ACKNOWLEDGMENT
The authors thank F. C. Anson and J. H. Christie for helpful discussions. LITERATURE CITED
(1) Anson, F. C., ANAL. CHEM.36, 932
(1964).
(2) Anson, F. C., Christie, J., Osteryoung, R., J . Electroanal. Chem., in press. (3) Barnartt, S., J . Electrochem. sot. 108,
102 (1961). (4) Booman, G., Holbrook, W., ANAL. CHEM.37, 795 (1965). (5) Christie, J., Lauer, G., Osteryoung,
R., J . Electroanal. Chem. 7, 60 (1964). (6) Laitinen, H., Chambers, L., ANAL. cHEM. 36, 7881 (1964). (7) Laitinen, H. A., Randles, J., Trans. Faraday SOC.51, 54 (1955). (8) Lingane, P., Christie, J., J . Electroanal. Chem. 10, 284 (1965). (9) Lauer, G., Osteryoung, R., ANAL. CHEM.38, 1137 (1966). (10) Murray, R. W., Gross, P. J., I b i d , 38, 392 (1966). (11) Nemec, L., J . Electroanal. Chem. 8 , 166 (1964). (12) Oldham, K. B., Ibid., 11, 171 (1966). (13) Oldham, K. B., Osteryoung, R. A., Ibid., 11, 397 (1966).
(14) Osteryoung, R., unpublished results. (15) Osteryoung, R., Lauer, G., Electroanalytical Symposium, Winter Meeting, ACS, Phoenix, January 1966. (16) Pouli, D., Huff, J., Pearson, J. C., ANAL.CHEM.38, 382 (1966). (17) Smith, ,I)., Electroanalytical Symposium, Winter Meeting, ACS, Phoenix, January 1966. RECEIVEDfor review April 19, 1966. Accepted June 10, 1966. Electroanalytical Instrumentation Symposium, Winter Meeting, ACS, Phoenix, Ariz , January 1966.
Kinetic Current Measu reme nts with Control I ed PotentiaI Application to the Uranium(V) Disproportionation Reaction DALLAS T. PENCE’ and GLENN L. BOOMAN’ Phillips Petroleum Co., Atomic Energy Division, Idaho Falls, Idaho
b The correct interpretation of kinetic current measurements that are obtained by the use of controlled potential polarography at short times requires a thorough understanding of the electrical characteristics of the cell and of the potentiostat. This paper demonstrates the practical application of transfer function measurements to test the design and to determine the response time of a controlled potential polarograph. Errors involved in the measurement of kinetic currents at short times and methods proposed for their correction are discussed. The uranium(V) disproportionation reaction was studied over the time region of 1 msec. to 10 sec. using the controlled-potential method. The determined value of the rate constant is 192 liter* mole-* sec.-l for 1 M perchloric acid medium.
T
LARGE AMPLITUDE, potentialstep method of controlled potential polarography has developed slowly compared to other electrochemical techniques. The controlled-potential method has been compared with other methods by Gerischer and Staubach (9) and more recently by Reinmuth (21). The use of any of the large amplitude methods to obtain kinetic current data a t short times is limited by a lack of knowledge of the effect of the doublelayer capacitance. To a degree, the controlled-potential method depends less on a precise understanding of the double-layer capacitance at short times than do other methods. A second advantage is that data interpretation is simplified because in cases involving parallel reactions, the directly measured current is the linear sunimation of the individual reaction currents. A third advantage is that nearly all the informaHE
1 1 12
ANALYTICAL CHEMISTRY
tion about a reaction can be obtained for a set of selected conditions with a single voltage pulse. Finally, absorption of interfering ions a t the electrode surface is a lesser problem with the controlled-potential method because kinetic current effects are more easily discriminated from adsorption current effects. The greatest single factor that has limited the popularity of the large amplitude, potential-step method of controlled potential polarography for short time measurements has been the complexity of the required instrumentation. The stringent bandwidth, current, and gain requirements for the opcrational amplifiers also have been limiting factors. Lastly, data collection has been a tedious task. The continued improvement of commercial operational amplifiers and automatic data acquisition equipment, and recent advances in the theory of controlled potential polarography undoubtedly will lead to its increased use. The purpose of this paper is to illustrate the important aspects in the collection and interpretation of kinetic current data taken a t short times by use of controlled potential polarography. As a practical demonstration, data are presented and discussed for the uranium(J’) disproportionation reaction in 1X perchlorate media. Also the use of transfer function measurements is demonstrated as a means of determining system response time. EXPERIMENTAL
Instrumentation. The block diagram showing the functional breakdown of the various components used to time, control, measure, and print out the collected data has been presented in Figure 1 of Reference 1 .
