Real-TimeComputer Prediction of End Points in Controlled-Potential Coulometry F. B. Stephens, Fredi Jakob,’ L. P. Rigdon, and J. E. Harrar Chemistry Department, Lawrence Radiation Laboratory, University of California, Livermore, Calif. 94550 Two real-time computer techniques have been developed for predicting the final value of the charge in a controlled-potential coulometric determination. One method, for which a suitable instrument has been designed, is an analog-type computation. The other technique employs an on-line digital computer programmed to perform the calculation. A procedure has been developed for handling background current corrections for predictive coulometry so that the technique can be used as an absolute method. Both analog and digital systems yield analytical results with an accuracy and precision of 0.1-0.2%, in one-third the electrolysis time required for conventional controlledpotential coulometry. The systems also incorporate a readout of the electrolytic rate constant. This feature is useful for testing an electrochemical system for conformance to the simple exponential rate law, which is necessary for accurate prediction, and for detecting kinetic complications in the electrochemical reaction.
DURINGTHE
EARLY development of controlled-potential coulometry, applications of the technique were limited because the duration of a typical electrolysis was 1 hr or more ( I , 2). To shorten the time of a determination, several calculational and extrapolation procedures were proposed (3-6) for locating the end point of the electrolysis without actually carrying it to completion. Unfortunately, the extrapolation techniques, when used off-line, shorten the electrolysis time only at the expense of operator time (7), and they generally sacrifice accuracy. The advent of improved cell designs (8-IO), leading to higher speed electrolyses, has made extrapolation procedures unnecessary for most analyses. Nevertheless, there remain several important cases for which end point prediction techniques would be of advantage. For example, it is desirable in certain determinations involving irreversible reactions to carry out the electrolysis at less than the limiting current, and thus at a slower rate, to avoid a high background current or the interference of another species. Calculation of the final value would also be useful in cases where a high background current prevents an accurate determination by complete electrolysis. In addition to these incentives for examining prediction 1 Present address, Department of Chemistry, Sacramento State College, Sacramento, Calif. 95819.
(1) J . J. Lingane, “Electroanalytical Chemistry,” 2nd ed., Interscience, New York, N. Y., 1958, Chapter XIX. (2) P. Delahay, “New Instrumental Methods in Electrochemistry,” Interscience, New York, N. Y., 1954, pp 287-289. (3) W. M. MacNevin and B. B. Baker, ANAL. CHEM.,24, 986 ( 1952). (4) L. Meites, ibid., 31, 1285 (1959). (5) S. Hanamura, Talanra, 2, 278 (1959). (6) Zbid.,9, 901 (1962). (7) W. D. Shults in “Standard Methods of Chemical Analysis,” 6th ed., Vol. 3, Part A, F. J. Welcher, Ed., D. Van Nostrand, Princeton, N. J., 1966, p 466. (8) G. L. Boornan, ANAL.CHEM., 29, 213 (1957). (9) A. J. Bard, ibid., 35, 1125 (1963). (10) G. C. Goode and J. Herrington, Anal. Chim. Acra, 33, 413 (1965). 764
ANALYTICAL CHEMISTRY, VOL. 42, NO. 7, J U N E 1970
methods, the present availability of suitable computational instrumentation affords the possibility of performing the calculations on-line. Such a technique could effect a real saving in operator time and effort, even for analyses that are already rapid. Two different approaches have been investigated for the real-time calculation of the final value of charge in a controlled-potential coulometric experiment. One approach involved the use of a multiplier/divider function module to evaluate the predicted value of charge as a function of the measured accumulated charge and the instantaneous current. The other technique was to use an on-line digital computer to calculate the final value at frequent intervals during the electrolysis. For the digital system, an equation of Meites (4), giving the predicted value in terms of the accumulated charge at three different times during the electrolysis, was modified and programmed for computation. Both the digital and analog systems were designed for rapid, continual readout of the predicted analytical result in milligrams. The two systems were tested and compared with respect to accuracy, precision, convenience of use, and time savings.
