Anal. Chem. 1998, 70, 445-454
A Phase Boundary Potential Model for Apparently “Twice-Nernstian” Responses of Liquid Membrane Ion-Selective Electrodes Shigeru Amemiya, Philippe Bu 1 hlmann, and Yoshio Umezawa*
Department of Chemistry, School of Science, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan
A model that describes divalent cation responses of liquid membrane ion-selective electrodes based on acidic ionophores and ionic sites is presented. Response slopes for membranes with ionophore and anionic sites are predicted to change from Nernstian to apparently “twiceNernstian” and then back to Nernstian again as the pH of the sample solution decreases. A maximum measuring range for apparently “twice-Nernstian” responses is expected for membranes with 50 mol % anionic sites relative to the ionophore. On the other hand, membranes with ionophore and cationic sites are expected to give only Nernstian responses, either to divalent cations at high pH or to H+ at low pH. The validity of the present model has been confirmed experimentally with the two Ba2+-selective carboxylate ionophores monensin and lasalocid and the Ca2+-selective organophosphate ionophore bis(2-heptylundecyl) phosphate. Addition of anionic sites gave apparently “twice-Nernstian” slopes for monensin at pH 7.0 (56.6 mV/decade), for lasalocid at pH 4.0 (53.3 mV/ decade), and for bis(2-heptylundecyl) phosphate at pH 3.5 (53.6 mV/decade). Membranes with cationic sites showed only pH responses at the respective pH. The apparently “twice-Nernstian” responses as discussed here are the first examples of super-Nernstian responses that can be explained with a quantitative model based on thermodynamic equilibria.
Conventional use of liquid membrane ion-selective electrodes (ISEs) is based on equilibrium partitioning of analyte ions at the phase boundary between the sample and the membrane, which results in potentiometric responses following the well-known Nernst equation.1,2 The slopes of such responses, called Nernstian slopes, are proportional to the inverse of the charge number of the analyte ions. As a result, sensitivities for multivalent ions are smaller than those for monovalent ions, which makes selective detection of the former in the presence of the latter difficult. Many liquid membrane electrodes for divalent ions have been developed * Corresponding author. E-mail:
[email protected]. (1) Morf, W. E. The Principles of Ion-Selective Electrodes and of Membrane Transport; Elsevier: New York, 1981. (2) Buck, R. P.; Lindner, E. Pure Appl. Chem. 1994, 66, 2528-2536. S0003-2700(97)01018-4 CCC: $15.00 Published on Web 01/06/1998
© 1998 American Chemical Society
by incorporating highly selective ionophores,3,4 by successfully applying optimized ionic site/ionophore ratios,5-7 and by using plasticizers with high dielectric constants,1,7 but there have been very few approaches to improve selectivities for multivalent ions by increasing response slopes. An example of the latter is the detection of biomedically important polyions, such as heparin or protamine, with ionophorefree ion-exchanger electrodes by measuring steady-state electromotive force (emf) responses.8 Recording emf values after contact of the membrane with the samples for a short, fixed period provides sigmoidal-shaped calibration curves with sufficient sensitivity and reproducibility. On the other hand, only very small potentiometric responses with slopes of less than 1 mV/decade are obtained for these polyions after phase boundary equilibration of the sample and ISE membrane. However, the necessity for regeneration of such ISEs poses a limitation to their use. Similar steady-state responses have been well-known for ionophore-free ion-exchanger electrodes9 and neutral carrier-based electrodes10 and have, so far, been the only examples of super-Nernstian responses that are theoretically fully understood. Also very interesting examples of super-Nernstian responses were reported by Suzuki and co-workers for ISEs based on natural polyether antibiotics with carboxyl groups (e.g., ionomycin, monensin, and salinomycin).11-13 Surprisingly, response slopes of 58 mV/decade to Ca2+ and Ba2+ have been obtained with these electrodes. This is twice as much as expected for Nernstian responses to these alkaline earth ions. The response times of these electrodes are within 30 s, and their calibration curves are linear in wide dynamic ranges,12 suggesting a response mechanism (3) Umezawa, Y. Handbook of Ion-Selective Electrodes: Selectivity Coefficients; CRC Press: Boca Raton, FL, 1990. (4) Bu ¨ hlmann, P.; Bakker, E.; Pretsch, E., submitted. (5) Meier, P. C.; Morf, W. E.; La¨ubli, M.; Simon, W. Anal. Chim. Acta 1984, 156, 1-8. (6) Eugster, R.; Gehrig, P. M.; Morf, W. E.; Spichiger, U. E.; Simon, W. Anal. Chem. 1991, 63, 2285-2289. (7) Bakker, E.; Bu ¨ hlmann, P.; Pretsch, E. Chem. Rev. 1997, 97, 3083-3132. (8) Meyerhoff, M. E.; Fu, B.; Bakker, E.; Yun, J.-H.; Yang, V. C. Anal. Chem. 1996, 68, 168A-175A. (9) Maj-z´urawska, M.; Sokalski, T.; Hulanicki, A. Talanta 1988, 35, 281-286 and references cited therein. (10) Bakker, E. Anal. Chem. 1997, 69, 1061-1069. (11) Suzuki, K.; Tohda, K.; Sasakura, H.; Shirai, T. Anal. Lett. 1987, 20, 3945. (12) Suzuki, K.; Tohda, K.; Aruga, H.; Matsuzoe, M.; Inoue, H.; Shirai, T. Anal. Chem. 1988, 60, 1714-1721. (13) Suzuki, K.; Tohda, K. Trends Anal. Chem. 1993, 12, 287-296.
