There are some inconsistencies in the comparison. A notable example is the ester (class 2)-alcohol (class 4) question. Despite the very high recognition (399 of 400 are correctly classified), M.I. is only moderately high, 1.71 bits. The inconsistency results from two factors: 1)the invalidity of the statistical independence approximation for the infrared spectral intervals and 2) the scatter in the plot of recognition vs. the square root of M.I. when statistical independence is true. The first factor may be illustrated by the dependence of the C=O and 0-H stretches in carboxylic acids. The second factor is seen in Figure 2. Thus far, we have no clue to the amount of mutual information that is necessary to absolutely determine the correct class if the statistical independence assumption is valid. When there is complete inter-class information (i.e., no two samples in different classes have the same spectrum), the M.I. is given by M.I. (indep) = -
P(c,) log P ( c j )
(15)
j
[P(cj)again is the probability of occurrence of class j . ] For the two class equiprobable case, P ( c l ) = P ( c z ) = 1h and M.I. (indep.) is 1.0 bit. This implies that 1bit of mutual information is necessary to distinguish separable, equi-probable classes. A value of M.I. greater than 1.0 bit implies that there is redundant information in the data. This, however, is consistent with the statistical dependence of the data. If the mutual information is to adequately rate a classifier, then t,he degree of statistical independence must be similar for each question (this is factor 1 above). Since a high correlation is found in the infrared problem, this is a good supposition and M.I. does act as a guide to the predictive ability of the maximum likelihood estimator. CONCLUSION The mutual information is shown to be directly related to the classification ability of a maximum likelihood classifier.
First, the relationship is demonstrated on an artificially generated set of probability schemes. In this problem, the linear correlation between recognition and the square root of M.I. is given to be 0.98. This high degree of correlation is further tested on a collection of binary infrared spectra. In this case, the linear correlation between recognition and M.I.'/* is 0.83. The decrease in the degree of correlation is caused by the invalidity of the statistical independence assumption for the infrared spectral intervals. The order of magnitude of the mutual information for the infrared problems is a direct function of the probability scheme and the degree of statistical independence. Since the magnitude of the latter component cannot be estimated beforehand, M.I. serves best as a comparative measure of recognition ability. This best describes the role of information theory for predicting classification results.
LITERATURE CITED (1) R . D. Levino and R . Kosloff, Chern. Phys. Lett., 28, 300 (1974). (2) H. P. Yockey. J. Theor. Biol., 46, 369 (1974). (3) A. Eskes, F. Dupuis, A. Dijkstra, H. DeClerq, and D. L. Massart, Anal. Chem., 47, 2166 (1975). (4) C. E. Shannon, Bell Syst. Tech. J., 27, 379 (1946). (5) H. C. Andrews, "Mathematical Techniques in Pattern Recognition". Wiley-lnterscience, New York. 1972. (6) H. B. Woodruff, S. R. Lowry and T. L. Isenhour, Anal. Chem. 46, 2150 (1974). (7) C. K. Chow, /E€€ Trans. Electron. Cornput., EC-14. 66 (1965). (8) H. B. Woodruff, G. L. Ritter, S. R. Lowry, and T. L. Isenhour, Technornetrics, 17, 455 (1975). (9) F. M. Reza, "An Introductionto Information Theory", McGraw-Hill, New York, N.Y., 1961.
RECEIVEDfor review September 8,1975. Accepted March 15, 1976. This work was supported by the Materials Research Center, University of North Carolina under contract number DAHC-15-73-69 with the Advanced Research Projects Agency. The financial support of the National Science Foundation is also appreciated.
Transport Properties of Neutral Carrier Ion Selective Membranes Werner E. Morf, Peter Wuhrmann, and Wilhelm Simon* Department of Organic Chemistry, Swiss federal Institute of Technology, Universitatstrasse 16, 8006 Zurich, Switzerland
A simple theoretical model is evolved for the description of the transport of ions through thick carrier membranes. The transference of mono- and divalent cations as well as monovalent anions is studied in detail. In agreement with experimental data, a correlation is found between the selectivity in the ion transport (permeability selectivity) and the ion selectivity observed potentiometrically. Some extensions of the model demonstrate the applicability of the given expressions and, in addition, offer more insight into the electrical properties of cation-permselective membranes.
26) [see also (17)l.The response characteristics of these sensors can be rationalized to a large extent if it is assumed that the ligands exhibit carrier properties for the ions in question (18, 29) and therefore induce permeability selectivity in the corresponding membranes (20). As a consequence, there should be a correlation between the ion selectivity of membranes observed potentiometrically and the ion transport selectivity of the same membranes, as has indeed been found between conductances and potentials in bilayer membranes (28). Here we report on a simple bulk-membrane model describing quantitatively such relationships, which are in agreement with experimental evidence.
