A Generalized Model for Apparently âNon-Nernstianâ Equilibrium

Anal. Chem. 2003, 75, 3329-3339

A Generalized Model for Apparently “Non-Nernstian” Equilibrium Responses of Ionophore-Based Ion-Selective Electrodes. 1. Independent Complexation of the Ionophore with Primary and Secondary Ions Shigeru Amemiya,*,† Philippe Bu 2 hlmann,‡ and Kazunori Odashima§

Department of Chemistry, School of Science, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan, Department of Chemistry, University of Minnesota, 207 Pleasant Street Southeast, Minneapolis, Minnesota 55455, and Faculty of Pharmaceutical Sciences, Nagoya City University, Tanabe-dori, Mizuho-ku, Nagoya, 467-8603, Japan

A generalized model that describes apparently “nonNernstian” equilibrium responses of ionophore-based ionselective electrodes (ISEs) is presented. It is formulated for primary and secondary ions of any charges that enter the membrane phase and independently form complexes with the ionophore, respectively. Equations for the phase boundary potential model were solved numerically to obtain whole response curves as a function of the sample activity of the primary ion, and analytical solutions could be obtained for apparently non-Nernstian response sections in these response curves. Ionophore-based ISEs can give three types of apparently non-Nernstian equilibrium responses, i.e., apparently “super-Nernstian”, “invertedNernstian”, and “sub-Nernstian” responses. The values of the response slopes depend on the charge numbers of the primary and secondary ions and on the stoichiometries of their complexes with the ionophore. The theoretical predictions for super-Nernstian responses agree well with the experimental results obtained with ISEs based on acidic ionophores or metalloporphyrin ionophores. Also, theoretical response curves with inverted-Nernstian slopes were found to be similar in character to the pH responses of Ca2+-selective electrodes based on organophosphate ionophores, which have been known to exhibit a so-called “potential dip”. The quantitative understanding of apparently non-Nernstian response slopes presented here provides an insight into ionophore-analyte complexation processes in ISE membranes and should be helpful for the design of new ionophores. During the past four decades, a large number of potentiometric ion-selective electrodes (ISEs) based on electrically charged or neutral ionophores have been developed and applied in many * Corresponding author. Address: Department of Chemistry, University of Pittsburgh, 219 Parkman Avenue, Pittsburgh, PA 15260. Phone: 412-624-1217. Fax: 412-624-5259. E-mail: [email protected]. † The University of Tokyo. ‡ University of Minnesota. § Nagoya City University. 10.1021/ac026471g CCC: $25.00 Published on Web 06/03/2003

© 2003 American Chemical Society

fields.1-3 Extensive studies of the potentiometric response mechanism revealed that Nernstian ISE responses are based on selective equilibrium partitioning of analyte ions between the sample and the membrane at their phase boundary.4,5 This phase boundary potential model was recently extended to describe equilibrium potentiometric responses of ISEs in mixed-ion solutions,6,7 which has been a long-standing problem of practical importance in the field.8 There are also several examples of apparently “non-Nernstian” responses that have been understood theoretically.9 On one hand, some of them have been explained as nonequilibrium steady-state non-Nernstian or Hulanicki-type responses.10 They are also based on a so-called mixed potential at the sample/membrane interface.11 In these cases, ion-exchange or coextraction processes at the membrane interface induce a local concentration polarization of the ions that can be partitioned between the two phases. As a result, quasi-steady-state fluxes of the ions across the membrane interface are achieved even though the sum of the currents carried by all ions together is always zero. Because of the concentration polarization, such potentiometric responses deviate from the Nernstian responses expected from the analyte concentrations in the bulk of the sample. This response mechanism helped to understand selectivities12 and detection limits13,14 of ISEs and has been applied for potentiometric polyion sensors. 15,16 (1) Umezawa, Y. Handbook of Ion-Selective Electrodes: Selectivity Coefficients; CRC Press: Boca Raton, FL, 1990. (2) Bu ¨ hlmann, P.; Pretsch, E.; Bakker, E. Chem. Rev. 1998, 98, 1593-1688. (3) Umezawa, Y.; Bu ¨ hlmann, P.; Umezawa, K.; Tohda, K.; Amemiya, S. Pure Appl. Chem. 2000, 72, 1851-2082. (4) Morf, W. E. The Principles of Ion-Selective Electrodes and of Membrane Transport; Elsevier: New York, 1981. (5) Bakker, E.; Bu ¨ hlmann, P.; Pretsch, E. Chem. Rev. 1997, 97, 3083-3132. (6) Bakker, E.; Meruva, R. V.; Pretsch, E.; Meyerhoff, M. E. Anal. Chem. 1994, 66, 3021-3030. (7) Na¨gele, M.; Bakker, E.; Pretsch, E. Anal. Chem. 1999, 71, 1041-1048. (8) Guilbault, G. G.; Durst, R. A.; Frant, M. S.; Freiser, H.; Hansen, E. H.; Light, T. S.; Pungor, E.; Rechnitz, G.; Rice, N. M.; Rohm, T. J.; Simon, W.; Thomas, J. D. R. Pure. Appl. Chem. 1976, 48, 127-132. (9) Bakker, E.; Meyerhoff, M. E. Anal. Chim. Acta 2000, 416, 121-137. (10) Maj-Zurawska, M.; Sokalski, T.; Hulanicki, A. Talanta 1988, 35, 281-286. (11) Kakiuchi, T.; Senda, M. Bull. Chem. Soc. Jpn. 1984, 57, 1801-1808. (12) Yoshida, Y.; Matsui, M.; Maeda, K.; Kihara, S. Anal. Chim. Acta 1998, 374, 269-281.