Analytical Chemistry, Vol. 70, No. 3, February 1, 1998 445
Table 1. pH Dependence of the Response Slopes of Ba2+-Selective oNPOE-PVC (2:1) Membranes Based on Monensin or Lasalocid and Ionic Sites monensina + KTpClPBb pH 1.0c 2.0d 4.0e 4.5e 5.5e 7.0f 8.0f d
slope (mV/decade)
linear range (M)
∼0 39.9 ( 0.4
10-3.1-10-2.3
57.8 ( 1.8 56.6 ( 0.5 56.4 ( 0.3
10-3.1-10-1.6 10-4.4-10-2.4 10-4.3-10-2.4
lasalocid + KTpClPBb slope (mV/decade)
linear range (M)
28.5 ( 0.2 30.1 ( 0.1 53.3 ( 0.3 51.6 ( 0.3 46.0 ( 0.5 29.1 ( 0.1 30.7 ( 0.1
10-4.4-10-2.4
10-4.2-10-2.3 10-4.1-10-2.3 10-4.1-10-1.6 10-5.1-10-2.3 10-4.4-10-2.4 10-4.3-10-2.4
lasalocid + TDDMAClb slope (mV/decade)
linear range (M)
∼0 15.4 ( 1.3
10-3.1-10-2.3
28.8 ( 0.2 28.1 ( 0.1 29.8 ( 0.3
10-3.2-10-2.3 10-4.4-10-2.4 10-4.3-10-2.4
a Anionic responses were obtained in the presence of TDDMACl (see text). b 50 mol % relative to the ionophore. c Measured in 0.10 M HCl. Adjusted with diluted solutions of HCl. e Measured in 1.0 mM (CH3COO)2Mg/HCl buffer solutions. f Measured in 0.10 M Tris/HCl buffer solutions.
that is not based on steady-state phase boundary potentials. To explain such apparently “twice-Nernstian” slopes, Suzuki and coworkers presented a schematic response mechanism based on ionophore-analyte complexation at the sample/membrane interface.12 The formation of a 1:1 complex between the deprotonated ionophore and the primary ion was suggested, requiring that one H+ is released from the membrane into the aqueous sample phase for every Ba2+ or Ca2+ forming a complex with the ionophore. However, no quantitative model has been presented for the explanation of the response of such ISEs. This may have hindered the wider application of this attractive phenomenon in the development of ISEs for multivalent ions. The quick and stable, apparently “twice-Nernstian” responses of ISEs based on acidic ionophores made us wonder whether this unusual phenomenon can be explained with a model for equilibrium phase boundary potentials.7,14 This model was recently used by Schaller et al. to account for monovalent cation responses of ISEs with acidic ionophores and ionic sites.15 In this paper, we discuss the slopes of responses to divalent cations as measured with ISEs based on acidic ionophores and ionic sites. Our phase boundary potential model newly includes the hydrogen iondivalent cation exchange equilibrium at the sample/membrane phase boundary and shows the conditions needed for the observation of apparently “twice-Nernstian” slopes. This model predicts that, in the case of membranes with acidic ionophores and anionic sites, the slopes of the responses to divalent cations depend on the pH of the sample solution. As the pH decreases, response slopes are found to change from Nernstian to apparently “twiceNernstian” and then back to Nernstian again. On the other hand, membranes with acidic ionophores and cationic sites are found to give Nernstian responses to divalent cations at high pH and to H+ at low pH. To confirm the validity of this model, the influence of the sample pH on the response slopes was determined