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W A L Y T I C A L CHEMISTRY, VOL. 51, NO. 3, MARCH 1979
Chronopotentiometry with Programmed Current Li Hang Chow' and Galen W. Ewing" Department of Chemistry, Seton Hall University, South Orange, New Jersey 07079
An instrument has been constructed for chronopotentiometry with current that varies as any selected power of time, tq, where gcan lie anywhere between 0.2 and 5.0. The ambiguity due to charging current is minimized by an initial brief operation at constant current. The determination of reducible species alone or in the presence of other species can be carried out to concentrations as low as 2.5 X M. Relative standard deviations are of the order of 1% at 5 X M.
Table I. Relation between Concentration and Transition %me for Various Exponents of Timea T
0 112
1 312 2 51 2 a
(s),
100 10 4.7 3.2 2.5 2.2
M
c (M), T =
IOOS
10-3
10"
loo
Calculated from Equation 2, assuming h =
mol.dm-3.S(9
Although constant-current electrolysis (as in chronopotentiometry) has been practiced for near15 100 years ( I , 2), its potential as an analytical tool was pointed out by Gierst and Juliard (3) only in 1953. and brought to the general attention of analytical chemists through the work of Delahay ( 4 , 5 ) ,Reilley (61, and their respective co-workers in 1955. It appeared to these authors that chronopotentiometry would develop into a highly sensitive analytical tool; Reilley (6) mentioned 5 X 10-jM as a probable lower attainable concentration limit. Since then, however, most electroanalytical chemists have taken a much dimmer view, seeing no advantage and many disadvantages to the method; see for example, Lingane ( 7 ) . T h e objections to chronopotentiometry have stemmed largely from the inherent difficulty in distinguishing between the component of the current responsible for chemical effects (the faradaic current) and that utilized in charging the double-layer capacitance. This dilemma has been attacked by several means. The first of these consists of some type of graphical correction of the observed voltage-time curve to allow for the effect of the charging current. The several attempts in this direction have been summarized previously (7-9); the most successful is the method of Laity and McIntyre (10). T h e second method requires the use of two duplicate electrolysis cells, one containing the supporting electrolyte alone, to act as a reference. T h e sample cell is connected conventionally to an operational amplifier acting as a galvanostat, while the reference cell is controlled by a potentiostat which forces the reference electrodes in the two cells to the same potential, while both working electrodes are maintained a t virtual ground. The current passing through the reference cell is converted to a corresponding voltage and summed with the constant current as normally fed to the sample cell. The result is that the faradaic current is held constant to a good approximation. This method was introduced by Shults et al. ( 1 1 ) and modernized by Bos and Van Dalen (8). This procedure has two drawbacks: the possibility of instability and resulting oscillation due to the inherent positive feedback, and the questionable validity, particularly in the presence of adsorption, of the implicit assumption that the capacitance of the double layer is independent of the presence of the electroactive material ( 7 ) . Another approach to the problem depends on the fact that the rate of double-layer charging is a function of the derivative d E / d t . The potential of the working electrode increases
lo6
+
rapidly a t both the beginning and the end of the transition period; hence, it is primarily in these regions that the charging current is important. A plot of the derivative against time gives, for a single electroactive species, a curve with two well-defined maxima. Takemori et al. (12) employed an ac modulation technique, analogous to ac polarography, which was shown mathematically t o produce a voltage proportional to d E / d t . These authors estimated the transition time by measuring the time between maxima. Iwamoto (13) and Peters and Burden ( 2 4 ) achieved a similar result by electronic differentiation of the dc potential. The coordinates of the intermediate minimum are shown (14) to be functionally related to the transition time, and a measurement of this quantity is essentially unaffected by charging current. Sturrock and co-authors (15-18) have modernized this treatment by providing positive feedback to add to the applied current a signal proportional to the derivative, so that the current impressed upon the system constitutes faradaic current only. The chief disadvantage to analog differentiation lies in the inherent degradation of the signal-to-noise ratio, a defect not found in the ac modulation technique. The latter, however, has not to our knowledge been pursued further than the paper cited (12). Advantages of Programmed Current. Since the double-layer charging is independent of the faradaic current, its importance is much less when the faradaic current is relatively large. As pointed out above, the charging current depends on the time derivative of the potential, and this is greatest a t both the start and the close of the transition period. The rise at the end of this period can thus be made less significant by the application of current programmed to increase as some positive power of time. However, this also enhances the charging effect a t the start, when the current is rising from zero (19). Programmed current alters the functional relation of the transition time T as related to bulk concentration C. If the faradaic current IF is forced to follow the relation I,(t) = btq (1) in which b is a constant. it can be shown by means of a mathematical treatment analogous to that of Oldham ( 2 0 ) , presented in Appendix A of this paper (Equation 23, for p = 01, that the governing relation is + 9 + 1 21 = kC (2)
'Present address, h'arner-Lambert Co.. hlorris Plains. N.J. 07950. 0003-2700/79/0351-0322$01.00/0
c=
4
where h is another constant. The calculated transition times C
1979 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 51, NO. 3, MARCH 1979
323
workers (23,24),if the current varies with the square root of time, the transition time of one species becomes independent of the presence of other reducible species, a major convenience.