THEORY Prediction Equations. In a controlled-potential coulometric determination, the quantity of electricity for a complete electrolysis, Qm,can be written as (11): Qm
= Qt
+
QR
where Q t is the charge accumulated after t seconds of electrolysis and Q R is the charge remaining to be accumulated. The systems developed for predicting Q mduring the course of an electrolysis were designed to determine Qn and add this quantity of charge to the corresponding value Q t . The value of Q c is found by conventional, analog integration of the current. The analog and digital prediction systems differ in the manner in which QR is evaluated. The prediction of Qmis based on the assumption that, for the calculation of QR, the electrolysis current obeys the Lingane equation (12): i = ioe-kt
(2)
where i is the current at time t , i o is the initial current, and k is the electrolytic rate constant. On integrating this equation, one readily obtains
(3) and Q m = Qt
-
i2 dijdt ~
(4)
Because the current decreases with time, the derivative is negative and the second term on the right-hand side of Equa(11) R. I. Gelb and L. Meites, J. Phys. Chem., 68,630 (1964). (12) Reference 1 , pp 224-229.
tion 4 adds to the first term. This equation is the basis of the computation in the analog predictor system. Because of the violent stirring in the coulometric cells, the electrolysis current signals are generally quite noisy and must be heavily filtered before computation. Thus there is a potential advantage in calculating Qmin terms of Q-values only, and for this purpose an equation derived and tested off-line by Meites ( 4 ) is most suitable. The full equation is
I
I
I
I
I
I
A
where Ql, Qz,and Q3are the quantities of charge accumulated after tl, t2, and t 3 seconds, respectively. This equation is also based on the assumption that Equation 2 is valid, and it is derived by setting ( t 2 - t,) = ( t 3 - tz). Meites improved the calculations involved in using Equation 5 by subtracting Ql from each Q-value and calculating a value of QR to be added to Ql. Even more satisfactory for the online prediction is the calculation of a value of QR to be added to the most recent value of charge, Q3, For this mode of evaluation, Equations 1 and 5 become:
,
I
-1
B
zo.1
8
I
i/io
e
This equation is the basis of the digital computer calculation; values of Q are stored in memory as the electrolysis proceeds and recalled as required for the evaluation of Qm. Here the term Q3 corresponds to the Qt term in the analog system equation, and the second term of Equation 6 is Q R . The larger the time interval, ( 1 2 - tl) = ( t 3 - tz) = A t , the more accurate is the calculation using Equation 6, because the differences in Q-values are larger. If the interval At were fixed, these differences would diminish as the electrolysis progresses. For these reasons the digital computer algorithm was designed to periodically increase At as the on-line calculation is repeated. Effect of Variations in k on the Accuracy of Prediction. The success of a coulometric prediction method will depend only upon how closely the electrolysis obeys Equation 2 for the period of time pertaining to the calculation of QR. For an electrolysis that does not conform initially to Equation 2, it is only necessary to wait until it does to obtain an accurate prediction. Writing QR in terms of i and k , Qm
= Qg
+ ki
-
(7)
shows that a change of k , or an error in its estimation, is reflected as an error in Qm according to the proportion of electrolysis that has taken place. For example, at t0.9, when 90 % of the electrolysis has occurred, an error in k will result in one tenth as great an error in Qm. The diminishing accuracy requirements on the measurement of i, the evaluation of k, and their conformance to Equation 2, aid in the convergence of the predicted value. If the rate constant k changes discontinuously to a new value during the electrolysis, the prediction system will accurately adjust to this change within a period of time dependent upon the system's response speed. On the other hand, a continuously varying k-value, arising for example because of uncompensated cell resistance or kinetic complications in the electrochemical reaction, will cause erroneous predictions during the time that k is varying. To illustrate this effect, the theoretical errors in prediction for two cases of rather large variations in k have been calculated and plotted in Figure 1. Also shown are normalized log i us. t curves that would be obtained under
Y
7
Pe9,, 0
100
200
300
400 t
500
600
I 700
' 800
0.01
f 0.001
- rec
Figure 1. Error in prediction for electrolyses with linearly varying rate constant k A . k = 0.005 + 0.008 sec-l in 800 sec B. k = 0.016 + 0.010 sec-' in 700 sec