for ISEs based on ionic sites and monensin, lasalocid (Ba2+-selective carboxylate ionophores), or bis(2-heptylundecyl) phosphate (a Ca2+-selective organophosphate ionophore). The good agreement of the theoretical predictions with the experimental results confirms a response mechanism for the apparently “twice-Nernstian” responses according to which the concentration of the free primary ion in the membrane phase changes with the inverse of the primary ion activity in the pH-buffered sample solution. This contrasts to conventional Nernstian responses of carrier-based ISEs, for which the concentration of the free primary ion in the 446 Analytical Chemistry, Vol. 70, No. 3, February 1, 1998
membrane phase is buffered with the ionophore and remains sample-independent. EXPERIMENTAL SECTION Reagents. All reagents were of the highest grade commercially available and were used without further purification unless mentioned otherwise. Deionized and charcoal-treated water (18.2 MΩ‚cm specific resistance) obtained with a Milli-Q PLUS reagent-grade water system (Millipore Corp., Bedford, MA) was used for preparing all sample solutions. Potassium tetrakis(4-chlorophenyl)borate (KTpClPB), 2-nitrophenyl octyl ether (oNPOE), and dioctyl phenylphosphonate (DOPP) were obtained from Dojindo Laboratories (Kumamoto, Japan). Poly(vinyl chloride) (PVC, high molecular weight) was purchased from Fluka AG (Buchs, Switzerland). The free acid forms of monensin and lasalocid were prepared from their sodium salts (Sigma Chemical Co., St. Louis, MO) as described elsewhere.12 Bis(2-heptylundecyl) hydrogen phosphate (BHU-PO4H) was synthesized as described elsewhere.16 Tridodecylmethylammonium chloride (TDDMACl, Aldrich Chemical Co., Milwaukee, WI) was recrystallized from ethyl acetate prior to use. 2-Amino-2-(hydroxymethyl)-1,3propanediol (Tris) was purchased from Wako Pure Chemical Industries (Osaka, Japan). Membranes. Solvent polymeric membranes, containing 20 mmol‚kg-1 ionophore in its acid form, ionic sites (for their concentrations see below), 64-65 wt % plasticizer (oNPOE or DOPP), and 33 wt % PVC, were prepared according to a procedure reported previously.17 The membranes were then cut into small disks of 7 mm diameter and mounted onto Philips electrode bodies (model IS-561, Philips Electronic Instruments Co., Mahwah, NJ). For overnight conditioning and as internal filling solutions, 0.1 M BaCl2 or CaCl2 solutions pH-buffered the same way as the sample solutions (see Tables 1 and 2, respectively) were used. Emf Measurements. Philips-type electrode bodies, doublejunction-type Ag/AgCl reference electrodes (Denki Kagaku Keiki (DKK) Co., Tokyo, Japan), and an ion meter (model IOL 50, DKK) were used for all emf measurements. The cell assembly for the (14) Bakker, E.; Na¨gele, M.; Schaller, U.; Pretsch, E. Electroanalysis 1995, 7, 817-822. (15) Schaller, U.; Bakker, E.; Pretsch, E. Anal. Chem. 1995, 67, 3123-3132. (16) Bu ¨ hlmann, P.; Amemiya, S.; Umezawa, Y., manuscript in preparation. (17) Amemiya, S.; Bu ¨ hlmann, P.; Tohda, K.; Umezawa, Y. Anal. Chim. Acta 1997, 341, 129-139.