INSTRUMENTATION
TIME
Flgure 1. Time sequencing diagram, showing both potential of the reference electrode and cell current as functions of time. The comparator flips at t,; switching from constant to varying current occurs at t 2 . Transition time is given by t , - t , . The instrumental constants must be chosen to make the two shaded segments equal in area
for a series of q values are given in Table I, together with values of concentration that will give a constant transition time, T = 100 s. Hence, if a method were available to permit selection of q at will, a wide range of concentrations could be measured on a convenient time scale. It should be noted that with constant-current chronopotentiometry a t an electrode of constant area, a similar selection of suitable transition times can be made only by altering the current level, which permits far less accessible range. T o remove substances that are more easily reduced than the species of interest, a number of workers (8,19,21,22) have utilized a potentiostat to hold the potential of the working electrode a t a point slightly positive of the potential where the desired species begins to be reduced a t a significant rate. The potentiostat is disconnected at the instant of application of the (constant) electrolysis current. This procedure, however, results in the passage of a finite amount of current a t a level not related to the subsequent constant current. Switched Current. We propose that, instead of a preelectrolysis at constant potential, a constant current be passed through the cell for a period of time At short compared to T , followed by current increasing as a positive power of time. The cell must be switched electronically from constant to increasing current at the instant a t which the quantity of electricity passed a t the constant rate is exactly equal to the quantity that would have been passed by the increasing current starting from zero, assuming only faradaic current to be present. This will have the effect of almost completely eliminating interference due to charging current a t both the start of the transition period, where the potential will assume its plateau value very quickly, and at the end, where the current will have increased to the point where the charging current is effectively swamped out. The time relations involved will be clarified by reference to Figure 1. At zero time, the electrodes are switched to a source of constant current (typically a few tens of microamperes). This current continues to flow until (at time t l )the potential becomes negative enough to start reduction of the species of interest, a t which preselected potential a voltage comparator trips a relay, accomplishing two things: it starts the recording of voltage as a function of time, and it turns on a generator of the desired power of time, though this does not yet control the cell current. At time t 2 , the constant current is terminated and replaced by the time varying current. Electrolysis then continues until the voltage rises abruptly a t the end of the transition period. The transition time is the interval t3 - t l , where t , is established by the intersection of tangents (10). In the work described in this paper, the exponent of time, q , is selected as because, as has been shown by previous
We now describe an instrument designed to implement power-of-time chronopotentiometry with constant-current preelectrolysis. The chronopotentiometer (25)consists of three segments: (1) conventional circuitry for driving the electrochemical cell a t constant current, (2) circuitry for driving the cell a t a selectable power of time, and (3) timing circuits for semiautomatic changeover from constant current to the selected power. The power-of-time circuit is based on the multifunction module, type 4335 (Analog Devices). This is a logarithmic multiplier-divider capable of producing an output potential
