Anal. Chem. 1989, 61, 453-458 (28) aassi, J.; Mclouf, J.; Pradelles P. Hendbook of Experimental Phermacology; Patrono, C., Peskar, E. A., Eds.; Springer-Verlag;Berlin-HeMelberg, 1987; Vol. 82, pp 91-141.
RECEIVEDfor review February 11,1988. Accepted November
453
21,1988. This work was supported by financial aid from the Institut de Recherche Fondamentale (Commissariat Ii 1’Energie Atomiaue. France) and the Fondation de la Recherche Midicale, Fiance.
Comparison of Proposed Response Mechanisms of Precipitate-Based Ion-Selective Electrodes in the Presence of Interfering Ions Thomas R. Berube and Richard P. Buck* Department of Chemistry, William Rand Kenan Laboratories, University of North Carolina, Chapel Hill, North Carolina 27599 Ern0 Lindner, Miklbs Gratzl, and Ern0 Pungor
Institute for General and Analytical Chemistry, Technical University of Budapest, GellBrt tBr 4, 11 11 Budapest XZ, Hungary
New experlmental results on nonmonotonlc translents are reported here to conflrm and extend published results. The new data support the notlon that both adsorptlon/desorptlon and surface reactlons of the crystal membrane are Important processes determlnlng transient selectivity coeffIclents. The former Induces local changes In exterlor surface concentratlons of responslve Ions. The latter change the composition of the membrane. Both affect the measured potential. Two proposed models for the response of a preclpltate-based Ion-selectlve electrode (ISE) In the presence of primary and Interfering Ions are discussed. The mathematlcal expressions for the models are examlned in llght of ISE thermodynamlcs, flnlte lonsxchange klnetlcs, and non-steady-state ditfuslon. The adsorptlon/desorptlon model descrlbes the overshoot/ decay portlon of the potentlal-tlme curves more accurately than the mlxed-phase formation model, but a klnetlc version of the latter applles at very long times. The comblnatlon of the expanded models was shown to describe all features of the experlmental curves.
INTRODUCTION Lindner et al. (1)have shown that precipitate-based ionselective electrodes (ISEs) can respond nonmonotonically to a sudden step change in the activity of interfering ions. The experimental potential-time curves were made up of three regions: a rapid potential excursion, a slower decay, and a region of very slow potential change (1,2). The occasion was an AgI membrane exposed on one side to a step increase in Br- interference at a constant level of background I-. The potential excursion was negative going, as though the electrode were exposed to excess I-. Likewise, a step decrease of Brfrom high to low or zero activity, again at constant I-, produced a reverse, positive-going overshoot/decay. The authors explained that a layer of chemisorbed primary ions, I-, was partially desorbed and replaced by interfering ions. Interfering ions diffused to the electrode surface, displaced adsorbed primary ions, and caused a rapid local increase of primary ion concentration at the surface, which was sensed by the electrode. The potential decay occurred as excess primary ions diffused toward the bulk. In the reverse case (negative activity 0003-2700/89/0361-0453$01.50/0
step), the instantaneous decrease in the bulk concentration of interfering ions caused their desorption, replacement by primary ions, and a rapid decrease in the concentration of primary ions in solution at the electrode surface. The third, slowly changing portion of the curve was attributed to surface transformation. This adsorption/desorption model was criticized by Morf (3), who disregarded the time dependence of the solution composition and pointed, instead, to the mixed phase formation (MPF) theory generalized by Hulanicki and Lewenstam (4) from the earlier work of Buck (5). Hulanicki and Lewenstam showed that variation in measured selectivity coefficients could be a result of the time dependence of the formation of a mixed phase AgY,XI, at the surface of the membrane. Morf derived the time dependence of the apparent coverage factors, defined as the surface mole fraction of AgY formed on the AgX surface by ion exchange, and fitted these equations to Lindner’s data. The third region of the curves was not discussed. Gratzl et al. (6) provided a quasi-quantitative description of the adsorption/desorption model and showed the resultant equations fit the data better than Morf’s equations. The authors concluded that for solid-state membranes, the MPF model was applicable for large values of the theoretical selectivity coefficient ( K x y >> l),but not valid for the case treated by Lindner et al. (1) (KXY 1, may be different, but both can be viewed as describing nearly the same phenomena. Interpretation of data from precipitate-based ISE’s in flowing solutions requires determination of the applicability of these theories (7). This paper therefore presents new experimental data for comparison of the existing models and introduces a kinetic model to give a more comprehensive description of the two-ion transients, including the previously unexplained long-time transients. THEORY For precipitate-based ISE’s,the interfacial potential differences are generated by ion exchange of the principal ions. For AgX 0 1989 American Chemical Society
454
ANALYTICAL CHEMISTRY, VOL. 61, NO. 5, MARCH 1, 1989
4 m e m b - +soh
=
(RT/mIn
1
k+(Ag+ s o h )
k ( X - soln) = (-RT/F) In (X- memb) L
