2112
Anal. Chem.
1980, 52, 2112-2116
Chirurgie Cardiovasculaire of the Centre Hospitalier Universitaire of Toulouse-Rangueil for his help with the experiments on animals.
LITERATURE CITED (1) (2) (3) (4) (5) (6) (7) (8) (9)
(10) (1 I )
Hess, B. Biochem. 2. 1956, 328, 110-116. Williams. D. L.; Doig, A. R.; Korosi, A. Anal. Chem. 1970, 42, 118-121. Racine, P.; Mindt. W. Experientia. Suppl. 1971, 18, 525-529. Racine, P.; Engelhardt, R.; Higelin, J. C.; Mindt, W. Med. Instrum. 1975, 9 , 11-14. Shinbo, T.; Sugiura, M.; Kamo, N. Anal. Chem. 1979, 51, 100-104. Biaedei, W. J.; Jenkins, R . A. Anal. Chem. 1976, 48, 1240-1247. Durliat. H.; Comtat, M.; Mahenc, J.; Baudras, A. J . Electroanal. Chem. 1975, 66, 73-76. Baudras, A.; Spyridakis, S. Biochimie 1971, 53, 943-955. Duriiat, H.; Corntat, M.; Mahenc, J.; Baudras, A. Anal. Chim. Acta 1976, 85, 31-40. Duriiat. H.; Comtat, M.; Baudras, A. Clin. Chem. 1976, 22, 1802-1805. Duriiat, H.; Corntat, M.; Mahenc, J. ~ n a l .chim. Acta 1979, 106, 13 1-135.
(12) Boucays, A. These Madecine, UniversRB Paul Sabatier, Touiouse, 1978. (13) Boccaion, A.; Puel, P.; Comtat, M.; Mahenc, J. Communication to the 2nd Congress on Hemodynamics of the Limbs, Scottsdale, AZ, Feb 1979. (14) Betso, S. R.; Kbpper, M. H.; Anderson, L. B. J. Am. Chem. Soc. 1972, 94, 8197-8204. (15) Mor, J. R.; Guarnaccia, R. Anal. Biochem. 1977, 79, 319-328. (16) Lerner. H.; Giner, J.; SoeMner, J. S.; Cotton. C. K. J . Electrochem. Soc. 1979, 726 (2), 237-242. (17) Marincic, L.; Soeldner, S.; Giner, J.; Cotton, C. K. J. Electrochem. Soc. 1979. 126 (10). 1687-1692. (18) Skou; E. €/ect&him. Acta 1977, 22(4), 313-318. (19) Duriiat, H.; Comtat, M. J. Elecfroanal. Chem. 1978, 89, 221-229.
RECEIVED for review February 5,1980. Accepted July 9,1980. This research was financially supported by the Ddegation GBnBrale la Recherche Scientifique et Technique within the framework of Contract No. 78.7.2932 of the Biomedical Engineering Committee.
Corrections for Systematic Errors from Analog Integration in Controlled-Potential Coulometry Thomas L. Frazzini, Michael K. Holland,* Jon R. Weiss, and Charles E. Pietri New Brunswick Laboratoty, U S . Department of Energy, 9800 South Cass Avenue, Argonne, Illinois 60439
The output signal from an ideal analog integrator should be proportional to the integrated current. Deviation from this ideal analog integrator response in a state-of-the-art controiledpotential coulometer has introduced systematlc errors of up to -0.1 % In the measurement of electroactive species. An extenslve evaluation of a controlled-potentlal coulometry system, widely used In the nuclear fleld, resulted in the identification of the sources of these errors. The analog integrator studied suffered from voltage output signal drift and offset caused by the operational amplifier. Furthermore, shlfts In this voltage output signal occurred because of leakage and dielectric absorption by the Integrating capacitor. Equatlons which describe the leakage and offset phenomena have been derived. Instrument operating methods which compensate for the drift and dielectrlc absorption effects have been developed. Corrections based on these equations and on the modlfled instrumental operatlng methods have slgnlflcantly decreased these systematic errors. These improvements permit the coulometer to be used in coulometric measurements whlch are directly based on a derived physical constant, the Faraday. The proposed corrections are applicable to other systems using analog lntegratlon circults.
