Anal. Chem. 1982, 5 4 . 207-212
gel layer, then one would expect a slowing down of the processes which determine the transient signal (16,19). This can partly be attributed to the increase of the diffusional layer thickness and partly to the decrease in value of the diffusion coefficient compared to that in the solution phase. LITERATURE CITED (1) K. Tbth, K.; Gavaller, I.; Pungor, E. Anal. Chim. Acta 1971, 57, 131. (2) Rechnltz, G. A.; Hameka, H. F. Fresenlus‘ 2. Anal. Chem. 1965, 214, 252. (3) Lindner, E.; Tbth, K.; Pungor, E. Anal. Chem. 1978. 4 8 , 1051. (4) Morf. W. E.; Llndner, E.; Slmon, W. Anal. Chem. 1975, 4 7 , 1596. (5) Johansson, G.; Norberg, K. J. Electroanal. Chem. 1988, 18, 239. (6) Buffle, J.; Parthasarathy, N. Anal. Chim. Acta 1977, 9 3 , 111. (7) Parthasarathy, N.; Buffle, J.; Haerdi, N. Anal. Chim. Acfa 1977, 93, 121. (8) Hawkings, R. C.; Corriveau, L. P. V.; Kushneriuk, S. A.; Wong, P. Y. Anal. Chim. Acfa 1978, 102, 61. (9) Shatkay, A. Anal. Chem. 1978, 48, 1039. (10) Lindner, E.; Tbth, K.; Pungor, E.; Morf, W. E.; Simon, W. Anal. Chem. 1978, 50, 1627. (11) Stover, F. S.; Brumleve, T. R.; Buck, R. P. Anal. Chim. Acta 1979, 109, 259. (12) Buck, R. P. I n “Ion-Selective Electrodes In Analytlcal Chemistry”; Freiser, H., Ed.; Plenum Press: New York, 1978. (13) Buck, R. P. I n "Ion-Selective Electrodes”; Pungor, E., Ed.; Akademiai Kiadb: Budapest, 1978. (14) Llndner. E.; Tbth, K.; Pungor, E. Anal. Chem. 1982, 5 4 , 72.
(15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30)
(31) (32)
207
Fleet, B.; Ryan, T. H.; Brand, M. J. D. Anal. Chem. 1974, 4 6 , 12. Rechnitz, G. A,; Kugler, 0. C. Anal. Chem. 1967, 3 9 , 1682. Bagg, J.; Vinen, R. Anal. Chem. 1972, 4 4 , 1773. Mathls, D. E.; Stover, F. S.; Buck, R. P. J . Membr. Scl. 1979, 4 , 395. Karlberg, B. J . Electroanal. Chem. 1973, 42, 115. Reinsfelder, R. E.; Schultz, F. A. Anal. Chlm. Acfa 1973, 65, 425. Denks, A.; Neeb, R. Fresenlus’ 2.Anal. Chem. 1977, 285, 233. Denks, A,; Neeb, R. Fresenlus’ 2.Anal. Chem. 1979, 297, 121. Denks, A,; Neeb, R. Fresenius’ 2.Anal. Chem. 1979, 298, 131. Cammann, K. “Das Arbeiten mlt ionenselektleven Elektroden”; Spinger-Verlag: Berlin, Heidelberg, New York, 1977. Morf, W. E. Anal. Lett. 1977, 10, 87. Rechnitz, G. A. I n “Ion-Selective Electrodes”; Durst, R. A,, Ed.; Washington, D C NBS. Spec. Publ. 314. Cammann, K. In Ion-Selective Electrodes”; Pungor, E., Ed.; Akademial Kiadb: Budapest, 1978. BellJustin, A. A.; Valova, I. V.; Ivanovskaja, I. S. In "Ion-Selective Electrodes”; Pungor, E., Ed.; Akademial Kiadb: Budapest, 1978. Doremus, R. H. In “Glass Electrodes for Hydrogen and Other Cations”; Elsenman, G., Ed.; Marcel Dekker: New York, 1967,. , Moody, G. J.; Thomas, J. D. R. ”Selectlve Ion Sensitive Electrodes”; Merrow: Watford, Herts, England, 1970. Robinson, R. A.; Stokes, R. H. “Electrolyte Solutlons”; Butterworth: London, 1955. Hulanicki, A.; Lewenstam, A. Talanta 1977, 24, 171.
