Anal. Chem. 1991, 63,946-953
946
Coupled Diffusion/Adsorption Model for Response of Precipitate-Based Iodide-Selective Electrodes to Primary-Ion Activity Steps Thomas R. Berube' and Richard P. Buck* Department of Chemistry, CB # 3290, University of North Carolina, Chapel Hill, North Carolina 27599
Em6 Lindner, KlPra Tbth, and Em6 Pungor Technical University of Budapest, Institute of General and Analytical Chemistry, GellBrt tdr 4, Budapest H-1111, Hungary
primary-ion adsorption at the ISE surface. The diffusional contribution to the response can be dominant for high activities of I-, since the adsorption layer is nearly saturated. At low activities adsorption processes retard the response. Since the equations for diffusion with such concentrationdependent boundary conditions cannot be generally solved, some simple systems are analyzed and arguments extended to account for the observed dependences on experimental conditions.
Swltched wall-jet streams of iodlde soiutlons Impinging on AgI-based Ion-selectlve electrodes (ISEs)glve potential responses after the first 20 ms. The responses to iodide actlvlty steps of constant ratlo a (final)/a (initial) are hlghly dependent on the starting actlvlty. This Is contrary to most present theories of ISE response, partlcularly those descrlbing reverslble surface ion exchange controlled by diffuslon to a surface or first-order reactlon klnetlcs. Experimentally,dlffusion Is Involved over all concentratlon ranges but is coupled with some surface adsorption isotherm. At low actlvitles, steps up or down are slower than dlffuslon because of the need to form or remove some of the adsorbed layer. At high activltles, step responses approach pure diffusion control, slnce adsorption Is nearly saturated. Dependence on the starting actlvlty requlres a nonlinear adsorptlon Isotherm, which is consistent wlth adsorption data showing logartthmic dependence of surface Ion concentration on bathing actlvlty. Experimentally, rate-controlilng dlssolution/preclpRatlon processes are not Ilkeiy.
THEORY Morf, Lindner, and Simon (8) proposed a useful model describing the time-dependent potential measured by an ISE following an activity step in the presence of only primary ions. Local equilibrium was assumed at the membrane/solution interface; i.e., ion exchange between solution species at the surface and the solid surface species is more rapid than the change in solution surface concentration. Thus, the Nernst equation can describe the time-dependent potential: E ( t ) = E"
INTRODUCTION Previous studies of solid-state ion-selective electrode (ISE) responses to activity steps have shown that the responses can vary with a number of experimental conditions, including the rate of mixing and initial activity. However, the reported results have often been conflicting; response curves and the associated response times for a given electrode have varied greatly. (The reader is directed to several excellent reviews on the subject for further details (1-3)J These observations of transient characteristics accordingly led to various models of potential response; the equations are summarized in Table I. Most models assumed that a single process, such as diffusion through a thin film or the kinetics of a surface precipitation/dissolution reaction ( 4 , 5 ) ,controls the response. A combination of any two or more processes is difficult to model mathematically and therefore difficult to use for interpreting experimental results, although some attempts have been made (6, 7). However, none of the physical models previously proposed could account for all observed variations in ISE response curves due to changes in experimental conditions. The aim of this paper is presentation of new response data using high-speed activity steps that show new dependences on experimental conditions in addition to those previously reported. The results suggest a new physical model for the ISE response, in which film diffusion is coupled with nonlinear
(1)
where a ( t ) is the time-dependent activity of the primary ion (i.e., Ag+ or I-) a t the interface and Eo is E(0). The time dependence of a ( t ) can be represented as a function of the initial and final activities and an arbitrary function of time: = a. + (a, - a,) f ( t ) (3) where a, = a(O), a, = a(..), and f ( t ) is the function describing the change in solution activity at the electrode surface with time, with f(0) = 0 and f ( m ) = 1. These equations have formed the basis of a number of proposed equations for ISE responses; some examples of the time functions from these models are listed in Table IA. The logarithmic form of eqs 2 and 3 accounts for the dependence of ISE response on the direction and relative magnitude (measured by a,/a,) of activity step. Steps from al to a higher activity a2 reach a given percent of the total potential change more rapidly than do responses to downward-going steps from a2 to al. For steps of different relative magnitudes from the same initial activity, the time to reach a given percent of the total potential change decreases with the step size. The form of the equation obtained by combining eqs 2 and 3 is useful for evaluating the dependence of ISE response on the starting activity level. The presence or absence of an activity level dependence in a given model is dictated by the time function f(t) chosen to describe the controlling process, because the combined equation contains only ratios of ac-