Because details of the fast data acquisition system are given in the same reference, only a brief description is presented here. The start signal, initiated by depressing the start switch in the digital timing and control section, gates the pulse from the digital-toanalog waveforin generator to the POtential control circuit and the cell. This also initiates the timing routine which controls the read signals to the analog-to-digital converter. The timing routine is preprogrammed on a patchboard in the digital timing and control section. The analog-to-digital converter measures and converts the analog voltages from the current-measuring circuit. The digital information is stored in the core memory unit. At the completion of the measurements, the stored information may be recalled and printed out by operating a manual switch in the digital timing and control section. The potentiostat, shown in Figure 1, is a simplified version of the one described by Booman and Holbrook (3) and contains the same operational amplifiers, plug-in units, and modifications. Because the various components of the potentiostat and their relationships to each other are discussed below at some length, the modified circuit diagram of the potentiostat is presented here. The control amplifier is represented by -4-1 with R1, R 2 , and C1 comprising the stabilization network. The voltage follower for the reference electrode is A-2. The current-measuring circuit consists of the current follower, A-3, and the load resistor Re. The bias and pulse voltages are applied to the control amplifier through the summing resistors R3 and Ri, respectively. Resistor Rs is a series cell resistance and resistor R? is additional uncompensated cell resistance. 1 Present address, Idaho Nuclear Corp , CPP, Kational Reactor Testing Station, P 0 Box 1845, Idaho Falls, Idaho 83401
+ Figure 1.
Electrochemical cell and potentiostat
Electrochemical Cell and Reagents. The electrochemical cell was a conventional three-electrode type. The counter electrode, a helically-wound platinum wire, surrounded the reference electrode mobe and the measuring electrode. The reference electrode was an H-cell with three compartments, each separated by an ultrafine, fritted glass disk. The reference cell was a silver-silver chloride electrode, with O.1M hydrochloric acid electrolyte, enclosed in a porousVycor-tipped, 7-mm. glass tube. The reference cell was fitted in one arm of the H-cell. Contamination from the reference cell was prevented by the continuous passage of supporting electrolyte, 1M perchloric acid, through the center compartment and the compartment containing the reference cell. The supporting electrolyte overflowed into a large glass reservoir through an outlet located in the compartment containing the reference cell. Contamination of the reference cell by the supporting electrolyte was prevented by maintaining a positive head of pressure on the hydrochloric acid solution in the reference cell. The reference electrode probe, also a porous-Vycor-tipped glass tube filled with supporting electrolyte, was connected to the other arm of the H-cell with Tygon tubing. The measuring electrode, a two-drop, hanging drop mercury electrode (HDXIE) was similar to that described by Shain (23). A commercial 6- to 12-second dropping mercury electrode (DME) delivered the drops to a glass scoop for transfer to the H D X E . The potential of the DME was maintained at a constant value, equal to that of the reference electrode, during the collection of the mercury drops. This was accomplished by connecting the output of the voltaee follower to the DME mercury reservoir and grounding the measuring and counter electrodes. An accurately timed, automatic drop knocker, installed on the DME, improved the reproducibility of the drop size. The cell container was a 100-ml. weighing bottle surrounded by a glass seal for circulating water thermostated to 25' + 0.1' C. The lid, made of Teflon, provided a tight seal on the weighing bottle. Holes were drilled in the lid for the various electrodes,
degassing dispersion tube, and scoop. The cell, DME mercury reservoir, DME drop knocker assembly, and reference electrode were attached to a framework which was shock-mounted with several layers of firm sponge rubber sandwiched between sheet lead. The uranium perchlorate stock solution was prepared by dissolving U308, XBS standard sample 950a, in 71% perchloric acid. The solution was diluted to be 0.1M in uranium perchlorate and 1 J i in perchloric acid. The test solutions were prepared from the diluted stock solution as needed. Measurements. The uranium perchlorate test solutions were purged for 20 to 30 minutes with argon before each set of measurements. The amplitude of the potential step was 0.6 volt, and the bias potential was set a t 0.2 volt anodic to the half-wave potential of the uranium(V1) reduction wave. To obtain the desired signal-to-noise ratio and to retain the fastest possible system response time, the kinetic current measurements, taken over four decades of time, were split into two parts. The load resistor] Rsin Figure 1, was a six-decade] o.O5y0 unit. The data for the first 1.5 decades of time (1 to about 15 msec.) were obtained with a low resistance value which was increased for the final 2.5 decades. The maximum values of the load resistor for the first 1.5 decades of time and the last 2.5 decades were 4000 and 8000 ohms, respectively. A 0.05-pfd. capacitor was placed in parallel with the load resistor during the final 2.5-decade measurements. The effect of this shunt capacitance was determined to be negligible. The current-time transient responses of the uranium perchlorate-perchloric acid system were calculated from the recorded voltage-time responses of the current follower amplifier, A-3 in Figure 1. Twelve sets of 41 voltage measurements were obtained for each of three concentrations of uranium perchlorate, 0.002.11, 0.005V. and 0.01OJf. in liM perchloric acid. Two sets of' voltage measurements were required for each test solution to give a complete set of measurements for the desired 4 decades of time because of the necessary load resistance change after about 1.5 decades. Six complete sets of measurements, all obtained the same day, are '