RECEIVED for review May 28,1981. Accepted September 8, 1981. The work was partly supported by the Hungarian Academy of Sciences in the frame of a HASc-NSF project.
Potentiostat with a Positive Feedback iR Compensation and a High Sensitivity Current Follower Indicator Circuit for Direct Determination of High Second-Order Rate Constants Jean-Jacques Meyer, Dominique Poupard, and Jacques-Emile Dubois’ Instltut de Topoiogie et de Dynamlque des Systgmes de I’Universlt6 Paris V I I , associ6 au C.N.R.S., 1, rue Guy de la Brosse, 75005 Paris. France
A statlonary potentlostatlc technlque Is described for dlrect measurement of high klnetlc rate constants ( N 10’ M-’ s-’) under very small amounts of reagents and an excess of sal. To control the influence of the charglng current induced by the varlatlon of the faradaic current flowing through the uncompensated cell resistance, a potentiostat with a positive feedback IR compensatlon and a high sensitivity current follower lndlcator clrcuit has been designed. An analytlcal evaluatlon of the accuracy of the potentlal control and of the amplitude of the Induced charglng current, followed by the estlmatlon of the Influence of other eddy components In the total Indicator current, allows calculation of the error on the rate constants of fast klnetlc reactions. For the bromlnation of 4-methyl-a-methoxystyrene In methanol it Is shown experlmentaily that the rate constant Is underestlmatedby about 6 % If the cell resistance Is uncompensated.
The presence of the charging current of the capacitive double layer formed at the working electrode-solution interface is one of the most limiting factors in electrochemical kinetics. In couloamperometry (I), a stationary potentiostatic technique for measuring high kinetic rate constants (=lo8 M-l s-l), chronoamperometric measurements are not distorted by the usual charging currents induced by variation of the applied potential. However, there is another source of charging current 0003-2700/82/0354-0207$01.25/0
(2);indeed, under potentiostatic conditions, varying the faradaic current flowing through the uncompensated cell resistance causes an ohmic drop which in turn induces a change in potential across the double layer at the electrode-solution interface and results in an induced charging current iI which subsequently adds to the faradaic current iF. Under TFCR-EXSEL conditions [very small amounts M) of reagents and excess of salt] (31, it is possible by couloamperometryto monitor fast second-orderreactions, with a half-life time lasting a few seconds. This stationary technique is used to record the rate of disappearance of an electroactive species by the variation vs. time of its diffusionlimited current. Specially, in the case of fast variations of ip, the iR drop can no more be neglected and must be compensated. Various iR compensation techniques have been discussed hitherto. Some of them consist in rectifying the iR drop arithmetically (4)or in using rather sophisticated mathematical treatments such as the convolution procedures used in convolutive voltammetry (5). Other approaches involve instrumental methods (6);the most efficient of these achieves 100% iR compensation which leads to a stable potentiostatic circuit by means of damped positive feedback, although this reduces the accuracy of the system at high frequencies. To reduce the influence of the induced charging current iI in couloamperometry,we have designed a potentiostat with (i) a high sensitivity current follower indicator circuit and (ii) a positive feedback iR compensation loop. 0 1982 American Chemical Society
208
ANALYTICAL CHEMISTRY, VOL. 54, NO. 2, FEBRUARY 1982 @
P
Counter Electrode
l+zrV-;v
1.1: z2
c,
r'tl
Working Electrode Flgure 1. Equlvalent cell clrcuit with current follower amplifier A3 and 1, compensating current source (CS): R , , reslstance of the solution between counterelectrode and reference electrode; R 2, uncompensated cell resistance; C,,Cprdouble layer capacitances; Z,, faradaic Impedance.