'Current address: Ethyl Technical Center, P.O.Box 14799, Baton Rouge, L A 70898. 0003-2700/91/0363-0946$02.50/0
+ RT - In [a(t)] ZF
C
1991 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 63, NO. 10, MAY 15, 1991
947
Table I. Selected Quantitative Models for ISE Response to Activity Steps A. Time Functions for Models Based on Nernst Equation (Eq 2) with Time-Dependent Activities (Eq 3) no. eq model ref (-1)" T-1 f ( t ) ' = 1exp[-(Zn + 1)%t] diffusion 8. 9 n-o(Zn + 1) 8, 9, 12, 13 T-2 f ( t ) = 1 - exp(-kt) first-order kz exp(-k,t) - k l exp(-k,t) T-3 f(t) = 1consecutive first-order 1 kz - kl 1 T-4 f(t) = 1second-order 3, 4, 5 1 + la, - ao)kt
-x-
T-5 T-6
1 1 + (kt)'/Z 1 f(t) = 1 - 1 + (kt)"
f(t) = 1-
liquid membrane
3, 8
empirical hyperbolic
2
B. Models Not Based on Nernst Equation (Eq 2) no. T-7
T-8
model
eq
E ( t ) = E, + In
(G) t
E ( t ) = Eo + CPj[l - exp(-kjt)l
ref
simple hyperbolic
23
multiple processes
6
I
tivities and no absolute activity dependence. As a result, experimental observations of activity-level-dependent ISE response indicate that in the model describing the response, f ( t )must contain some dependence on the absolute activity level. The opposite is also true; observed activity-level-independent responses imply time functions that are independent of absolute activity level, although these may contain a ratio of the initial and final activities. Markovic and Osburn (9) used eqs 2 and 3 to describe an activity-level-independent model based on diffusion through a planar film of solution adhering to the electrode surface (Nernst layer). The result was eq T-1 with a rate constant (or time constant 7) defined by (4)
where D is the ion diffusion coefficient with supporting electrolyte or the salt diffusion coefficient when no supporting electrolyte is present and 6 is the thickness of the Nemst layer. Since the thickness of the latter is affected by the solution flow rate (IO), flow-rate-dependent response curves point to classical diffusion control as a limiting factor (3, 11). The diffusion model equation has been presented in a simplified form (8,9)using only the first term of the summation (eq T-2), which corresponds to steady-state diffusion in a finite cell. Tbth and Pungor (12, 13) independently obtained this equation by using a first-order reaction model based on ion desolvation. An activity-level-dependent time function f ( t )(eq T-4) for solid-state ISE response was derived by Buffle and Parthasarathy (4) using an empirical "adsorption layer" theory, based on observations of Davies and Jones (14).The latter showed that the rate of dissolution or precipitation of silver chloride precipitates could be described by an equation of the form
d A / d t = -kA2 = ICAt) - C,(m)l
(5) (6b)
where C ( t ) and C(m) are the reacting ion concentrations at
a given time after instantaneous mixing and at equilibrium (the saturation value). Therefore, A is the number of moles per liter of available precipitating salt, and this quantity is "controlled" by the ion in least concentration; i.e. the ion in least concentration suffers the larger percentage concentration change. Buffle and Parthasarathy, however, erroneously equated these concentrations with those of the stepped I- in deriving eq T-4. In our experiments, these equations only describe the fate of nonequilibrium Ag+ after the iodide concentration step is complete. Therefore, eq T-4 does not describe the dissolution/precipitation of the ISE membrane as stated by Buffle and Parthasarathy. Nevertheless, with Davies and Jones results, an equation functionally similar to eq T-4 can be derived for the change in Ag+ activity (2). Therefore, if the ISE truly detects only Ag+ and not I-, the dissolution/precipitation model can still hold. The results presented here, however, dismiss this possibility. Coupling of diffusion processes with surface reactions cannot usually be solved analytically. A few exceptions, such as the consecutive first-order reaction model presented by Lindner et al. (I) (eq T-3), do exist, but these typically predict activity-level-independent responses. Instead, empirical equations have been used to describe the response curves by one of two methods. The first (6),given by eq T-8, assumes independent first-order electrode processes and attributes a fraction of the total response to each process. The second (2) approximates the response due to a pair of coupled known processes by inserting the time-dependent functions from Table IA into (7) a ( t ) = a, + (a, - ao) f i ( t ) f&) and inserting the result into eq 2. Equation 7 arises from the assumption that the "fiial" activity a, for the second process a t any given time is equivalent to the value of the function a ( t ) from the first process at that time, assuming that the second process immediately follows (and therefore is coupled to) the first. In essence, the boundary conditions are applied to a result, so the equation obtained can only be considered quasi-quantitative. Nevertheless, this method has advantages. It is not limited to activity-level-independent forms and can