Table 2. pH Dependence of the Response Slopes of Ca2+-Selective DOPP-PVC (2:1) Membranes Based on BHU-PO4H and Ionic Sites KTpClPBa pH
slope (mV/decade)
linear range (M)
1.0b 2.5c 3.5c 4.0d 4.5d 5.0d 5.5d 7.0e
∼0 25.9 ( 5.1 53.6 ( 0.9 51.5 ( 0.5 49.6 ( 0.5 45.7 ( 0.1 42.8 ( 0.7 31.8 ( 0.2
10-2.3-10-1.6 10-2.3-10-1.6 10-2.7-10-1.6 10-3.6-10-1.6 10-4.0-10-1.6 10-5.0-10-1.6 10-5.0-10-2.3
follows:
E)
TDDMACla slope (mV/decade)
linear range (M)
14.2 ( 0.4
10-3.1-10-1.6
27.8 ( 0.9 29.8 ( 0.2
10-2.7-10-1.6 10-4.0-10-1.6
a 50 mol % relative to the ionophore. b Measured in 0.10 M HCl. Adjusted with diluted solutions of HCl. d Measured in 1.0 mM CH3COOK/HCl buffer solutions. e Measured in 1.0 mM Tris/HCl buffer solutions.
c
potentiometric measurements was as follows:
Ag|AgCl|3 M KCl||outer filling solution||sample solution|membrane|internal filling solution|AgCl|Ag The sample solutions were prepared by dissolving appropriate amounts of BaCl2‚2H2O or CaCl2‚2H2O in buffer solutions (see Tables 1 and 2, respectively). As outer filling solutions, 1 M LiCl was used for the lasalocid-based electrodes and 1 M KCl for all other electrodes. Response times (∆E/∆t < 0.1 mV/30 s)18 were less than 5 min. Calculations. Mathematica 3.0 (Wolfram Research Inc., Champaign, IL) was used to calculate the curves of the figures in the theoretical part. THEORY The description of divalent cation-selective membranes with an acidic ionophore and ionic sites as presented below is based on the following assumptions:7,14 (1) The phase boundary potential at the sample/membrane interface changes with the composition of the sample solution. All other contributions to the measured emfsmost importantly, the membrane internal potential difference (diffusion potential) and the phase boundary potential at the membrane/internal filling solution interfacesremain constant. (2) Local chemical equilibrium is achieved at the phase boundary between the membrane and the sample solution. (3) The formation of ion pairs between ionic sites and their counterions is neglected. (4) Activity coefficients are constant for all ionic species in the membrane phase, and therefore concentration values instead of activities are used. (5) The doubly charged primary ion, M2+, forms substantially stronger 1:1 complexes with the charged form of the ionophore, L-, than with its neutral form, LH (giving complexes LM+ and LHM2+, respectively), but no complexes of higher stoichiometry are formed. Based on these assumptions, the phase boundary potential E at the membrane/sample solution interface is described as (18) Uemasu, I.; Umezawa, Y. Anal. Chem. 1982, 54, 1198-1200.
with
kMaM RT ln 2F [M2+]
(1)
kM ) exp({µ0M(aq) - µ0M(mem)}/RT)
where kM is the so-called single ion distribution coefficient describing the distribution of M2+ between the sample solution and membrane phase, aM and [M2+] are the activity of M2+ in the sample solution and the concentration of M2+ in the membrane phase, respectively, and µ0M is the chemical standard potential of M2+ in the respective solvent (“aq” denotes species in the sample solution phase, “mem” species in the membrane phase). The following eqs 2-5 describe the various equilibria into which the membrane components are involved. They are defined by the formation constants, βLHM and βLM, for the complexes of the respective form of the ionophore, LH and L-, and the primary ion, M2+,
[LHM2+]
βLHM ) βLM )
[LH][M2+] [LM+] [L-][M2+]
(2)
(3)
the deprotonation constant, Ka, of the acidic ionophore,
Ka )
[L-][H+] [LH]
(4)
and the constant, KM,H, for the cation exchange of the free H+ and free primary ion, M2+, between the sample solution and the membrane phase,