Certain lipophilic, electrically neutral, organic complexing THEORETICAL agents are attractive components for highly cation-selective liquid-membrane electrodes. At the present time, such comThe primary aim of a theoretical analysis of the ion transponents are used for the determination of K+ (2-7), Na+ (8), port through membranes is to describe the ionic fluxes (i.e., Li+ (9),NH4+ (10, I Z ) , Ca2+ (22, 23), Sr2+ (14),and Ba2+ (15, contributions to the electrical current) as a closed function of ANALYTICAL CHEMISTRY, VOL. 48, NO. 7, JUNE 1976
1031
branes [see also Lev et al. (28)]. In contrast, the bilayer models created by the groups of Eisenman (18,21) and Lauger (22) assume permeability for only one ion, thereby excluding a priori any other ions. The boundary values of concentrations and electrical potential, as required for the evaluation of e.g., Equation 2, may be determined by analyzing the ionic fluxes across the phase boundaries. Rigorously, the fluxes between the aqueous solutions and the membrane (for the notation see Figure 1) should be described according to Eyring’s rate theory. Thus, t
I
O
d
- x
Ji =
Figure 1. Schematic representation of the electrochemical cell
ion activities in contact with the membrane and the applied voltage. Generally, a suitable solution can be best obtained for the steady-state, where the ionic fluxes have become constant throughout the cell. Here, the cell consists of two aqueous solutions separated by a nonporous membrane (Figure 1). The ionic fluxes within the membrane of thickness d , which is treated as an ideal phase, are given most correctly by the Nernst-Planck Equation 1 as a function of concentration gradients and electrical potential gradient: [0 < x
11 (35b) zui uy Theories of the zero-current membrane potential of a corresponding membrane electrode (23, 27, 31, 34) lead to the following results:
+
E = const
+ RT -In a ZF
[Ka > 11
(36b)
ANALYTICAL CHEMISTRY, VOL. 48, NO. 7, JUNE 1976
Using Equation 35, these expressions may be condensed to one formula:
E = const +
(+ ti - 1) y l n a
(37)
Obviously, in terms of the model discussed here, a high cation permselectivity is required in order that membrane electrodes yield a cationic EMF-response. Saturation effects, which are observed with carrier membrane electrodes in the presence of highly lipid-soluble sample anions, such as thiocyanate or picrate, are described elsewhere (27, 31,34,35). ' 3. Extensions of the Model. a. Nonsymmetrical Cell Assembly. For a description of membranes in contact with two aqueous solutions of different composition, the simple model is not appropriate and, thus, an extended theory is required. The most comprehensive treatment of electrodiffusion through aqueous diffusion layers or bulk membranes was offered by Schlogl (23) in a pioneering but, unfortunately, scarcely cited work. Based on the steady-state assumption, the electroneutrality condition and the assumption of interfacial equilibria, this model permits an implicit description of the ionic fluxes. For general applications, however, the numerical evaluation becomes rather grueling. In the limiting case of two classes of permeating ions and high voltages (values in the order of 1V), Schlogl's extended theory leads to a description of the ion transference which is in analogy to the relationships derived above for symmetrical cells. Thus, the selectivity in the cation transport (cases a. and b. in Section 2) is generally given by Equations 25 and 26, respectively, Equations 29 and 30, where the ion activities refer to the aqueous solution on the side of the cation uptake into the membrane. For the description of the anion transference (case c.), Equation 33 can also be applied without corrections. Here, the anion concentration refers to that membrane boundary where the anions enter, respectively, the cations leave the membrane. Accordingly, the anionic transference number is determined by the composition of the adjoining aqueous solution, but is independent of the opposite solution. These theoretical results are confirmed by experimental evidence (see below). b. Deviation from Interfacial Equilibrium (Kinetic Limitations). For thin membranes, the ionic fluxes may become large at rather low values of A+, respectively E , as is suggested by Equation 13. Thus, large displacements from the equilibrium state must be considered to occur at the membranesolution interfaces. As a consequence, Equations 8 and 9 of