948
ANALYTICAL CHEMISTRY, VOL. 63, NO. 10, MAY 15, 1991
Table 11. Sample Results of Nonlinear Curve Fits to Response Curves in the High-Concentration Range (Equations Fitted to 400 Equally Spaced Points from 0-1 s)
-
A. Activity Step,
eq T-4b T-5'
I-; Linear Flow Rate, 3.1 m/s
S, mV
k 1" (1.18 f 0.05) X (1.59 f 0.22) x (2.89 f 0.03) X (2.55 f 0.03) X (3.45 f 0.15) X (4.68 f 0.27) X (3.35 f 0.04) X (3.08 f 0.02) X
k2"
a, mV
(6.86 f 0.71) X lo2 (2.95 f 0.09) X lo2 (3.95 i 0.21) x 105 (2.59 f 0.23) X lo5
1.45 2.15 0.55 0.64 0.53 0.52 0.39 0.36
lo5 104 lo2
T-3b T-1/T-2' T-1IT-4' T-1IT-5'
-58.14 f 0.08 -60.22 f 0.23 -57.37 f 0.03 -57.38 f 0.03 -57.37 f 0.03 -57.37 f 0.03 -57.63 f 0.02 -58.13 f 0.04
eq
S, mV
kl"
a
a, mV
T-6'
-57.41 f 0.02
(2.70 f 0.04) X lo2
2.68 f 0.05
0.48
T-lb T-2'
PI,mV
eq
T-8
56.38
f
0.18
T-4' T-5b
T-lb T-2'
T-3bd T-1/T-2' T-1/T-4' T-1IT-5'
k 2"
145 f 3
1.53 f 0.16
2.63 f 0.90
-
S, mV
T-6'
-57.48 f 0.02
104 loz lo2 lo2 lo2 lo2 lo2 lo2
(3.01 f 4.74) X (4.45 f 0.56) X (6.04 f 0.40) X (5.93 f 0.73) x
k I" (1.87
0.72
a, mV
k2"
(2.24 f 0.08) x (4.70 f 0.55) X (1.69 f 0.02) X (1.22 f 0.03) X (3.00 f 4.66) X (1.95 f 0.07) X (2.43 f 0.06) X (1.99 f 0.03) X
f 0.12 f 0.02 f 0.03 f 0.02 f 0.02 f 0.02 f 0.03
mV
I-; Linear Flow Rate, 3.1 m/s kl"
f 0.05
eq
a,
Pz, mV
S, mV -57.89 -59.16 -57.47 -57.49 -57.46 -57.46 -57.63 -57.97
lo2 lo2 lo2 lo2
kl"
B. Activity Step,
eq
lo2
* 0.02) X lo2
loz lo2 10' 103
0.86 1.32 0.43 0.64 0.43 0.39 0.33 0.33
a
a, mV
2.26 f 0.04
0.35
eq
PI,mV
kl"
Pz,mV
kZ"
T-8
-56.32 f 0.09
249 k 3
-1.44 f 0.08
4.40 f 0.54
a,
mV
0.33
"Units are s-* except for eq T-4, where units are M-I 8-l. 'The time functions in these equations were used with eqs 2 and 3 to calculate potential. The time functions in these equations were used with eqs 2 and 7 to calculate potential. Results indicate kl and k2 are highly correlated. therefore be used to approximate the effect of combining single-process physical models. In addition, it does not completely ignore the coupling of the different physical processes.
E 3
EXPERIMENTAL SECTION Activity step measurements were performed by using the instrumentation described previously (15). The I- ISEs tested were made by incorporating pressed pellets of AgI, Ag2S,and mixtures of the two into Philips IS-550 electrode bodies. Contact from the ISE to the saturated calomel single-junction reference electrode (Radelkis OP-083OP) was made by using a flowing 0.1 M KNO, solution. Iodide ion activity steps were performed in high (l@-lV3 M) and low (lo-'LIOd and 1@-104 M)KI concentration ranges. Experiments were repeated several times over several months to ensure day-to-day reproducibility of the results. All solutions were maintained at 0.1 M ionic strength with the addition of KNO,. Nonlinear least-squares fittings of theoretical and empirical equations to the data were performed by using a Marquardt algorithm (16) programmed in ASYST on a Zenith ZF-158-43 PC.
Figure 1. Flow-rate dependence of response curves in the highconcentration range. Curves have been superimposed for comparison. M I-; linear Conditions: membrane, AgI; activity step, lo-* flow rates (a) 1.5 m/s, (b) 2.0 m/s, (c) 3.7 m/s.
RESULTS AND DISCUSSION High-Concentration Range. Figure 1 gives typical short-time (1 s) response curves to activity steps in the high-concentration range for the 100% AgI membrane ionselective electrode ISE.These curves, as well as those recorded by using the 1:l and 7:3 Ag,S/AgI mixed-precipitate ISEs, show a rapid change in the measured potential followed by a slower asymptotic approach to the equilibrium value. The curves also show a flow-rate dependence, which indicates a
contribution by diffusion through the Nernst layer to the measured response. The results of multiparametric curve fittings to the full film diffusion model (eqs 2, 3, T-1)or the one-term form (eqs 2,3, T-2) show that these simple models give adequate fits with low residuals (Table 11). However, examination of long-time (10 or 25 s) response curves show that the systems take much longer to reach equilibrium than any simple diffusion model predicts. Therefore, a second process must control the asymptotic portion of the response.
1
0 3
tJ
s
tJ
a0 I 0
Ln
a
a
I
Time ( s )
-
ANALYTICAL CHEMISTRY, VOL.
03,NO. 10, MAY 15, 1991 949
I
60
s 3 2
1
a
trb r,
1
:
U
C
u
0
a I
I
a ti
a
0
ti Time
0
0
ti
ti
(SI
Figure 2. Effect of ISE membrane surface condition on response curves in the high-concentration range. Curves have been superimposed for comparison: (a) freshly polished ISE; (b) ISE conditioned M K I for 2 h. Conditions: membrane, AgI; linear flow rate, in 1.6 mls; activity step, lo-* M I-.