Analytical Chemistry, Vol. 75, No. 14, July 15, 2003 3329

On the other hand, we have recently shown that apparently non-Nernstian responses as observed with certain liquid membrane ISEs can be explained on the basis of thermodynamic equilibrium models17-19 and are not related to ion fluxes. The apparently “twice-Nernstian” responses of ISEs based on acidic ionophores20-22 were the first examples of apparently nonNernstian responses that could be explained on the basis of equilibrium phase boundary potentials alone.17 In this system, the acidic ionophores in their deprotonated form can bind both to dications as primary ions and to H+ ions as secondary ions. As a result of multiple equilibria at the membrane/sample interface, the response of membranes containing the acidic ionophores and anionic sites depends simultaneously on the two types of ions over several orders of magnitude of the sample primary ion activity. Indeed, it was theoretically predicted and then experimentally confirmed that these ISEs give Nernstian, apparently twiceNernstian, and again Nernstian responses to the primary dications when samples of high, intermediate, and low pH are used, respectively. More recently, this mechanism of apparently twiceNernstian responses to dications observed with ISEs based on monensin was confirmed experimentally by ion-transfer voltammetry at liquid/liquid interfaces.23 Analogously, apparently “superNernstian” responses to monoanions as observed with metalloporphyrin-based ISEs were shown to result from the formation of OH- -bridged metalloporphyrin dimers in the ISE membranes.19,24-26 Apparently super-Nernstian responses to F- with slopes from -70 to -85 mV/decade as obtained with membranes based on GaIIIoctaethylporphyrin and anionic sites were quantitatively explained as responses codetermined by F- as the primary and OH- ions as the secondary ion. A recent model took into account the contribution of diffusion potentials to apparently superNernstian responses.27 A general model for apparently nonNernstian responses of ionophore-free ion-exchanger electrodes was developed18 and applied to the case of a Ca2+-selective electrode based on a cationic ionophore.28 However, a more indepth understanding of the responses of ionophore-based ISEs in mixed-ion solutions and the control of response slopes was needed to give a general answer to the question of where (13) Ceresa, A.; Bakker, E.; Hattendorf, B.; Gu ¨ nther, D.; Pretsch, E. Anal. Chem. 2001, 73, 343-351. (14) Bakker, E.; Pretsch, E. Trends Anal. Chem. 2001, 20, 11-19. (15) Fu, B.; Bakker, E.; Yun, J. H.; Yang, V. C.; Meyerhoff, M. E. Anal. Chem. 1994, 66, 2250-2259. (16) Meyerhoff, M. E.; Fu, B.; Bakker, E.; Yun, J.-H.; Yang, V. C. Anal. Chem. 1996, 68, 168A-175A. (17) Amemiya, S.; Bu ¨ hlmann, P.; Umezawa, Y. Anal. Chem. 1998, 70, 445454. (18) Bu ¨ hlmann, P.; Umezawa, Y. Electroanalysis 1999, 11, 687-693. (19) Steinle, E. D.; Amemiya, S.; Bu ¨ hlmann, P.; Meyerhoff, M. E. Anal. Chem. 2000, 72, 5766-5773. (20) Suzuki, K.; Tohda, K.; Sasakura, H.; Shirai, T. Anal. Lett. 1987, 20, 39-45. (21) Suzuki, K.; Tohda, K.; Aruga, H.; Matsuzoe, M.; Inoue, H.; Shirai, T. Anal. Chem. 1988, 60, 1714-1721. (22) Suzuki, K.; Tohda, K. Trends Anal. Chem. 1993, 12, 287-296. (23) Dassie, S. A.; Baruzzi, A. M. J. Electroanal. Chem. 2000, 492, 94-102. (24) Malinowska, E.; Niedzio´lka, J.; Meyerhoff, M. E. Anal. Chim. Acta 2001, 432, 67-78. (25) Malinowska, E.; Niedzio´lka, J.; Rozniecka, E.; Meyerhoff, M. E. J. Electroanal. Chem. 2001, 514, 109-117. (26) Malinowska, E.; Go´rski L.; Meyerhoff, M. E. Anal. Chim. Acta 2002, 468, 133-141. (27) Peyre, V.; Baillet, S.; Letellier, P. Anal. Chem. 2000, 72, 2377-2382. (28) Bu ¨ hlmann, P.; Umezawa, Y.; Rondinini, S.; Vertova, A.; Pigliucci, A.; Bertesago, L. Anal. Chem. 2000, 72, 1843-1852.

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Analytical Chemistry, Vol. 75, No. 14, July 15, 2003

apparently non-Nernstian equilibrium responses should be expected and how they can be explained. Existing theories for equilibrium potentiometric responses of ionophore-based ISEs in mixed-ion solutions6-8 were not general enough to model apparently non-Nernstian equilibrium responses. Here we report on a general model for apparently nonNernstian equilibrium responses of ionophore-based ISEs. It is an extension of the model that was recently developed for describing apparently non-Nernstian responses of ionophore-free ion-exchanger electrodes.18 In the case of the ionophore-based ISEs described here, the primary and secondary ions, IzI and JzJ, respectively, enter the membrane phase and form 1:nI and 1:nJ complexes with the ionophore. This situation can be represented as

IzI + nILzL h ILn(zII+nIzL)

(1a)

JzJ + nJLzL h JLn(zJJ+nJzL)

(1b)

where LzL is the ionophore and ILn(zII+nIzL) and JLn(zJJ+nJzL) are the complexes of the ionophore with the primary and secondary ions, respectively. In the second generalized case, the ions may simultaneously form complexes with the ionophore, which is represented as

nIIzI + nJJzJ + nLLzL h InIJnJ Ln(nLIzI+nJzJ+nLzL)

(2)

where InIJnJ Ln(nLIzI+nJzJ+nLzL) are mixed complexes that the ionophore simultaneously forms with the primary and secondary ions. Depending on the binding sites of the ionophore, both of these situations17 or only one of them occurs in the membrane phase.18,19 We consider here only the situation of eq 1. Binding of primary and secondary ions in the same complex, as described by eq 2, is particularly relevant for acidic and basic ionophores and will be discussed separately. 29 For the former case discussed here, analytical and numerical calculations based on a phase boundary potential model show that ionophore-based ISEs can give three types of apparently nonNernstian equilibrium responses, i.e., apparently super-Nernstian, “inverted-Nernstian”, and “sub-Nernstian” responses. Some of the responses that follow from our general discussion have already been observed experimentally by us and others, supporting the validity of our generalized model. GENERAL EQUATIONS FOR APPARENTLY “NON-NERNSTIAN” EQUILIBRIUM RESPONSES Model Assumptions. The general description of apparently non-Nernstian responses of ionophore-based ISEs is based on the calculation of the phase boundary potential at the interface between the aqueous sample solution and the ISE membrane.5,30 The usual assumptions, which we previously described in detail,31,32 were presumed in this study, too. The equilibrium (29) Bu ¨ hlmann, P.; Amemiya, S., manuscript in preparation. (30) Bakker, E.; Na¨gele, M.; Schaller, U.; Pretsch, E. Electroanalysis 1995, 7, 817-822. (31) Bu ¨ hlmann, P.; Amemiya, S.; Yajima, S.; Umezawa, Y. Anal. Chem. 1998, 70, 4291-4303.

to eqs 1a and 1b, respectively. It should be noted that, consistent with IUPAC recommendations,34 these formation constants can be defined on a concentration basis as long as the activity coefficients are constant (see figure captions for units of the constants). The combination of eq 3 applied for IzI and eq 3 applied for JzJ gives35 Figure 1. Composition of the membranes discussed in this study. The primary and secondary ions, IzI and JzJ, respectively, occur both in the membrane and in the sample solution. They form 1:nI and 1:nJ complexes, ILn(zI I+nIzL) and JLn(zJJ+nJzL), respectively, with the ionophore, LzL, in the membrane phase. Lipophilic ionic sites, RzR, are added to the membrane phase.