1
(5)
(la)
Ob)
Eo values are defined in terms of single-ion partition coefficients (ki)or in terms of standard free energies of the ions in each phase. The interfacial potential is determined by activities of primary ions in solution adjacent to the membrane, and in the bulk membrane at its surface contacting the solution. All processes that generate changes in bulk primary ion composition or change activities in solution affect the interfacial potential. Physical adsorption, however, does not affect the interfacial potential when phase compositions remain constant. Differences between activities in solution bulk and near the surface can occur in the transient state by imperfect mixing in the stagnant layer, or by generation of a slug of ions previously (in time) adsorbed on the surface. In addition, if AgY were a pure phase and more stable than a second pure phase AgX, then the chemical reaction AgX + Y- = AgY + X- has a metathetical equilibrium constant Kxy that is positive and >> 1, specifically given by K,o(AgX)/K,,O(AgY) (5). In the case where solid-solution or mixed-phase (AgX1,Ys) formation (4, 5) is possible, Kxy is given by
Of course, ax’ and ay’ apply to the solution composition at the membrane surface, and s and (1 - s) do not necessarily have to be uniform throughout the film. It is sufficient (for local equilibrium) to convert the surface composition to the equilibriumvalue, if the underlying layers are converted much more slowly. It is therefore important to consider both adsorption effects and surface transformation in an attempt to describe the transients, since both affect the interfacial potential. The “conflicting”theories are in reality limiting cases which stress one of these phenomena over the other. Limiting cases occur when (a) the primary ion or interferent does not appreciably adsorb, (b) there is no surface conversion, or (c) the rates of adsorption and surface conversion are appreciably different. Morf treated the case of constant solution activity at the surface. The equilibrium expression in eq 2 is used locally with surface activities ax’ and ay’ that are determined by balancing inward and outward fluxes through the stagnant layer of thickness 6 by using the steady-state approximation ( 3 , 4 ) . The equation for instantaneous electrode response is (3)
and C is an experimentallydetermined parameter (3) defined as
C = Dy’A/n,,6
(6)
where A is the area of the membrane surface and ntotis the total number of moles of AgX converted to AgY in AgY,,X, by mixed-phase formation. In Gratzl’s treatment of the adsorption/desorption model (6),surface transformation was considered to be kinetically slower than sorption. The equations described the limiting case of no surface transformation. These arise from a consideration of concurrent transient diffusion of the two ions with similar diffusion coefficients in the Nernst layer. As an initial simplifying assumption, the interfering ion diffusion to the surface of the electrode and the resulting desorption of primary ion were considered to be instantaneous. The time dependence of the released primary ion concentration at the electrode surface was only time-dependent quantity determining the potential-time response. It was
where ci is the concentration of species i, MXis the quantity of primary ions forced to desorb from the electrode surface, and D, is the mean diffusion coefficient of the ions in the diffusion layer. The potential-time dependence was obtained by inserting eq 7 into
E ( t )= Exo - S log [cXO + cx(x=O,t) + K x y ~ y ( ~ = O , t = m (8) )] where E ( t ) is the time-dependent cell potential, S is the experimentally determined slope of the electrode calibration curve, cxo is the initial concentration of primary ion (prior to desorption), and cx(x=O,t)is that arising from desorption (or adsorption). In the more detailed formulation of this model, finite interfering ion diffusion was taken into consideration. Due to the unknown nature of the isotherm for the competitive adsorption of the two ions, the isotherm was assumed to be linear (eq 14 in ref 6) with slope K. The equation for this detailed model contained several printing errors, as well as a timedependent selectivity term. However, no surface transformation takes place in the time frame of the sorption processes, according to the initial assumptions. The resultant equations (8 and eq 17 in ref 6) should therefore not contain a timedependent Kxycy term, as Lewenstam et al. have pointed out (7), although it may contain a constant selectivity term (kXyaPPcy[ t = O ] ) due to prior surface transformation. The corrected form of the equation for the detailed model, written to emphasize the unknown nature of the isotherm slope, is
where ax and a y are bulk solution activities of the primary and interfering ion, respectively, and D/ = Di/ yi. Time dependence of the potential arises from the time dependences of apparent coverage factor s ( t ) and electrolyte surface concentrations. For an AgX membrane with an initial apparent coverage factor of so, this time dependence is given by (ax + Kxyay)Ct = [Kxy - (Dy’/Dx’)I(s - SO) a x + (Dy’/Dx’)ay In (4) KXY a x + KxYaY
(-)
where seq is the equilibrium value of the apparent coverage factor
where Acy = cy(x=O,t=m) - cy(r=O,t=O) and kxynPP is the apparent selectivity coefficient at t = 0.