-
There is also evidence for surface roughness effects. Figure 2 shows that the asymptotic response of an ISE with a freshly polished surface is faster than one which has been conditioned in a solution of primary ion. During conditioning, dissolution/precipitation and ion-exchange equilibria can cause an increase in the surface roughness. It is apparent that some surface-dependent process is responsible for the slow asymptotic region of these curves. However, it is difficult to isolate the process at these high concentrations, since the effect is limited to the last few millivolts of the response. Possibilities include diffusion into pores or grain boundaries, adsorption effects, dissolution effects, or any combination of these. Lindner (17)has had some success using eqs 2 , 3 , and T-5 to describe the response of precipitate-based membranes following conditioning in solutions which can enhance surface roughness. Our attempts to fit these long-time iodide response curves with hyperbolic eq T-4 also gave excellent fits. These results indicate that the form of the time-dependent function for this second process is hyperbolic in nature rather than the exponential form that results from most activity-level-independent processes and thus may be a result of activity-level-dependent processes. Further support for this conclusion is given by the short time range fitting results using the multiple-process equations. The use of eq 7 to couple the time functions for diffusion (eq T-1)and the hyperbolic forms (eq T-4 and T-5) gave fittings of equal or better quality than those using the independent first-order process equation (eq T-€9, even though the latter required additional variable parameters. Low-Concentration Range. Response curves measured by using a pure AgI membrane ISE in the low-concentration range are given in Figure 3. The responses varied with the absolute activity level, pointing to the use of an activity-level-dependent model to describe these curves. The hyperbolic eq T-4 from the dissolution model of Buffle and Parthasarathy ( 4 , 5 ) ,in fact, gave excellent fits to the curves for steps down in activity, as seen in Figure 4 and Table 111. However, the equation poorly fit the ISE response to steps up in activity. The response to these upward steps was faster than that predicted by using the parameters from fittings to the corresponding downward steps. Also, these curves showed more of an exponential shape than a hyperbolic shape at short times, indicating that the process controlling the response is described by a time function which is itself dependent on the direction of step. Theoretical considerations have not totally ruled out dissolution and precipitation kinetics as controlling processes but showed that the model can only hold if the ISE measures the kinetics of Ag+ change induced by the I- step, and not the Istep itself. Since the steps down were fit well by the equation
-
0
Time ( s )
--
Figure 3. Concentration dependence of response curves in the lowconcentratbn range. Curves have been superimposed for comparison. Conditions: membrane, AgI; activity steps (a) lo4 lo-' M I-, (b) lo-' lo-' M I-, (c) lo-' M I-, (d) lo4 lo-' M I-.
-
-
l
(a)
"
"
"
"
"
1
1
0 '3
30
u t
-
PI
u
a0 0
0
0
d
d
-Y
Time ( s )
l " " " " " 1
(b) 0
3 E
b 1
0 '3
-30
u
c
PI
u
a0
-60 D
Ln
d
d
Time
0
(SI
Flgure 4. Nonlinear curve fits to response curves in the iow-concentration range: (0)experimental data; (- - -) eqs 2, 3,T-3; (--) eqs 2, 3,T-5; (-) eqs 2, 3,T-4. Fitting parameters are given in Table 11. Conditions: membrane, AgI; linear flow rate, 1.5 m/s; activity lo-' M I-, (b) lo-' M I-. steps (a)
-
-