the simple model have to be replaced by the complete description, Equations 3 and 4. I t is immediately seen that the validity of Equations 17 and 18 is limited in this case. Nevertheless, the residual framework of the theory presented above may be applied without further corrections when restricting considerations to a symmetrical cell containing only one permeating cation. Then, a generalized form of Equation 8 may be found by combining Equations 3-7: (8a)
nonlinearity of the current-voltage characteristic, the ionic fluxes are no longer given by Equation 18 but may be approximated by Equation 42. Hence, the transference number ratio is found to be:
(43) For cations of the same valency, the potential term vanishes and, in this case, we may characterize the ion transport selectivity by:
Together with Equation 16, we get: pil2
(44)
(38)
G e z Ci iF ( E - A d ) / 4 R T
Inserting this potential function into Equation 3, one obtains the following description of the flux across the phase boundary: Ji
= - 2 Riai
@ Kiai sinh
1z.F
-
(39) [ 4 R T (E A4)] I
Recalling Equation 13 for the flux within the membrane, we may derive a relationship between the internal and the total membrane potential difference, A$ respectively, E: A
[
2.F 1 ziF A 4 = sinh -- (E - Am)] RT 4 RT
(40)
where
Equation 40 is formally identical to an expression derived by Lauger’s group (22) from a different model which is based on the equilibrium assumption but considers deviations from electroneutrality. Hence, a different factor A was found. However, the dependence on the parameters d and ai is the same for both approaches. For a discussion of Equation 40, we consider two limiting cases. For relatively thick membranes and sufficiently fast reaction kinetics a t the membrane boundaries, we have A > 1, which is fulfilled for thin membranes, respectively for slow interfacial reactions, it is the ion transfer from the aqueous phase into the membrane that is rate-determining. Here, we obtain A 4 = 0 and, according to Equation 39, we expect a pronounced nonlinearity of the current-voltage characteristic [see also (2211. For higher voltages, i.e., -E > 0.1 V, Equation 39 reduces to a pure exponential relationship:
-
2/%Kiai
The same limiting value was derived by Eisenman’s group (21) for the permeability ratio of carrier membranes; the parameters k were found to be determined by the rate constants of the complexation reaction. The pronounced ion specificity in the complex formation by carriers, however, is to a large degree given by the rate of the decomplexation reaction (36, 37). Therefore, a certain loss of ion selectivity has to be tolerated in the “kinetic domain”. In contrast, a drastic change in the ion selectivity due to kinetic limitations must be expected if cations of different charge are present. The reason is that the potential function p, in analogy to Equation 38, is generally related to the term exp[F(E A 4 ) / 2 R T ] . Therefore, we may conclude from Equation 43 that the cation of the highest valency is strongly preferred in the “kinetic domain” when the applied voltage is increased. c. Limitations Resulting from Closed-Circuit Flux of the Carrier within the Membrane. For thick carrier membranes, we may usually apply the assumption of a thermodynamic equilibrium at the phase boundaries as a good approximation (see above): On the other hand, i t should be considered that a positive flux of cationic complexes within the membrane (as induced by an applied voltage E < 0) leads to a certain accumulation of free carriers a t z = d, respectively to a depletion at x = 0. Thus, for a more rigorous treatment, we have to modify Equation 11as follows:
-
Restricting ourselves to a symmetrical cell containing only one permeating species, we then find from Equation 8: (46) It is evident that, in this case, Equations 9,17, and 18 of the simple model are no longer valid. Combination of Equations 16 and 46 clearly shows that any departure from a uniform carrier distribution leads to a deviation between the internal and the total membrane potential difference:
7
Ji
= kiai
Recalling Equation 38, we find immediately: Ji
(47)
,-ziFE/4RT
= giaiF-zi/2
(42) A comparison of this result with Equation 3 reveals that, in the present case, the ionic flux is given by the rate of the forward reaction alone. It is obvious that large displacements from the interfacial equilibria should lead to serious changes in the selectivity behavior of a membrane, as is observable when two or more cations are simultaneously involved in the partitioning reactions. Thus, in the “kinetic domain” of membrane properties [see also (21)] which is generally recognizable by a strong