E,
Y(Z/X)m
a
(3)
where X,Y , and 2 are analog inputs. T h e exponent m is continuously variable from 0.2 to 5.0 via a resistive network (26). For our purposes, the 4335 is fed a linear ramp (generated by analog integration a t amplifier A-1, Figure 2 ) through the 2-input (i.e., 2 c: t ) ,both X and Yare given fixed potentials, and the exponent is set a t 'Iz, so that
E , = Kt112
(4)
where K is an adjustable constant with the units of V.S-'/~. The voltage output of the 4335 is inverted by amplifier A-2 if a reductive process is to be studied, and fed to the control amplifier A-3. The sequencing circuitry operates in the following manner. Prior to t = t o ,the power supply (not shown) is ON, switch S-3 is in the position shown, both relays are unenergized, FET-switch Q-2 is ON (but no current is flowing through it), Q-3 is OFF, and the output of comparator A-5 is high ( f 5 V). At t = to,when S-3 is reversed manually, nominally constant current starts to flow through Q-1and Q-2 to the cell, and the recorder starts. The potential of the reference electrode increases rapidly until (at t = tl) it fires the comparator A-5, which has several simultaneous effects: it fires the monostable (74121) thereby energizing relay L-2 so that constant current flows to the cell without going through the poorly-defined channel resistances of the FET-switches, and it closes Q-3 so that the power-of-time voltage starts to be generated, even though it cannot yet result in a cell current. This state of things continues until the pulse from the monostable has expired (several seconds) a t which time ( t = t z ) the relay relaxes and the current switches from constant to powerof-time. This is allowed to continue until the recorder trace rises abruptly, when S-3 is opened by hand to stop the experiment (except that if more than one reducible species is present, the current may be allowed to continue as needed). The necessary constant current for a selected switching time ( t 2- t J is calculated as follows: Equation 1 is written twice, with the exponent q = 0 for the constant case and ' I 2for the ramp:
The surface concentrations for the two cases are shown by Murray and Reilley (27) to be, respectively,
C(0,t)=
c
C ( 0 , t )=
2 bot112 -
c
nFAD'I2dI2 -
2nFAD'f
(7)
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ANALYTICAL CHEMISTRY, VOL. 51, NO. 3, MARCH 1979
Figure 2. Schematic of power-of-time amperostat. FET switches Q-1, 2, 3 are segments of AD-751 1 (Analog Devices) analog switch. Operational amplifiers A-1 and A-3 are FET-input types; all others are 741. The monostable is 74121; N-1, 2, 3 are 7400. Power supplies are conventional. The points marked x , x connect to a remote start switch on the x-y recorder
Table 11; Transition Time as a Function of Switching Time* switching time, s 2.0 3.0 5.0 constant current, g A run 44.5 54.4 70.3 1 16.40 16.04 16.36 2 16.20 15.72 15.88 3 16.36 15.a4 15.88 4 16.20 15.96 15.92 5 16.00 16.24 15.92 16.12 15.92 6 16.16 15.98 16.22 15.99 mean 0.17 0.17 0.13 std. dev., u 1.1 1.1 0.80 rel. std. dev., ?& a Solution: 1.00 x M P b 2 +in 0.5 M KNO,. Electrode: Mercury pool, 0.99-cm diameter. b , 40 I.1 A.
-1
/ 2
Since the switch-over is to take place a t the moment when C(0,t) is the same as calculated either way, Equations 7 and 8 can be equated and solved for bo, and hence the constant current becomes
So, for a known b1I2 and a selected switching time, the required constant current can be determined and dialed into the apparatus a t R-1. F u n c t i o n Generation. For settings of the exponent m between 0.4 and 2.5, the output voltage from the 4335 function generator was found to be in error by an average of only 0.32% (relative) with a spread from 0.00 to 0.72, which compared rather well with the manufacturer's specification of M . 3 7'6.
