of the membrane properties including fluxes, profiles, and current-voltage relationships for the limiting case of strong association between sites and counterions. This is a case of practical importance, since most
A liquid ion-exchange membrane consists of a waterimmiscible solvent in which is dissolved an ionic component with a high affinity for the membrane phase.a The properties of such a membrane a-resimilar in many ways to those of a solid ion-exchange membrane, e.g., with respect to the restriction of ionic m ~ v e m e n t the ,~ large degree of s e l e ~ t i v i t y ,and ~ * ~the ability to function as an ion-specific electrode.6 On the other hand, a liquid exchange membrane is also distinguished from (1) This work was supported by research grant GB-4039 from the its solid counterpart by its partial solubility in the National Science Foundation and was aided by U. S. Public Health Service General Research Support Grant FR-5367 and Training aqueous phases4 and by the movement of dissociated Grant 5-T1-GM-833. ion pairs in the membrane interi0r.57~ (2) Department of Physiology and Medical Biophysics, University Although they have attracted considerable interest, of Uppsala, Uppsala, Sweden. liquid ion-exchange membranes have mostly been (3) K. Sollner and G. M. Shean, J . Amer. Chem. Soc., 86, 1901 (1964); G. M. Shean and K. Sollner, Ann. N . Y . Acad. Sci., 137, studied under such experimental conditions that a 759 (1966). steady state has not been attained,4,5J’-10which is the (4) K. F. Bonhoeffer, M. Kahlweit, and H. Strehlow, Z . Phys. Chem. (Frankfurt am Main), 1, 21 (1954). reason why a complete description of the steady-state (5) 0. D. Bonner and J. Lunney, J . Phys. Chem., 70, 1140 (1966). properties analogous to that which has been developed (6) J. W. Ross, Bulletin 92-81, Orion Research Inc., 1967. for fixed-site is presently lacking. (7) M. Kahlweit, Arch. Ges. Physiol., 271, 139 (1960). The first attempt, however, a t describing the steady(8) G. Eisenman, Anal. Chem., 40, 310 (1968). state behavior of liquid membranes was made by (9) M. Dupeyrat, J . Chim. Phys., 61,306, 323 (1964). Conti and Eisenman,I6 who studied a membrane con(10) H. L. Rosano, P. Duby, and J. H. Schulman, J. Phys. Chem., taining mobile sites with complete dissociation between 65, 1704 (1961). sites and counterions. Their theory was later ex(11) T. Teorell, Progr. Biophys. Biophys. Chem., 3, 305 (1953). tended by Sandblom, Eisenman, and Walker,17 who (12) R. Schlogl, Z. Phys. Chem. (Frankfurt am Main), 1, 305 (1954). (13) F. Helfferich, “Ion Exchange,” McGraw-Hill Book Co., Inc., included the effects of association by assuming that New York, N. Y., 1962. sites and counterions combined to form electroneutral (14) Y. Kobatake, J. Chem. Phys., 28, 146 (1958). ion pairs within the membrane. This added consider(15) E’. Conti and G. Eisenman, Biophys. J., 5 , 247 (1965). able complexity to the mathematical treatment, al(16) F. Conti and G. Eisenman, ibid., 6 , 227 (1966). though explicit solutions were obtained for the current(17) (a) J. Sandblom, G. Eisenman, and J. L. Walker, Jr., J . Phys. voltage relationship in the case of a single c ~ u n t e r i o n l ~ ~Chem., 71, 3862 (1967); (b) J. Sandblom, G. Eisenman, and J. L. Walker, Jr., ibid., 71, 3871 (1967). and for the membrane potential at zero current in the (18) J. L. Walker, G. Eisenman, and J. Sandblom, ibid., 72, 978 The & case of two strongly associated c ~ u n t e r i o n s . ~ ~ (1968). Volume 73, Number 1 January 1969
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of the conventional liquid ion-exchange membranes have functional groups which are weakly ionized, owing partly to the relatively low dielectric constants of the organic diluenka A strong association between sites and counterions is therefore expected to be a characteristic property of the unifunctional weakly acidic and basic liquid ion exchangers (e.g., alkylphosphoric acids or alkyl amine^)'^ in solvents of low dielectric constants (