KIJ )

kIzJ kJzI

)

[IzI]zJ[JzJ]-zI aIzJ aJ-zI

(5a)

which is equivalent to response, E, of a liquid membrane ISE to any ion izi that is distributed between the sample and membrane phases is described by

[IzI][JzJ]-zI/zJ ) aIaJ-zI/zJ/K1J1/zJ

RT kiai E ) E′ + ln ziF [izi]

If IzI and JzJ have opposite charge signs, the coefficient KIJ describes salt partitioning between the sample and the membrane phase. Alternatively, if IzI and JzJ have the same charge sign, KIJ describes an ion-exchange equilibrium. The mass balance for the ionophore (eq 6) and the electroneutrality condition for membranes with ionic sites (eq 7) are described as

(3)

with

ki ) exp({µ0i (aq) - µ0i (mem)}/RT)

nI[ILn(zII+nIzL)] + nJ[JLn(zJJ+nJzL)] + [LzL] ) Ltot where the second term on the right-hand side of eq 3 represents phase boundary potentials at the sample/membrane interface,33 E′ is a constant that represents all contributions to the measured emf other than the phase boundary potential at the sample/ membrane interface, ai and [izi] are the activity of izi in the sample solution and the concentration of izi in the membrane phase, respectively, µ0i is the chemical standard potential of izi in the respective phase (“aq” denotes species in the sample solution phase, “mem” species in the membrane phase), and R, T, and F have their usual meanings. The constant ki has been referred to in the literature, perhaps deceptively, as the single-ion distribution coefficient. The following eqs 4a, 4b, and 5a describe the various equilibria into which the membrane components are involved (Figure 1). The formation constants, βILnI and βJLnJ, for complexes of the ionophore with the primary and secondary ions, respectively, are defined as

βILnI )

βJLnI )

[ILnI(zI+nIzL)] [IzI][LzL]nI [JLnJ(zJ+nJzL)] [JzJ][LzL]nJ

(4a)

(4b)

where the bracketed terms indicate the concentrations of the species in the membrane phase. Equations 4a and 4b correspond (32) Amemiya, S.; Bu ¨ hlmann, P.; Pretsch, E.; Rusterholz, B.; Umezawa, Y. Anal. Chem. 2000, 72, 1618-1631. (33) Buck, R. P.; Lindner, E. Acc. Chem. Res. 1998, 31, 257-266.

(5b)

(6)

(zI + nIzL)[ILn(zII+nIzL)] + (zJ + nJzL)[JLn(zJJ+nJzL)] + zL[LzL] + zI[IzI] + zJ[JzJ] + zR[RzR] ) 0 (7)

where Ltot and [RzR] are the total membrane concentrations of the ionophore and the ionic sites with charge zR, respectively. Numerical and Analytical Approaches. The combination of eqs 3-7 allows the numerical determination of the response curves and concentrations of membrane species as a function of the activities of the primary and secondary ions in the sample (as similarly described previously17 for less general cases). This cannot be done analytically in most cases because the resulting equations are high-order equations. The numerically calculated response curves show that there can be up to three linear response ranges for each curve and that the middle linear ranges represent one of three types of apparently non-Nernstian responses. While this numerical approach is useful to express whole response curves, we use further assumptions to obtain analytical equations that describe only the apparently non-Nernstian response range of each response curve. Results of both approaches will be compared to confirm the validity of the further assumptions. To obtain analytical expressions for apparently “non-Nernstian” responses, we use two assumptions: (1) The free species, IzI, JzJ, and LzL, are not membrane species of high concentrations. (2) Only the ionophore complexes, ILn(zII+nIzL) and JLn(zJJ+nJzL), as well as the ionic sites are major membrane components. (34) Mills, I.; Cvitas, T.; Homann, K.; Kallay, N.; Kuchitsu, K. Quantities, Units and Symbols in Physical Chemistry; Blackwell Science: Oxford, U.K., 1993; p 50. (35) Vanysek, P.; Buck, R. P. J. Electroanal. Chem. 1991, 297, 19-35.


3331

The following considerations justify these assumptions:17-19 Because the response slope, S, is defined as

S)

∂E ∂log aI

(

(

)

z

)

∂ log[I I] RT ln 10 1zIF ∂ log aI

(9)

()

zJ 1 (12) nJzI - nIzJ nJ

(8)

inserting eq 3 applied for the primary ion into eq 8 gives

S)

()

zI zR[RzR] 1 1 >zL + > nJzI - nIzJ nI nJzI - nIzJ Ltot

Equation 12 shows the charge sign and concentration of the ionic sites that are required for apparently non-Nernstian equilibrium responses. Additionally, eqs 10-12 show that apparently nonNernstian responses require

nIzJ * nJzI Equation 9 shows that Nernstian responses are obtained when the membrane concentration of the free primary ion does not depend on the primary ion activity in the sample phase and that apparently non-Nernstian responses are obtained if the membrane concentration of the free primary ion changes as a function of the primary ion activity in the sample solution. Moreover, the concentrations of the major components in the membrane phase have been assumed to be almost constant, excluding significant diffusion potentials, very slow response times due to a major membrane reconditioning, or both,17-19 i.e., nonequilibrium responses. IzI and JzJ cannot be membrane species of high concentrations because significant changes in the membrane concentration of the former ion are a prerequisite for apparently non-Nernstian responses. In contrast, ILn(zII+nIzL) and JLn(zJJ+nJzL) must be the major components in the membrane phase because phase boundary potentials are independent of the activity of an ion izi in the sample phase unless the membrane phase contains either the free ion or its complexes with the ionophore.36 Furthermore, because both [ILn(zII+nIzL)] and [JLn(zJJ+nJzL)] cannot change significantly over the range of the apparently non-Nernstian response, eq 6 is satisfied only if [LzL] is also constant or if [LzL] is negligibly small. However, constant concentrations of the free ionophore and its complexes make [IzI] constant (eq 4a), which would result in the usual Nernstian responses. Thus, [LzL] must be negligibly small and only ILn(zII+nIzL) and JLn(zJJ+nJzL) can be the major components in membranes that give apparently non-Nernstian responses. On the basis of these assumptions, analytical expressions for apparently non-Nernstian responses can be obtained. As the membrane concentrations of IzI, JzJ, and LzL are very small and can be neglected in eqs 6 and 7, the two equations lead to two equations with two unknowns. Solving them for the membrane concentrations of ILn(zII+nIzL) and JLn(zJJ+nJzL) gives