Table 111. Reproducibility of Transition Times* concentration, M 5.00 x 10.00 x run 10-4 1 5.36 s 10.76 s 2 5.36 10.76 3 5.36 10.92 4 5.40 10.80 5 5.40 10.80 6 5.40 10.68 7 5.48 10.68 8 5.44 10.76 9 5.52 10.76 10 5.52 10.76 10.7 6 11 10.68 12 13 io. a 4 14 10.88 15 10.84 10.80 16 17 10.80
ia
10 72
5.42 10.78 mean std. dev., u 0.06 0.07 rel. std. dev., % 1.11 0.65 a Solution: TIC1 in 0 . 1 M KC1. Electrode: Mprcury pool, 0.99-cm diameter. b , , > : 40 p A s"" ; b o : 44.5 u A . Switching time: 2.0 s. The measured error includes any nonlinearity in the integrator circuit. T r a n s i t i o n - T i m e Constancy. Ideally, the measured transition time should be independent of the selected switching time over a fair range. T o test this relation, experiments were made with millimolar lead solutions, as summarized in Table 11. The %second time was selected for routine use. In another test (Table III), the transition times
ANALYTICAL CHEMISTRY, VOL. 51, NO. 3, MARCH 1979
>
- I300
5
10
15
20
25
,
30
325
, I
35
T I M E , SECONDS
Figure 3. Square-rootaf-time current chronopotentiograms of cadmium and zinc in 0.5 M KNO,. = 44.5 PA; p,,2 = 40 @A;switching time
Po
I
= 2 s
5
were measured on two concentrations of thallous chloride. The linearity is seen to be valid within the limits of experimental error. Transition Times in a Multicomponent System. A typical series of chronopotentiograms, reproduced in Figure 3, shows the expected direct additivity of transition times, as well as the proportionality with concentration. The transition time in each curve is measured from the time ( t l ) a t which the potential is 150 mV below the plateau to the time ( t 3 ) defined by the intersection of the plateau with the subsequent rise, both extrapolated. I t should be noted that the rising portions of the curves are much closer to vertical than in previous methods (see, for example, Figure 8 of ref. 7). Sensitivity. Many experiments were performed with bl,2 (Equation 6) taken as 40.0 ~ A - S - 'which ! ~ , was satisfactory for concentrations down to about 5 X M. For more dilute solutions, the transition time is too short to measure accurately, and a smaller current must be chosen. With b l j z = 3.0 ~ A - S - ~concentrations /~, down to about 2.5 X M could be measured with adequate precision. Figure 4 shows a curve M Cd+' in KNO,, with b I l 2 = 3.0. Note that for 5.00 X at this low current, a time of about 10 s was required to charge the double layer to the point where the comparator flipped ( t l ) . The dashed curve shows that cadmium in the supporting electrolyte could not have been greater than about 1 X
T I M E , SECONDS
Figure 4. Square-root-of-time current chronopotentiogram of cadmium in 0.5 M KNO,. Conditions as in Figure 3
In Oldham's treatment, the current I is allowed to vary independently only linearly with time. This can be generalized by means of two modulating relations:
A(t) =
atP
(11)
I ( t ) = bt4
(12)
and
Following Oldham, we can now introduce new variables as follows:
M. In this paper, we have explored experimentally only the utility of current varying as the square root of time. The apparatus, however, can be used with equal ease to produce other powers of time, and some of these will be described in further publications.
APPENDIX A Oldham (20) has presented equations for chronopotentiometry (among other procedures) under conditions where the area of the electrode surface can be varied as a function of time, but he does not treat in detail the equally significant case providing for programming the current. He has shown that the concentration of an electroreducible species at the surface of an electrode under conditions of semi-infinite diffusion control can be predicted from the very general equation,
where Y replaces the independent variable t , L( y)replaces the current I ( t ) ,and t ( Y ) replaces the rationalized potential AE(t). G and H merely stand in for the awkward time-invariant multipliers. Using Oldham's procedure, we can show that Equation 10 can be replaced by the following relation in which the symbol L{. . .) indicates Laplace transformation with respect to the variable Y
1 + C(V 1
u)\(16)
-L{y[-2P/(2P+l)lL(
&. The Reversible Case. For this case, Equation 16 simplifies to
(symbols are identified in Appendix B).
Substituting from Equation 13 and 14 and performing the
326
ANALYTICAL CHEMISTRY, VOL. 51, NO. 3, MARCH 1979
indicated Laplace transformation gives:
Table IV. Selected Parameters 4 P 2 (21" - 1) 0
0
112
3.
constant current
112
0 0
1 3/2 112
1.
sq. rt. of time
0.587 3.