The maximum value of the compensation factor for which the feedback system is stable was found to be ,8 = 0.8. The accuracy of the measured rate constants was estimated by analyzing the major error sources under optimum experimental conditions. The influence of the induced charging current i~ on the value of a high kinetic rate constant (lz = 7 X lo7 M-l s-l for the bromination of 4-methyl-a-methoxystyrene in methanol) was demonstrated experimentally. POTENTIOSTAT-CELL SYSTEM DESCRIPTION The size and arrangement of the electrodes in the cell were chosen so as to obtain reproducible diffusion-limitedcurrents of halogen and full potential control accuracy in a minimum cell volume (ca 40 cm3). A glass-sealed platinum wire ( 1 = 1 cm; 4 = 0.04 cm) was chosen for the counterelectrode primarily to keep the hydrodynamic flow laminar at the working electrode. Experimentally, this seems to be an important condition for the obtainment of reproducible measurements of very small indicator currents. The reference electrode was a saturated calomel electrode (Tacussel,type C8) in methanol or water. As the working electrode we used a 2 or 5 mm diameter platinum disk ATacussel ED1 49335) with a rotating speed set at 2000 rpm by an electronic speed controller (Controvit BNC 49335). The three electrodes were in the same plane and the reference electrode tip was located as close as possible to the working electrode. The cell transfer function A' [ratio: (ERE - EWE)/(ECE E W E ) ] is expressed by
A'=
(")(+ C1
C2
1 + R2Cg 1 + R'Clp
)
(1)
where
R ' = R1+ R2 p = jo
To control the electrical cell model, the calculated cell attenuation was compared with its frequency response measured with a PAR JB4 lock-in amplifier (for R1,R2, C1,and Cz values, cf. Table I). The results show that at low frequencies up to 1-2 kHz, the cell can reliably be represented by the equivalent circuit in Figure 1. The current iT which flows through the electrochemical cell during the kinetic process can be split into several components (Figure 1) and expressed as iT = iF il i, iI (2) where iF is the faradic diffusion current proportional to the concentration of halogen, il is the leakage current at node 3,
+ + +
Flgure 2.
Potentiostat-cell system: amplifiers A,, A, A, Texas Instruments Model TLOB1 A (K, = 2 X IO5, r , = 3.3 X lO-'s); ampllfler A, Analog Devlces Model AD 45 K ( K 3 = 5 X lo4, 7, = 4 X loa3s);R = 20 kQ, C = 4700 pF, R,* = 5 MQ f 0.01%, C, = 800 pF; P , , P,, IO-turn wire-wound potentiometers (P = 50 kQ; P, = 100 Q).
,
i, is the residual cell current due to impurities present in solution, and iI is the induced current resulting from the presence of uncompensated resistance Rz between the working and reference electrodes. Any change in the iF level gives rise to an ohmic drop at R2 As the potential of the working electrode (VI - V3)is held constant with respect to that of the reference electrode (POtentiostatic conditions), the voltage across the double-layer capacitor C2 changes and a current iI given by (3)
flows through C2. The analytical expression of iI and its relative value vs. iF are shown later in the text. To accurately measure low level faradaic currents and to compensate the effects of eddy currents (4 i,, iI),the feedback control system shown in Figure 2 was designed and built in this laboratory. It comprises an operational amplifier A3wired in the current follower mode and designed with a guarding ring around the current sensing input to minimize the leakage currents il, a current source (CS), continuously adjustable in the 0-200 nA range, which compensates the residual current i, (i, depends upon the nature and quality of the solvent; its amplitude is commonly about 10 nA),and a positive iR compensation feedback loop at the output of A3 to reduce the effects of the induced current iI. Owing to the low level of iF (2-3 nA min) an operational amplifier A3 with an input bias current which is very small in comparison to iF was selected. The high resistance Rf (ca. 5 MQ max.) which is necessary for adequate sensitivity induces a nonzero input resistance of the current follower AB. Kanazawa and Galwey (8)showed that Rf is reflected in the input impedance 2, of the current follower as a series inductance arm. When uncompensated, this