Theoretical discussions of controlled-potential coulometry typically state its application permits the determination of electroactive species directly from the quantity of charge used to electrolyze the desired species and that chemical standard reference materials are not required ( I , 2). In application, the use of controlled-potential coulometry based on the Faraday is not simple. Many of the special considerations required for this application have been well researched: accurate corrections for background current ( 3 , 4 ) ,correction for the fraction of the sample electrolyzed within the selected potential span ( 5 ) ,the uncertainty in the value of the Faraday
( 6 ) , and the general requirements of an integrator for controlled-potential coulometry (7, 8). Although analog integrators have been designed which, in practice, are close to meeting the integration requirements, no corrections for the significant deviations from ideal integrator response in controlled-potential coulometry have been proposed. A stateof-the-art controlled-potentid coulometer widely used in the nuclear field contains a highly reliable solid-state analog integrator with nearly ideal response which should behave according to
where Ek and Eoutare the integrator input and output signal voltages, respectively, and R and C are the integrating resistor and capacitor values as shown in Figure 1. Our evaluation of this integrator resulted in the development of corrections for previously recognized integrator behavior (8,9):(1)the voltage output signal is subject to amplifier drift and offset; (2) the output signal which is held on the integrating capacitor is subject to dielectric absorption in the capacitor; (3) the output signal is subject to leakage. These deviations from ideal response have introduced a systematic error of approximately -0.1% in the measurement of electroactive species such as plutonium. We report, in this paper, an improved methodology which dramatically reduces these systematic errors and permits the coulometer to be used for measurements directly based on Faraday’s constant. The impact of drift,offset, dielectric absorption, and leakage phenomena upon analog integration of the exponentially decaying dc electrolysis current produced during a controlled-potential coulometry determination can be predicted. Drift is the rate of change of integrator output, measured under conditions where the input voltage from the potentiostat is zero. Since uncompensated input offset in the integrator will cause drift, the operational amplifier of the integrator must be adjusted to minimize the drift which may, in turn,
This article not subject to U S . Copyright. Published 1980 by the American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 52,
NO. 13, NOVEMBER 1980 ~-
0012,
2113
0 06
hteqrut Inq
0000
L
'
~-
0
I00
I
200
300
000
__-i__
400
500
600
T I M E IICCI
L-.
.............. '..._. ....
.
Flgure 1. Typical analog integration circuit. For the M-T Model 3 coulometer, R = 1 MQ and C = 10 pF. result in an offset voltage at the output of the integrator. Since integration is initiated from the integrator offset voltage rather than from zero volts, the final integrated voltage is in error. Dielectric absorption is a change in capacitor characteristics such that the field produced by the charge separation induces an alignment of molecular dipoles of the insulating material in the direction of the electric field. This change results in a decrease in the voltage drop across a capacitor with a corresponding decrease in integrator output. Models accurately depicting this phenomenon describe t h e decrease in voltage drop as (1)an alignment, at the conducto-dielectric interface, of a small fraction of the charge with t h e molecular dipoles, which results in t h e aligned charge no longer contributing to the voltage drop, i.e., absorption of charge (IO), or (2) a change in t h e effective capacitance due t o t h e polarization at t h e conductor-dielectric interface (11). Both models are equivalent insofar as the proposed corrections are applied to analog integration of dc current in controlled-potential coulometry. T h e proposed corrections are presented herein in terms of the first model but are consistent with either model. Leakage is t h e loss of charge on a capacitor through the unavoidable insulation resistance, RL,between the capacitor terminals, which results in a decreased integrator output (11).
EXPERIMENTAL SECTION Apparatus. An M-T Model 3 controlled-potential coulometry system (M-T Electronics, San Leandro, CA), consisting of a potentiostat and an analog integrator, was evaluated. A circuit diagram for analog integrators representative of the type used in the M-T Model 3 coulometer is shown in Figure 1. In this evaluation, the analog integrator was connected to the potentiostat in the same manner as for electrical calibration and sample determination. The analog integrator was evaluated for drift, offset, dielectric absorption, and leakage. Drift and Offset. Testing of the d r i i was performed according to the manufacturer's procedure (9) on each day the coulometer was used. This testing demonstrated that of the problems encountered, drift was the simplest to eliminate. The trim control on the operational amplifier, which can be set to minimize either drift or offset, was adjusted as required to maintain a drift of 99% completion. The second source of systematic error due to dielectric absorption results from using the integrator when absorbed charge from the previous integration is still present in the capacitor dielectric. This source of error can be eliminated by placing the integrator module in the discharge position for at least 5 min between current integrations (9). Frequently, preliminary sample electrolysis, where the integrated current values are not used, exceed the required 5-min discharge period and may be conveniently utilized (8).