defined as a run. A fresh mercury drop electrode was used for each transient response measurement. The voltage measurements were taken on a logarithmic time scale a t a rate of 10 measurements per decade of time. One-molar perchloric acid blanks were measured using the same load resistance values as were used on the test solutions. After the voltage measurements for a test solution were averaged for each of the 41 times and the averaged background voltages were subtracted, the current values for each time of the run were determined by dividing the voltages by the appropriate load resistance values. Voltage values for a typical run are shown in Table I along with the current values of all the runs. The three runs of each of the uranium perchlorate solutions were taken at approximately one week intervals. The average radii and surface areas of the hanging mercury drops were calculated from the weight of 50 DME drops, which were collected in the test solution after each run. The current values for each run were normalized to a standard drop surface area (0.0519 cm.z) for overall evaluation and for precision calculations. The per cent relative standard deviations were calculated to obtain a measure of the within-run reproducibility of the mercury electrode drop size and of the electronic measurement equipment. The precisions of the six sets of voltage measurements were determined for each time of the various runs. As expected, the precisions were lower for both short times and for long times as compared to the intermediate time region, particularly at long times where the signalto-noise ratios were small. At both ends of the time scale (about 1 to 10 msec. and 1 to 10 seconds), the precisions were as low as several per cent. Within the range of 1 msec. to 1 second, however, the precision values were all less than 1%, with an average value of slightly less than 0.5%. Within-run precisions for one of the runs is shown in Table I. RESULTS AND DISCUSSION
Determination of System Response Time. To evaluate the measurement of kinetic currents a t short times, the response time of the system must be determined. The term system includes the potentiostat, the electrochemical cell, and the current-measuring circuit. The best method of determining the response time is to measure the transfer function of the system. The use of transfer function measurements for the selection of components and for the general design of operational amplifier circuits for controlled-potential instrumentation has been discussed by Booman and Holbrook (2, 3). A table of feedback network values is available (3) for a variety of cell characteristics which VOL. 38, NO. 9, AUGUST 1966
1 113
h
6
B
0
3
3
+. +. o o o o o o o o o ~ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o ~
11 14
ANALYTICAL CHEMISTRY
io3
io3
I
1
I
IO‘
~
~
_-_
IO s i 0
16‘ lo-2
100 Hz
I kHz
IO kHr Frequency
100 kHz
I MHz
Figure 2. Measured cell, control circuit, and system trinsfer functions
provides for optimum stability and frequency response for a system of the type presented in the present paper. Because frequency response and stability are sensitive to many factors, a t least minor adjustments to the feedback network usually are necessary for optimum performance. Transfer function measurements provide a fundamental test of any electronic system, and from these measurements the stability of the control system can be analyzed, the design and proper operation of the various components can be verified, and the system response can be determined. Because of the interaction between the various components of an electroanalytical system, particularly between the control circuit and the cell, the determination of the system transfer function is the most practical method to obtain the stability characteristics and the frequency response of the system. The application of transfer function measurements to verify instrument performance and to determine system response is illustrated below. Because the transfer function of the overall system usually cannot be determined by one set of measurements, it is convenient to measure the cell and current-measuring circuit transfer function separately from the potential control circuit transfer function. Inasmuch as the two sections of the overall system chosen for transfer function measurement are not electrically independent, the individual transfer functions must be measured with the system in operation. The product of the two individual transfer functions approximately represents the overall system transfer function. For convenience, the term “cell transfer function” is defined to include the effects of the double-layer capacitance, uncompensated cell resistance, series cell resistance, reference electrode impedances, and the currentmeasuring circuit. Similarly, the “control circuit” is defined to include the
10‘~100 Hz
I kHz
IO kHz Frequency
100 kHz
I MHz
Figure 3. Calculated cell, control circuit, and system transfer functions