for this model and since these two processes could possibly be described by different equations, which would be consistent with these results, a second set of experiments was conducted. The ISE response was measured by using solutions of I- which were preequilibrated (saturated) with AgI. On the basis of the precipitation kinetics hypothesis, these experiments should have yielded different response curves than those using nonsaturated solutions. The Ag+ activity and I- activity were stepped simultaneously; as a result, the ISE response should have reflected diffusion rather than dissolution kinetics. The curves, however, were identical to those with nonsaturated
950
ANALYTICAL CHEMISTRY, VOL. 63, NO. 10, MAY 15, 1991
Table 111. Sample Results of Nonlinear Curve Fittings to Response Curves in the Low Concentration Range (Equations Fitted to 400 Equally Spaced Points from 0-1 s) A. Activity Step, lo-' eq T-4b T-5b
T-lb T-2b T-3b T-1/T-2' T-1/T-4' T-1/T-5'
-
S, mV -60.01 -68.65 -56.78 -56.89 -56.89 -56.89 -59.75 -64.29
I-; Linear Flow Rate, 1.8 m/s k la
f 0.05 f 0.46 f 0.13
lo8
(1.94 f 0.02) X (6.54 f 0.49) X (6.44 f 0.13) X (5.40 f 0.10) X (5.40 f 0.11) X i.36d x 103 (2.32 f 0.04) X (1.04 f 0.02) X
f 0.11 f 0.11 f 0.11 f 0.03 f 0.16
u, mV
k2'
lo2 10' 10' 10'
lo2 lo2
6.22d X 10' (5.40 f 0.10) X 10' (2.12 f 0.01) x 108 (1.78 f 0.07) X lo3 u,
eq
S, mV
k 1'
a
T-6b
-58.81 f 0.03
(1.26 f 0.01) X lo2
1.234 f 0.005
eq
Pl, mV
T-8
44.27 f 0.26
47.5 f 0.5
eq T-4b T-5b
T-lb T-2b T-3b T-1/T-2' T-1/T-4' T-1/T-5'
-
S, mV -59.11 -62.92 -57.77 -57.92 -57.83 -57.85 -58.45 -59.29
a, mV
13.89 f 0.25
6.61 f 0.13
0.32
lo4 I-; Linear Flow Rate, 1.8 m/s u,
k2'
(5.42 f 0.15) X (4.18 f 0.45) X (5.17 f 0.04) X (3.33 f 0.05) X (4.33 f 0.07) X (1.91 f 0.06) x (9.86 f 0.09) X (6.87 f 0.04) X
f 0.04 f 0.03 f 0.02 f 0.01 f 0.03
0.26
k za
k la
f 0.08 f 0.32 f 0.03
mV
Pz, mV
kl'
B. Activity Step,
0.71 2.32 2.42 2.13 2.13 2.13 0.32 0.80
lo6 10' 10' 10' 10' 102 lo1 10'
(3.06 f 0.17) X (3.84 f 0.04) X (1.07 f 0.01) X (5.37 f 0.17) X
mV
1.26 2.44 0.60 0.81 0.48 0.42 0.17 0.18
102 10'
10s 102
eq
S, mV
k 1'
4
a, mV
T-6b
-58.02 f 0.01
(5.29 f 0.02) X 10'
1.712 f 0.010
0.26
eq
P1,mV
k la
Pz, mV
T-8
-54.90 f 0.18
76.5 f 0.7
-3.28 f 0.16
C. Activity Step, eq T-4b T-5b
T-lb T-2b T-3b T-1IT-2' T-1/T-4' T-1/ T-5'
S, mV -60.21 -93.28 -50.94 -51.38 -51.38 -51.38 -60.20 -84.00
S, mV
T-6b
-60.75 f 0.06
eq
P1, mV
T-8
27.14 f 0.18
(3.64 f 0.09)
T-4b T-5b T-lb
T-2b T-3b T-l/T-2' T-l/T-4' T-l/T-5'
-60.86 -70.82 -57.51 -57.96 -57.70 -57.77 -59.07 -60.40
0.15 0.76 0.05 f 0.06 f 0.02 f 0.03 f 0.08 f 0.18 f f f
-
0.35
u,
lo2 10'
X
10'
u,
0.974 f 0.003
mV
0.19 1.68 3.81 3.21 3.22 3.22 0.18 0.84
1 . 4 x~ 107 (1.58 f 0.03) X 10' (3.99 f 0.01) x 106 (4.34 f 0.15) X 10' a
mV
0.17
Pz,mV
kza
a, mV
28.31 f 0.15
3.31 f 0.03
0.22
k la
S, mV
lo6 10' 10' 10' 10'
kl"
22.9 f 0.2
* 0.42
k2'
(3.98 f 0.01) X (2.50 f 0.14) X (1.91 f 0.04) X (1.58 f 0.03) X (1.58 f 0.03) X 7 . x 103 ~ (3.68 f 0.27) X (5.05 f 0.10) X
D. Activity Step, lo+
5.97
10" I-; Linear Flow Rate, 1.5 m/s
k la
f 0.03 f 1.08 f 0.22 f 0.19 f 0.19 f 0.19 f 0.02 f 0.52
eq
eq
-
a, mV
k2'
10" I-; Linear Flow Rate, 1.5 m/s kl'
(1.52 f 0.04) X 4.52 f 0.49 (1.70 f 0.01) X (1.10 f 0.01) x (1.43 f 0.01) X (6.65 f 0.15) X (3.30 f 0.08) X (2.30 f 0.04) X
kZ0
a, mV
(1.02 f 0.03) X lo2 (1.25 f 0.01) X 10' (3.14 f 0.12) X l@ (1.09 f 0.13) X lo2
1.83 3.48 0.96 1.03 0.40 0.55 0.78 0.76
lo6 10' 10' 10' LO' 10' 10'
eq
S, mV
kl'
4
T-6'
-58.22 f 0.03
(1.71 f 0.01) X 10'
1.697 f 0.009
u,
mV
0.36
ANALYTICAL CHEMISTRY, VOL. 63, NO. 10, MAY 15, 1991
951
Table 111 (Continued) eq
ha
PI, mV
Pz,mV
ha
a,
mV
T-8 -54.64 f 0.16 24.5 f 0.1 -3.46 & 0.13 3.59 f 0.21 0.19 Units are s-* except for eq T-4, where units are M-I sd. *The time functions in these equations were used with eqs 2 and 3 to calculate potential. 'The time functions in these equations were used with eqs 2 and 7 to calculate potential. dStandard deviation indicates parameter is not important.
1
(a)
60
b
30
0
I
I 0
Ln
0
v,
Q
d
d
d
d
1
Time
Time ( s )
(SI
r
r I \
\u/
I
I I
IL\
I
I
I
I
0
3 E
b 1
0 '1
-30
U
c
0.