inductance can resonate with the double layer capacitance of the cell and generate destabilizing phase shifts (9). The inductive element can be eliminated by shunting feedback resistor Rf with a capacitor Cf the capacitance of which must be chosen so that RfCf equals r3 (r3is the characteristic response time of amplifier A3). Hence the capacitance Cfmust be high enough to keep the feedback control system stable; C, r3/Rfand Cf lo0 MQ) (TL081 A). The destabilizing phase shift introduced by the “RC”network of the reference electrode resistance and its capacitance to ground (6) was reduced by soldering the shielding of the reference electrode and the braid of its connection to the input of amplifier A2, at the voltage follower output. For Al, a low drift operational amplifier with a uniform 20 dB/decade frequency response (TL081 A, Texas Instruments) was selected. During a kinetic process, the potentiostat works in the following manner: The rapid injection of the reagent in the solution induces a decrease in the concentration of halogen and a consecutive change in the faradaic impedance 2., The resulting voltage change A(Vl - V3) is fed to the voltage follower output A2 (point 4, Figure 2). The control amplifier AI then delivers a current in the electrochemical cell so that the voltage (Vl - V,) is set back to its initial control value. The ability of the feedback system to react to a rapid halogen concentration change is electrically equivalent to the response of the potentiostat to a voltage step applied a t its control input (ein). In order to analyze the stability and transient response of the potentiostat-cell system, its open-loop transfer function is calculated. THEORETICAL ANALYSIS OF THE STABILITY The potentiostat-cell system comprises a forward path with control amplifier Al (transfer function T F = Q)and half-cell C1 - RI (TF= A), and the following two feedback loops (Figure 2): a unity gain loop (1,4,6) fitted with voltage amplifier A2 (TF = 1); an iR drop positive feedback compensation loop (5, 3, 7, 6) with T F = B. According to graph theory (10) and reverting to the canonical feedback system (with a forward TF = G and a single feedback TF = H), the closed loop transfer function ( C / R )of our system is given by
C/R =
QA 1- &(A
+ B)
(4)
in which G = QA and H = ( A + B ) / A . Consequently, the open-loop transfer function (GH)is expressed by GH = &(A B) (5)
+
The expressions of the open-loop transfer functions with [TF,(p)] and without positive feedback [TF(p)] are given in the Appendix. The notations are the same as those used by Brown et al. (11). EXPERIMENTAL MEASUREMENTS The following experimental tests are performed with a ”standard”bromination reaction of ethylenic compounds in methanol (usual solvent). Bromine is generated in situ by electrolysis of a 0.1 M sodium bromide solution, acting as supporting electrolyte. The cell parameters are measured and the stability of the cell-potentiostat system is discussed via the Bode phase and attenuation plots. Then the step response of the system is examined. Measurement of Cell Parameters. Bulk solution resistances R1 and R2 are measured by the current interrupt method (12).The resistance is calculated from the steep iR voltage drop [recorded on the screen of a high sensitivity storage oscilloscope (Tektronix 5115) fitted with a differential amplifer plug-in 5A22NI arising when the constant electrolysis current is switched off. Bromine concentrations correspond to those encountered during kinetic experiments. The electrolysis current intensities (0.1-1 PA) are selected so that the voltage between the working and reference electrodes coincides
4
I
l0OL-1
I
I
1
41
209
10
102
103
w (rd/s I
I 04
Figure 3. Bode plot of potentlostat-cell system open-loop gain for various values of p: 1, 0 = 0; 2, p = 0.5;3, p = 0.8;4a, = 0.85; 5, 6 = 0.9; 6a, = 1; NaBr 0.1 M iMeOH; working electrode diameter, 5 mm; R 2 = 6 kQ; R 1 = 8.8 kQ; C2= 3.1 pF; C, = 1.7 /AF;R f * = 5 MQ; Cf = 800 pF; R = 20 kQ; C = 4700 pF; K , = 2 s. s; K , = 5 X 10‘; T~ = 4 X X IO6; T~ = 3.3 X
1 401
,
, 1 . . 1 . J
0,1
,..A, , , . , . , I I
1
10
4,,,.,,.I
102
,‘,..A
10)
‘‘‘...A 10)
w (rd/5) Flgure 4. Bode plot of potentlostat-cell system phase for various values of 0: 4a, p = 0.85; 6a, /3 = 1.