remaining portion of the circuit comprising the voltage follower, summing network, and control amplifier with its feedback network. It is recognized that the system transfer function as just described is not the true system transfer function, The exact definition of the transfer function of a system is the Laplace transform of the system output divided by the Laplace transform of the system input at zero initial conditions. This definition, applied to the above system, is the Laplace transform of the voltage across the double-layer capacitance divided by the Laplace transform of the voltage supplied to the summing resistor (R4 in Figure 1) of the control amplifier input network. Even though it is not possible to directly measure the voltage across the double-layer capacitance of an actual system, the procedure outlined below gives a value that closely approximates the true system transfer function. The double-layer capacitance was measured a t various potentials with a n a x . bridge for a 1M perchloric acid electrolyte. Based on the geometry of the cell, the uncompensated cell resistance (the resistance between the measuring electrode and the reference electrode) was calculated to be about 0.85 of the resistance between the measuring electrode and the counter electrode. This last resistance was measured and the uncompensated cell resistance was determined. The value of the double-layer capacitance, a t a potential corresponding to a point at the foot of the first cathodic uraniuni(V1) wave and a t a frequency of 20 kHz., was determined to be 1.2 pf. for a mercury drop of radius 0.052 cm.; and the uncompensated cell resistance was estimated to be 4 ohms. From a consideration of stability, frequency response, and current-measuring sensitivity @), a 1000-ohm series cell resistance, Rs, was added in the circuit (Figure 1). For the same reasons, resistance, R7, was added in series with
the uncompensated cell resistance at the measuring electrode to increase the total uncompensated cell resistance to 15 ohms. The stabilization network, first selected from Table I1 of Reference 3, then was modified to provide for minimum voltage rise time across the capacitor representing the double-layer capacitance in a dummy cell. The cell transfer function, with 1M perchloric acid as the electrolyte, and the control circuit transfer function, are shown in Figure 2 as a Bode diagram, a logarithmic plot of the transfer function us. frequency (2, 3). Also shown is the system transfer function which is the product of the cell and control circuit transfer functions. The cell transfer function was obtained by applying an a.c. signal to a resistor in the summing network, then measuring the input voltage (10 mv. or less) a t the output terminal of the control amplifier and the output voltage a t the input terminal of the voltage follower. The control circuit transfer function was obtained by applying the a.c. signal to a resistor a t the input terminal of the voltage follower and measuring the voltage gain between the input of the voltage follower and the output of the control amplifier for a series of frequencies. Care was taken to ensure that the input resistor, through which the a.c. signal was applied, was of sufficient value not to change the loop gain of the potentiostat. The calculated transfer functions of the cell, control circuit, and the system are shown in Figure 3. The following approximations were used for these calculations: ideal performance (defined as all summing points maintained a t zero potential) for all the amplifiers; electrical independence of the cell and control circuit from each other; representation of the electrochemical cell as a series capacitance and resistance (3); and representation of the control circuit transfer function by the combined transfer functions of the voltage follower and control amplifier. The VOL. 38, NO. 9, AUGUST 1966
11 15
c
.-C0
W
u
Frequency
Figure 5. Calculated and measured cell transfer function with nonideal current follower effects included
Frequency
Figure 4. Measured cell, control circuit, and system transfer functions showing effects of nonideal current follower behavior
details of the transfer function calculations are given elsewhere (2, 3, 24). A comparison of Figures 2 and 3 shows that the measured and calculated transfer functions closely agree up to about 10 kHz. and that the unity-gain crossover frequencies agree within about 1 kHz. Several important conclusions can be drawn from these results. First, the approximations that were made in the calculated transfer functions are valid to a t least 10 kHz. Second, the agreement between the measured and calculated transfer functions of the cell excellently justifies the use of the electrical equivalent model to represent the electrochemical cell even well above 10 kHz. Finally, the system functioned as designed with the best possible frequency response for the amplifiers that were used. The deviation of the measured transfer functions from the calculated values, starting about 10 kHz., indicates that one or more of the approximations are beginning to fail. From a knowledge of the amplifier characteristics, the approximation first likely to become invalid is the one that assumes ideal amplifiers. This is supported by Figure 4 which shows the result of placing a large load resistor in the current-measuring circuit. As stated earlier, the cell transfer function calculations and measurements include the current follower. By placing a 4000-ohm load resistor in the current-measuring circuit, the unity-gain crossover frequency is reduced almost by a factor of two (Figure 4). The system response time, which is directly related to the unitygain crossover frequency, is therefore also decreased. When the nonideal characteristics of the current follower amplifier are included in the calculations of the cell transfer function, the agreement between the measured and calculated values is quite satisfactory (Figure 5). If large load resistances 11 16