[ILn(zII+nIzL)] ) [JLn(zJJ+nJzL)] )

(zJ + nJzL)Ltot + nJzR[RzR] nJzI - nIzJ

(zI + nIzL)Ltot + nIzR[RzR] nJzI - nIzJ

(10)

(11)

Equations 10 and 11 confirm that [ILn(zII+nIzL)] and [JLn(zJJ+nJzL)] are constant. Furthermore, because [ILn(zII+nIzL)] and [JLn(zJJ+nJzL)] must have positive values, the right-hand sides of eqs 10 and 11 must be positive too. Rearrangement of the resulting equations gives (36) Lindner, E.; Cosofret, V. V.; Kusy, R. P.; Buck, R. P.; Rosatzin, T.; Schaller, U.; Simon, W.; Jeney, J.; To´th, K.; Pungor, E. Talanta 1993, 40, 957-967.

3332


(13)

By using eqs 10 and 11, potentiometric responses, E, can be expressed as a function of aI as described in the following. Elimination of [LzL] from eqs 4a and 4b gives

(

) (

[ILn(zII+nIzL)] βILnI[IzI]

nJ

)

)

[ILn(zII+nIzL)] βJLnJ[JzJ]

nI

(14)

Elimination of [JzJ] by combination of eqs 5b and 14 gives

{ (

[IzI] ) (aIkI)nIzJ

)( nJzI

βILnI

[ILn(zII+nIzL)]

)}

[JLn(zJJ+nJzL)] aJkJβILnJ

1/(nIzJ - nJzI)

n

z

(15)

The apparently non-Nernstian responses to IzI can be obtained by replacing [IzI] in eq 3 with the right-hand side of eq 15

E ) E′ +

(

) {(

RT 1 ln F nJzI - nIzJ

kIβILnI

)( nJ

[ILn(zII+nIzL)]

(

)} )

[JLn(zJJ+nJzL)] aJkJβILn

J

nI

+

nJ RT ln aI (16) F nJzI - nIzJ Because the concentrations of ILn(zII+nIzL) and JLn(zJJ+nJzL) do not depend on the sample activities of IzI and JzJ (eqs 10 and 11), the first and second terms on the right-hand side of eq 16 can be represented by a constant E′′ as long as the sample activity of the secondary ion is constant. The response slopes can be obtained by replacing E in eq 8 with the right-hand side of eq 16, giving

S)

(

)

1 RT ln 10 zIF 1 - nIzJ/nJzI

(17)

Equation 17 shows that a linear response occurs and that the response slope depends on the charge numbers of the primary and secondary ions and on the stoichiometries of their complexes with the ionophore. It should also be noted that eq 17 does not depend on the charge of the ionophore. This implies that apparently non-Nernstian responses are expected for ISEs based on charged ionophores as well as for ISEs based on neutral

ionophores if the charge sign and concentration of the ionic sites allow it (eq 12). This result is consistent with our previous studies, which showed that the influence of ionic sites on selectivities,32 co-ion interference (Donnan failure),31 and lifetime28 of ISEs based on charged ionophores can be explained in complete analogy to the corresponding properties of ISEs based on neutral ionophores. Equation 17 is useful to evaluate the non-Nernstian slopes when only one primary or one secondary ion is bound by one or more ionophore molecules. However, this equation can also be extended to the more general case, where more than one primary and secondary ion is bound by the ionophore, as it is, for example, conceivable for anion binding to metalloporphyrins.37 The equilibria for such a case can be represented by

mIIzI + nILzL h ImILn(mI IzI+nIzL)

(18a)

mJJzJ + nJLzL h JmJLn(mI JzJ+nJzL)

(18b)

In this case, a more general form of eq 17 can be obtained as (see Appendix)

S)

(

)

RT ln 10 1 zIF 1 - mJnIzJ/mInJzI

(19)

Table 1. Apparently Super-Nernstian Response Slopes Calculated for the Primary Ion and Secondary Ion, with the Charges, z, and Complex Stoichiometries, n response slope chargea

stoichiometry

zI

zJ

nI

nJ

slope, mV/decade

3 3 2 2

2 1 3 1

1 1

2 1

1 1 1 1 1 1 1 1 1 2

1 1 2 1 2 3 2 3 4 3

+58 +29 +116 +58 +39 +174 +116 +87 +77 +174

normalized slopeb 3 3/2 4 2 4/3 3 2 3/2 4/3 3

a An analogous table for anion-selective electrodes is obtained upon replacing zI, zJ, and the slope with -zI, -zJ, and -slope. b Obtained by division of the calculated response slope with the Nernstian slope (58/zI mV). According to eq 17, (normalized slope) ) 1/(1 - nIzJ/nJzI).

Apparently “Super-Nernstian” Responses. If the charge numbers of the primary and secondary ions and the stoichiometries of their complexes are such that

0 < nIzJ/nJzI < 1

(21)

where

mJnIzJ * mInJzI

(20)

eqs 19 and 20 are equivalent to eqs 17 and 13, respectively, when mI ) mJ ) 1. While the introduction of these two parameters, mI and mJ, complicates the mathematial procedures, such an extension is required only for special cases. Therefore, the cases as represented by eqs 1a and 1b rather than eq 18a and 18b will be discussed in the following in order to introduce the concept of apparently “non-Nernstian” equilibrium responses. THREE TYPES OF APPARENTLY “NON-NERNSTIAN” EQUILIBRIUM RESPONSES According to eq 17, three types of apparently non-Nernstian responses are expected. When 0 < nIzJ/nJzI zL + > nJ|zI| - nI|zJ| nI nJ|zI| - nI|zJ| zI Ltot

()

|zJ| 1 (22) nJ|zI| - nI|zJ| nJ Rearrangement of eq 21 to

nI|zJ| < nJ|zI|

(23)

allows the simplification of eq 22 to

(

)

|zI| zR[RzR] |zJ| |zI| >zL + > nI zI Ltot nJ

(24)