KM,H )
kM
aH2[M2+]
kH
[H+]2aM
) 2
(5)
where kH is the so-called single ion distribution coefficient describing the distribution of H+ between the sample solution and membrane phase. Consequently, the mass balance for the ionophore (eq 6) and the electroneutrality condition for membranes with anionic or cationic sites (eqs 7 and 8, respectively) are described as
LT ) [L-] + [LM+] + [LH] + [LHM2+]
(6)
[L-] + [R-] ) [LM+] + 2[M2+] + 2[LHM2+] + [H+] (7) [L-] ) [R+] + [LM+] + 2[M2+] + 2[LHM2+] + [H+] (8) where LT, [R-], and [R+] are the total concentrations of the ionophore and the anionic and cationic sites in the membrane phase. In the following, the concentration of the free primary ion in the membrane phase, which is derived from eqs 2-8, and the corresponding phase boundary potentials are expressed as Analytical Chemistry, Vol. 70, No. 3, February 1, 1998
447
functions of aM. For simplicity, it is assumed that [M2+], [H+] , [R-], [R+] < LT. Membranes with Anionic Sites. Neglecting the very small concentrations of [M2+] and [H+] in eq 7, combination of eqs 2-7 gives an equation of the third degree in [M2+]:
βLHM2(2LT - [R-])2[M2+]3 - Ka2KM,HβLM(LT -
(
[R-])2
(
aM
aH2
)
+ 2A [M2+]2 + Ka2KM,HβLM2(LT2 -
)
aM aM [R-]2) 2 2 + B [M2+] - Ka2KM,H(LT + [R-])2 2 ≈ 0 (9) aH aH with
A) B)
βLHM[R-](2LT - [R-]) Ka2KM,HβLM2(LT - [R-])2 [R-]2 Ka KM,HβLM(LT2 - [R-]2) 2
The equilibrium concentrations of the membrane components can be calculated as functions of [M2+] by combining eqs 2, 3, 6, and 7 (for [H+], see eq 5):
[L-] )
βLHM[M2+](2LT - [R-]) - [R-] γ
[LM+] ) βLM[M2+][L-] [LH] )
(11)
LT + [R-] - βLM[M2+](LT - [R-]) (12) γ
[LHM2+] ) βLHM[M2+][LH] with
(10)
(13)
γ ) βLMβLHM[M2+]2 + (3βLHM - βLM)[M2+] + 1
By solving eq 9 for [M2+] and inserting the solution into eq 1, the phase boundary potential can be obtained as a function of aM. Figure 1 shows response curves thus calculated for a representative set of parameters (βLM, βLHM, Ka, KM,H, LT, [R-]) and various values of pH. Similar responses have been calculated for a number of other parameter sets. In general, the response curves consist of three linear ranges with Nernstian, apparently “twiceNernstian”, and again Nernstian slopes (29.1, 58.2, and 29.1 mV/ decade at 20 °C, respectively). A change of the pH by 1 unit shifts the response curve horizontally by 2 log aM units, while the shape of the response curve remains unchanged (Figure 1). For understanding the response mechanism, equilibrium concentrations of the free H+, the ionophore, and its complexes were calculated as a function of aM by inserting the solution of eq 9 into eqs 5 and 10-13. Figure 2 shows the concentrations of the membrane components as obtained for the same parameter set as that used for Figure 1 (pH 7.0). The composition of the membrane changes with log aM, as can be seen from Figure 2. For a given range of aM (e.g., 10-6-10-1 M), the membrane composition, and concomitantly the type of emf response, depend on the pH. The response mechanisms for three limiting cases, which correspond to the respective linear response ranges, and the conditions for obtaining a maximum range for the apparently 448
Analytical Chemistry, Vol. 70, No. 3, February 1, 1998
Figure 1. Calculated phase boundary potentials as a function of the activity of a divalent cation in the sample solution for a membrane with an acidic ionophore and anionic sites. Responses are shown for sample solutions of pH 4.0, 7.0, and 10.0. For the calculation of these phase boundary potentials, the following parameters were chosen: βLM ) 1010 kg‚mol-1, βLHM ) 103 kg‚mol-1, Ka ) 10-12 mol‚kg-1, KM,H ) 1, LT ) 20 mmol‚kg-1, and [R-] ) 10 mmol‚kg-1.