ANALYTICAL CHEMISTRY
are necessary to obtain the desired current sensitivities, the frequency response restrictions must be accepted or the current follower amplifier must be replaced by one with better gain characteristics. The major cause of the deviation of the control circuit transfer function above about 10 kHz. is believed to be inadequate gain characteristics of the control amplifier. To verify this by calculations would be a time-consuming task that was not deemed to be warranted. For all experimental short time measurements, the largest load resistance in the current-measuring circuit, 4000 ohms, resulted in a unity-gain crossover frequency of about 4.5 kHz. (Figure 4). This gave a control circuit time constant of about 35 ksec. The system response time, defined as the 90% voltage rise time across the double-layer capacitance, was calculated to be about 106 psec., using the control circuit time constant, the cell time constant (19 psec.), and Table I of Reference 5. Uranium(V) Disproportionation Rate Constant. The ratio of limiting current to diffusion current is plotted vs. the logarithm of time in Figure 6 for the disproportionation reaction of 0.002M uranium perchlorate in 144 perchloric acid. The solid curve represents the experimental data drawn from 10 data points per decade of time. The broken line is the theoretical curve obtained by a geqeralized numerical solution of the second-order rate equations that describe the reaction (4). The experimental curve was calculated with the aid of a computer, programmed to include a shielding factor. This factor corrects for the loss in current due to the reduction in the surface area of the drop caused by its physical attachment to the HDME, but does not take into account the depletion effect caused by the shielding from the HDME. Figures 7 and 8 show the same type of
plot for 0.005M and 0.010M uranium perchlorate in 1M perchloric acid, respectively. To determine the experimental formal rate constants from the theoretical working curves (4), data for about two decades of time were used. These data include the region of the curve between the ratios of the limiting to diffusion current of about 1.02 to 1.4. The deviations of the experimental curves from the theoretical ones, noted a t the longer times for the 0.005M and 0.010M uranium perchlorate solutions, are probably attributable to the depletion of the bulk concentration near the HDME caused by shielding. The observed errors from this effect agree with those obtained from approximate calculations which include the shielding effect and increase with uranium concentration. Other concentration dependent errors also may be involved but were not studied. The formal rate constants for the uraniumw) disproportionation reaction for various concentrations of uranium perchlorate in 1M perchloric acid are given in Table 11. A value of 0.71 x 10-5 cm.* set? for the diffusion coefficient was used for the calculations. This value was obtained by fitting the experimental curves to the unity base line of the theoretical working curves (4). Precisions for the kinetic current values are given in Table I. The precision of the voltage measurements is considerably better than that obtained by the conventional, oscilloscope photographic method.
Table II. Formal Rate Constants for Various Concentrations of Uranium Perchlorate in 1 M Perchloric Acid
Concn. ( M )
k(l* mole-* sec.
0.002 0.005 0.010
192 192 212
Formal rate constant normalized to unity hydrogen ion concentration.
I
I
> I
I
1.5
1.0
I
I -
1.1
I
-3--
I
IO
IO
Time (seconds)
Figure 6. Ratio of limiting current to diffusion current vs. time for 0.002M uranium perchlorate in 1M perchloric acid
Time (seconds)
Figure 7. Ratio of limiting current to diffusion current vs. time for 0.005M uranium perchlorate in 1 M perchloric acid
The accuracy of the formal rate constants for the 0.002iM and 0.005M uranium perchlorate solutions (Table 11) is estimated to be better than 10%. The accuracy of the theoretical working curves is estimated to be better than 5% and because the experimental curves were visually fit to the theoretical working curves t o determine the rate constants, the 10% value is a conservative estimate. This accuracy could be improved by a least-squares fit of the experimental curves to the theoretical working curves by a computer technique. The accuracy of the rate constant obtained for the 0.010M uranium perchlorate solution is estimated to be 20%. The increased error is believed to result from the more pronounced shielding effect of the HDME a t the higher uranium concentration. The formal rate constant for the uranium(V) disproportionation reaction in 3M perchloric acid was determined earlier with a slower instrument. Three concentrations of uranium perchlorate, O . O O l M , 0.005.44, and O.O1OMJ were used. The kinetic currents were measured with an oscilloscope and were recorded with a Polaroid Camera. The experimental data were analyzed in the same manner as described above. The average value, normalized to unity hydrogen ion concentration was 665 & 100 liter2 mole-2 SeC. -1
The formal rate constants for the uranium(V) disproportionation reaction reported in this paper cannot be compared directly with previously reported values because of differences in ionic strength and supporting electrolyte. A rough comparison was made by plotting the logarithm of the disproportionation rate constant, normalized to unity hydrogen ion concentration, us. the square root of ionic strength, as used by hfasters and Schwartz (15). Included in the comparison were the data of Masters and Schwartz, other older data ( 8 ) ,and more recent data (7, 16). The experimental data presented in this paper agree with the comparison data.