For practical purposes, the carriers are confined to the membrane phase in the case of ideal bulk membranes [in contrast, a supply of carriers from the outside solutions is stipulated for transport studies on bilayers (21, 38)]. Thus, the following steady-state conditions hold:
,
Ctot
= nci
+ c8 = const
et= nJi + J , = 0
(48) (49)
Here, the symbols i and s signify the complexed cation and the free carrier, respectively. The term cB in Equation 48 denotes the mean concentration of free carriers which is equal to the concentration in the center of the membrane: ANALYTICAL CHEMISTRY, VOL. 48, NO. 7, JUNE 1976
1035
lim J7at = 0 CQ-0
A similar trend may be derived from the bilayer treatment of Eisenman's group (21); because of differing model assumptions, however, the corresponding results are more complicated. In order to discuss the electrical properties of thick carrier membranes at low voltages, we consider two limiting cases. For membranes with predominantly free carriers: uici
5
0
10
20
15
I S E I
Figure 4. Theoretical current-voltage characteristics of thick carrier membranes with different charge concentrations c = zic; Curves computed using Equations 48,52,53, and 55 with n = 1 and u, = us.The values for c a r e arbitrary and increase from curve A to D
We may conclude from Equations 52 and 53 that A$ = E as long as I Jil < Accordingly, the formalism of the simple model gives a satisfactory description of the electrical properties in this case. From Equation 53, respectively Equation 18,we expect a pronounced ohmic behavior of the membrane in the lower voltage range. The membrane conductance is found to be determined by the concentration and mobility of the cationic complexes. Obviously, when the charge concentration c within the membrane is increased, the ohmic conductance increases by the same factor (see curves A and B in Figure 4). A contrasting behavior is found for membranes with predominantly complexed carriers where: uici
Using Equations 49 and 2, one obtains the following relation between the ionic flux and the induced concentration gradient of free carriers:
With the help of Equations 47 and 50, the unknown concentration values in Equation 51 may finally be replaced by a function of the boundary potential difference: Ji=
__-2 u,RTc,
1 ZiF tanh -- ( E - A $ ) [ 2 n RT
]
(52) n d As a second relation between Ji and A$, we recall the reduced Nernst-Planck flux Equation 13 written in the form:
Together with Equation 48, a numerical evaluation of the function J i ( E ) is possible. Inspection of these results shows that a saturation of the ionic flux, respectively the electrical current, is predicted a t high voltages: 2 u,RTc, J y = f --
-
[E =F a] (54) n d The existence of such an asymptotic value is due to the fact that the back-diffusion of free carriers within the membrane cannot be accelerated beyond a certain limit. This may be seen more clearly from Equation 51, where the boundary concentrations, according to Equation 50, can never exceed the limiting values 2csr respectively 0. Evidently, if the fraction of complexed carriers is negligibly small, then the flux at extreme voltages may attain the highest possible value: (55)
When the charge concentration c = zici within the membrane is increased, the saturation flux, respectively the saturation current, decreases to the same extent (see Figure 4). For membranes with nearly all carriers complexed, finally, the circulation of carriers collapses and hence we find: 1036
ANALYTICAL CHEMISTRY, VOL. 48, NO. 7, JUNE 1976
> usc,
From Equations 52 and 53, we get A+ = 0 in this case. Hence, the ionic flux may be described by the approximation:
1
1 z ~ F Ji = J y t tanh --E (57) 12nRT An equivalent expression holds for liquid membranes based on dissociated ion-exchangers (39). It is evident that the current-voltage characteristic of such membranes becomes strongly nonlinear (saturation is attained at rather low voltage values). The electrical properties are directly related to the mean concentration and the mobility of the free carriers (cf. Equation 54). Thus, in contrast to the behavior predicted above, the zero-current membrane conductance is here found to increase with decreasing complex concentration (i.e., with increasing carrier concentration). This implies that a maximum must exist for the zero-current conductance of the membrane at intermediate charge concentrations c (see curves C and D in Figure 4). Summarizing this section, we can state that the parameter c largely determines the shape of the current-voltage curve of a carrier membrane.