with the diffusion plateau of the halogen (+200 mV vs. SCE in methanol), A bipolar current source switches the electrolysis current by means of a mercury wetted contact relay (Clare HGOM 10008). The results obtained with 2 and 5 mm diameter disk working electrodes (in methanol and other protic or aprotic solvents) are reported in Table I. The high measured resistances are probably relevant to (i) the low supporting electrolyte and halogen concentrations and (ii) the electrode dimensions and arrangement. The double layer capacitances C1 and Cz were measured with a Hewlett-Packard 4800 A vector impedancemeter. To avoid polarization effects, the voltage applied to the electrodes never rose beyond 2.2 mV peak to peak. To measure the capacitance of the working electrode, we used a large surface platinum grid instead of the counterelectrode so that only the capacitance of the working electrode contributes to the measured impedance magnitude. During this experiment, the cell only contains a 0.1 M sodium bromide solution in methanol. The same operating process was repeated for the measurement of the counterelectrode capacitance. Results are summarized in Table I. Experimental Control of the System Stability. The cell parameters used are those measured for the “standard” bromination reaction (complete cell and potentiostat parameters are given in the caption of Figure 3). Consider the Bode phase angle and magnitude plots of Figures 3 and 4 obtained from the frequency response functions given by expressions A-2 and A-4. Without iR compensation (/3 = 0, curve 1,Figure 3), a slope of -20 dB/frequency decade at the unity gain cross-over frequency and a phase margin of about 90” are both indicative of a stable system. With a fraction iR compensation factor p set at 0.85, curve 4a (Figure 3) shows that the gain roll-off changes from -40 to -20 dB/decade just after the unity gain point. The corresponding phase diagram (curve 4a, Figure 4) indicates a nominally stable system with a phase margin of about 35O.
210
ANALYTICAL CHEMISTRY, VOL. 54, NO. 2, FEBRUARY 1982
Table I. Cell Parameters R,,R,, C,, and C, for Different Halogenation Solventsa fixed currents, PA -1
-0.1
solvents supporting salt (0.1 M) Me0H:NaBr
working electrode potential, mV t 200
disk electrode diameter, mm 2 5 2 5 2 5
LiCl
+400
H,O(pH 5):NaBr
+400
bleCN:(C,H,),NBr
-350
2
R,, k a 8 8 9.4 9.4 9 9 14
Me0H:NaBr
t 200
2 5
8.8 8.8
R,,kn 35
C,,PF
C,,PF
1.7 1.7 1.4 1.4 1.0 1.0 1.4
0.5 3.1 0.4 2.5 0.3 2.3 0.4
1.7 1.7
0.5 3.1
5
42 6
40 6 60
40 6
VTq
a ( A R I R = f 10%;AC/C = +15%). The reference electrode depends on the solvent: MeOH, saturated calomel electrode in methanol; H,O, saturated calomel electrode in water; MeCN, Ag/Ag+,NO,- 0.01 M in MeCN.