1.7
1 I
I .6
1.5 1.4
1.3
1.2
L=aza
1.1 in I."
rd3
lo-2
1O-l
Time (seconds)
I
IO
Figure 8. Ratio of limiting current to diffusion current vs. time for 0.01 OM uranium perchlorate in 1 M perchloric acid
Newton and Baker (16) studied the uranium(V) disproportionation reaction by spectrophotometric means. They detected a stable uranium(V1)-uranium (V) complex which tends to inhibit the uranium(V) disproportionation. Because some of this complex is undoubtedly present during the electrochemical reduction of uranium(V1) to uraniumv) and its subsequent disproportionation, Newton and Baker suggested that the disproportionation rates obtained from polarographic techniques are low. They extrapolated the rate constant to zero uranium(V1) concentration for several perchloric acid solutions of low concentration. The only polarographic data available which may be used to compare the formal rate constants extrapolated to zero uranium(V1) concentration are those of Fischer and Dracka (8) and those of the present paper. If the rate constant obtained by extrapolating the data of Newton and Baker (16) to 0.5M perchloric acid concentration is compared with the rate constant of 195 liter2 mole-2 sec.-l reported by Fischer
and Dracka for the same conditions, the latter rate constant value is lower, by an amount slightly exceeding the estimated experimental error. However if the rate constant obtained by extrapolating the data of Newton and Baker to 1M perchloric acid concentration is compared with the rate constant reported in the present paper, agreement is obtained within the limits of experimental error. The fact that the data of Fischer and Dracka did not agree with the extrapolation data but the data presented in this paper did, may be explained on the basis of the respective polarographic techniques used. The chronopotentiometric method used by Fischer and Dracka is probably more susceptible to interference caused by the formation of the uranium(V)-uranium(V1) complex than is the controlled-potential, stationary electrode method. However, the extrapolation of the data presented by Newton and Baker to 1M perchloric acid concentration is open to question. The disproportionation formal rate constant of 417 liter2 mole-2 sec.-l in 2M perchloric acid reported by VOL 38, NO. 9, AUGUST 1966
11 17
Fischer and Dracka (8) and the 3M perchloric acid data reported in the present paper definitely indicate that the extrapolation is not valid for concentrations of perchloric acid much higher than lM. Because Newton and Baker limited their investigation of the disproportionation rate constant as a function of uranium(V1) concentration, to perchloric acid solutions of low ionic strength, a direct comparison to data obtained by polarographic techniques is not available. Newton and Baker did attempt to duplicate the condition in the DME polarographic investigations of Imai (11) and Koryta and Koutecky (12) for comparison purposes. However, the range of 2 to 4 for U(VI)/U(V) may not be realistic in terms of the DRIE experiments. Further work is needed to evaluate the errors involved in the polarographic determination of the uranium(V) disproportionation reaction due to the uranium(V)-uranium (VI) complex. From the limited data available, the errors do not appear to be too great. The data collected with use of the controlled-potential method are probably less affected than other polarographic methods because the uranium (VI) is completely depleted near the electrode surface. Errors Involved in Interpretation of Kinetic Currents Measured at Short Times, T o categorize the errors involved in the interpretation of kinetic current measurements is difficult, especially those taken a t short times, because most are interrelated and are dependent on the instrument and cell characteristics. However, it is interesting to note that a nearly identical classification to that given below, though discussed in a different manner, recently was proposed by Oldham (20). The errors involved result from the finite rise time of the potentiostat, the voltage drop caused by the flow of Faradaic and double-layer charging currents, through the uncompensated cell resistance, and the difficulty in separating the Faradaic current component from the double-layer capacitance charging current component in total cell current measurements. The deviation of the experimental curves from the theoretical curves a t short times in Figures 6, 7 , and 8 result from these errors. The finite rise time of the potentiostat is directly related to the system response time which has been defined to include the control-circuit time constant and the cell time constant. The effect of the finite rise time of the potentiostat on the Faradaic current can be considered as a special case of linear scan, controlled potential polarography (3). From basic theoretical relationships (18), it is evident that the net effect of the linear scan on the Faradaic current is 1 1 18
ANALYTICAL CHEMISTRY
a displacement of the current toward longer times. These relationships also show that the Faradaic current can be accurately calculated for all times despite the displacement. One would expect that the time displacement effect can be minimized by use of fast scan rates. A calculation similar to that of Booman and Holbrook (S), on the data of Xicholson and Shain (18), shows that the time displacement or time delay, t ’ , can be approximated by t’ = 0.5T