EXPERIMENTAL Cell Assemblies. In the electrodialysis experiments, the cell consisted of two electrolyte compartments (cylinders of 2.5-cm diameter and 5-cm length) separated by a cation-permselective membrane (disk of 0.5-cm diameter and -0.Ol-cm thickness). Both compartments were equipped with Ag, AgCl electrodes (disks of 2-cm diameter) and provided with magnetic stirring. The cell was placed in a thermostated Teflon block (25 "C). Far the EMF measurements, the cell: Hg,Hg~Clz; KCl(satd)/O.l M NH4NOs/sample solution//membrane//O.Ol M KC1; AgC1, Ag (25 "C) was used. The membrane was mounted in a Philips electrode body type IS 560. Two types of carrier-based PVC membranes were used for the experimental studies. The calcium-selective membrane was prepared from a mixture of synthetic Ca2+-carrier (12) (3%),o-nitrophenyl octyl ether (65%),and PVC (3%). The composition of the sodium-selective membrane was: Na+-carrier (8) (3%), dibutyl sebacate (65%), and PVC (32%).The membranes were prepared as described earlier (40). Procedures. For the measurement of the current-voltage curves of the Ca2+-permselectivemembrane, the two compartments of the electrodialysis cell were filled with identical CaC2 solutions (ranging from to lo-' M). A maximal voltage of about 25 V was applied
1000
I
nNaM
I
600
l I
1-
I
1
TRANSPORT
p.: lO-‘M
200
10-1M
EMF I
1 0
-5
-10
-15
-20
-25 ECV)
Figure 5. Experimental current-voltage characteristics of a PVC membrane with a synthetic Ca*+-carrier Measured curves for different concentrations of the external CaClpsolutions and the respective steady-state current was attained after 5 to 15 min. Further current values were obtained by a stepwise reduction of the applied voltage. The standard deviation of a single current-respectively voltage-determination was less than 1%. For the determination of the transport selectivity between alkali ions for the Na+-carrier membrane, the anode compartment of the electrodialysis cell was filled with a solution of 5 X M NaCl and 5X M MC1 (M+ = Li+, Na+, K+, NH4+, Rb+, Cs+) whereas a lo-* M KCl solution (respectively RbCl for M+ = K+) was used in the cathode compartment. A constant voltage was applied over the membrane so that an electrical current of about 2 KA (equivalent to 0.1 kA/mm2) was induced. The transfer of the cations Na+ and M+ through the membrane led to a concentration increase of the respective species in the cathode compartment which was studied by flameless atomic absorption spectroscopy. T o this end, 5 probes of 10 ~1 were collected every 30 min up to 90 min. The standard deviation of such a concentration determination was about 5%. Finally, the transference number of an ion was obtained as the ratio of the charge equivalent of transferred species and the current-time integral. The latter was measured by means of an integrator (41). For the ion transport studies on the Ca2+-carrier membrane, the general procedure was the same as above. When investigating the selectivity between the cations Ca2+ and Na+, we used 1:l mixtures of the respective chlorides in the anode compartment of the cell and KC1 solutions of the same total cation concentration in the cathode compartment. For the study of the transport selectivity between permeating cations and anions, Ca2+ and SCN-, equimolar solutions of CaC12 and KSCN were used in the anode and cathode compartment, respectively. For the determination of the potentiometric ion selectivity of carrier membranes, EMF measurements were carried out with the respective membrane electrodes, using an impedance converter (input impedance: lOI4 Q)in conjunction with a digital voltmeter and a printer. The alkali ion selectivity of the Na+-sensitive membrane electrode was M solutions of the alkali obtained from measurements on 5 X chlorides MC1. The monovalenMivalent cation selectivity of the Ca2+ electrode was also determined by the separate solution method, using equimolar solutions of NaCl and CaC12, respectively The standard deviation of a single EMF-determination was 0.1 mV. Reagents. Synthetic carriers: for preparation, see (42).PVC: SDP Hochmolekular, Lonza AG, Basel, Switzerland. o-Nitrophenyl octyl ether: for preparation, see ( 4 3 ) .Dibutyl sebacate: Fluka AG, Buchs, Switzerland. Salts: Merck AG, Darmstadt, West Germany. Apparatus. Power supply: type 401, Weir Electronics Ltd., Bognor Regis, Sussex, England. Digital voltmeter: type 7100A, Fairchild Instruments, Mountain View, Calif. 94040. Amphremeter: type 150B, Keithley Instruments, Cleveland, Ohio 44139. Impedance converter: type 71S, Firma Knick, Berlin, West Germany. Printer: type DT 21 Mk 11, Wenger Datentechnik, Basel, Switzerland. Atomic absorption spectrometer: type 300 (graphite cell type HGA 7 2 ) ,Perkin-Elmer, Ueberlingen, West Germany.