“twice-Nernstian” response (∆ log aM in Figure 1; for definition, see below) are discussed in the following. Emf Response at High pH (Charged Carrier Mechanism). At sufficiently high pH, where aM/aH2 . B (>A), eq 9 becomes
βLHM2(2LT - [R-])2[M2+]3 - Ka2KM,HβLM2(LT aM aM [R-])2 2[M2+]2 + 2Ka2KM,HβLM(LT2 - [R-]2) 2[M2+] aH aH aM Ka2KM,H(LT + [R-])2 2 ≈ 0 (14) aH which can be simplified to (see Appendix)
aM - Ka2KM,H 2{βLM(LT - [R-])[M2+] aH (LT + [R-])}2 ≈ 0 (15) Solving eq 15 shows that the concentration of the free primary ion in the membrane is given as
[M2+] ≈
LT + [R-] βLM(LT - [R-])
(16)
Inserting eq 16 into eq 1 gives a Nernstian response to the divalent cation M2+, corresponding to the linear range on the right side of each response curve in Figure 1. Combination of eq 16 with eqs 10-13 gives the concentrations of the ionophore and its complexes in the membrane as [L-] ≈ (LT - [R-])/2, [LM+] ≈
≈ [LM+] ≈ 0. This membrane composition is typical for the conventional response mechanism of neutral carrier-based membranes, in which the vast majority of the ionophore molecules are protonated and in this form bind the primary ions (Figures 2a and 3b). Emf Response at Intermediate pH (Apparently “Twice-Nernstian” Response Mechanism). In an intermediate pH range where B . aM/aH2 . A, eq 9 becomes
βLHM2(2LT - [R-])2[M2+]3 aM Ka2KM,HβLM2(LT - [R-])2 2[M2+]2 + [R-]2[M2+] aH aM Ka2KM,H(LT + [R-])2 2 ≈ 0 (20) aH which can be simplified to (see Appendix)
{
}
aM [M2+] Ka2KM,HβLM2(LT - [R-])2 2[M2+] - [R-]2 ≈ 0 aH (21) Because [M2+] * 0 (see Appendix), eq 21 gives
Figure 2. Calculated equilibrium concentrations of (a) the ionophore and its complexes and (b) the deprotonated ionophore and the free primary and hydrogen ions in the membrane phase of a divalent cation-selective electrode based on an acidic ionophore and added anionic sites. These concentrations were calculated for pH 7.0 and the same parameter set as that used for Figure 1.
(LT + [R-])/2, and [LH] ≈ [LHM2+] ≈ 0. This membrane composition is typical for charged carrier-based membranes, in which the acidic ionophore occurs almost exclusively in its deprotonated form and binds the primary ion in this form (Figures 2a and 3a). Emf Response at Low pH (Neutral Carrier Mechanism). At sufficiently low pH, where aM/aH2 , A ( 0, it follows from eqs 10 and 12
βLHM[M2+](2LT - [R-]) - [R-] g 0
(A1)
LT + [R-] - βLM[M2+](LT - [R-]) g 0
(A2)
and for γ < 0 analogously
βLHM[M2+](2LT - [R-]) - [R-] e 0
(A3)
LT + [R-] - βLM[M2+](LT - [R-]) e 0
(A4)
Combining eqs A1 and A2 gives
{
βLM e βLHM 1 + 2
(LT/[R-])2 LT/[R-] - 1
}
(A5)
which contradicts the assumption that βLM is substantially larger Analytical Chemistry, Vol. 70, No. 3, February 1, 1998
453
than βLHM unless R ≈ 0 or LT/[R-] ≈ 1. The eqs A3 and A4 are in the following used to deduce eqs 15, 18, and 21. Simplification of Eq 14 to Eq 15. At high pH, where aM/ 2 aH . A, it follows for C, which is defined as the quotient of the first term on the left-hand side of eq 14 divided by its second term, that
fourth term on the left-hand side of eq 17 divided by its third term, that
D,D
aH2B ) aM β
LT + [R-] LM[M
](LT - [R-])
2+
(A8)
Furthermore, eq A4 can be rearranged to
C,C
aM aH2A
)
βLHM[M2+](2LT - [R-]) -
[R ]
βLM[M2+](LT - [R-])
Because eq A3 can be rearrangeed to
βLHM[M2+](2LT - [R-]) [R-]
e1
(A7)
combination of eqs A6 and A7 shows that C , 1 and that eq 15 can be obtained by neglecting the first term on the left-hand side of eq 14. Simplification of Eq 17 to Eq 18. At high pH, where aM/ aH2 , B, it follows for D, which is defined as the quotient of the
454
LT + [R-]
(A6)
Analytical Chemistry, Vol. 70, No. 3, February 1, 1998
e1
(A9)
It follows from eqs A8 and A9 that D , 1 and that eq 18 can be obtained by neglecting the fourth term on the left-hand side of eq 17. Simplification of Eq 20 to Eq 21. C and D can be used analogously in eq 20. Because B . aM/aH2 . A for an intermediate pH, eqs A6-A9 are also valid in the apparently “twiceNernstian” response range. Therefore, eq 21 can be obtained by omitting the first and fourth terms on the left-hand side of eq 20. Received for review September 15, 1997. November 24, 1997. AC9710184
Accepted