ANALYTICAL CHEMISTRY, VOL. 61, NO. 5, MARCH 1, 1989
455
-
0. 0
I
I 0
d
v,
0
d
1
T i m e (s) b -
85. 0 65. 0
45. 0 25. 0
0
v,
0
d
d
1
T i m e (s) Figure 1. Experimental transients recorded with an I- ISE. Potentials are in millivolts and have been shifted for comparison. (a) Positive activity steps: bottom, lo-' M I-, 0 lo-' M B r ; middle, lo4 M I-, 0 lo-' M Br-; top, lo3 M I-, 0 10' M Br-. (b) Negative activity steps: top, lo-' M I-, lo-' 0 M Br-; middle, lo-' M I-, lo-' 0 M I-, 10' 0 M Br-. M Br-; bottom,
-
--
-
RESULTS AND DISCUSSION To evaluate the contributions of the two models to the analysis of the transients, new experiments were performed with improved instrumentation capable of measuring very fast transients (8,9).Measurements were made with a homemade silver iodide pellet fixed in a Phillips IS-550 electrode body as an iodide selective electrode. Activity steps were performed as shown in Figure 1 with KN03 as supporting electrolyte. A family of transient responses to Br- and fiied I- are given in Figure 1. The different peak heights result, in the adsorption/desorption model, from relative surface concentration change in comparison with the preexisting I- level in solution. The experimental curve shape (rapid rise in potential to a peak, followed by a slower decay, for either direction of the activity step) is a simple peaking function. There is no apparent potential arrest or an inflection. The new data were taken by using higher solution flow rates than the previously published results of Lindner et al. (1) because we wished to compare the short-time rise and decays of potential with both models. Figure 2a is an expanded view of one transient, and Figure 2b shows a fit to the long form of Gratzl's equation (eq 9) assuming a linear isotherm slope. The fit is not perfect, and it is probable that the unknown isotherm for this competitive tweion adsorption limits this equation from providing a better description of the overshoots. It is clear from the slope term that a nonlinear competitive adsorption isotherm would introduce additional time-dependent concentration terms into the convolution integral of eq 9. This omission may explain why eq 9 gives fits with a slightly slower rise and faster decay than the experimental. It also may explain why eq 9 predicts identical curves for any ax and ay in a constant ratio a y / a x , in contrast to the concentration dependence shown in Figure
8
2
8
m o g
d
0
0
0
4
't 0,
-.
v
p.
0
n
1
j
n
j
Time ( s )
Figure 2. Overshoot portions of potential-time curves for positive and negative activity steps. (a) Experimental curve: [I-] = lo4 M; [Br-] = 0 lo-' 0 M. (b) Detailed adsorption/desorption model (eq 9). The slope of the adsorption isotherm is assumed to be constant and equal to 3.398 X lo-'. Other parameters: 6 = 8.686 X lo-' cm-'; S = 59.16 mV; D, = 1.86 X cm'/s; E ( 0 ) = 0 mV.