For /3 = 0.9, however, the system is unstable. Experimentally, the system becomes unstable when /3 = 0.83-0.85. This result indicates that, despite uncertainties, the theoretical model is not too different from the actual behavior of the potentiostat-cell system. Stable operation of the system with ideal 100% iR compensation is possible using different stabilization techniques none of which are free from drawbacks, particularly regarding the high-frequency accuracy of the potentiostat control across the double layer which, unfortunately, is important in the present case. Therefore a fraction iR compensation of ca. 0.80 was chosen for these experiments. Step Response. To ascertain the type of response of the system (overdamped, critically damped, underdamped) and to check that the instrument response time is much shorter than the mixing time (estimated as 0.1-0.2 s) of the reacting species, we estimated the step response of the feedback system. The response was measured by feeding the potentiostat to the dummy cell in Figure 2. The circuit presenta an underdamped response with an overshoot of 3% and a response time of a few milliseconds. The experimental results well agree with the rough indications given by the open-loop gain vs. frequency curve of the feedback system (Figure 3). Moreover, Carter (23) has shown that the reciprocal open-loop unity gain cross-over frequency approximately gives the closed-loop response time to a step function input. For the “standard” bromination reaction with /3 set at 0.8, a measured unity gain frequency of about 100 Hz corresponds to a potentiostat response time of approximately 10 ms. Applying Carter’s relationship between phase margin and percent overshoot to this case results in an overshoot estimate of 2.54%. ANALYSIS OF THE RATE CONSTANT MEASUREMENT ACCURACY An attempt was made to determine the main error sources encountered in the determination of the kinetic rate constants. Error due to the Current Measurement Method. In the first step, only the faradaic component of the cell current is taken into account on the premise that the feedback control system is ideal. The remaining inaccuracy then depends on the faradaic current measurement method and on the graphic rate constant determination procedure. The worst accuracy in measurement of the faradaic current iF is due to the determination of resistance Rf (0.01%),to the current flowing through C,in the case of rapid kinetics (0.4%), and to the offset and drift characteristics of the AD 45K operational amplifier (0.2%) and lies around 0.6%. Error due to the Induced Charging Current iI. To calculate the analytical expression of iI vs. time and to compare its magnitude with iF, we adjusted the theoretical model
IF
ti
in
c,
@In
Flgure 5. Schematic diagram of the half electrochemical cell (potentiostatic conditions).
proposed by Fratoni and Perone (2) to fit our case. To simplify calculations, the case of a pseudo-first-order kinetic process is considered. From the relation between the pseudo-first-order rate constant k’and the eecond-order constant k (k’ = k[olefin]), a value of k’ (which will be introduced into expression 9 to calculate i ~corresponding ) to a high kinetic rate constant k (ca. lo8M-l s-l) can be chosen. If the concentration of halogen used is ca. M, the concentration of olefin will approximately be 20 times greater than that of halogen (pseudofirst-order conditions): the value of k’chosen would be then 2 s-1.
X2 + olefin
k
products
The time dependence of the concentration of the electroreducible species X2 is given by [X,] = [Xz]oe-k’there, [XZIO is the initial concentration of halogen; and k’ = k[olefin]. Let a be the proportionality coefficient between the diffusion-limited current and the concentration of halogen; then the faradaic current iFis expressed as (it is assumed (3)that any variations in the double layer voltage do not exceed the limits of the diffusion plateau, which is the case in our “standard conditions”)
iF(t) = 6e-k’t
(6)
in which 6 = [X2]o/a. The schematic diagram of a half-cell including the working and reference electrodes across which the control voltage eh is applied is shown in Figure 5. Kirchhoff s current law at node 2 yields iT = iF + iI (7) The Laplace transform of
iT
gives
iT03) = [ein/P - V~(P)I/R2 Likewise, if Cz is charged to the voltage ein at t = 0, then i103) = [V203) - e i J ~ l 0 3 C J The Laplace transform of eq 6 becomes iF(P) = 6/03 + k?
ANALYTICAL CHEMISTRY, VOL. 54, NO. 2, FEBRUARY 1982
211
Table 111. Potentiostat Accuracy vs. Frequency for Various Values of p (Eo)act/ein
(Eo)id/ein
p
f, Hz
I I
9.738 97.389 5 436.945 0.7 0.1 9.738 97.389 1 0.8 0.1 9.738 1 97.389
0
0.1 1
phase I I 9.737 90" 96.97 90" 90" 417.93 90" 9.739 97.30 90" 9.745 90" 90" 97.404
accuphase racy, % 89"18 0.01 5 81"72 0.7 51'71 4.3 89'64 0.004 86"35 0.09 89"70 0.067 87'01 0.015
of the current follower is calculated. Kirchhoffs current law applied to node 6 (Figure 2) becomes (ei, Ef Ei - 3e,)/R = (e, - eo)Cp (10)
+ +
I
0.5
I
I
1
1.5
'(4
I
I
2
-
Flgure 8. Plot of calculated currents /At), ir(t),and / A t ) during a fast kinetic reaction: k' = 2 s-' (see text); R2C2 1.86 X s; 6 = 0.5.