Equation 24 describes the charge sign and the concentration of the ionic sites that give apparently super-Nernstian responses. Table 1 shows response slopes for different sets of nI, nJ, zI, and zJ, as obtained from eqs 17 and 21. Examples 4 and 7 in Table 1 show the consistency of this theory with the results from our previous studies.17,19 In the case of apparently twice-Nernstian responses to alkaline earth cations, as obtained with ISEs based on acidic ionophores, it was assumed that both primary dications (zI ) 2) and H+ ions (as secondary ions with zJ ) 1) form 1:1 complexes with the ionophores (nI ) nJ ) 1).17 This assumption Analytical Chemistry, Vol. 75, No. 14, July 15, 2003

3333

agrees with the conditions in example 4, where apparently twiceNernstian responses to dications are expected. On the other hand, the origin of apparently super-Nernstian responses to monoanions as observed with ISEs based on metalloporphyrins was ascribed to the formation of OH- -bridged metalloporphyrin dimers in the membrane phase.19,24-26 Indeed, example 7 in Table 1 suggests that apparently twice-Nernstian responses are expected if a metalloporphyrin ionophore forms 1:1 complexes with a primary monoanion (zI ) -1 and nI ) 1), and 1:2 complexes with OH- as secondary ion (zJ ) -1 and nJ ) 2). For certain examples of Table 1 and otherwise representative parameters, we calculated numerically the dependence of the potentiometric responses and concentrations of membrane species on the sample activity of the primary ion. The membrane concentrations of the free primary and secondary ions, the free ionophore, and its complexes with the primary and secondary ions were obtained from eqs 4a, 4b, and 5-7. Equation 3 was used to calculate the potentiometric responses as a function of the sample activity of the primary ion.38 Figure 2A shows the response curves of a monocation-selective electrode based on a negatively charged ionophore (zL ) -1). The charge number of the primary and secondary ions and the stoichiometries of their complexes with the ionophore were chosen so that the dashed, solid, and dotted curves in Figure 2A correspond to examples 6-8 in Table 1, respectively. The same charges and concentrations of ionic sites were used for the three curves. They fulfill eq 24. Figure 2A shows that all three response curves consist of three linear ranges. The calculated responses are at first constant, then apparently super-Nernstian, and finally Nernstian as the sample activity of the primary ion increases. Also, panels B and C of Figure 2 show that the three linear ranges of the response curve represented in Figure 2A with a solid line correspond to three different response mechanisms. At low activities of the primary ion in the sample solution, the major membrane components are the free secondary ion and its complexes with the ionophore (Figure 2B and C). This indicates that the membranes respond to the secondary ion in an ionophoreindependent mechanism32 because the ionophore concentration is not sufficient to bind the vast majority of the secondary ions in the membrane. Consequently, the membrane contains a high concentration of uncomplexed secondary ions, and Nernstian responses are observed as with ionophore-free ion-exchanger electrodes.39 Thus, the constant potential in this range is due to the constant sample activity of the secondary ion. As the sample activity of the primary ion increases, apparently super-Nernstian slopes of +174, +116, and +87 mV/decade for zI ) 1 are obtained for zJ ) 2, 1, and 1, respectively. The apparently super-Nernstian slopes agree with the slopes predicted from eq 17 (Table 1). As depicted in Figure 2B and C, the major membrane components in this response range are ILn(zII+nIzL) and JLn(zJJ+nJzL), and their concentrations are constant. On the other hand, the membrane concentrations of the free primary and secondary ions and the free ionophore are relatively low and change significantly in this response range. Note that the membrane concentration of (38) Mathematica 3.0 (Wolfram Research Inc., Champaign, IL) was used to perform the numerical calculations. (39) For a critical description of “ionophore-free ion-exchanger electrodes”, see ref 2, p 1595.

3334 Analytical Chemistry, Vol. 75, No. 14, July 15, 2003

Figure 2. ISEs based on a charged ionophore, L-. (A) Emf responses to the primary ion, I+, at a constant activity of the secondary ion, JzJ, in the sample solution. The three curves (s, ‚‚‚, and - - - ) represent the results for (zJ, nJ, βJLnJ) ) (1, 2, 1020 kg2/mol2), (1, 3, 1030 kg3/mol3), and (2, 3, 1030 kg3/mol3), respectively. nI ) zI ) 1, zL ) -1, zR ) 1, βIL ) 1015 kg/mol, Ltot ) 0.02 mol/kg, [R+] ) Ltot/6 were used for all curves. (B) Membrane concentrations of I+ (s), J+ (‚‚‚), and L- (- - ) as a function of the I+ activity in the sample at a constant activity of J+; parameters as for the solid line under (A). (C) Concentrations of IL (s), JL2- (‚‚‚), L- (- - ), and R+ (-‚-) in the corresponding membrane as a function of the I+ activity in the sample at a constant activity of J+; parameters as for the solid line under (A).

the free primary ion decreases as its activity in the sample solution increases. This causes the apparently super-Nernstian responses (see eq 9). Finally, conventional Nernstian responses are obtained at higher activities of the primary ion in the sample phase. There, the major membrane components are the free ionophore and its

Table 2. Apparently Inverted-Nernstian Response Slopes Calculated for the Primary Ion and Secondary Ion, with the Charges, z, and Complex Stoichiometries, n response slope chargea

stoichiometry

zI

zJ

nI

nJ

slope, mV/decade

normalized slopeb

2 1

3 3

1

2

1

1

1 1 1 1 2 2 2 3 3

1 2 1 1 3 1 1 1 2

-58 -116 -29 -58 -174 -19 -58 -29 -116

-2 -2 -1/2 -1 -3 -1/3 -1 -1/2 -2


complexes with the primary ion, indicating that the membrane responds to the primary ion in an ionophore-based mechanism32 because the total concentration of the analyte in the membrane is low enough that the membrane contains a considerable concentration of free ionophore. As a result of this high concentration of the free ionophore, the analyte can be efficiently complexed and buffered. Thus, the resulting constant concentration of the analyte in the membrane phase leads to Nernstian responses. Apparently Inverted-Nernstian Responses (“Potential Dip”). If the charge numbers of the primary and secondary ions and the stoichiometries of their complexes are such that