RESULTS AND DISCUSSION The current-voltage characteristics of thick Ca2+-selective carrier membranes are shown in Figure 5 [see also (44)]. A comparison with Figure 4 reveals a striking parallelism be-
1.5
1 rO
Figure6. Transport selectivity and potentiometric selectivity of a Na+carrier membrane Experimental coefficients K ~ obtained M with Equation 26, respectively,Equation 27, given as a function of the ionic radius of M+
tween theoretical and experimental curves. It is evident that, in practice, the membrane parameter c is not a real constant but increases to a certain degree with increasing activity of the exterl solutions. A more detailed experimental investigation of the electrical properties of thick carrier membranes was performed by Lev et al. (28);the auors also offered an interesting theoretical approach, which, however, is limited to 1:l electrolytes. For polar membranes with the carrier valinomycin, the internal membrane conductance in the ohmic limit was found to vary moderately (in analogy to Figure 5 ) ; whereas, for nonpolar carrier membranes, the electrical properties were nearly independent of the composition of the aqueous solutions contacting the membrane (28). This is in striking contrast to the findings for bilayer membranes (21, 22,38) where the zero-current conductance is usually directly proportional to the outside activity of the permeating ion as well as to its overall partition coefficient, which mimics the ion selectivity of the neutral carrier. So far, there is no cogent explanation for the “unusual extraction behavior” (28) observed for thick carrier membranes. However, a qualitative understanding of many phenomena may be obtained by postulating that the membrane behaves as a phase with fixed negative charges of roughly constant concentration c (see Section 1.). For thin carrier membranes, the transport selectivity between ions of the same charge can usually be determined by simple conductance measurements (21, 38). Obviously (see Equations 18, 19, 22-24), such a procedure would lead to rather poor apparent selectivities when applied to thick membranes (28). Nevertheless, a pronounced ion specificity can be observed in electrodialysis experiments if different ions are permeating simultaneously and if their transference number is studied (see Section 2, a,). In Figure 6, a correlation obtained potentiobetween selectivity parameters K N ~ M metrically and the corresponding values found in electrodi’alysis experiments is given. For both experiments, the same type of thick PVC membrane containing a synthetic neutral carrier in dibutyl sebacate has been used (see Experimental Section). Although widely different methods have been used to determine the ion selectivity, the agreement between the two sets of data is surprising .and corroborates the model presented. The deviations for Rb+ and Cs+ may possibly be due to kinetic limitations suggesting a loss in transport selectivity (see Section 3, b,), For cations of different charge, Figure 2 predicts a pronounced concentration dependence of the transference numbers. This trend is confirmed by experimental evidence (see Figure 7). The slight variations of KcsNa a t higher concentrations may easily be rationalized by certain changes in ANALYTICAL CHEMISTRY, VOL. 48, NO. 7, JUNE 1976
1037
TRANSFERENCE NUMBER
logKCaNa
5-
't
4-
3I
I
I
,
-5
-4
-3
-2
,
-1
2-
,
0 logrn
1-
Flgure 7. Transference number for Ca2+, as obtained In the electrodlalysls of Ca2+ and Na+ through a Ca2+-carrler membrane
0-
Experimental polnts as a function of the concentration mot each cation in the anode compartment. Theoretical curves accordlng to Flgure 2
-1
-5
the parameter c mentioned above. A drastic change in the observed transport selectivity occurs at low sample concentrations (Figure 7 ) .The same behavior is, however, found in the potentiometric study of ion-selective liquid membranes and is related to the processes determining the lower detection limit of such systems. Accordingly, there is a close correlation between the experimental selectivity parameters obtained potentiometrically and those found by transport studies (see Figure 8). The model presented here suggests that anions of high membrane solubility (lipophilic anions; large values of k,, respectively K) may participate in the ion transport. The theoretical approach, however, indicates a substantial concentration dependence of the anion transference (see Figure 3). Clearly such a behavior is found experimentally when the lipophilic anion SCN- is present in the cathode compartment of the electrodialysis cell (Figure 9). The deviations between experimental points and theoretical curves are similar to those in Figure 7 . In agreement with the outlines given in Section 3, a,, the anion transference is found to be controlled exclusively by the composition of the aqueous solution in the cathode compartment. Accordingly, the same value of 0.68 f 0.03 was found for tcawhen using a solution of M KSCN in the cathode compartment and CaC12 solutions of varying concentration in the anode compartment (40,45). In addition to a more fundamental understanding of the ion selectivity of membrane electrodes, the theoretical and experimental results presented here suggest attractive applications of neutral carriers for highly selective ion separations. They show that one given ionophore may act as carrier for different cations; which one is preferentially transported depends on the composition of the solutions contacting the membrane, on the properties of the membrane [see ( 4 6 ) ] ,as well as on the magnitude of the electrically induced ion fluxes. The carrier-membrane model may offer an explanation for some transport phenomena observed in biological systems.