these conditions. Because these electrolyses are over 99 % complete before the prediction reaches the error level of conventional coulometry, these cases represent the extremes at which predictive methods would be useful. Calculation of Electrolytic Rate Constant k. An experimental determination of k as a function of time is useful in assessing whether a given electrolysis is amenable to prediction. As has been pointed out by Gelb and Meites (11, 13), a nonconstant value of k is revealed much more clearly by a direct evaluation of k, rather than by log i us. t plots. Provision for on-line determination of k has therefore been incorporated into both the analog and digital systems. To compute k with the analog system, the circuitry is rearranged to evaluate
For the digital computer system, because the instantaneous current is not measured directly, k must be determined in a different manner. Restricting the measurement to values of charge only, the following equation of Gelb and Meites (II), (9)
(13) L.Meites, Pure Appl. Chem., 18,35 (1969). ANALYTICAL CHEMISTRY, VOL. 42, NO. 7, JUNE 1970
765
lOOL
-
Figure 2. Error in prediction due to constant background current Dashed lines indicate magnitude of background current correction in conventional coulometry (id,t = t 0.999)
I
I
I
I
I
I
I
I
I
ae
I e L
Y
kt
could be used. However, to avoid computing logarithms and measuring the elapsed time accurately, a different approach is used, which is based on the fact that from Equations 1 and 7
k = -i
Substitution of this expression for itotsl into Equation 4 and adding the term ibt for the accumulated charge due to the background current yields Qm
QR
The average current at time tz is computed from the Q values at times tl and t 3 and is then divided by the calculated charge remaining at time t2. The equation used is
= Qt+
i 2ib -+id+-
+ ib2ek1 __
k k kio The sum of the first two terms on the right-hand side of this equation is the predicted value that would be obtained in the absence of a background current. Writing this sum as iolk and expressing the contribution of the other three terms as a fraction of i o / k ,one obtains the following equation: 10
This computation is carried out at time t a : thus the readout values of k lag behind the indicated values of Q m . This method of estimating i from values of Q is accurate to the extent that the variation of Q us. t is essentially linear in the interval At. However, as noted above, At increases during a run, and this causes an increasing error in the digital computation of k . The program is designed and used in such a way that the theoretical error in estimating k at to.$ is +1.3x. This error is easily tolerable in determining the constancy of k for prediction purposes. Effect of Background Currents. Electrolytic background currents influence the predicted final values by contributing to both the accumulated charge, Q t , and the estimate of QR. Three types of background current are present in all controlled-potential coulometric determinations (14) : the charging current; a faradaic current that decays according to the Lingane equation ; and a continuous, constant faradaic current. In predictive coulometry, the only effect of the charging current, which decays rapidly to zero, would be a simple additive contribution to Q t , as in conventional coulometry. This also would be the only effect of an extraneous faradaic current, if it becomes negligible by the time QR is computed. A limiting case of particular interest, where the current does not decay to zero, is the continuous faradaic background current (14). Assuming that the total current in an electrolysis is the sum of a regular, exponentially decreasing current i and a constant background current ib, itotal
= i
+ ib = ioe-" +
ib
(12)
(14) L. Meites and S. A. Moros, ANAL.CHEM., 31,23 (1959).
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ANALYTICAL CHEMISTRY, VOL. 42, NO. 7, JUNE 1970
This error ratio, as a percentage, is plotted as a function of the dimensionless quantity kt for three different ratios of i b / i o in Figure 2. The curves show that for ib/io ratios of about 0.01 or less, and for t = or less, there is less error in the predicted value than would be present in the actual charge obtained in conventional, complete electrolysis. It is interesting to note that the largest portion of the error for times up to about t o s scomes from the fourth term in Equation 13, not from the accumulated charge, i b t . The minimum error, at zero time, is equal to 2 i b l i o . The fourth and fifth terms represent the error in the computation of QR due to the background current. The effect of a continuous background current on the digital computer prediction can be assessed by adding an ibt term to each of the Q terms in Equation 6. This equation then becomes
Qm =
Q3
+ + ibta
KQz
~(Qz
+ i d z ) - + id3)]* + + +
+ ibh) -
(Q3
(Ql
id1
Q3
(1 5 ) ibfd
When this equation is simplified, with t 3 - t2 = At, it can be written Qm
=
Q3
+ +id3 + QR
The sum of the first two terms on the right-hand side of Equation 16 is again the value that would be predicted in the absence of a background current. The effects of the fourth and fifth error terms in Equation 16 are identical to the corresponding terms of the continuous data Equation 13.