- -
1 and reported by Lindner (1). As a result, eq 9 cannot be considered a final equation describing the detailed adsorption/desorption model until the competitive adsorption has been characterized, and curve fittings that assume a linear isotherm are only useful to visualize the effect of diffusion/ sorption phenomena. The MPF model predicts either peaked responses or peaks followed by plateau, an inflection, and a sudden potential fall to equilibrium. The shape of the response for a particular system is dependent upon the direction of the activity step and the magnitude of KXV The peaking function shown in Figure 3a is expected for the case of AgI bathed in constant I- with a step to Br- + I-, as well as for AgBr with an AgI layer when the bathing solution is stepped from Br- + I- to Bralone. The second form of response in Figure 3b occurs for AgI with a thin AgBr layer when Br- + I- is changed to Ialone; the equivalent case is AgBr in Br- stepped to I- + Br-. Figure 4a shows the MPF model function from eq 3 and 4 applied to the experimental data from Figure 2a. These functions with the same numerical parameters cannot fit both negative and positive activity steps. First of all, the experimental curves do not show the slow potential change followed by a more rapid change to near the equilibrium value, i.e. do not show a long-time inflection point. In an attempt to explain this, Lewenstam et al. (6) suggested that not enough of the surface had been converted to participate in local equilibrium. (See Figure 6 in ref 7, noting that the labels on the curves should be transposed.) However, their conclusion that this lack of surface conversion occurred in Lindner's experiments must be incorrect, based on the values of 6 and n,determined by Morf (3). These predict the minimum surface conversion required is on the order of s = far below the amount predicted by the fittings of eq 3 and 4 to the data of Lindner et al. An alternative explanation is that the region of slow potential change is not seen because of non-steady-state
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 5, MARCH 1, 1989
Table I. Two-Parameter Curve-Fitting Results for the Simplified Adsorption/Desorption Model exptb
6,
cm
M x , mmol/cm2
torr'
RSSd
A1 A2 A3 A4 A5 A6 A7 A8
(7.548 f 0.207) X (8.511 f 0.385) X (6.371 f 0.119) X loe3 (7.456 f 0.287) X (5.485 f 0.098) X (6.489 f 0.110) x 10-3 (5.781 f 0.111) X (6.209 f 0.097) X
(1.530 f 0.005) X (1.446 f 0.005) X (1.358 f 0.009) X (1.244 f 0.008) X (1.304 f 0.012) X (2.021 f 0.011) x 10-8 (1.255 f 0.011) X (1.810 f 0.011) X
-0.451 -0.418 -0.481 -0.447 -0.502 -0.504 -0.493 -0.505
4.162 4.745 11.758 11.415 22.500 14.681 18.753 14.456
B1
(2.145 f 0.072) X (3.304 f 0.096) X (3.352 f 0.101) X (3.002 f 0.109) x 10-3 (2.814 f 0.094) X (4.356 f 0.156) X (4.063 f 0.146) X
(4.069 f 0.128) X (4.992 f 0.127) X (5.035 f 0.131) X (4.025 f 0.130) X (4.291 f 0.131) X (4.241 f 0.102) X (4.075 f 0.105) x
-0.464 -0.446 -0.442 -0.447 -0.458 -0.429 -0.413
27.168 31.460 35.506 34.109 32.068 27.278 29.195
B2 B3 B4 B5 B6 B7
-
-
a From eq 7 and 8 with K x y = 0, S = 59.16 mV, and D, = 1.86 X lo-&cm2/s, using 101 points from about 25 ms to 1 s. b A = Br- activity step: 0 M in M I-. B = Br- activity step: 0 lo-' M in 10"' M I-. 'Correlation between fitting parameters. dSquare root of sum of sauared deviations.
70
I
I
~
m
s
4
.-. .
.-.
p.
p.
j
r\j
0
Time
d Time
d
d
d
n
Time ( s )
Flgure 3. Shape of the MPF model curve (eq 3 and 4) for the fobwing cases: (a) s < sq and K, < D y ' I D i or s > sq and K, > D y ' l D i ; (b) s > seq and K,, < D,'/Dx' or s < s w and K,, > Dy'lD;.