where T is the system time response. This calculation assumes a potential step of 0.3 volt starting about 0.15 volt before the half-wave potential. It was also shown (3) that the time required for the time displacement error t o reduce to 5% is lot’ and is 50t’ for a 1% error. With a system having a response time of about 100 psec. (such as the one described in the present paper) the times required for the time displacement error to be reduced to 5y0 and 1% are about 500 bsec. and 2.5 msec. Thus, shorttime current measurements require a fast system-response time or the use of correct ions. The voltage drop caused by the flow of the Faradaic and double-layer capacitance charging currents through the uncompensated cell resistance results in a difft-rcnce in the potential across the double-layer capacitance and that maintained by the potentiostat. The magnitude of this error is directly proportional t o the uncompensated cell resistance, which, in turn, is a function of cell design and of the solution conductance. The placement of the reference electrode and the magnitude of the double-layer capacitance are considered to be included in the cell design. Nicholson (17) and De Vries and Van Dalen (6) independently derived and numerically solved the equations describing the electrode response in the presence of a voltage drop due to the flow of Faradaic current through the uncompensated resistance for the case of linear scan, stationary electrode polarography at controlled potential. De Vries and Van Dalen suggest that the voltage drop due to the double-layer charging current is probably negligible with the linear scan technique a t scan rates up to about 100 volts per second. However, when using the large potential-step method where the effective scan rates during the initiation of the potential step may be from several thousand to over ten thousand volts per second, the charging current effects cannot be ignored. Hence, a satisfactory treatment of this type for the potential-step method must include the double-layer charging current as well as the Faradaic current. Unfortunately, the problem is complicated by the fact that the double-layer capacitance is a function of the applied
potential. In most cases, this variation is probably significant. For example, in the 1M perchloric acid solution the double-layer capacitance changed about 25% over the range of the potential step. If the potentiostat rise time and the voltage drop caused by the presence of uncompensated resistance were negligible, the current component due to the charging of the double-layer capacitance could be calculated rather easily. However, a t short times the competition for the available current between the Faradaic current and double-layer charging current is complicated by the finite rise time of the potentiostat. The accurate determination of the current component due to the charging of the double-layer capacitance requires a knowledge of the voltage function across the double-layer capacitance and of the voltage dependence of the doublelayer capacitance. Most of the methods proposed (14, 19, 20) for the interpretation of large pulse, controlled potential polarographic data a t short times are variations of the graphical extrapolation method first proposed by Gerischer and Vielstich (10). The recent approach by Oldham (20) is the most comprehensive and is the first to clearly recognize the limitations of the extrapolation technique. The use of positive feedback to correct for the error in potential across the double-layer capacitance due to the uncompensated cell resistance has been investigated by a number of workers (5, 9, IS, 21). This method has considerable appeal, but it requires a system with an excellent response time to effectively correct short time data. In some systems, such as the one used in this investigation, it is necessary to add resistance in series with the uncompensated cell resistance to improve the frequency response ( 3 ) . The use of positive feedback in a system such as this is equivalent to lowering the total uncompensated resistance. The net effect is to decrease the overall system response time. The use of amplifiers of sufficient bandwidth, current, and gain capabilities would allow the reduction of the uncompensated cell resistance to the lowest possible value limited only by cell geometry. Complete compensation would require amplifiers having infinite gain-bandwidth characteristics. A method which appears to be the most useful for the correction of data taken a t short times is an extension to the approach used by Nicholson (17) and De Vries and Van Dalen (6). If the nonconstant double-layer capacitance can be expressed in a relatively simple general form, this development, including the voltage drop due to the double-layer charging current and the correction for the charging current itself, does not appear to be too difficult. This method could be used in conjunc-
tion with positive feedback and doublepulse instrumentation. Of course, the use of better amplifiers will improve the effectiveness of all the methods for interpretation of the data taken a t short times. with advanced instrumentation, full advantage Of the potentiostatic technique will be realized only by application of correction methods. LITERATURE CITED
( 1 ) Booman, G. L., ANAL. CHEM.38, 1141 (1966). ( 2 ) Booman, G. L., I-Iolbrook, W. B., Ibid., 35, 1793 (1963). ( 3 ) Ibid., 37, 795 (1965). ( 4 ) Booman, G. L., Pence, D. T., Ibid., 37, 1366 (1965). ( 5 ) Brown, E. R., McCord, T. G., Smith, D. E., DeFord, D. D., Ibid., 38, 1119 (1966).
(6) De Vries, W. T., Van Dalen, E., J. Ezectroanal* “9 lS3 (1965). (7) Feldberg, S. W., Auerbach, C., ANAL. 36, 505 (1964). (8) Fischer, O., Dracka, O., Collection C‘zech. Chem. C‘ommun. 24, 3046 (1959). (9) Gerischer, H., Staubach, K. E., 2. Electrochem. 61,789 (1957). (10) Gerischer, H., Vielstich, W., 2. Physik. Chem. (Frankfurt)3, 16 (1955); 4, 10 (1955). (11) Imai, H., Bull. Chem. SOC.Japan 30, 873 (1957). (12) Koryta, J., Koutecky, J., Collection Czech. Chem. Commun. 20, 423 (1955). (13) Lauer, G., Osteryoung, R. A,, ANAL. CHEM.38, 1106 (1966). (14) Lingane, P. J., Christie, J. H., J. Electroanal. Chem. 10, 284 (1965). (15) Masters, B. J., Schwartz, L. L., J . Am. Chem. SOC.83, 2620 (1961). (16) Newton, T. W., Baker, F. B., Znorg. Chem. 4, 1166 (1965). (17) Nicholson, R. S., ANAL. CHEM.37, 667 (1965).
cHEM.