1 < nIzJ/nJzI

(25)

eq 17 indicates that the sign of the response slopes is opposite to that of Nernstian slopes (apparently inverted-Nernstian responses). Because eq 25 can only be fulfilled if 0 < zJ/zI, it can be concluded that apparently inverted-Nernstian responses are obtained when the charge sign of the primary ion is the same as that of the secondary ion. In Table 2, examples of apparently invertedNernstian responses with slopes of (-1/3), (-1/2), (-1), (-2), and (-3) × (Nernstian slope) are shown for different sets of nI, nJ, zI, and zJ. Because eq 25 can be rearranged to

nI|zJ| > nI|zJ|

(26)

eq 22 can be simplified to

(

)

|zI| zR[RzR] |zJ| |zI| 1). This results in the apparently invertedNernstian responses (see eq 9). One may notice that eqs 26 and 27, which represent the requirements for apparently inverted-Nernstian responses, are equivalent to the corresponding equations for apparently superNernstian responses (eqs 23 and 24, respectively) upon exchanging the parameters of the primary ion with those of the secondary ion. Indeed, comparison of Tables 1 and 2 shows that examples 1, 3, 4, and 6-9 in Table 2 are, in such a way, equivalent to examples 1, 2, 4, 5, 7, 8, and 10 in Table 1. This means that if membranes give apparently super-Nernstian responses to a primary ion solution containing a constant concentration of a secondary ion, the same membranes are expected to give apparently inverted-Nernstian responses to the secondary ion in sample solutions containing a constant concentration of the primary ion. The latter situation seems to have been often observed with Ca2+-selective electrodes based on organophosphate ionophores.40-44 The pH responses of such electrodes in sample solutions containing a constant activity of Ca2+ were constant at high pH, decreased as the pH decreased, and then increased at very low pH. These unusual pH response curves, which have been called potential dips, are similar to the calculated response curves in Figure 3A. Indeed, it can be predicted from eqs 17 and 27 that an apparently inverted-Nernstian response slope of -58 mV/ decade is expected if the organophosphate ionophore (zL ) -1) forms 1:1 complexes with H+ and with Ca2+ (see example 4 in Table 2) and if the membranes contain anionic sites in the concentration range between 0 and 100 mol % relative to the ionophore (eq 27). In real membranes that exhibited potential dips, the anionic sites were apparently anionic impurities from the PVC matrix and plasticizers.45-48 Interestingly, to the best of our knowledge, there is only one direct report on ISEs that give apparently super-Nernstian responses to a primary ion and (40) Bagg, J.; Vinen, R. Anal. Chem. 1972, 44, 1773-1777. (41) Ruzicka, J.; Hansen, E. H.; Tjell, J. C. Anal. Chim. Acta 1973, 67, 155178. (42) Cattrall, R. W.; Drew, D. M. Anal. Chim. Acta 1975, 77, 9-17. (43) Egorov, V. V.; Lushchik, Y. F.; Pavlovskaya, E. A. J. Anal. Chem. 1992, 47, 388-392. (44) Mikhelson, K. N.; Lewenstam, A.; Didina, S. E. Electroanalysis 1999, 11, 793-798. (45) Horvai, G.; Gra´f, E.; To´th, K.; Pungor, E.; Buck, R. P. Anal. Chem. 1986, 58, 2735-2740. (46) Van den Berg, A.; van der Wal, P. D.; Skowronska-Ptasinska, M.; Sudholter, E. J. R.; Reinhoudt, D. N.; Bergveld, P. Anal. Chem. 1987, 59, 2827-2829. (47) Bu ¨ hlmann, P.; Yajima, S.; Tohda, K.; Umezawa, Y. Electrochim. Acta 1995, 40, 3021-3027. (48) Watanabe, M.; Toko, K.; Sato, K.; Kina, K.; Takahashi, Y.; Iiyama, S. Sens. Mater. 1998, 10, 103-112.

3336


apparently inverted-Nernstian responses to a secondary ion.49 The authors observed apparently super-Nernstian responses to Ca2+ in sample solutions containing constant concentrations of H+ or Mg2+ (the response slopes were 40 and 39 mV/decade, respectively) and apparently inverted-Nernstian responses to the secondary ions (the response slope to H+ at pCa ) 3.1 is ∼-43 mV/ decade as read from Figure 1a in ref 49). Unfortunately, the slopes are not perfect as expected for twice-Nernstian and invertedNernstian responses, probably because the secondary ion activity in the sample solution was not optimized,17 because the binding constants are not large enough,19 or because not all the requirements for apparently non-Nernstian responses discussed below were fulfilled. The responses to Ca2+ and H+, however, can be qualitatively explained as a combination of the examples 4 in Tables 1 and 2. On the other hand, as the ratio of zI and zJ determines apparently non-Nernstian slopes (see eq 17), respective responses to Ca2+ and Mg2+ can be explained as a combination of examples for zI ) zJ ) 1 in Tables 1 and 2 (for instance, example 7 in both tables; 1:1 and 1:2 complexes with Ca2+ and Mg2+, respectively). The negative responses in potential dips have been so far attributed to the change of the membrane diffusion potential4,44,49 or to the responses to anionic impurities leaching from the membrane phase into the sample solution.43 Although further studies are necessary to describe the response mechanisms in more detail,50 our model suggests that such responses can be explained on the basis of phase boundary potentials. In particular, it also accounts for the complementary relationship between apparently super-Nernstian and inverted-Nernstian responses. Apparently Sub-Nernstian Responses. If the charge numbers of the primary and secondary ions and the stoichiometries of their complexes are such that

nIzJ/nJzI < 0

(28)

eq 17 shows that apparently sub-Nernstian responses can be obtained. In contrast to the other two types of apparently nonNernstian responses, eq 28 shows that such sub-Nernstian responses can be obtained when the charge sign of the secondary ion is opposite to that of the primary ion. Indeed, in the case of cation-selective electrodes, apparently sub-Nernstian slopes from (1/4) to (3/4) × (Nernstian slope) are obtained for different sets of nI, nJ, zI, and zJ (Table 3) by using eqs 17 and 28. These apparently sub-Nernstian responses can be obtained only when the charge sign and concentration of the ionic sites in the membrane phase satisfy eq 12, which is equivalent to

(

)

|zI| |zI| zR[RzR] |zJ| >zL + >nI zI Ltot nJ

(29)

because zJ/zI < 0. Numerical calculations of potentiometric responses and concentrations of membrane species again agree well with the (49) Didina, S. E.; Shilin, A. G.; Grekovich, A. L.; Veshev, S. A.; Materova, E. A.; Mikhelson, K. N. Sov. Electrochem. 1987, 23, 544-550. (50) Barefoot M. B.; Amemiya, S.; Bu ¨ hlmann, P.; Schaller, U.; Pretsch, E., unpublished results.