linear ramp (DME)
1 312
By combining terms and taking the inverse transforms, we obtain:
notes
312
Treating this equation by a similar sequence of steps to that used for the reversible case leads to
1.( p 2p " + 1)
4
2p
+ 2q + 3 4P+ 2
)*
y[(2q-2P+11/(4P+2)1 (19)
This, when solved for variables, gives
AE =
t(Y)
and converted back to the original
RT - In (nFCD'/2Qt-' - 1) nF
(20)
in which Q represents the numerical constants:
and 2
q -p
+ 1/2
(22)
It must be noted that although the electrolysis current does not appear explicitly in Equation 20, it is nevertheless involved through its equivalent btQ (Equation 12). The next step is to derive a general expression for the transition time, T , defined as the time required for the concentration of the electroactive substance at the electrode surface to decrease from its initial value, C, to zero. This is the time a t which the logarithmic term of Equation 20 becomes zero, so that ' T
= nFCD1I2Q = KC
(23)
Substitution of this value into Equation 20 gives:
Taking the logarithmic argument as zero and solving for t (which is replaced by T), gives an expression identical with that of Equation 23. Equation 23 can be used to predict the relative values of the transition times for successive reductions taking place in a multicomponent system. As pointed out by Delahay and Mamantov ( 4 ) and by Reilley, Everett, and Johns (6),at the conclusion of the transition time associated with the most easily reduced species, the potential jumps to a more negative value where a second species begins to be reduced, but the oxidized form of the first species continues to diffuse to the electrode and be reduced, so that the impressed current is split between the two components. The authors cited were concerned only with constant current, but their conclusions can be generalized as follows: From Equation 23, taking il as the transition time for component 1, TIZ
= FQ = constant for any value of t (28)
nlC1Dl1l2
This constant, FQ, can also be equated to the difference between the combined transition times of two components and that of the first alone:
Hence these two can be set equal. If we simplify by assuming the two species to be equal with respect to n, C, and D , then TiZ
RT
T2- t Z
(24)
=
(25) The Irreversible Case. If the system is irreversible, Equation 16 reduces to
+ 72)'
- Ti*
(30)
from which, 72
Note that for conventional chronopotentiometry, where p = q = 0, Equation 24 becomes the well-known Karaoglanoff relation ( 2 8 ) ,and Equation 23 is the Sand equation (21,
(Ti
= (2l"
- 1)T1
(31)
In Table IV are listed values of ( 2 ' ! * 1) for various z corresponding to potentially interesting physical situations. The value for q = p = 0 is identical to that given by the previous authors ( 4 , 6); that for q = 1 / 2 , p = 0 agrees with the predictions of Murray and Reilley (27) for current varying as the square root of time; the value for q = 1, p = 0 corresponds to Oldham's Equation 53 (24). The last entry is predicted for chronopotentiometry at the dropping mercury electrode under conditions that give constant current density. ~
U
A(t) b
APPENDIX B Electrode area a t unit time Electrode area at time t Electrolysis current a t unit time
ANALYTICAL CHEMISTRY, VOL. 51, NO. 3, MARCH 1979
Bulk concentration of active species Concentration of active species at surface of electrode at time t Diffusion coefficient of active species Voltage o u t p u t from 4335 function generator Potential of electrode at time t , minus a reference potential Faraday's constant Numerical constant defined in Equation 13 Numerical constant defined in Equation 14 Electrolysis current a t time t Faradaic current Reference r a t e constant corresponding t o t h e selected reference potential; a constant A constant Laplace operator with respect t o Y Exponent of time Number of electrons transferred per ion Power of time by which electrode area increases Power of time by which t h e current increases Numerical constant defined in Equation 21 Universal gas constant D u m m y variable of Laplace transformation Time; for subscripts, see Figure 2 Kelvin temperature Variables used t o describe action of 4335 Y - P + 112 Cathodic transfer coefficient Dimensionless analog of potential, defined Equation 15 Dimensionless analog of current, defined Equation 14 Transition time
n n
327