Flgure 4. Overshoot portions of potential-time curves for positive and negative activity steps. (a) MPF model cwve (eq 3 and 4): C = 85000 M-' s-'; so = 0; Dy'lD,' = 1; K,, = 1.66 X lo-'. (b) MPF model with buildup to steady-state diffusion: 6 = 9.5 X lo4 cm.
diffusion following the activity step. If steady-state diffusion is only achieved in the region of the inflection point, i.e. when the potential has begun to change rapidly (Figure 4a), the MPF model curve would look more peaked, much like the experimental curves. Figure 4b shows such an example. It also shows that, although the consideration of non-steady-state diffusion makes the rising portion of the curves look more like the experimental curves, the predicted potential decay is much faster than the experimental curves show for the negative activity step; in fact, it is nearly instantaneous. In addition, the experimental curves recorded a t high flow rates (Figures 1 and 2) show that the potential nearly returns to the initial value after the overshoot, unlike those recorded at lower flow rates (1). The MPF model equations (eq 3 and 4) are not capable of describing this return to the initial
potential and the subsequent long-time variation in the potential seen by Lindner et al., since they predict a decay to the final equilibrium potential given by the Nikolsky equation and offer no explanation for the origin of the third part of the transients. To investigate Lewenstam's claim that Gratzl's curve fittings were only better than Morfs due to the use of more fitting parameters, several experimental curves were fitted to the simplified sorption model equations (Table I) and the MPF model equations (Table 11) by using a modified Marquardt algorithm (10). The results show that the sorption model equations generally fit the data slightly better than the MPF equations. The estimated correlation between parameters (11)does indicate that the quality of fittings with the MPF model may not suffer if only one parameter were varied.
ANALYTICAL CHEMISTRY, VOL. 61, NO. 5, MARCH 1, 1989
457
Table 11. Two-Parameter Curve-Fitting Results for the MPF Model exptb
C, M-' s-l
AI A2 A3 A4 A5 A6 A7 A8
(1.049 f 0.044) X (1.056 f 0.051) X (8.902 f 0.427) X (1.003 f 0.043) X (7.947 0.390) X (8.120 0.445) X (8.662 f 0.429) X (8.101 f 0.429) X
B1 B2 B3 B4 B5 B6 B7
(5.575 f 0.644) X (6.936 f 0.566) X (7.553 f 0.588) X (6.950 f 0.545) X (6.118 f 0.551) X (1.079 f 0.066) X lo-" (1.111k 0.078) X IO-"
* *
lo-" IO-' IO-" lo-" lo-'
KXY
corrC
RSSd
(2.473 f 0.055) X (2.299 f 0.058) X lo4 (1.936 f 0.054) X lo4 (1.913 f 0.046) X lo-" (1.690 f 0.052) X lo-" (2.805 f 0.086) X lo4 (1.727 f 0.051) X lo4 (2.472 f 0.075) X IO-"
0.845 0.814 0.851 0.825 0.879 0.881 0.863 0.880
65.140 85.360 75.312 60.215 69.875 121.912 73.790 106.169
(2.311 f 0.241) X (4.450 f 0.298) X (4.756 f 0.300) X (3.476 f 0.230) X (3.249 f 0.252) X (5.654 f 0.258) X (5.099 f 0.274) X
0.961 0.924 0.918 0.933 0.944 0.865 0.871
32.448 44.523 44.847 28.142 33.586 33.706 39.322
10" 10" loT6
IO"
-
-
OFrom eq 3 and 4 with so = 0 and Dy'/Dx' = 1, using 101 points from about 25 ms to 1 s. bA = Br- activity step: 0 lo-* M in 10" M I-. B = Br- activity step: 0 lo-' M in lo-" M I-. cCorrelation between fitting parameters. dSquare root of sum of squared deviations. In addition, fitting Kxy to the data resulted in erroneous values of this parameter (Table 11, experiments Bl-B7). Lewenstam et al. suggested iodide adsorption isotherm saturation as an additional problem with Lindner's model. However, it is not clear how saturation of a single ion isotherm would affect the competitive adsorption in question. Surface Transformation as a Kinetic Process. The resolution of the conflict between the models lies in the time frame of the surface transformation. Lindner et al. ( I ) have proposed that surface conversion is a long-time process unrelated to the overshoots, while Morf (3) and Lewenstam et al. (7) suggested that it is a fast process and the cause of the overshoots. The equations of Hulanicki and Lewenstam and of Morf must therefore be examined closely to determine the role of surface transformation. Morf has referred to the work of Schwab (12) and Jaenicke (13, 14) in assuming that the rate of surface conversion is limited by anion diffusion through the aqueous boundary layer. While this assumption is accurate for the thermodynamically favorable case treated by these authors (Kxy >> l), further justification is necessary when it is applied to the opposite case (Kxy