LITERATURE CITED 2 . Stefanac and W. Simon, Chlmla (Switzerland), 20, 436 (1966). L. A. R. Pioda, V. Stankova, and W. Simon, Anal. Lett.,2, 665 (1969). I. H. Kruli, C. A. Mask, and R. E. Cosgrove, Anal. Lett.,3, 43 (1970). 0. Ryba and J. Petranek, J. Electroanal. Chem., 44, 425 (1973). (5) 0. A. Rechnitz and E. Eyai, Anal. Chem., 44, 370 (1972). (6) J. Koryta, "ion-Selective Electrodes", Cambridge University Press, Cambrldge-London-New York-Melbourne, 1975. (7) E. Pretsch, D. Ammann, and W. Simon, Res./Dev., 25, 20 (1974). (8) D. Ammann, E. Pretsch, and W. Simon, Anal. Lett.,7, 23 (1974). , E. Pretsch, and W. Simon, Anal. Len.,8(12), 657 (9) M. G ~ Q QU.~ Fiedier, (1975). (IO) R. E. Cosgrove, C. A. Mask, and I. H. Kruil, Anal. Lett.,3, 457 (1970). (11) R . P. Scholer and W. Simon, Chlmla(Swltzerland), 24, 372 (1970). (12) D. Ammann, E. Pretsch, and W. Simon, Anal. Lett.,5, 843 (1972). (13) D. Ammann, M. Guggi, E. Pretsch, and W. Simon, Anal. Lett.,8(10), 709 (1975). (14) E. W. Baumann, Anal. Chem., 47, 959 (1975). (1) (2) (3) (4)
1038
ANALYTICAL CHEMISTRY, VOL. 48, NO. 7, JUNE 1976
-4
-3
-2
-1
0
log rn
Flgure8. Transport selectivity and potentiometric selectlvity of a Ca2+carrier membrane Experimental coefficients Kc.N. obtalned wlth Equation 30, respectively, Equation 31, given as a function of the catlonlc concentration m
TRANSFERENCE NUMBER 'ca
uCa/uSCN= 0 25
1-
1 -1,
-3
-2
-I
0
log rn
Flgure 9. Transference number for Ca2+, as obtained in the electrodlalysls of Ca2+ and SCN- through a Ca2+-carrler membrane Experimental points as a function of the molarity of the external solutions (45). Theoretical curves similar to Figure 3
(15) R. 1. Levlns, Anal. Chem., 43, 1045 (1971); 44, 1544 (1972). (18) W. Slmon, E. Pretsch, D. Ammann, W. E. Morf, M. Wggi, R. Bissig, and M. Kessler, Pure Appl. Chem., 44 (3), 613 (1975). (17) D. Ammann, R. Bissig, 2. Clmerman, U.Fiedler, M. Guggi, W. E. Morf, M. Oehme, H.Osswald, E. Pretsch, and W. Slmon, in "Proceedlngs of the international Workshop on ion Selectlve Electrodes and on Enzyme Eiectrodes in Biology and in Mediclne", Urban & Schwarzenberg, MunichBerlin-Vienna, 1975, In press. (18) G.Elsenman, S.M. Ciani, and G. Szabo, Fed. Roc., 27, 1289 (1968). (19) S.M. Ciani, G. Elsenman, and (3. Szabo, J. Membr. Blol., 1, 1 (1969). (20) N. Lakshminarayanaiah, "TranspMt Phenomena in Membranes", Academic Press, New York, 1969. (21) S.M. C h i , (3. Elsenmen, R. Laprade, andG. Szabo, In "Membranes", Vol. 2, G. Elsenman, Ed., Marcel Dekker, New York, 1973. (22) P. Uuger and B. Neumcke, in "Membranes", Vol. 2, G. Elsenman, Ed., Marcel Dekker, New York, 1973. (23) R. Schibgi, 2.Phys. Chem. (FrankturtamMeln),1, 305 (1954), (24) M. Pianck, Ann. Phys., 38, 161 (1890): 40, 561 (1690). (25) T. Teoreii, 2.Elektrochem., 55, 460 (1951). (26) 0. Kedem, M. Perry, and R. Bloch, IUPAC International Symposium on Selective Ion-Sensitive Electrodes, paper 44, Cardlff, 1973. (27) J. H. Boles and R. P. Buck, Anal. Chem., 45, 2057 (1973). (28) A. A. Lev, V. V. Malev, and V. V. Osipov, in "Membranes", Vol. 2,G. Eisenman, Ed., Marcel Dekker, New York, 1973. (29) J. A. V. Butler, Trans. Faraday Soc., 18, 729 (1924). (30) T. Erdey-Gruz and M. Volmer, 2.Phys. Chem. (Lelpzlg),150, 203 (1930). (31) W. E. Morf, G. Kahr, and W. Simon, Anal. Len., 7, 9 (1974). (32) 0. Elsenman, in "ion-Selective Electrodes", R. A. Durst, Ed., NaN. Bur. Stand. Spec. Pub/., (U.S.A.),314, Washington, D.C., 1969. (33) W. E. Morf. D. Ammann, E. Pretsch, and W. Slmon, Pure Appl. Chem., 36, 421 (1973). (34) W. E. Morf and W. Simon, in "Ion-Selective Electrodes", M. S.Frant and J. W. Ross, Ed., in press. (35) W. E. M , D . Amrnann,and W. Slmon, Chlmkr(Swkerkrnd),26,65(1974).