U
U
0
a
0
Q
-60
-60 I 0
VI
d
d
I
Ln
0
d
d
3
rime (s)
Time ( s )
Figure 5. Flow-rate dependence of response curves in the low-concentration range. Curves have been superimposed for comparison. lo-' M I-, (b) Conditions: membrane, AgI; activity steps (a) lo-' M I-; linear flow rates (A) 1.6 m/s, (B) 2.7 m/s, (c) 3.4 m/s. solutions, indicating that the mechanism controlling the response cannot be the proposed dissolution/precipitation model. As an alternative to control by kinetics of dissolution processes, control by adsorption has not previously been considered, even though the coupling of adsorption and diffusion has been shown to control the response of the AgI ISE to iodide in the presence of interferents (18). A model based on primary-ion diffusion/adsorption can qualitatively account for the slow ISE response at low activities. In this model, primary ions diffuse away from the ISE surface following a step down in activity. As ions leave the surface region, the adsorbed layer is no longer in equilibrium with the solution, causing ions to desorb. This desorption of primary ions keeps the activity at the surface higher than would be expected with pure diffusion control, thus retarding the observed response. Following a step to higher activity, ions diffuse to the surface, where some of them become part of the adsorption layer. This also slows the response, since the concentration of ions in solution a t the surface is kept low until equilibrium is established between the adsorbed layer and the solution at a,. This model assumes that diffusion plays an important role in the response of these ISEs at low concentrations, which is
-
1
I 0
-
Figure 6. Effect of ISE membrane surface condition on response curves in the low-concentration range: Curves have been superimposed for comparison: (A) freshly polished ISE; (B) ISE conditioned in lo3 M K I for 25 h. Conditions: membrane, AgI; linear flow rate, lo-' M I-, (b) lo-' M I-. 1.5 m/s; activity steps (a) in apparent contradiction to the earlier findings of Lindner, T6th, and Pungor (11). However, experiments performed by using high-speed activity steps (Figure 5) show that a dependence on flow rate does exist in this concentration range, indicating that diffusion must be considered. This apparent contradiction of data is a result of the use of initial slopes as a measure of overall response in the earlier studies. While the curves in Figure 5 do show a flow-rate dependence, their initial slopes are quite similar. As already reported for high-concentration steps, surface conditions of the membrane can influence the response. The curves in Figure 6 reveal that the surface effect is not as large for downward activity steps (desorption-driven responses) as for upward steps, in which a greater surface area could accelerate adsorption. Response curves recorded by using the mixed-precipitate membrane ISEs provide further evidence of the importance of surface effects on iodide ISE response. These curves show a slower potential change than those using the pure AgI membrane ISE. Curve-fitting results indicated that the responses to downward steps generally deviated from eq T-4, unlike the AgI ISE experimental results. The empirical hyperbolic eq T-6 gave excellent fits to these curves, with the exponent (Y falling in the range 0.6-0.8. The value
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ANALYTICAL CHEMISTRY, VOL. 63, NO. 10, MAY 15, 1991
of this exponent may give an indication of the relative degree of surface roughness, as well as the homogeneity of the surface of the membrane or adsorbed layer on the membrane.
DESCRIPTIONS OF TRANSPORT/ADSORPTION COUPLING The proposed model involves the coupling of diffusion with an activity-level-dependent equation (or equations) describing the adsorption and desorption of I- on the ISE membrane surface. In such a model, Fick’s laws apply to each diffusing-reacting ion, regardless of the surface reaction rate, assuming there is supporting electrolyte to eliminate electrostatic coupling and there are no homogeneous reactions such as ion pairing. Fluxes of ions, instead of being zero at the electrode surface (an assumption used in the simpler theories), are set equal to the rate of adsorption J = -D dC/dx at all x > 0 (8) J = -dr/dt at x = 0 (electrode surface) (9) with r as the adsorbed layer concentration in mol/cm2. As long as there is an adsorbed layer of ions whose concentration r varies by some isotherm with the ambient concentration C, the flux of ions following an activity step will not be zero at the surface until the new equilibrium coverage is reached. Determination of the equation for the iodide isotherm is necessary for a complete solution of these equations. Experimentally (19),the isotherm for iodide adsorption is neither Henry’s law (linear) nor Langmuir, but logarithmically dependent on bathing I- concentration. In addition, other factors influence the shape of the isotherm. For example, adsorption a t constant I- is a function of the concentration of inert salt, such as KNOB(19). As a result, it is difficult, if not impossible, to derive an equation describing the new model, although numerical solutions or approximations may be possible. Examination of the solutions for those cases which can be solved can indicate the suitability of the model assumptions. One such case is diffusion and Henry’s law adsorption with the boundary condition dI’/dt = kfC (at x = 0) - kbr (10) When the adsorption equilibrium constant is written p = k,/kb = r / C (at x = 0), the solution is like combined eqs 3 and T-1, but the time constant becomes
where /3 has the dimension of length. When adsorption is extensive (0is large), the time to reach a given coverage is larger than in the absence of adsorption. However, the process remains first-order at long times and cannot show absolute concentration dependences. Nevertheless, the time constant for Henry’s law adsorption shows that adsorption slows the rate by which surface concentrations can readjust after a step up or down in activity. There is some evidence that nonlinear isotherms affect ISE responses. Reinmuth (20) calculated time responses for steps with Langmuir isotherm adsorption. His series solution was not aimed a t long times, but his calculations show concentration-dependent times to reach a given, low coverage. These are expected intuitively for Langmuir adsorption. In effect, B is replaced by
(12) for competitive adsorption between i and j (21). The apparent time constants for very dilute solutions should be independent of bathing concentration. But, as concentrations are increased, time constants get shorter, until the limiting value of eq 4 is all that remains.