'_______----------I
I I I
Controlledpotential coulometer system
5x2 =--
x O
I
x3
x4
I
Figure 3. Analog predictor system block diagram
Voltage amplifier
a Digital voltmeter
An important advantage of early prediction vs. conventional coulometry is that a higher constant background current can be tolerated before its contribution becomes significant. I n conventional coulometry, the measured values of charge are usually corrected for the constant background current either by calculating i b t , by extrapolating the final current to zero time (Z4, or by carrying out a blank determination. For predictive coulometry it would be impossible in principle to determine the background correction for the continuous background current by running a blank, because dijdt or 2Q2 - Ql - Q3 would become zero and the quantity QRwould become infinite. The best procedure for background current correction in predictive coulometry is to determine first whether the error in prediction due to ib is negligible. This can be ascertained from the approximate relative magnitudes of io and ib by the use of Equation 14 or Figure 2. If it is negligible, a blank correction can be determined by conventional coulometry for the period of electrolysis for which the predicted value is taken. This experimental blank value then would include, as well as ibtr the appropriate corrections for the charging current and any other extraneous faradaic current. If the ratio ib/io is large enough to cause a prediction error, a correction must be calculated from the error equations and added to the blank that is determined experimentally by conventional coulometry. This is relatively easy because, for t < to.Q,the fifth terms of Equations 13 and 16 are negligible compared with the fourth terms. Thus, the corrections for prediction error in the analog computation require a knowledge of k and can be calculated as 2 ib/k. For the digital computation, the correction term to be calculated is 2 ib AtQEj(Qs- Q2); however this requires values of quantities not readily available during the prediction. Thus, it is much easier to calculate 2 ib/kas the correction for prediction error in the digital computation as well, especially since the values of k are automatically computed on-line. The value of k used for the background correction must be taken from the digital or analog system readout before the accuracy of the k-value itself is significantly degraded by the presence of the background current. The error in the digital computation of k at t = t o . gwhen ib/io = 0.01 is approximately 2%, which is readily tolerable in background calculations.
techniques have been described previously (15,26). Details of the construction and operation of the mercury-pool electrolysis cell are also available elsewhere (17). Two types of platinum working-electrode cells were used during the course of this work. The type routinely used in this Laboratory (16) is designated cell A. The second cell, which produced a low-frequency noise current level approximately one fourth to one half that of cell A, is designated cell B. Cell B was a beaker-type cell (without a stopcock at the bottom for drainage); solutions in this cell were stirred by a magnetic stirring bar. The electrodes and their relative placement were the same as for cell A. Experiments were carried out at ambient temperatures ranging from 20 to 24 "C,with the variation during a measurement run not exceeding 1 "C. Analog Predictor System. FUNCTIONMODULE. A detailed block diagram of the instrumentation for prediction with analog components is shown in Figure 3. The heart of the system is a Bell and Howell, CEC Control Products Division, Model 19-302 function module, which was designed to evaluate an equation quite similar in form to that of Equation 4 :
EXPERIMENTAL
(15) J. E.Harrar and E. Behrin, ANAL.CHEM., 39,1230 (1967). (16) J. E.Harrar and L. P. Rigdon, ibid.,41,758 (1969). (17) J. E. Harrar, U. S. Ar. Energy Commission Rept. UCRL50335 (1967).
Coulometric Apparatus and Instrumentation. The controlled-potential coulometer system and the general analytical
(17)
The system is arranged so that a voltage proportional to i is applied to the XIand X 2 inputs, obtaining i 2 ; similarly, X 3 and Xd become dijdt and - Q l , respectively. The readout value of X O then corresponds to Q m . The a's are scaling coefficients established by resistors connected to the module. Constraints on the use of the module are that X a must be positive, alX1/X3 must not exceed 1 1 . 2 , and X O must not exceed ~ t 1 2V. Either the dijdt ( X , ) or the Q t ( X 4 )signal is inverted as required. The function module coefficients and the differentiator time constants are selected on the basis of the expected electrolysis current level and electrolytic rate constant. To read out the k-value on-line, the system is switched to the configuration shown in Figure 4, to compute Equation 8. The function module is a small solid-state unit of the pulseheight, pulse-width modulation type characterized by high accuracy at moderate bandwidth (100 Hz). Static accuracy tests with various combinations of input voltages, polarities, and magnitudes in the 50-mV to 10-V range revealed a maxi-
ANALYTICAL CHEMISTRY, VOL. 42, NO. 7, J U N E 1970
767
2x.
-----__
1
Controlledcoulorne ter
Figure 4. Analog system arranged to compute electrolytic rate constant k (k) D ifferentiator Digital voltmeter
Printer
Table I.
Time for 99.9% electrolysis, rnin >25 15-25 6-15
k , sec-l
‘
Analog Predictor Differentiator Parameters Ri,
kQ 82.5
Ci, PF
400
10
200
40
10
100
20
0.022 0.022 0.022