(18) Nicholson, R. S., Shain, I., Z b d , , 36, 706 (1964). (19) Okinaka, Y., Toshima, S., Okaniwa, H., Talanta 11, 203 (1964). (20) Oldham, K. B., J. Electroanal. Chem. 1 1 , 171 (1966). (21) Pouli, D., Huff, J. R., Pearson, J. C., ANAL.CHEM.38,382 (1966). (22) Reinmuth, W. H., Zbid., 36, 211R (1964). (23) Shain, I., “Treatise on Analytical Chemistry,” I. M. Koltoff and P. J. Elving, Eds., Part I, Vol. 4, p. 2560, Interscience, New York, 1963. (24) Shain, I., Harrar, J. E., Booman, G. L., Zbid., 37, 1768 (1965).
RECEIVEDfor review April 18, 1966. Accepted June 9, 1966. In part, Division of Analytical Chemistry, Winter Meeting, ACS, Phoenix, A r k , January 1966. Work supported by the U. S. Atomic Energy Commission under Contract No. AT(10-1)-205 through the Idaho Operations Office.
Some Investigations on Instrumental Compensation of Nonfaradaic Effects in Voltammetric Techniques ERIC R. BROWN, THOMAS G. McCORD, DONALD E. SMITH, and DONALD D. DeFORD Department o f Chemisfry, Northwestern University, Evanston, 111. The feasibility of achieving accurate, direct readout of the faradaic component in voltammetric techniques under conditions where nonfaradaic effects are substantial has been reinvestigated. compensation of ohmic potential loss was effected with the aid of the addition of a positive feedback loop to a conventional operational amplifier potentiostat. Double-layer charging current compensation was carried out b y direct subtraction of the current obtained with a solution of supporting electrolyte from the current obtained with a solution of supporting electrolyte and electroactive component. Application was made to cyclic voltammetry, fundamental harmonic a.c. polarography, and higher harmonic a.c. polarography. Readout of the faradaic component was possible with an apparent high degree of accuracy for at least moderately demanding conditions. All measurements were performed with the dropping mercury electrode. Successful application of the dropping mercury electrode was facilitated by a timing circuit which performed a number of functions including controlling and synchronizing mercury drop growth and fall in the two cells; controlling a sample-and-hold readout operation in a.c. polarographic measurements; controlling application of a triangular wave impulse in cyclic voltammetric measurements.
T
WIDELY recognized problems associated with the contributions of ohmic potential loss (iR drop) and double-layer charging current in voltammetric techniques remain a source of much concern in experimental electrochemistry (1-11, 16-20, 25-28, 31-34, 39-42, 46, 51, 52, 56, 57, 62, 64-67, HE
69-71).
In one way or another these nonfaradaic influences limit the scope of voltammetrio methods in kinetic and mechanistic studies of electrode reactions as well as in analytical applications. Particularly profound are their effects in modern electrochemical relaxation methods such as cyclic voltammetry (16, 20, 25, 42), a.c. polarography (55, 57, 64, 70, 7 l ) , and high-speed potentiostatic measurements (10, 11, 37, 55). The contributions of iR drop and double-layer charging current in these latter techniques can be made minimal in measurements involving time scales (the period of the alternating potential in a s . polarography and cyclic voltammetry, the measurement time in chrono-amperometry, etc.) which are large. However, with a small time scale their contributions to instrument readout frequently are sufficiently large that even the most simple-minded mechanistic conclusions are impossible until correction for these effects has been accomplished. The complexity associated with the corrections vanes, depending
on the nonfaradaic effect and experimental technique in question, but the tedium associated with these operations is seldom insignificant. Numerical correction for the doublelayer charging current often is accomplished by performing two experiments: one on a solution containing supporting electrolyte and electroactive component (the sample solution) and one on a solution containing only supporting electrolyte (the reference solution) (17). The current observed with the reference solution is subtracted from that observed with the sample solution. The subtraction operation may involve simple scalar subtraction, as in cyclic voltammetry, or the mathematically more cumbersome vectorial subtraction, as in a x . polarography. Due recognition must be given to the effects of iR drop before this subtraction can be effected accurately. In cases where the electroactive species significantly alter the double-layer capacity, a more sophisticated correction scheme must be employed (21, 60). To correct for the effects of iR drop, one must perform the additional experiment of measuring the effective ohmic resistance (10, 11, 17). Once this is accomplished, the data may be corrected for iR drop in some cases by simply considering its effect on the magnitude of the applied potential, as in d.c. and a s . polarography (17, 62, 67). In other techniques, the iR drop may VOL. 38, NO. 9, A U G U S T 1966
0
1 1 19