Although detailed calculations of coupled diffusion/adsorption-controlled response times have not been done, the sense of Reinmuth’s results for nonlinear adsorption isotherms is that surface concentrations may well show absolute dependences on step size, especially a t low concentrations. At high concentrations, the contribution of adsorption to the response decreases. As a result, adsorption processes cannot be ignored in describing ISE response, since they can cause results similar to the observed activity-level-dependent responses.
CONCLUSIONS The response of the AgI precipitate-based I- ISE to primary-ion activity steps has been shown to be dependent upon the starting activity level, the flow rate (at all activity levels), and preconditioning of the membrane surface. There is also an apparent change of functional dependence a t low concentrations; steps down in activity show predoininantly hyperbolic curves, while steps up in activity include an exponential contribution, similar to, but slower than, the high concentration responses. These phenomena can be qualitatively described by using the diffusion/adsorption model presented, in which diffusion is coupled with a nonlinear adsorption isotherm. This new model can also explain other published results. Since the solution of zero charge (equal adsorbed Ag+ and I-) is about 2.8 X lo4 M Ag+ (19), the concentrations of these two ions in the adsorbed layer are different for all typical step concentrations. A difference in the adsorption kinetics for these two ions a t a given concentration level is therefore expected, which may explain the reported differences between responses of AgI ISEs to Iactivity steps and Ag+ activity steps (10, 22). Although Fick’s laws with boundary conditions for this model system cannot be solved analytically, we have shown from the exact solution for diffusion with equilibrium Henry’s law adsorption that adsorption processes can indeed affect the surface concentrations measured by the ISE. For steps down in activity, desorption of ions is capable of temporarily “buffering” the activity at the surface, so the response curves are relatively slow. For steps up in activity, adsorption slows the rate of surface equilibration by consuming a portion of the ions in solution, essentially converting them into a nonpotential determining form. We cannot specify the exact mathematical form for potential response under couple$diffusion/nonlinear adsorption conditions. We have ruled out precipitation/dissolution kinetics, but curve fittings indicate that equations which describe the ISE response may have a contribution from a hyperbolic form similar to that of Buffle and Parthasarathy (eq T-4). While it is difficult in practice to isolate and identify sorption processes coupled with diffusion from experimental responses, numerical solutions for Fick’s laws with boundary conditions, as well as nonlinear curve fitting, may lead to a better understanding of how the coupling of diffusion and sorption processes affect ISE response. ACKNOWLEDGMENT We thank Paul Troiano for the computer program that translated the data files and IrGn Budai for the pressed pellets and solutions used in this work. Registry No. AgI, 7783-96-2; Ag2S, 21548-73-2; I, 20461-54-5; Ag, 7440-22-4; IK, 7681-11-0.
LITERATURE CITED (1) Lindner, E.; T&h, K.; Pungor, E. Dynamic CheracferisficS of Ion-Selecfive Nectrodes; CRC Press: Boca Raton, FL, 1988. (2) Berube, T. R. Ph.D. Dissertation, Unlversity of North Carolina at Chapel Hill, 1989. (3) Morf, W. E. The Principles of Ion-Selecfive Elecfrodes and of Membrane Transport,Elsevier: New York, 1981; Chapter 14. (4) Buffle, J.; Parthasarathy, N. Anal. Chim. Acta 1977, 93, 111-120.
Anal. Chem. 1991, 63, 953-957 (5) Partheserathy, N.; Buffle, J.; Haerdi, W. Anal. Chim. Acta 1977, 93,
121-128. (6) Shatkay, A. Anal. Chem. 1976, 48, 1039-1050. (7) Mod, W. E.; Simon, W. Ion-Selectlve Electrodes in Analytical Chemistry; Freiser, H., Ed.; Plenum': New York, 1978; Vol. 1, Chapter 3. (8) Mort W.: Lindner, E.;Simon, W. Anal. Chem. 1975, 47, 1596-1601. (9) Markovic, P. L.; Osburn, J. 0. AIChE J. 1973, 19, 504-510. (10) Lindner, E.; TBth. K.; Pungor, E. Anal. Chem. 1978, 48, 1071-1078. (11) L$dner, E.; TBth, K.; Pungor, E. Anal. Chem. 1982, 54. 72-76. (12) Toth, K.: Pungor, E. Anal. Chim. Acta 1973, 64, 417-421. (13) Tbth, K. Ion-Selectlve Electrodes; Pungor, E.; BuzBs, I., Eds.; Akad6miai Klado: Budapest, 1973;pp 145-164. (14) Davies, C. W.; Jones, A. L. Trans. Faraday SOC. 1955, 51. 812-817. (15) Lindner, E.; Tbth, K.: Pungor, E.; Berube, T. R.; Buck, R. P. Anal. Chem. 1967, 59, 2213-2216. (16) Nash, J. C. Numerlcal Methods for Computers: Linear Algebra and Function Minimization;Halsted Press: New York, 1979;pp 170-178.