Table 3. Apparently Sub-Nernstian Response Slopes Calculated for the Primary Ions and Secondary Ion, with the Charges, z, and Complex Stoichiometries, n response slope chargea

stoichiometry

zI

zJ

nI

nJ

slope, mV/decade

3 2 1 1 1

-1 -1 -3 -2 -1

1 1 1 1 1 1 1 2 3

1 1 1 1 1 2 3 1 1

+14.5 +19.3 +14.5 +19.3 +29.0 +38.7 +43.5 +19.7 +14.5

normalized slopeb 3/4 2/3 1/4 1/3 1/2 2/3 3/4 1/3 1/4


predictions from the analytical equations (eqs 10, 11, 15, and 16). Panels A-C of Figure 4 show examples of membranes based on a neutral ionophore (zL ) 0) that forms 1:1 complexes with the primary monocation and with secondary anions (zJ ) -3, -2, and -1; corresponding to examples 3-5 in Table 3, respectively). Figure 4A shows the response curves of membranes based on the neutral ionophore and 50 mol % anionic sites. All three response curves consist of three linear ranges. The membranes give a Nernstian response to the primary ion at the low activities, explained by an ionophore-based mechanism. As the sample activity of the primary ion increases, the response slopes decrease to apparently sub-Nernstian slopes. The slopes of +29, +19.3, and +14.5 mV/decade for zI ) 1 are obtained for zJ ) -1, -2, and -3 in Figure 4A, respectively. In the linear response range, the major components in the membrane phase are the complexes of the ionophores with the primary and secondary ions. Importantly, as the sample activity of the primary ion increases, its membrane concentration increases so moderately (0 < ∂ log[IzI]/∂ log aI < 1) that apparently sub-Nernstian responses are obtained (see eq 9). At higher activities of the primary ion in the sample phase, the response slopes are the same as the apparently sub-Nernstian slopes that are observed for the intermediate sample activities of the primary ion. However, their response mechanisms are different. At high activities, the free primary and secondary ions as well as the ionophore-primary ion complexes are major membrane components. Thus, the apparently sub-Nernstian responses at the high sample activities of the primary ion are due to salt partitioning of the free primary and secondary ions between sample and membrane31 and are independent of the ionophore. Note that membranes with the neutral ionophore and 50 mol % cationic sites instead of 50 mol % anionic sites also give apparently sub-Nernstian responses (see eqs 17 and 29). In this case, no emf responses instead of Nernstian responses are obtained at low sample activities of the primary ion because the secondary anion determines the phase boundary potential in this range (see Supporting Information for details). When Are Apparently Non-Nernstian Responses Observed Experimentally? It follows from eq 13 that apparently non-Nernstian responses are possible when nIzJ * nJzI. However,

Figure 4. ISEs based on a neutral ionophore L. (A) Emf responses to the primary ion, I+, at a constant activity of the secondary ion, JzJ, in the sample solution. The three curves (s, ‚‚‚, and - - -) represent the results for zJ ) -1, -2, and -3, respectively. nI ) nJ ) zI ) 1, zL ) 0, zR ) -1, βIL ) 108 kg/mol, βJL ) 106 kg/mol, Ltot ) 0.02 mol/ kg, [R-] ) Ltot/2 were used for all curves. (B) Membrane concentrations of I+ (s), J- (‚‚‚), and L (- - -) as a function of the I+ activity in the sample at a constant activity of J-; parameters as for the solid line under (A). (C) Concentrations of IL+ (s), JL- (‚‚‚), L (- - -), and R- (-‚-) in the corresponding membrane as a function of the I+ activity in the sample at a constant activity of J-; parameters as for the solid line under (A).

the number of experimental examples of apparently non-Nernstian equilibrium responses in the literature is still limited. On one hand, more such responses may have been observed but were not reported because their origin was not understood. On the other hand, nIzJ * nJzI is a necessary but not the only requirement for apparently non-Nernstian responses. In the following, we discuss other factors that determine whether apparently non-Nernstian responses can be observed experimentally. Analytical Chemistry, Vol. 75, No. 14, July 15, 2003

3337

Binding Sites of Ionophores. Most ionophores bind only anions or only cations.2 ISEs based on such ionophores may give apparently super-Nernstian or inverted-Nernstian responses to the primary ion in sample solutions containing a secondary ion with the same charge sign as the primary ion. On the other hand, ionophores must be able to independently form complexes with cations and with anions to give apparently sub-Nernstian responses. Thus, it is unlikely that apparently sub-Nernstian responses based on the mechanism described above are observed with many ionophores. Furthermore, if ionophores have binding sites for the independent complexation with cations and anions, it is also possible that the ionophores simultaneously form complexes with the cations and anions, as represented by eq 2. This case will be discussed separately.29 Ionic Sites. The properties of ISEs based on electrically charged and neutral ionophores are strongly influenced by ionic sites in their membranes. This is also true for apparently non-Nernstian equilibrium responses. To obtain optimum selectivities of an ionophore-based ISE, the concentration and charge sign of the ionic sites must be chosen so that the membrane responds to the primary ion in an ionophore-based mechanism.32 This occurs when the concentration and charge of the ionic sites satisfy

(

)

|zI| |zI| zR[RzR] >zL + >0 nI zI Ltot


∆ log aI )

( )(

βJLnI[JzJ]IID βILnI[IzI]ID n J zI nI -1 log (z +n z ) - log (z +n z ) nIzJ nJ [JLnJJ J L ] [ILnII I L ]

)

(31)

(30)

which is equivalent to eq 10 in ref 32. This relationship is fulfilled by the membranes that give apparently super-Nernstian or subNernstian responses (eqs 24 and 29, respectively) but not by membranes that give apparently inverted-Nernstian responses (eq 27). Indeed, only the former two types of membranes respond to the primary ion in an ionophore-based mechanism at high and low sample activities of the primary ions, respectively (Figures 2A and 4A). Apparently inverted-Nernstian responses are observed when the concentration and charge sign of ionic sites are not optimized for the primary ion and are, therefore, not analytically useful. Binding Constants. It follows from the above discussion about ionic sites and ionophore binding sites that apparently superNernstian responses are most probable in real experiments when ionophores independently form complexes with the primary and secondary ions. Indeed, it is important to point out that eqs 23 and 24, which represent requirements for apparently superNernstian responses, are the same as the equations that represent requirements for obtaining membranes that respond in “separate solution experiments” to the primary ion in an ionophore-based mechanism and to the secondary (interfering) ion in an ionophoreindependent mechanism (see eqs 15 and 16 in ref 32). For such membranes, an optimum selectivity is expected in “separate solution experiments”.32 Thus, membranes whose selectivities are optimized for separate solution experiments can give apparently super-Nernstian responses in mixed solutions of the primary and interfering ions. The range of the apparently super-Nernstian response (∆log aI) is determined by extrapolation of the three linear segments of the response to aI (Figure 5). Combination of equations representing each linear responses gives (see Supporting Information) 3338