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1975.
RECEIVEDfor review November 10,1975. Accepted February 19,1976. This work was partly supported by the Schweizerischer Nationalfonds zur Forderung der wissenschaftlichen Forschung.
Transient Potentials in Ion-Specific Electrodes Adam Shatkay
Israel Institute for Biological Research, Ness-Ziona, Israel
Three aspects of the transient potentials in Ion-specific electrodes are Considered: the various physical models for the mechanism of the transient; the approximations and assumptions implicit In the mathematical treatment of the models; and the agreement of experimental results with the theory. Detailed analysis is carried out for the various “energy barrier” models and the “diffusion through stagnant layer” models. Some improvements in treatment are suggested, and some new Implications of the models are considered. Additional experimental data are presented and analyzed, suggesting that the transients are a complex phenomenon, and that several models have to be applied simultaneously to account for them satisfactorily.
STATEMENT OF THE PROBLEM When ion-specific electrodes are employed (either in a finite volume of some solution or in a flowing system), it is assumed that on a step-change in the concentration (activity) of the measured sample a fast response follows, and once this response is completed a time-independent reading can be safely postulated. The speed of response is of interest to the analyst, and should be considered as one of the chief characteristics of the electrode. Ion-specific electrodes are recommended for the monitoring of flow processes ( I ) , for continuous analyses (2),and for the measurement of reaction kinetics (3-5). In all these cases it is assumed that the electrode response is fast as compared with the velocity of the process investigated. This assumption is not always justified (6);for instance Brand and Rechnitz (7) criticize on this ground the experimental results of Johansson and Norberg (8) to which we shall refer later. It may cause some surprise to note that the subject has been comparatively little studied, despite the extensive interest in ion-specific electrodes in general (9-13). While well over 1000 publications appear annually treating ion-specific electrodes, we estimate that less than ten papers per year deal with even a single aspect of the transient potentials. This appears to stem from two causes: first, the theoretical treatment is fairly complicated; next, the experimental test of the theory is also difficult. On sifting the literature dealing with transient potentials it is possible to come to the following general conclusions:
a. Duration of the Response Time. The response times studied cover a very wide time-scale. By response time we mean the time taken by the emf to change from its initial value to within a given limit of the “final” value. Some authors use this term when referring to a shift of 50% of the difference between the two emf‘s ( t 1 / 2 ) , others use a shift of 95% ( t 9 5 ) , or even of 99% ( t g g ) . One finds that the response times can be very short, from hundredth of a second to a few seconds [Ca2+specific membrane electrode (14), Ca2+liquid membrane electrode ( 5 , 1 5 ) , cation sensitive glass electrodes (16), AgI membrane electrode ( I 7), cyanide-sensitive solid-state membrane electrode (1), silver sulfide single crystal electrode (18),glass pH electrodes (15),and C1- liquid membrane electrodes ( 1 5 ) ] . The response time can also be of the order of minutes [Na+ liquid membrane electrodes (19), silver sulfide membrane electrode (20), Na+ specific glass electrode (21), pH glass electrode (8), I- membrane electrode (22), silver halide membrane electrodes (23)]. Response times of the order of hours have also been reported [cation-sensitive glass electrodes (24, 25), silver sulfide membrane electrode as shown in our results further in the present paper, Cu2+specific solid state electrode (26),collodion membrane electrodes (27)]. Even longer response times can be encountered, though they are rarely reported (28). b. Effect of Concentration on the Response Time. Concentration of the ion to which the electrode responds affects the response time. First, there is the question what happens when the ionic concentration C1 changes to a concentration C * = aC1,assuming that in a series of experiments C1 varies, while a remains constant. For instance when a = 2, we might consider the transfer of the electrode from C1 = M solution to 2 X M, or from C1 = to 2 X 10-3 M solution. Some authors note that changes in C1 do not affect the response-time in such experiments (11, 14, 18, 22, 29). Others note a change in the response time with the change of C1;as C1 decreases, the response time increases (8,17,19,26, 28,30, our results presented further in this paper). Next, there is the related question of C1 remaining constant in a series of experiments, while a changes. According to some authors the size of a has but little effect on the response time (11,221. ANALYTICAL CHEMISTRY, VOL. 48, NO. 7, JUNE 1976
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