ANALYTICAL CHEMISTRY, VOL. 52, NO. 13, NOVEMBER 1980
Leakage. As in the case of dielectric absorption, the leakage error results in a positive systematic error during calibration and in a negative systematic error during sample electrolysis. Unlike the dielectric absorption errors, these two leakage errors are not equal in magnitude and, hence, do not cancel even when the calibration interval is longer than 300 s. Evaluation of the leakage phenomenon revealed that it is a function of both the integration time and the instantaneous voltage output signal and not simply the final voltage output signal. Thus, the leakage error is a function of the shape of the integrator charging curve. This dependence upon the integrator charging curve is consistent with the finding that the signal loss due to leakage vs. the voltage output signal is a constant. These findings are in agreement with Ohm’s law and the capacitor discharge equation. The leakage-corrected integrator voltage output signal, VL, is calculated by using eq 4, where Vf is the = Vf L ( t ) (4)
+
v,
final voltage output signal and L ( t ) is the signal loss due to leakage which depends upon the time, t , for a given Vf. Equations for determining the loss in the voltage output signal due to leakage for an analog integrator used in controlledpotential coulometry may be derived from a form of the capacitor discharge equation, eq 5, where V(t) is the voltage on 1 -d V - -V(t) (5) dt RLC the integrating capacitor as a function of time. The correction for the leakage error requires the use of the appropriate V(t), which specifically describes the integrator charging curves in controlled-potential coulometry. For electrical calibration Vf L ( t = t f ) t - L(t) V(t) = tf where tf is the time a t which the final voltage output signal was obtained. Substituting eq 6 into eq 5 and integrating yields Vftf L(t=tf)tf L ( t = t f ,electrical calibration) = 2RLC 2RLC
+
~
+
design. In the particular integrator design tested a t this laboratory, the RLC of the integrating capacitors ranged from 1 X lo6 to 6 X lo6 s. Depending upon the capacitor quality and intended application, RLC for different types of capacitors can be very small. Corrections for leakage due to RLC = 2000 s will be on the order of several percent which is not a practical value for analog integration in high-quality controlled-potential coulometers. Corrected Integrator Voltage O u t p u t Signal. The integrator voltage output signal corrected for both leakage and offset, VL’, is calculated by using eq 12. Substitution of eq (12) = VL - Voffset 4 and 10 into eq 12 gives the leakage and offset correction for electrical calibration. vLr
VL’(electrica1 calibration) = Vf
Vftf +2RLC - Voffset
[Vf + L ( t = t f ) l [ l - e x p ( - t / ~ , ) l V(t) =
[1 - exp(-tf/~,)l
-
L(t)
(8)
where T~ is the time constant of the exponentially decreasing sample electrolysis current which is being integrated. Substituting eq 8 into eq 5 and integrating yields
For analog integrators used in controlled-potential coulometry, typically: RLC >> tf > 7, and Vf >> L(t=tf) 2 L(t). Under these conditions, 1- exp(-tf/T,) approaches unity, greatly simplifying eq 7 and 9. Approximations of L(t) for electrical calibration and sample determination are Vftf 2RLC
L(t=tf,electrical calibration) = L ( t = t f , sample determination) =
Vf(tf-
7,)
(10)
(11) RLC The value of RLC varies for individual integrators of the same
(13)
Substitution of eq 4 and 11 into eq 12 gives the leakage and offset corrections for coulometric sample determination. VLr(sampledetermination) = Vf
+ VAtf RLC
7,)
- Voffset
Typical data obtained by using the modified instrumental operating methods described in this paper are as follows: Vf = 5.78431 V; tf = 500 S; T , = 35 S; RLC = 1.6 X IO6 S; Voffsst = -0.000 29 V. Applying these data to eq 14 results in V L ~ = 5.786 28 V for a correction of 0.034% for leakage and offset. Application of eq 14 is not limited to integration of current which closely follows the exponential decay equation. Since the magnitude of the correction is small and changes in 7, only affect the term tf - T ~ a, large uncertainty in 7Ccan be tolerated. Using 7C = 105 s in the above sample calculation yields V,’ = 5.78603 V for a correction of 0.030% for leakage and offset. Thus, a threefold change in 7 cresults in an uncertainty of only 0.004% in VL‘. If changes in electrolysis current kinetics result in unac, eq 15 can be used in place ceptably larger changes in T ~ then VL’bample determination) = Vf
For sample determination
2115
VfK + RLC -
-
Voffset
(15)
of eq 14, where K is the average fraction of the voltage measured on the capacitor during integration. DISCUSSION With the proper corrections for nonideal integrator response, a coulometer with an analog integrator may be calibrated electrically without introducing a systematic error from the integrator electronics. The proposed procedural and mathematical corrections (temperature control to *1 OC,drift adjustment to f10 pV/min, minimum equilibration interval of 300 s, minimum electrical calibration time of 300 s, and the equations for leakage and offset) provide the M-T Model 3 analog integrator with the capability of being employed in coulometric measurements based directly on the Faraday. Without any compensation for the nonideal integrator responses of the M-T Model 3 coulometer, the associated systematic error was determined to be as large as -0.1 7’. Even when chemical calibration is used but the corrections for nonideal integrator responses are not made, errors as large as 0.05% can be introduced into the analysis. For example, if the chemical calibration and the sample analysis are performed by using the 30-pA electrolysis current cutoff, but the sample electrolysis time or the curve shape do not match those of the calibration standard, for any reason, then complete compensation for the systematic leakage error is not achieved. Judicious selection of experimental parameters, which min-
2116
Anal. Chem. 1980, 52, 2116-2123
Table 111. Controlled-Potential Coulometric Determination of Plutonium Neglecting the Proposed Corrections for Analog Integration Errors no. of mean Pu preparation aliquotsa recovery, % A
18
B
28
C
27
D
9
E F
15 17
G
9
H
12
99.91 99.84 99.83 99.93 99.90 99.91 99.93 99.92
RSD, % 0.07
0.08 0.07 0.07 0.08
0.06 0.05 0.06
a Aliquots were prepared from National Bureau of Standards SRM 9 4 9 e plutonium metal.