Figure 5. ISEs based on a charged ionophore L-. Emf responses to the primary ion, I+, at a constant activity of the secondary ion, J+, in the sample solution. Same parameters for the solid line of Figure 2A except for βIL ) 1017 kg/mol for curve a and βJL2 ) 1024 kg2/mol2 for curve c. The dashed lines overlapping with the linear parts of curve b were obtained on the basis of eqs 16, S1a, and S2a (see Supporting Information) with the same parameters as used for curve b.

Although [ILn(zII+nIzL)], [JLn(zJJ+nJzL)], and [JzJ] are independent of βILnI and βJLnJ, [IzI]ID is inversely proportional to βILnI (see eq S1b in Supporting Information). Thus, eq 31 shows that ∆log aI depends on βJLnJ but not on βILnI. This prediction agrees well with the results of numerical calculations (Figure 5). The range of apparently super-Nernstian response is not influenced by βILnI since a change in the binding constant shifts the upper and lower limits simultaneously (curves a and b). On the other hand, it follows from eq 31 that the larger βJLnJ is, the wider the measuring range becomes. This is also confirmed by numerical calculations (curves b and c). This allows the conclusion that less selective ionophores have wider ranges of apparently super-Nernstian responses. CONCLUSIONS In this study, we presented a general model for apparently nonNernstian equilibrium responses of ionophore-based ISEs by considering simultaneous responses to two ions (i.e., the primary and secondary ions) that independently form complexes with an ionophore in the membrane phase. It was shown that depending on the charges of the two ions and stoichiometries of their complexes with the ionophore, three types of apparently nonNernstian responses can be obtained in mixed solutions, i.e., apparently super-Nernstian, inverted-Nernstian, and sub-Nernstian responses. Interestingly, none of the three types of apparently nonNernstian equilibrium responses have been predicted by previous general models that describe potentiometric responses of ISEs in mixed-ion solutions.6-8 The comparison between our model and the previous models6,7 points to a key assumption in previous models. To simplify the mathematical treatment, those models were restricted by the assumption that the concentration of free ionophore was always high. However, we have found that the membrane concentration of free ionophores is extremely small and changes significantly in magnitude when apparently nonNernstian responses are obtained. Besides our previous, less

general models on apparently non-Nernstian equilibrium responses,17,19 significant changes of the membrane concentration of free ionophores were considered only in one model.51 However, in that latter case, large fluxes of primary ions into and interfering ions out of the sensor membrane significantly changed the membrane concentration of the complexes within the observed range of primary ion activities. This is not the case for the responses discussed in this article. Our model also shows that apparently non-Nernstian responses observed under equilibrium conditions (i.e., no significant diffusion potentials and ion fluxes52) indicate that the sample solutions contain a secondary ion. In such a case, Nernstian responses can be obtained by identifying the secondary ion and by decreasing or increasing the sample activity of that ion. Nernstian responses thus obtained and corresponding non-Nernstian responses in mixed-ion solutions will allow the characterization of potentiometric selectivities on the basis of thermodynamic equilibria,50,52 allowing the determination of very useful information for the ionophore design (such as stoichiometries and binding constants of the analyte-ionophore complexes).53,54 ACKNOWLEDGMENT We thank Dr. M. E. Meyerhoff, Department of Chemistry, University of Michigan, Ann Arbor, for helpful discussion. S. A. thanks the Japan Society for the Promotion of Science (J.S.P.S.) for a postdoctoral fellowship.

on phase boundary equilibria, the concentrations of the major components in the membrane phase can be assumed to be constant, excluding significant diffusion potentials, very slow response times due to major membrane reconditioning, or both.17-19 Therefore, [ImI Ln(mI IzI+nIzL)] and [JmJ Ln(mI IzI+nIzL)] must be independent of the sample activity of the primary ion. Thus, the partial differential of the logarithms of eqs A1a and A1b with respect to log aI gives

mI

∂ log[IzI] ∂ log[LzL] + nI )0 ∂ log aI ∂ log aI

(A2a)

mJ

∂ log[JzJ] ∂ log[LzL] + nJ )0 ∂ log aI ∂ log aI

(A2b)

Combining eqs A2a and A2b gives

∂ log[JzJ] mInJ ∂ log [IzI] ) ∂ log aI mJnI ∂ log aI

On the other hand, the partial differential of the logarithm of eq 5a with respect to log aI gives

zJ APPENDIX Equation 19 can be derived as follows. When the primary and secondary ions form mI:nI and mJ:nJ complexes with the ionophore (eqs 18a and 18b), respectively, the formation constants, βImILnI and βJmJLnJ, for the complexes are defined as

βImILnI ) βJmJLnJ )

[ImILn(mI IzI+nIzL)] [IzI]mI[LzL]nI [JmJ Ln(mI IzI+nIzL)] [JzJ]mJ[LzL]nJ

(A3)

∂ log[IzI] ∂ log[JzJ] - zI - zJ ) 0 ∂ log aI ∂ log aI

(A4)

Combining eqs A3 and A4 gives

∂log [IzI] 1 ) ∂logaI 1 - mInJzI/mJnIzJ

(A1a)

(A5)

Equation 19 can be obtained by combing eqs 9 and A5.

(A1b)

In the range of an apparently non-Nernstian response based purely (51) Eugster, R.; Spichiger, U. E.; Simon, W. Anal. Chem. 1993, 65, 5, 689695. (52) Bakker, E.; Pretsch, E.; Bu ¨ hlmann, P. Anal. Chem. 2000, 72, 1127-1133. (53) Mi, Y.; Bakker, E. Anal. Chem. 1999, 71, 5279-5287. (54) Ceresa, A.; Pretsch, E. Anal. Chim. Acta 1999, 395, 41-52.

SUPPORTING INFORMATION AVAILABLE Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org. Received for review December 27, 2002. Accepted April 25, 2003. AC026471G


3339

A Generalized Model for Apparently âNon-Nernstianâ Equilibrium

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