LITERATURE CITED (1) H. F. Weber, Wed. Ann., 7, 536 (1879). (2) H. J. S. Sand, Phil. Mag., I,45 (1901). (3) L. Gierst and A . Juiiard, J , Phys. Chem., 57, 701 (1953). (4) P. Delahay and G. Mamantov, Anal. Chem., 27, 478 (1955). (5) P. Delahay, "New Instrumental Methods in Electrochemistry", Interscience. New York, 1954,Chap. 8. (6) C. N.Reilley, G. W. Everett, and R. H. Johns, Anal. Chem., 27, 483 (1955). (7) P. J. Lingane, Crlf. Rev. Anal. Chem.. 1, 587 (1971). (8) P. 60s and E. Van Dalen, J . Electroanal. Chem., 45, 165 (1973). (9) M. L. Olmstead and R. S. Nicholson, J . Phys. Chem., 72, 1650 (1968). (10) R. W. Laity and J. D. E. McIntyre, J. Am. Chem. Soc., 87,3806 (1965). (11) W. D. Shults, F. E. Haga, T. R. Muelier, and H. C . Jones, Anal. Chem., 37, 1415 (1965). 112) Y . Takernori. T. Kambara, M. Senda, and 1. Tachi, J . Phys. Chem.. 61, 968 (1957). (13) R. T. Iwamoto, Anal. Chem., 31, 1062 (1959). (14) D. G. Peters and S. L. Burden, Anal. Chem., 38, 530 (1966). (15) P E Sturrock. J. L. Huahes. B. Vandreuil. G. E. O'Brien. and R. H.Gibson, J . €lecfrochei. Socr, 122, 1195 (1975). (16) P. E. Sturrock, 6 .Vandreuil, and R . H. Gibson, J . Electrochem. Soc., 122, 1311 (1975). (17) P. E. Sturrock and R. H. Gibson, J. Electrochem. Soc.. 123, 629 (1976). (18) R. H. Gibson and P. E. Sturrock, J . Electrochem. Soc.. 123, 1170 (1976). (19) R. W. Murray, Anal. Chem., 35, 1784 (1963). (20) K. B. Oldham, Anal. Chem.. 41,936 (1969). (21) H. Hurwitz and L. Gierst, J . Elecfroanal. Chem., 2, 128 (1961). (22) F. JoviE and I. KontusiE, J . Elecfroanal. Chem., 50,269 (1974). (23) M. Senda. Rev. Polarogr., 4,89 (1965). (24) T. Kambara and I . Tachi, J . Phys. Chem., 61, 1405 (1957). (25) L. H. Chow, Ph.D. Dissertation,Seton Hall University, South Orange, N.J., 1977. (26) G. W. Ewing, Analog Dialogue, 8(l),19 (1974). (27) R. W. Murray and C. N. Reilley, J . Elecfroanal. Chem., 3, 64 (1962). (28) Z . Karaoglanoff, Z. Elekbochem., 12, 5 (1906). \
,
RECEIVEDfor review October 6, 1978. Accepted December 4, 1978. One of us (L.H.C.) was supported by a grant from the State of New Jersey under the Independent Colleges and Universities Utilization Act of 1972. Presented in part a t the 29th Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy, Cleveland, Ohio, February 28, 1978.
Experimental Evaluation of Recursive Estimation Applied to Linear Sweep Anodic Stripping Voltammetry for Real Time Analysis Paul F. Seelig' and Henry N. Blount' Brown Chemical Laboratory, The University of Delaware, Newark, Delaware 1971 1
The recursive estimation technique known as the Kalman filter has been experimentally verified for linear sweep anodic stripping voltammetry (LSASV) at hanging mercury drop (HMDE) and thin film mercury (TFME) electrodes. Synthetic lead samples were employed for a critical comparison between this real time recursive estimation technique and traditional nonreal time digital filtering techniques. The recursive technique generally returned concentration estimates which were of greater precision than the other digital methods for LSASV at an HMDE and of comparable precision for LSASV at a TFME. The limits of detection for these techniques (0.4 ppb) were found to be governed by the exogenous background concentration of the analyte rather than the technique per se.
Present address: Department of Chemistry, Reiss Science Center, Georgetown University, Washington, D.C. 20057. 0003-2700/79/0351-0327$0 1 .OO/O
Recent advances in digital hardware which permit more comprehensive data analysis and experimental control are being incorporated into chemical instrumentation. As hardware costs drop and expertise with these new devices increases, more sophisticated and "intelligent" microprocessors are appearing as integral parts of new instruments (1-7). Such systems become increasingly capable of executing complex algorithms designed to extract analytically significant information from the output of the transducers. Paralleling these hardware developments have been substantial improvements in numerical methods of data analysis. The classification of data and identification of subsets by methods such as factor analysis (8-12), principal components (13),and pattern recognition (24-16) have been demonstrated. Smoothing of data (17-19) has been shown to be advantageous in many situations (20,21),although caution must be exercised to maintain the information content of the signal. Transforms C 1979 American Chemical Society