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(17) E. Lindner, Technical University of Budapest, unpublished results, 1989. (18)Berube, T. R.; Buck, R. P.; Lindner, E.; Gratzl, M.; Pungor, E. Anal. Chem. 1989, 61, 453. (19)Mackor, E. L. Red. Trav. Chim. Pays-Bas 1951, 763-783. (20) Reinmuth, W. H. J. Phys. Chem. 1961, 6 5 , 473. (21) Bard, A. J.; Faulkner, L. R. Ektrochemlcal Methods, Fundamentals and Applications; John Wiley and Sons: New York, 1980; p 517. (22)Dencks, A.; Neeb, R. Fresenlus' 2. Anal. Chem. 1979, 297, 121-125. (23)Muller, R. H. Anal. Chem. 1969, 41(12),113A-116A.
RECEIVED for review January 24,1990. Acceptld February 22,1991. Support from the NSF and Hungarian Academy of Sciences is gratefully acknowledged.
Differential Pulse Voltammetric Determination of Cobalt with a Perfluorinated Sulfonated Polymer-2,2-Bipyridyl Modified Carbon Paste Electrode Zhiqiang Gao, Guangqing Wang, Peibiao Li,* a n d Zhofan Zhao
Department of Chemistry, Wuhan University, Hubei 430072, China
A highly selectlve method for the determlnathm of cobalt with a chemkally modifled carbon paste electrode Is proposed. It is based on the chemical reactlvlty of an knmoblllred modlfler, 2,2-blpyrklyl, toward cobalt( I I ) Ion to ylekl the corresponding cation complex ((Bpy)sCo)2+, which Is taken up by another modlfler, perfluorlnated sulfonated polymer (Naflon). Dlfferentlal pulse voltammetry Is employed, and the oxidation of the complex, at 4-0.1 V vs SCE, Is observed. For a 5-mln preconcentration perlod, a llnear callbratlon curve Is obtained for cobalt concentratlons ranglng from 7 X to 1 X mol/L, and the detection llmlt Is 3 X mol/L. A lower detection llmlt can be obtalned for longer preconcentratlon times. For elght preconcentratlon/determlnatlon/renewal cycles, the differentlal pulse voltammetric response could be reproduced wlth 4.3% relatlve standard deviation. Rapld and convenlent acld renewal allows the use of an lndlvldual modifled electrode surface In multiple analytical quantitatlons. Many coexisting metal Ions have llttle or no effect on the determination of cobalt. The procedure was applled to the determination of cobalt for four standard reference materials with relatlve standard devlatlon of 4.0-5.2%.
INTRODUCTION Chemically modified electrodes (CMEs) have received a great deal of attention (1, 2 ) , particularly to enhance the sensitivity and selectivity of electrochemical techniques (3). Generally, such procedures involve the immobilization of the chemically active species on the electrode surface. The target analyte is preconcentrated to the modifier, which has a particular affinity for the analyte. Subsequently, a measurement of the electrochemical response such as the current or charge is made. The magnitude of the analytical signal obtained by electrochemical oxidation or reduction of the accumlated analyte is correlated with the concentration of the analyte in
solution. Conceptually, this approach is analogous to conventional voltammetric stripping method in that a preconcentration step is used to enhance the sensitivity and selectivity. The difference is that the sensitivity and selectivity of the CME are determined by the chemical reactivity of the modifier rather than the oxidation/reduction potential of the analyte. The CME approach is considered more selective than conventional stripping analysis. Reports on the analytical utilization of chemically modified electrodes have included complexation (4),precipitation (5), electrostatic accumulation (6),bioaccumulation (7), potentiometric response (8), covalent attachment (9),and others (10-13). Chemically modified electrodes have been employed to enhance the selectivity or/and prevent electrode fouling in analtyical measurements of organic and biorganic substances such as dopamine (14),ascorbic acid (15),amine (16), and others (17,18). In addition, many modification methods have been used to introduce modifiers onto the electrode surface. Of these, the use of carbon paste electrodes appears especially advantageous, because only the addition of a sparingly soluble modifier to an otherwise conventional graphite powder/Nujol oil mixture is involved. This offers several attractive features to the prospective electrochemical analyst. The chemically modified carbon paste electrodes are exceedingly easy to fabricate and can be readily prepared with a varity of modifier loading levels. Furthermore, the surface of the freshly modified electrode can be generated rapidly and reproducibly in quantity. The low background current is a further advantage for practical analytical applications. To date, a number of studies on the use of chemically modified carbon paste electrodes for analytical determinations have been reported. One of the earliest examples was that of Cheek and Nelson (19), who employed carbon paste electrodes modified with aminosilanes for the determination of silver. Recently, Baldwin and co-workers employed modified carbon paste electrodes for the determination of nickel (4) and copper ( 5 ) with excellent sensitivity and reproducibility. A mi-
0003-2700/9l /0363-0953$02.50/0 0 199 l American Chemical Society