Table IV. Controlled-Potential Coulometric Determination of Plutonium Applying the Proposed Corrections for Analog Integration Errors no, of
preparation aliquotsa
I J
32
K
13 18 36 44 49
L M N 0
P
15
Q R
11 21 10
S
12
The proposed corrections were applied to the controlledpotential coulometric method used a t this laboratory for plutonium measurement. This method, which utilizes calibration based on the Faraday and corrections for background current and for fraction of the sample not electrolyzed within the selected potential, exhibited a -0.1% systematic error which was significantly reduced to a negligible level of 4.01 % by the proposed corrections (Tables I11 and IV). The average RSD was improved somewhat. These corrections were further verified by application to the controlled-potential coulometric determination of iron in high-purity (>99.98%) iron standards. The requirements of accurate and precise integration, although important in controlled-potential coulometry, are not limited to this field. Since the deficiencies observed were a result of the properties of operational amplifiers and capacitors which are found in all analog integrators, the nonideal responses described should be considered when using any type of analog integration circuitry.
LITERATURE CITED
mean Pu
recovery, %
RSD, %
99.99 99.99 99.97 99.98
0.05 0.03
100.00 99.99 99.98
(1) Lingane, J. J. "Electroanalytical Chemistry", 2nd ed.;Interscience Publishers: New York. 1958; p 450. (2) Willard, H. H.; Merritt, L. L., Jr.; Dean, J. A. "Instrumental Methods of Analysis", 5th ed.; D. Van Nostrand Co.: New York, 1974; p 702. (3) Meites, L.; Moros, S. A. Anal. Chem. 1959, 3 7 , 23. (4) Holland, M . K.; Weiss, J. R.; Pietri, C . E. In "Analytical Chemistry in Nuclear Fuel Reprocessing"; Lyon, W. S., Ed.; Science Press: Princeton, NJ, 1978; pp 142-150. (5) Holland, M. K.; Weiss, J. R.; Pletri. C. E. Anal. Chem. 1978, 5 0 , 236. (6) Koch, W. L.; Diehl, H. Talanfa 1976, 23, 509. (7) Harrar, J. E.; Behrin. E. Anal. Chem. 1967, 39, 1230. (6) Harrar, J. E. In "Electroanalytical Chemistry"; Bard, A. J., Ed.; Marcel Dekker: New York, 1975; Vol. 8, pp 54-56. (9) "Coneolled-Potential Coulometry System Model 3 Operating and Service Manual", M-T Electronics Co.: San Leandro, CA, 1970. (10) Von Hippel, A. In "Dielectric Materials and Applications": Von Hippel, A., Ed., Technology Press of M.I.T. and Wiley: New York, 1954; p 5. (11) Korn, G. A.; Korn, T. M. "Electronics Analog and Hybrid Computers"; McGraw-Hill: New York, 1964; pp 93-96. (12) Component Technology and Standardization; General Electric Co.Corporate Research and Development: Schenectady, NY, 1978; Vol. 1, Section 3, p 13. (13) Cornell, J. I.In "Electrical Engineers Handbook", 4th ed.;Pender, H., McIlwain, K., Eds.; Wlley: New York, 1950; Chapter 3, p 55.
0.03
0.05 0.05 0.05 0.07
100.00
0.06
100.01
0.04
99.98 100.00
0.06 0.02
Aliquots were prepared from National Bureau of Standards SRM 9 4 9 e plutonium metal. imize the systematic errors, is required if chemical calibration is to be employed. T h e proposed corrections are significant since several laboratories can perform plutonium analysis by controlled-potential coulometry with an uncertainty of