in which e, = eo/G1,where GI is the open-loop gain of amplifier Al. Substituting the expression of (Ei + Ef)vs. eo into eq 10 and writing the relationship between eo and Eogives
Table 11. Percent Values of l i I / i F I for Different Times during a Fast Kinetic Process a 0.1s
3.86
0.5s
1s
2s
3.86 3.86 3.86 k' = 2 s-'; R , C , = 1.86 X 10+ s.
3.28 a
0.3s
3s
4s
3.86
3.86
Substituting i&), iI(p), and iF(p) into eq 7 leads to V203), the inverse transform of which is given by
Then the following equations can be written:
and
Curves iI, iF, and iTare plotted vs. time for the "standard" experimental conditions in Figure 6. These curves indicate that the ohmic drop across R2 gives rise to an error (characterized by liI/ipl) in the measurement of ip Percent values of liI/iFI a t different times are given in Table 11. The error due to the induced charging current is greater, for rapid kinetics than the error introduced by the current measurement method. Errors due to i, and il. It was verified, under appropriate experimental conditions, that i, remains constant during a kinetic process. Consequently, it can be fully cancelled by means of the compensating course (SC) (Figure 2). The leakage current il was small in comparison to the faradaic diffusion current iF, because of the input characteristics of amplifier As. Accuracy of the Potentiostatic Control. The desired role of the potentiostat is to hold the voltage across the double layer capacitance C2equal to the input voltage eh (Figure 2). Ideally (21) the steady-state sinusoidal response of the current follower is (Eo)d= euJl&p. This would mean that the output of the current measuring amplifier will be -90° out of phase with respect to the input control signal e& To estimate the error of the feedback system, the actual output voltage (EO)ad
+
= K&' 3K,Cz/K1 T = RC 7' = R'C' 72 = R2C2 Tf = RfCf In thie calculation, we assumed G1 = -K1 and G3 = - K p In other words, the open-loop gain roll-off of the amplifiers was neglected. This approximation is valid for the frequency range below the potentiostat unity gain cross-over point. Values of (Eo)id/ein,(Eo)act/ein, and the percent accuracy of IEo/ehl for different values of 0are reported in Table 111. The data in Table I11 indicate that under the "standard" experimental conditions ( p = 0.8, f = 1Hz),the error due to the feedback system (0.067%) is small compared to the error due to the current measurement method and that due to the induced current. Accuracy of Rate Constant Determination. The logarithmic graph of the recorded kinetic curve allows calculation of the second-order rate constant k . Since ~4
in which i&) is the diffusion-limited current of the halogen at time t , i m ( t mis) the same current at the end of the kinetic process, and f(t) is a linear time function, then k is calculated from the slope m of f ( t ) : m = kai,(t,) (a was previously defined). The error on the rate constant determination is calculated by the classical least-squares method. The rate constant of 4-methyl-a-methoxystyrenebromination was statistically determined in methanol; the relative accuracy of the current iT is 1.4% for ,6 = 0.8 and 4.5% for p = 0. The worst case error analysis shows that with the herein described potentiostat, kinetic rate constants around lo8 M-l s-l are still determined with an accuracy better than 10%. To test the precision of k measurement, a series of experiments was carried out with the same reacting species at different time intervals. It is noteworthy that, in every case,
212
ANALYTICAL CHEMISTRY, VOL. 54, NO. 2, FEBRUARY 1982
Table IV. Rate Constants of 4-Methyl-a-methoxystyrene Bromination ( M e O H , 0.2 M NaBr) Measured without (p = 0) and with iR Compensation ( p = 0.8)
L ( L ) , 10-7k,
0.8
M
5.1 4.4 4.6 4.7 4.8 4.4 4.5 4.3 4.2 4.6
M-I s - l
6.66 6.75 6.70 6.67 6.60 6.90 6.98 7.04
u = R2C2RfCf
k,
1 0 9 ~
P 0
with
b = R2C2 + R&
M-1 s-l
G3)
in which G3is the open-loop gain of A3 6.7 X
lo’
c
RJfR‘C’
d = R‘C‘+ RfCf + RfC’/(1
R’= R1+
7.02 X 10’
7.10
G1 = -KI/(l
measurement precision is better than f5%, APPLICATION TO KINETIC MEASUREMENTS In order to determine experimentally the influence of the induced current iI upon the measured rate constants, two series of measurements with and without iR compensation were run. Each set comprises five brominations of 4-methyl-a-methoxystyrene in methanol with 0.2 M sodium bromide. The results, summarized in Table IV, are in good agreement with the expected values (14) and show that the rate constants measured with iR compensation (P = 0.8) are, as predicted by the theoretical curve of Figure 6, greater than those obtained without compensation. By applying the statistical t test to the data of Table IV we found that for P = 0 and ,f3 = 0.8 the mean rate constants lie in the intervals (6.575-6.825) X lo7M-ls-’ and (6.833-7.207) X lo7 M-’ s-l, respectively, with 99% confidence,thus showing a statistically significant difference. ACKNOWLEDGMENT The authors thank J. C. Fontaine, J. Aubard, and M.F. Ruasse for helpful suggestions and numerous informative discussions. APPENDIX By use of the symbols defined in Figure 2 and the same notations as those used by Brown et al. (11),the open-loop transfer function without iR compensation.(P = 0) is expressed as
-
G3)
R2
+ C,) + T$)
C’ = ClC2/(C1
7.08
The Laplace transform of eq AI gives
+ RfC2/(1-
where G1is the open-loop gain of Al. Similarly, the open-loop transfer function with iR compensation given by
leads to
TF&)
=-
(c1:c2) K1
RCT&
+ (KIRC + RC + 371)p + 3 > x
(
~ ( 1 P)p2 + ( b - PRZCJp cp2 + d p 1
+
+1
)
(A4)
LITERATURE CITED Dubois, J. E.; Aicais, P.; Barbier, G. J . Electroanal. Chem. 1984, 8 , 359-365. Fratoni, S. S.; Perone, S. P. Anal. Chem. 1976, 48, 287-295. Poupard, D.; Ruasse, M. F.; Dubois, J. E., unpublished work, 1980. Mumby, J. E.; Perone, S. P. Chem, Instrum. 1971, 3 , 191. Saveant, J. M.;Tessier, D. J . flecfroanal.Chem. 1977, 7 7 , 225-235. Britz, D. J . Electroanal. Chem. 1978, 88, 309-352. Electrochim. Acta 1980, 25, 1449-1452. Harrar, J. E.; Pomernacki, C. L. Anal. Chem. 1973, 45, 57-78. Kanazawa, K. K.; Gaiwey, R. K. Anal. Chem. 1977, 49, 877-678. Davis, F.; Toren, C. Anal. Chem. 1974, 4 6 , 647-650. Di Stefano, J. J.; Stubberud, A. R.; Williams, I. J. “Feedback and Control Systems”, Schaum’s series; McGraw-Hili: New York, 1967; p 141. Brown, E. R.; Smith, J. E.; Booman, G. L. Anal. Chem. 1988, 4 0 , 1411-1423. Besson, J.; Guitton, J. “Manipuiatlons d’Blectrochlmie, Introduction 6 la theorie et B la pratique de ia cineetique Blectrochimique”, Masson Ed. 1972; p 202. Carter, W. C. Instrum. Control Syst. 1987, 4 0 , 107-110. Argile, A. Doctoral Thesis, UniversitB Paris VIJ, 1980.
RECEIVED for review April 1, 1981. Accepted September 8, 1981.
